On the Boundary Value Problems of
$\Psi$-Hilfer Fractional Differential Equations

Ashwini D. Mali 1

<EMAIL_ADDRESS>
Kishor D. Kucche 2

<EMAIL_ADDRESS>
1,2 Department of Mathematics, Shivaji University, Kolhapur-416 004,
Maharashtra, India.

###### Abstract

In the current paper, we derive the comparison results for the homogeneous and
non-homogeneous linear initial value problem (IVP) for $\Psi$-Hilfer
fractional differential equations. In the presence of upper and lower
solutions, the obtained comparison results and the location of roots theorem
utilized to prove the existence and uniqueness of the solution for the linear
$\Psi$-Hilfer boundary value problem (BVP) through the linear non-homogeneous
$\Psi$-Hilfer IVP. Assuming the existence of lower solution $w_{0}$ and upper
solution $z_{0}$, we establish the existence of minimal and maximal solutions
for the nonlinear $\Psi$-Hilfer BVP in the line segment $[w_{0},\,z_{0}]$ of
the weighted space $C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$.
Further, it demonstrated that the iterative Picard type sequences that began
with lower and upper solutions respectively converges to a minimal and maximal
solutions, and that started with any point on a line segment converge to the
exact solution of nonlinear $\Psi$-Hilfer BVP. Finally, an example is provided
in support of the main results we acquired.

Key words: $\Psi$-Hilfer fractional derivative; Boundary value problems,
Existence and uniqueness; Upper and lower solutions; Extremal solutions,
Monotone iterative technique.
2010 Mathematics Subject Classification: 26A33, 34B15, 34A12.

##  1 Introduction

The boundary value problems (BVPs) of fractional differential equations (FDEs)
have significant applications [1, 2, 3, 4] in mathematical physics, financial
mathematics, mathematical biology, biochemical system, biomedical engineering
etc. Because of widespread applications, fractional-order BVPs were analyzed
by numerous researchers for the existence of solutions by utilizing various
procedures, for example, fixed point theorems [5, 6, 7, 8], fixed point index
theory [9, 10, 11, 12], measure of non-compactness method [13, 14, 15], upper
and lower solution method [16, 17, 18, 19], and so forth.

The technique of upper and lower solutions combined with the monotone
iterative approach utilized effectively to derive sufficient conditions about
the existence and uniqueness of solutions for the differential equations of
integer order [20, 21, 22] and fractional order [23, 24, 25, 26, 27, 28]
subject to initial or boundary condition. The primary thought behind this
method is to construct the monotonic sequences for the corresponding
differential equations, where the initial approximations taken are upper and
lower solutions. At that point, it will be demonstrated that the sequences
converge monotonically to the corresponding maximal and minimal solutions.

The fractional derivative with respect to another function is presented by
Kilbas et al. [29] and Almeida [30] respectively in the sense of Riemann-
Liouville and Caputo fractional derivative. Stimulated by the concept of
fractional derivative with respect to another function, in [31] Sousa and
Oliveira introduced $\Psi$-Hilfer fractional derivative operator
${}^{H}\mathcal{D}^{\alpha,\beta;\,\Psi}_{0+}(\cdot)$ and investigated its
important properties. Authors have proved that the $\Psi$-Hilfer derivative
operator is more generalized and incorporates numerous fractional derivatives
as its special cases. For more details, we refer the reader to the paper [31].
For the analysis of nonlinear $\Psi$-Hilfer FDEs about existence and
uniqueness of the solution, and its qualitative properties, such as Ulam-Hyers
stability and various types of data dependence results, we refer the reader to
the recent papers [32, 33, 34, 35, 36, 37, 38, 39, 40].

Motivated by the work mentioned above and the results obtained by Lin et al.
[28], in the present paper, we discuss the existence and uniqueness results
for linear and nonlinear BVPs of $\Psi$-Hilfer FDEs by the method of upper and
lower solutions combined with monotone iterative technique.

To discuss the existence and uniqueness of solutions for BVPs of $\Psi$-Hilfer
FDEs via method of upper and lower solutions, we need to derive the proper
fractional differential inequalities in the setting of $\Psi$-Hilfer
derivative as comparison results. Therefore, initially we obtain the
comparison results for the homogeneous linear initial value problem (IVP) for
$\Psi$-Hilfer FDEs of the form

$\displaystyle\begin{cases}&{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-My(t)=0,~{}t\in(0,\,T],\\\
&\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)=0,\end{cases}$ (1.1)

and non-homogeneous linear IVP for $\Psi$-Hilfer FDEs. Here
$M>0~{}(M\in\mathbb{R})$, $\Psi\in C^{1}([0,T],\mathbb{R})$ is an increasing
function such that $\Psi^{\prime}(t)\neq 0$, $t\in[0,T]$,
${}^{H}\mathcal{D}^{\alpha,\beta;\,\Psi}_{0+}(\cdot)$ is the $\Psi$-Hilfer
fractional derivative of order $\alpha~{}(0<\alpha<1)$ and type
$\beta~{}(0\leq\beta\leq 1)$, $\gamma=\alpha+\beta\left(1-\alpha\right)$ and
$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}(\cdot)$ is the $\Psi$-Riemann-Liouville
fractional integral of order $1-\,\gamma$.

With the help of acquired comparison results for the linear $\Psi$-Hilfer IVP
and the location of roots theorem [41], assuming existence of lower and upper
solution, we investigated the existence and uniqueness of the solution for the
following linear $\Psi$-Hilfer BVP

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-My(t)=g(t),~{}t\in(0,\,T],~{}$
(1.2)
$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(T),$
(1.3)

through the linear non-homogeneous $\Psi$-Hilfer IVP, where $g\in
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right),~{}J=[0,\,T]$ and
$0<r<\frac{1}{E_{\alpha,1}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)}.$ The two
parameter Mittag-Leffler function $E_{\alpha,1}(\cdot)$ and the weighted space
$C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ will be defined later in
the preliminary section.

Next, we consider the nonlinear $\Psi$-Hilfer BVP of the form

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-My(t)=f(t,y(t)),~{}t\in(0,\,T],~{}$
(1.4)
$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(T),$
(1.5)

where $f\left(\cdot,y(\cdot)\right)\in
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ for each $y\in
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$. Utilizing the existence
and uniqueness results that are obtained for the linear non-homogeneous BVP of
$\Psi$-Hilfer FDEs, and assuming that the nonlinear $\Psi$-Hilfer FDEs
(1.4)-(1.5) has lower solution $w_{0}$ and upper solution $z_{0}$, it is
proved that there exist minimal and maximal solutions on the line segment
$[w_{0},\,z_{0}]$ of the ordered Banach space (weighted space)
$C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$. Further, it is proved
that the Picard iterative sequences $\\{w_{n}\\}_{n=1}^{\infty}$ and
$\\{z_{n}\\}_{n=1}^{\infty}$ starting respectively with $w_{0}$ (lower
solution) and $z_{0}$ (upper solution) are monotonic in the ordered Banach
space $C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ and converges
respectively to minimal and maximal solutions of the nonlinear BVP of
$\Psi$-Hilfer FDEs (1.4)-(1.5).

Assuming that the function $f$ satisfies the one-sided Lipschitz condition, we
have shown that the Picard iterative sequences beginning with any arbitrary
point on the line segment $[w_{0},z_{0}]$ converges to the unique solution of
the nonlinear $\Psi$-Hilfer BVP (1.4)-(1.5). Further, the error bound between
the $n^{th}$ approximation $y_{n}$ and the exact solution $y^{*}$ of the
nonlinear $\Psi$-Hilfer BVP (1.4)-(1.5) is obtained with respect to the norm
on the weighted space $C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb R}\right)$.

Finally, an example is provided to illustrate the existence and uniqueness
results that we acquired through the method of lower and upper solution.

The outcomes acquired in the present are the generalization of the results
derived in [28] and can be achieved by taking $\beta=1$ and $\Psi(t)=t$. The
$\Psi$-Hilfer derivative
${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}(\cdot)$ is generalized
derivative operator that incorporates many notable fractional derivatives
recorded in [31] as its special cases including most widely used derivative
operators, such as, Riemann-Liouville derivative [29], Caputo derivative [29],
Hadmard derivative[29], Erd$\acute{e}$ly-Kober derivative [29], Hilfer
derivative [42], Katugampola derivative [43] etc. Along these lines, the
outcomes acquired in the current paper are likewise valid for the fractional
derivatives listed in [31] as particular cases of the $\Psi$-Hilfer
derivative.

The paper is composed as follows: In section 2, we collect a few definitions
and essential outcomes about $\Psi$-Hilfer fractional derivative. Further, we
have provided some results which assume a significant role in our analysis. In
Section 3, we prove the comparison results for $\Psi$-Hilfer FDEs. Section 4
deals with the existence and uniqueness of the linear $\Psi$-Hilfer BVPs. In
Section 5, the existence and uniqueness results are proved for the nonlinear
$\Psi$-Hilfer BVPs (1.4)-(1.5). An example is provided in section 6 to verify
the assurance of our primary outcomes.

##  2 Preliminaries

Let $[a,b]$ $(0<a<b<\infty)$ be a finite interval and $\Psi\in
C^{1}([a,b],\mathbb{R})$ be an increasing function such that
$\Psi^{\prime}(t)\neq 0$, $t\in[a,b]$. Consider the weighted space [31]

$\mathcal{X}:=C_{1-\,\gamma;\,\Psi}\left[a,b\right]=\left\\{\mathfrak{h}:\left(a,b\right]\rightarrow\mathbb{R}~{}\big{|}~{}\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\,\gamma}\mathfrak{h}(t)\in
C\left[a,b\right]\right\\},\text{ }0<\gamma\leq 1,$

endowed with the norm

$\left\|\mathfrak{h}\right\|_{C_{1-\,\gamma;\Psi}\left[a,b\right]}=\underset{t\in\left[a,b\right]}{\max}\left|\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\,\gamma}\mathfrak{h}(t)\right|,$
(2.1)

where $\gamma=\alpha+\beta(1-\alpha)$. Then,
$\left(\mathcal{X},\,\left\|\cdot\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\right)$ is a partially ordered Banach space with the partial order
relation $\preceq$ defined by

$x,\,y\in\mathcal{X},\,x\preceq y\,~{}\text{if and only if}~{}\,x(t)\leq
y(t),\,t\in(0,\,T].$

######  Definition 2.1 ([29])

Let $\mu>0~{}(\mu\in\mbox{\Bbb R})$, $\mathfrak{h}$ be an integrable function
defined on $[a,b]$. Then, the $\Psi$-Riemann–Liouville fractional integral of
a function $\mathfrak{h}$ of order $\mu$ with respect to $\Psi$ is given by

$I_{a+}^{\mu\,;\Psi}\mathfrak{h}(t)=\frac{1}{\Gamma\left(\alpha\right)}\int_{a}^{t}\Psi^{\prime}\left(s\right)\left(\Psi\left(t\right)-\Psi\left(s\right)\right)^{\mu-1}\mathfrak{h}(s)ds.$

######  Definition 2.2 ([31])

Let $0<\alpha<1$ and $\mathfrak{h}\in C^{1}([a,b],\mathbb{R})$. Then, the
$\Psi$-Hilfer fractional derivative of a function $\mathfrak{h}$ of order
$\alpha$ and type $\beta\left(0\leq\beta\leq 1\right)$, is defined by

$^{H}\mathcal{D}_{a+}^{\alpha,\beta\,;\Psi}\mathfrak{h}\left(t\right)=I_{a+}^{\beta\left(1-\alpha\right)\,;\Psi}\left(\frac{1}{\Psi^{\prime}\left(t\right)}\frac{d}{dt}\right)I_{a+}^{\left(1-\beta\right)\left(1-\alpha\right)\,;\Psi}\mathfrak{h}(t).$

######  Lemma 2.1 ([29, 31])

Let $\mu_{i}>0~{}(i=1,2)$ and $\delta>0$. Then,

  1. (a)

$\mathcal{I}_{a+}^{\mu_{1}\,;\,\Psi}\mathcal{I}_{a+}^{\mu_{2}\,;\,\Psi}\mathfrak{h}(t)=\mathcal{I}_{a+}^{\mu_{1}+\mu_{2}\,;\,\Psi}\mathfrak{h}(t)$.

  2. (b)

$\mathcal{I}_{a+}^{\mu\,;\,\Psi}\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{\delta-1}=\frac{\Gamma\left(\delta\right)}{\Gamma\left(\mu+\delta\right)}\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{\mu+\delta-1}.$

######  Lemma 2.2 ([31])

If $\mathfrak{h}\in C^{1}[a,b]$, $0<\alpha<1$ and $0\leq\beta\leq 1$, then

  1. (a)

$I_{a+}^{\alpha\,;\Psi}\text{
}^{H}\mathcal{D}_{a+}^{\alpha,\beta\,;\Psi}\mathfrak{h}\left(t\right)=\mathfrak{h}\left(t\right)-\frac{\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{\gamma-1}}{\Gamma\left(\gamma\right)}I_{a+}^{1-\gamma\,;\Psi}\mathfrak{h}(a)$.

  2. (a)

${}^{H}\mathcal{D}_{a+}^{\alpha,\beta\,;\Psi}I_{a+}^{\alpha\,;\Psi}\mathfrak{h}\left(t\right)=\mathfrak{h}\left(t\right).$

######  Lemma 2.3 ([35])

If $\mu>0$ and $0\leq\omega<1,$ then $\mathcal{I}_{a+}^{\mu\,;\,\Psi}(\cdot)$
is bounded from $C_{\omega\,;\,\Psi}\left[a,b\right]$ to
$C_{\omega\,;\,\Psi}\left[a,b\right].$ In addition, if $\omega\leq\mu$, then
$\mathcal{I}_{a+}^{\mu\,;\,\Psi}(\cdot)$ is bounded from
$C_{\omega\,;\,\Psi}\left[a,b\right]$ to $C\left[a,b\right]$.

######  Lemma 2.4 ([36])

Let $g\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ and
$\eta\in\mbox{\Bbb R}$. Then, the solution of the Cauchy problem for FDEs with
constant coefficient involving $\Psi$-Hilfer fractional derivative,

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-\eta
y(t)=g(t),~{}~{}t\in(0,\,T],$
$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)=y_{0},$

is given by

$\displaystyle y(t)$
$\displaystyle=y_{0}(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(\eta(\Psi(t)-\Psi(0))^{\alpha}\right)$
$\displaystyle+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(\eta(\Psi(t)-\Psi(s))^{\alpha}\right)g(s)ds,~{}~{}t\in(0,\,T].$

######  Lemma 2.5 ([44])

Let $n_{1},n_{2}>0,~{}(n_{i}\in\mathbb{R}),~{}i=1,2$. Consider the two
parameter Mittag–Leffler function defined by
$E_{n_{1},n_{2}}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(n_{1}k+n_{2})},\,z\in\mathbb{C}$.
Then, the power series defining $E_{n_{1},n_{2}}(z)$ is convergent for all
$z\in\mathbb{C}$.

######  Lemma 2.6 (Location of roots theorem ([41]))

Let $I=[a,b]$ and let $f:I\rightarrow\mbox{\Bbb R}$ be continuous on $I$. If
$f(a)<0<f(b)$, or if $f(a)>0>f(b)$, then there exists a number $c\in(a,b)$
such that $f(c)=0$.

######  Lemma 2.7 ([21])

Let $\mathcal{X}$ be a partially ordered Banach space,
$\\{x_{n}\\}\subset\mathcal{X}$ a monotone sequence and relatively compact
set, then $\\{x_{n}\\}$ is convergent.

######  Lemma 2.8 ([21])

Let $\mathcal{X}$ be a partially ordered Banach space, $x_{n}\preceq
y_{n}(n=1,2,3,\cdots)$ if $x_{n}\rightarrow x^{*},\,y_{n}\rightarrow y^{*}$ we
have $x^{*}\preceq y^{*}$.

###### Definition 2.3

Let $u\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$. We say that
$u$ is a lower solution of the linear $\Psi$-Hilfer BVP (1.2)-(1.3) if

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}u(t)-Mu(t)\leq
g(t)-a_{u}(t),~{}t\in(0,\,T],$

where

$a_{u}(t)=\begin{cases}0,~{}\text{if}\,\,r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\geq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)\\\
\frac{1}{r}\left({}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\xi(t)-M\xi(t)\right)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right),\\\
~{}\text{if}\,\,r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)<\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0),\end{cases}$
(2.2)

and

$\xi(t)=\frac{\Gamma(2+\delta-\gamma)}{\Gamma(\delta+1)}\frac{(\Psi(t)-\Psi(0))^{\delta}}{(\Psi(T)-\Psi(0))^{1-\,\gamma+\delta}},~{}\delta>0,\,t\in
J.$ (2.3)

Clearly, $\xi\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$.

###### Definition 2.4

Let $v\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$. We say that
$v$ is an upper solution of the linear $\Psi$-Hilfer BVP (1.2)-(1.3) if

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}v(t)-Mv(t)\geq
g(t)+b_{v}(t),~{}t\in(0,\,T],$

where

$b_{v}(t)=\begin{cases}0,~{}\text{if}\,\,r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(T)\leq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(0)\\\
\frac{1}{r}\left({}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\xi(t)-M\xi(t)\right)\left(r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(T)-\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(0)\right),\\\
~{}\text{if}\,\,r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(T)>\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(0),\end{cases}$
(2.4)

where $\xi(t)$ is defined as in the equation (2.3).

##  3 Comparison Theorems

###### Theorem 3.1

Assume that $y\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ and
satisfies

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-My(t)\leq
0,~{}t\in(0,\,T],~{}$ (3.1)
$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)\leq 0.$ (3.2)

Then, $y(t)\leq 0$, $\,t\in(0,\,T]$.

Proof: Consider the following linear IVP

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-My(t)=\sigma(t),~{}t\in(0,\,T],~{}$
(3.3) $\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)=y_{0},$ (3.4)

where $\sigma\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$. By
Lemma 2.4, the linear IVP (3.3)-(3.4) has a unique solution given by

$\displaystyle y(t)$
$\displaystyle=y_{0}(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)$
$\displaystyle+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)\sigma(s)ds,~{}t\in(0,\,T].$
(3.5)

From equations (3.2) and (3.4), we have $y_{0}\leq 0$. Since $\Psi$ is an
increasing function, we have $(\Psi(t)-\Psi(0))^{\gamma-1}\geq 0,\,t\in J$ and
$(\Psi(t)-\Psi(0))^{\alpha}\geq 0,\,t\in J$. This gives,
$E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)>0,\,t\in J$.
Therefore, from (3), we have

$y(t)\leq\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)\sigma(s)ds,~{}t\in(0,\,T].$
(3.6)

From equations (3.1) and (3.3), it follows that $\sigma(t)\leq 0,\,t\in J$.
Since $\Psi:J\rightarrow\mbox{\Bbb R}$ is an increasing continuous function,
we have $\Psi^{\prime}(t)>0,$ for $\,t\in J$,
$(\Psi(t)-\Psi(s))^{\alpha-1}\geq 0,$ for $t\geq s\geq 0$ and
$E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)>0,$ for $t\geq
s\geq 0$. Therefore,

$\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)\sigma(s)\leq
0,~{}\text{for}\,~{}0\leq s\leq t\leq T.$ (3.7)

From (3.6) and (3.7), we obtain

$y(t)\leq 0,~{}t\in(0,\,T].$

$\Box$

###### Theorem 3.2

Let $y\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ satisfies

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-My(t)\leq-
a_{u}(t),~{}t\in(0,\,T],~{}$ (3.8)
$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)\leq 0.$ (3.9)

Then, $y(t)\leq 0$, $\,t\in(0,\,T]$.

Proof: In the view of definition of $a_{u}$ given in (2.2), we give the proof
in following two cases.
Case 1: If
$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\geq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)$
then $a_{u}(t)=0,\,t\in(0,\,T]$. Thus, equations (3.8)-(3.9) reduces to
equations (3.1)-(3.2). Applying Theorem 3.1, we obtain $y(t)\leq 0$,
$t\in(0,\,T]$.
Case 2: If
$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)<\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)$
then

$a_{u}(t)=\frac{1}{r}\left({}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\xi(t)-M\xi(t)\right)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right),\,t\in(0,\,T].$

Define

$\rho(t)=y(t)+\frac{1}{r}\xi(t)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right),~{}t\in(0,\,T].$
(3.10)

By assumption

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)>0.$

Further, from equation (2.3), we have $\xi(t)\geq 0,\,t\in J$. Hence, we have

$\frac{1}{r}\xi(t)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right)\geq
0,\,t\in J.$ (3.11)

From (3.10) and (3.11), it follows that

$y(t)\leq\rho(t),~{}t\in(0,\,T].$ (3.12)

Next, using (3.10), we have

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\rho(t)-M\rho(t)$
$\displaystyle=\,^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\left[y(t)+\frac{1}{r}\xi(t)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right)\right]$
$\displaystyle\qquad-M\left[y(t)+\frac{1}{r}\xi(t)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right)\right]$
$\displaystyle=\,^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-My(t)+\frac{1}{r}\left[{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\xi(t)-M\xi(t)\right]\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right)$
$\displaystyle=\,^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-My(t)+a_{u}(t),~{}t\in(0,\,T].$

Using the inequality (3.8), above equation reduces to the following inequality

$^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\rho(t)-M\rho(t)\leq
0,~{}t\in(0,\,T].$ (3.13)

Further, using (3.10) and Lemma 2.1(b), we have

$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\rho(t)$
$\displaystyle=\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t)+\frac{1}{r}\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right)\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\xi(t)$
$\displaystyle=\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t)$
$\displaystyle+\frac{1}{r}\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right)\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\left[\frac{\Gamma(2+\delta-\gamma)}{\Gamma(\delta+1)}\frac{(\Psi(t)-\Psi(0))^{\delta}}{(\Psi(T)-\Psi(0))^{1-\,\gamma+\delta}}\right]$
$\displaystyle=\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t)+\frac{1}{r}\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right)\frac{(\Psi(t)-\Psi(0))^{1-\,\gamma+\delta}}{(\Psi(T)-\Psi(0))^{1-\,\gamma+\delta}}.$
(3.14)

From (3.9) and (3), it follows that

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\rho(0)\leq 0.$ (3.15)

By applying Theorem 3.1 to the inequalities (3.13) and (3.15), we have

$\rho(t)\leq 0,~{}t\in(0,\,T].$ (3.16)

Further, from inequalities (3.12) and (3.16), we obtain $y(t)\leq
0,\,t\in(0,\,T].$ $\Box$

Following similar type of steps as in the proof of Theorem 3.2, one can easily
prove the following theorem.

###### Theorem 3.3

Let $y\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ satisfies

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-My(t)\leq-
b_{v}(t),~{}t\in(0,\,T],~{}$ (3.17)
$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)\leq 0.$ (3.18)

Then, $y(t)\leq 0,\,t\in(0,\,T].$

##  4 Existence and uniqueness for the linear $\Psi$-Hilfer BVP

In this section, using the method of upper and lower solutions, we derive the
existence and uniqueness results for the linear $\Psi$-Hilfer BVP (1.2)-(1.3).

###### Theorem 4.1

Assume that there exist upper and lower solutions $v,u\in
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ respectively of the
linear $\Psi$-Hilfer BVP (1.2)-(1.3) such that $u\preceq v$. Then, the linear
$\Psi$-Hilfer BVP (1.2)-(1.3) has a unique solution $y\in
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ that satisfy $u\preceq
y\preceq v$.

Proof: We give the proof in following two parts.
Part I: In this part we prove that the linear $\Psi$-Hilfer BVP (1.2)-(1.3)
has a unique solution. Consider the functions $p,q\in
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ defined by

$p(t)=\begin{cases}r\,u(t),~{}\text{if}\,\,r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\geq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)\\\
r\,u(t)+\xi(t)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right),~{}\text{if}\,\,r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)<\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0),\end{cases}$
(4.1)

and

$q(t)=\begin{cases}r\,v(t),~{}\text{if}\,\,r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(T)\leq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(0)\\\
r\,v(t)-\xi(t)\left(r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(T)-\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(0)\right),~{}\text{if}\,\,r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(T)>\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(0),\end{cases}$
(4.2)

where $\xi(t)$ is defined as in the equation (2.3).
Case 1: If
$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\geq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)$
then $p(t)=r\,u(t),\,t\in(0,\,T]$. Therefore,

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(0)=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0).$

Further,

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\geq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)\\\
=\frac{\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(0)}{r}.$

Thus,

$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)\geq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(0).$

Case 2: If
$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)<\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)$
then
$p(t)=r\,u(t)+\xi(t)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right),$
$\,t\in(0,\,T]$. Therefore,

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(0)=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)+\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\xi(0)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right).$
(4.3)

But from (2.3) and Lemma 2.1(b), we have

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\xi(t)=\frac{(\Psi(t)-\Psi(0))^{1-\,\gamma+\delta}}{(\Psi(T)-\Psi(0))^{1-\,\gamma+\delta}},\,t\in
J.$ (4.4)

From (4.3) and (4.4), it follows that

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(0)=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0).$

Further,

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)+\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\xi(T)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right).$
(4.5)

Again from (4.4) and (4.5), it follows that

$\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)=\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)=\frac{\mathcal{I}_{0+}^{1-\gamma;\,\Psi}p(0)}{r}.$

Thus,

$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)=\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(0).$

From Case 1 and Case 2, we have

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(0)=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0),$
(4.6)
$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)\geq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(0).$
(4.7)

On the similar line for the function $q$ one can obtain the following
relations

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(0)=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(0),$
(4.8)
$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(T)\leq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(0).$
(4.9)

Next, our aim is to prove that $p\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)$ satisfies the following fractional differential inequality

$^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}p(t)-Mp(t)\leq
r\,g(t),~{}t\in(0,\,T].$ (4.10)

If
$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\geq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)$,
then $a_{u}(t)=0,~{}t\in(0,\,T]$. Further, $u$ is a lower solution of linear
$\Psi$-Hilfer BVP (1.2)-(1.3). Therefore, we have

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}p(t)-Mp(t)$
$\displaystyle=r\left[{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}u(t)-Mu(t)\right]$
$\displaystyle\leq r\left[g(t)-a_{u}(t)\right]$
$\displaystyle=r\,g(t),\,t\in(0,\,T].$

If
$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)<\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0),$
then

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}p(t)-Mp(t)$
$\displaystyle=\,^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\left[r\,u(t)+\xi(t)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right)\right]$
$\displaystyle\qquad-M\left[r\,u(t)+\xi(t)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right)\right]$
$\displaystyle=r\left[{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}u(t)-Mu(t)\right]$
$\displaystyle\qquad+\left[{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\xi(t)-M\xi(t)\right]\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\right),\,t\in(0,\,T].$

Using the definition of $a_{u}(t)$ and the fact that $u$ is a lower solution
of linear $\Psi$-Hilfer BVP (1.2)-(1.3) from above equation, we obtain

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}p(t)-Mp(t)\leq
r\left[g(t)-a_{u}(t)\right]+r\,a_{u}(t)=r\,g(t),\,t\in(0,\,T].$

In both cases
$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\geq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)$
and
$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)<\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0),$
we have proved that the function $p\in
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ defined in equation (4.1)
satisfies the fractional differential inequality (4.10). On the similar line,
one can prove that the function $q\in
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ defined in equation (4.2)
satisfies the following fractional differential inequality

$^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}q(t)-Mq(t)\geq
r\,g(t),~{}t\in(0,\,T].$ (4.11)

Define $\sigma(t)=p(t)-q(t),\,t\in(0,\,T].$ Then, $\sigma\in
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$. By using fractional
differential inequalities (4.10) and (4.11), we obtain

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\sigma(t)$
$\displaystyle=\,^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}p(t)-\,^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}q(t)$
$\displaystyle\leq\left[r\,g(t)+M\,p(t)\right]-\left[r\,g(t)+M\,q(t)\right]$
$\displaystyle=M\,\sigma(t),\,t\in(0,\,T].$

Further, by using equations (4.6) and (4.8) and hypothesis, we obtain

$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\sigma(0)$
$\displaystyle=\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(0)-\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(0)$
$\displaystyle=r\left[\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}v(0)\right]$
$\displaystyle\leq 0.$

We have proved that $\sigma\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)$ satisfies

$\displaystyle\begin{cases}&{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\sigma(t)-M\,\sigma(t)\leq
0,\,t\in(0,\,T],\\\ &\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\sigma(0)\leq
0.\end{cases}$

By applying Theorem 3.1, we obtain $\sigma(t)\leq 0,\,t\in(0,\,T].$ This gives

$p(t)\leq q(t),\,t\in(0,\,T].$

Next, for any $\lambda\in\mbox{\Bbb R}$, consider the following linear
$\Psi$-Hilfer FDEs subject to initial condition

$\displaystyle\begin{cases}&{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-My(t)=g(t),\,t\in(0,\,T],\\\
&\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t)|_{t=0}=\lambda.\end{cases}$ (4.12)

Then, by Lemma 2.4, it has a unique solution in
$C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ given by

$\displaystyle y(t,\lambda)$
$\displaystyle=\lambda\,(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)$
$\displaystyle+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)g(s)ds,\,t\in(0,\,T].$
(4.13)

Since $g\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$, the function
$(\Psi(\cdot)-\Psi(0))^{1-\,\gamma}y(\cdot,\lambda)$ is continuous on $J$ for
each $\lambda\in\mbox{\Bbb R}$.
Define $\chi(t)=p(t)-r\,y(t,\lambda),\,t\in(0,\,T],$ where $y(t,\lambda)$ is a
solution of (4.12). Then, $\chi\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)$. Take any $\lambda\in\mbox{\Bbb R}$ such that

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)\leq\lambda\leq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(T).$
(4.14)

From equations (4.10) and (4.12), we have

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\chi(t)-M\,\chi(t)$
$\displaystyle=\,^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\left[p(t)-r\,y(t,\lambda)\right]-M\,\left[p(t)-r\,y(t,\lambda)\right]$
$\displaystyle=\left[{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}p(t)-M\,p(t)\right]-r\left[{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t,\lambda)-M\,y(t,\lambda)\right]$
$\displaystyle\leq r\,g(t)-r\,g(t)=0,\,t\in(0,\,T].$

Further, by using inequalities (4.7) and (4.14) and initial condition in
(4.12), we obtain

$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\chi(0)$
$\displaystyle=\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(0)-r\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t,\lambda)|_{t=0}$
$\displaystyle\leq
r\left[\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)-\lambda\right]$
$\displaystyle\leq 0.$

Therefore, $\chi\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$
satisfies

$\displaystyle\begin{cases}{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\chi(t)-M\,\chi(t)\leq
0,\,t\in(0,\,T],\\\ \mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\chi(0)\leq
0.\end{cases}$

By applying Theorem 3.1, we obtain $\chi(t)\leq 0,\,t\in(0,\,T].$ This implies
$p(t)\leq r\,y(t,\lambda),\,t\in(0,\,T]$. On the similar line one can prove
that $r\,y(t,\lambda)\leq q(t),\,t\in(0,\,T]$. Therefore,

$p(t)\leq r\,y(t,\lambda)\leq q(t),\,t\in(0,\,T].$

Since $\Psi$-Riemann-Liouville fractional integral operator
$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}$ is monotonic, from above inequalities,
we obtain

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(t)\leq
r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t,\lambda)\leq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(t),\,t\in(0,\,T].$

Therefore, we can write

$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)\leq
r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(T,\lambda)\leq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(T),$
(4.15)

for any
$\lambda\in\left[\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T),\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(T)\right].$

Define

$g(\lambda)=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(T,\lambda)-\lambda,\,\,\lambda\in\left[\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T),\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(T)\right].$
(4.16)

Since two parameter Mittag-Leffler function is analytic, using it’s series
representation the equation (4) can be written as

$\displaystyle y(t,\lambda)$
$\displaystyle=\lambda\,(\Psi(t)-\Psi(0))^{\gamma-1}\sum_{k=0}^{\infty}\frac{\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)^{k}}{\Gamma(k\alpha+\gamma)}$
$\displaystyle\qquad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}\sum_{k=0}^{\infty}\frac{\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)^{k}}{\Gamma(k\alpha+\alpha)}g(s)ds$
$\displaystyle=\lambda\sum_{k=0}^{\infty}\frac{M^{k}}{\Gamma(k\alpha+\gamma)}\left(\Psi(t)-\Psi(0)\right)^{k\alpha+\gamma-1}$
$\displaystyle\qquad+\sum_{k=0}^{\infty}\frac{M^{k}}{\Gamma(\alpha(k+1))}\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha(k+1)-1}g(s)ds$
$\displaystyle=\lambda\sum_{k=0}^{\infty}\frac{M^{k}}{\Gamma(k\alpha+\gamma)}\left(\Psi(t)-\Psi(0)\right)^{k\alpha+\gamma-1}+\sum_{k=0}^{\infty}M^{k}\,\mathcal{I}_{0+}^{\alpha(k+1);\Psi}g(t),\,t\in(0,\,T].$

Using Lemma 2.1, from above equation, we obtain

$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t,\lambda)$
$\displaystyle=\lambda\sum_{k=0}^{\infty}\frac{M^{k}}{\Gamma(k\alpha+\gamma)}\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\left(\Psi(t)-\Psi(0)\right)^{k\alpha+\gamma-1}+\sum_{k=0}^{\infty}M^{k}\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}\mathcal{I}_{0+}^{\alpha(k+1);\Psi}g(t)$
$\displaystyle=\lambda\sum_{k=0}^{\infty}\frac{M^{k}}{\Gamma(k\alpha+\gamma)}\frac{\Gamma(k\alpha+\gamma)}{\Gamma(k\alpha+\gamma+1-\,\gamma)}\left(\Psi(t)-\Psi(0)\right)^{k\alpha}+\sum_{k=0}^{\infty}M^{k}\,\mathcal{I}_{0+}^{1-\,\gamma+\alpha(k+1);\Psi}g(t)$
$\displaystyle=\lambda\sum_{k=0}^{\infty}\frac{M^{k}}{\Gamma(k\alpha+1)}\left(\Psi(t)-\Psi(0)\right)^{k\alpha}+\sum_{k=0}^{\infty}M^{k}\,\mathcal{I}_{0+}^{1-\,\gamma+\alpha(k+1);\Psi}g(t)$
$\displaystyle=\lambda\,E_{\alpha,1}\left(M\left(\Psi(t)-\Psi(0)\right)^{\alpha}\right)$
$\displaystyle\qquad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-\gamma}\sum_{k=0}^{\infty}\frac{\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)^{k}}{\Gamma(k\alpha+\alpha+1-\,\gamma)}g(s)ds$
$\displaystyle=\lambda\,E_{\alpha,1}\left(M\left(\Psi(t)-\Psi(0)\right)^{\alpha}\right)$
$\displaystyle\qquad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(t)-\Psi(s)\right)^{\alpha}\right)g(s)ds,\,t\in(0,\,T].$

Therefore,

$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t,\lambda)|_{t=T}$
$\displaystyle=\lambda\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)$
$\displaystyle\qquad+\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)g(s)ds.$
(4.17)

Using equation (4) in equation (4.16), we get

$\displaystyle g(\lambda)$
$\displaystyle=r\left\\{\lambda\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right.$
$\displaystyle\qquad\left.+\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)g(s)ds\right\\}-\lambda.$

Differentiating above equation with respect to $\lambda$ and using the
condition on $r$, we obtain

$g^{\prime}(\lambda)=r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)-1<0.$

This implies $g$ is strictly decreasing function on the closed interval
$\left[\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T),\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(T)\right]$.
Therefore, we have

$g\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)\right)>g\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(T)\right).$

Next, we show that the equation $g(\lambda)=0$ has atmost one solution on ℝ.
Using equation (4.15), we obtain

$\displaystyle
g\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(T)\right)=\left[r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(T,\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(T))-\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(T)\right]\leq
0,$ (4.18)

and

$\displaystyle
g\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)\right)=\left[r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(T,\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T))-\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)\right]\geq
0.$ (4.19)

Since $g$ is continuous on the closed interval
$\left[\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T),\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(T)\right]$
and satisfies the conditions (4.18) and (4.19), using the location of root
theorem given in Lemma 2.6 coupled with strictly decreasing nature of $g$,
there exist at most one
$\lambda_{0}\in\left[\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T),\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}q(T)\right]$
such that

$g(\lambda_{0})=0.$ (4.20)

From equations (4.12), (4.16) and (4.20), we obtain

$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t,\lambda_{0})|_{t=T}=\lambda_{0}=\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t)|_{t=0}.$
(4.21)

From equations (4.12) and (4), it follows that

$\displaystyle y(t,\lambda_{0})$
$\displaystyle=\lambda_{0}\,(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)$
$\displaystyle\qquad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)g(s)ds$
(4.22)

is the unique solution of linear $\Psi$-Hilfer FDEs with initial condition

$\displaystyle\begin{cases}&{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-My(t)=g(t),\,t\in(0,\,T],\\\
&\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t)|_{t=0}=\lambda_{0}.\end{cases}$
(4.23)

Further, from (4.21) and (4.23) it follows that, $y(t,\lambda_{0})$ given in
(4) is the unique solution of linear $\Psi$-Hilfer BVP (1.2)-(1.3).
Part II: In this part we prove that the unique solution $y(t,\lambda_{0})$
obtained in the first part satisfies

$u\preceq y(\cdot,\lambda_{0})\preceq v~{}\text{
in}~{}C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right).$

Define $h(t)=u(t)-y(t,\lambda_{0}),\,t\in(0,\,T]$. Clearly $h\in
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$.
Case 1: If
$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)\geq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)$,
then $a_{u}(t)=0,\,t\in(0,\,T]$. Since $u$ is a lower solution of the linear
$\Psi$-Hilfer BVP (1.2)-(1.3) and $y(t,\lambda_{0})$ is the solution of linear
$\Psi$-Hilfer FDEs (4.23), we obtain

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}h(t)-M\,h(t)$
$\displaystyle=\,^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\left[u(t)-y(t,\lambda_{0})\right]-M\,\left[u(t)-y(t,\lambda_{0})\right]$
$\displaystyle=\left[{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}u(t)-M\,u(t)\right]-\left[{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t,\lambda_{0})-M\,y(t,\lambda_{0})\right]$
$\displaystyle\leq\,g(t)-a_{u}(t)-g(t)$ $\displaystyle=0,\,t\in(0,\,T].$

Further, using the equations (4.6) and (4.7) and initial condition in (4.23),
we have

$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}h(0)$
$\displaystyle=\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)-\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t,\lambda_{0})|_{t=0}$
$\displaystyle=\frac{\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(0)}{r}-\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(t,\lambda_{0})|_{t=0}$
$\displaystyle\leq\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}p(T)-\,\lambda_{0}$
$\displaystyle\leq 0.$

Applying Theorem 3.1 to the fractional inequalities

$\displaystyle\begin{cases}&{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}h(t)-M\,h(t)\leq
0,\,t\in(0,\,T],\\\ &\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}h(0)\leq
0,\end{cases}$

we obtain $h(t)\leq 0,\,t\in(0,\,T].$ This gives, $u(t)\leq
y(t,\lambda_{0}),\,t\in(0,\,T].$
Case 2: If
$r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(T)<\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}u(0)$
then $a_{u}(t)\neq 0,\,t\in(0,\,T]$ as defined in (2.2). Since $u$ is a lower
solution of the linear $\Psi$-Hilfer BVP (1.2)-(1.3) and $y(t,\lambda_{0})$ is
the solution of linear $\Psi$-Hilfer FDEs (4.23), we obtain

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}h(t)-M\,h(t)$
$\displaystyle=\,^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\left[u(t)-y(t,\lambda_{0})\right]-M\,\left[u(t)-y(t,\lambda_{0})\right]$
$\displaystyle=\left[{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}u(t)-M\,u(t)\right]-\left[{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t,\lambda_{0})-M\,y(t,\lambda_{0})\right]$
$\displaystyle\leq\,g(t)-a_{u}(t)-g(t)$
$\displaystyle=-a_{u}(t),\,t\in(0,\,T].$

Further, we have $\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}h(0)\leq 0.$ By applying
Theorem 3.2 to the fractional inequalities

$\displaystyle\begin{cases}&{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}h(t)-M\,h(t)\leq\,g(t)-a_{u}(t)-g(t)=-a_{u}(t),\,t\in(0,\,T],\\\
&\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}h(0)\leq 0,\end{cases}$

we have $h(t)\leq 0,\,t\in(0,\,T].$ This implies, $u(t)\leq
y(t,\lambda_{0}),\,t\in(0,\,T].$ From Case 1 and Case 2, it follows that

$u(t)\leq y(t,\lambda_{0}),\,t\in(0,\,T].$

Following the similar approach, one can show that

$v(t)\geq y(t,\lambda_{0}),\,t\in(0,\,T].$

Therefore, we have

$u(t)\leq y(t,\lambda_{0})\leq v(t),\,t\in(0,\,T].$

By Part I and Part II, it follows that $y(\cdot,\lambda_{0})\in
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ is the unique solution of
the linear $\Psi$-Hilfer BVP (1.2)-(1.3) that satisfies the condition

$u\preceq y(\cdot,\lambda_{0})\preceq v~{}\text{
in}~{}C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right).$

This completes the proof. $\Box$

##  5 Existence and uniqueness for the nonlinear $\Psi$-Hilfer BVP

###### Definition 5.1

We say that $w_{0}\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ and
$z_{0}\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ are the lower
and upper solutions respectively of the nonlinear $\Psi$-Hilfer BVP
(1.4)-(1.5) if

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}w_{0}(t)-Mw_{0}(t)\leq
f(t,w_{0}(t))-a_{w_{0}}(t),~{}t\in(0,\,T],$

and

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}z_{0}(t)-Mz_{0}(t)\geq
f(t,z_{0}(t))+b_{z_{0}}(t),~{}t\in(0,\,T],$

where

$a_{w_{0}}(t)=\begin{cases}0,~{}\text{if}\,\,r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}w_{0}(T)\geq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}w_{0}(0)\\\
\frac{1}{r}\left({}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\xi(t)-M\xi(t)\right)\left(\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}w_{0}(0)-r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}w_{0}(T)\right),\\\
~{}\text{if}\,\,r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}w_{0}(T)<\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}w_{0}(0),\end{cases}$

and

$b_{z_{0}}(t)=\begin{cases}0,~{}\text{if}\,\,r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}z_{0}(T)\leq\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}z_{0}(0)\\\
\frac{1}{r}\left({}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}\xi(t)-M\xi(t)\right)\left(r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}z_{0}(T)-\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}z_{0}(0)\right),\\\
~{}\text{if}\,\,r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}z_{0}(T)>\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}z_{0}(0),\end{cases}$

and $\xi$ is the function as defined in (2.3).

###### Theorem 5.1

Assume $z_{0}\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ and
$w_{0}\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ are the upper
and lower solutions respectively of the nonlinear $\Psi$-Hilfer BVP
(1.4)-(1.5) such that $w_{0}\preceq z_{0}$. Further, assume that:
(H1) the function $f$ satisfies

  1. (i)

$f\left(\cdot,y(\cdot)\right)\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)$ for each $y\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)$,

  2. (ii)

$f(t,\,y_{1})\leq f(t,\,y_{2}),~{}\text{for any }y_{1},\,y_{2}\in\mbox{\Bbb
R}\text{ with }y_{1}\leq y_{2}~{}~{}\text{and}~{}~{}t\in(0,\,T].$

Then, the nonlinear $\Psi$-Hilfer BVP (1.4)-(1.5) has a minimal solution
$w^{*}\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ and a maximal
solution $z^{*}\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ in the
line segment

$[w_{0},\,z_{0}]=\\{y\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right):w_{0}\preceq y\preceq z_{0}\\}.$

Further, if $\\{w_{n}\\}_{n=1}^{\infty}$ and $\\{z_{n}\\}_{n=1}^{\infty}$ are
the iterative sequences defined by

$\displaystyle w_{n}(t)$
$\displaystyle=\frac{r\,(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)}{1-r\,E_{\alpha,\,1}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)}$
$\displaystyle\qquad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(s))^{\alpha}\right)f(s,\,w_{n-1}(s))ds$
$\displaystyle\qquad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)f(s,\,w_{n-1}(s))ds$

and

$\displaystyle z_{n}(t)$
$\displaystyle=\frac{r\,(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)}{1-r\,E_{\alpha,\,1}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)}$
$\displaystyle\qquad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(s))^{\alpha}\right)f(s,\,z_{n-1}(s))ds$
$\displaystyle\qquad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)f(s,\,z_{n-1}(s))ds$

then $\\{w_{n}\\}_{n=1}^{\infty}$ and $\\{z_{n}\\}_{n=1}^{\infty}$ are the
monotonic sequences in $C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$
such that

$\displaystyle\lim\limits_{n\rightarrow\infty}\left\|w_{n}-w^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\mbox{\Bbb
R}\right)}=0$

and

$\displaystyle\lim\limits_{n\rightarrow\infty}\left\|z_{n}-z^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\mbox{\Bbb
R}\right)}=0.$

Proof: We give the proof in five parts.
Part 1: We denote $\mathfrak{D}=[w_{0},\,z_{0}]$. For any
$\varphi\in\mathfrak{D}$, we consider the following linear $\Psi$-Hilfer BVP

$\left.\begin{aligned}
&{}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}y(t)-My(t)=f(t,\varphi(t)),~{}t\in(0,\,T],\\\
&\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(T).\end{aligned}\right\\}$
(5.1)

Since $w_{0},z_{0}$ are lower and upper solutions respectively of the
nonlinear $\Psi$-Hilfer BVP (1.4)-(1.5), using hypothesis (H1)(ii), we have

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}w_{0}(t)-Mw_{0}(t)\leq
f(t,w_{0}(t))-a_{w_{0}}(t)\leq f(t,\varphi(t))-a_{w_{0}}(t),~{}t\in(0,\,T]$

and

${}^{H}\mathcal{D}^{\alpha,\,\beta\,;\,\Psi}_{0+}z_{0}(t)-Mz_{0}(t)\geq
f(t,z_{0}(t))+b_{z_{0}}(t)\geq f(t,\varphi(t))+b_{z_{0}}(t),~{}t\in(0,\,T].$

This implies $w_{0}$ and $z_{0}$ are the lower and upper solutions
respectively of the linear $\Psi$-Hilfer BVP (5.1). In the view of Theorem
4.1, the linear $\Psi$-Hilfer BVP (5.1) has a unique solution
$y\in\mathfrak{D}$, given by

$\displaystyle y(t)$
$\displaystyle=\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)\,(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)$
$\displaystyle\qquad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)f(s,\varphi(s))ds,~{}t\in(0,\,T].$
(5.2)

Following the similar steps as in the proof of the Theorem 4.1, from equation
(5), we have

$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(T)$
$\displaystyle=\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)$
$\displaystyle~{}+\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)f(s,\varphi(s))ds.$

Since
$\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(T)$,
we have

$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)$
$\displaystyle=r\,\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)$
$\displaystyle~{}+r\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)f(s,\varphi(s))ds.$

Hence,

$\displaystyle\mathcal{I}_{0+}^{1-\,\gamma;\,\Psi}y(0)$
$\displaystyle=\frac{r}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}$
$\displaystyle\quad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)f(s,\varphi(s))ds.$
(5.3)

Using equation (5) in (5), we get

$\displaystyle y(t)$
$\displaystyle=\frac{r\,(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}$
$\displaystyle\qquad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)f(s,\varphi(s))ds$
$\displaystyle\qquad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)f(s,\varphi(s))ds,~{}t\in(0,\,T].$

Consider the operator $\mathcal{A}:\mathfrak{D}\rightarrow\mathcal{X}$ defined
by

$\displaystyle\mathcal{A}\varphi(t)$
$\displaystyle=\frac{r\,(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}$
$\displaystyle\qquad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)f(s,\varphi(s))ds$
$\displaystyle\quad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)f(s,\varphi(s))ds,~{}t\in(0,\,T].$
(5.4)

It follows that, $\mathcal{A}\varphi(t)$ is the solution of the linear BVP
(5.1). Using the Theorem 4.1, we have

$w_{0}(t)\leq\mathcal{A}\varphi(t)\leq
z_{0}(t),~{}\text{for}~{}\varphi\in\mathfrak{D}=[w_{0},\,z_{0}]~{}\text{and}~{}t\in(0,\,T].$

In particular, we have $w_{0}\preceq\mathcal{A}w_{0}$ and
$\mathcal{A}z_{0}\preceq z_{0}$ in $C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)$.
Part 2: In this part, we prove that
$\mathcal{A}:\mathfrak{D}\rightarrow\mathcal{X}$ is completely continuous
operator.
Firstly we prove that, $\mathcal{A}(\mathfrak{D})$ is uniformly bounded. Since
$(\Psi(\cdot)-\Psi(0))^{1-\gamma}f(\cdot,y(\cdot))$ is continuous on compact
interval $J$, it is bounded. Hence, there exists constant $\mathfrak{K}>0$
such that

$\underset{t\in
J}{\max}\left|(\Psi(t)-\Psi(0))^{1-\gamma}f(t,\,y(t))\right|\leq\mathfrak{K}.$
(5.5)

Using increasing nature of $\Psi$, condition (5.5) and the Lemma 2.1(b), for
any $t\in J,$ we obtain

$\displaystyle\left|(\Psi(t)-\Psi(0))^{1-\,\gamma}\mathcal{A}\varphi(t)\right|$
$\displaystyle=\left|\frac{r\,E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}\right.$
$\displaystyle\left.\qquad\quad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)f(s,\varphi(s))ds\right.$
$\displaystyle\left.\qquad+(\Psi(t)-\Psi(0))^{1-\,\gamma}\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)f(s,\varphi(s))ds\right|$
$\displaystyle\leq\frac{r\,E_{\alpha,\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\,E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}$
$\displaystyle\qquad\quad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}(\Psi(s)-\Psi(0))^{\gamma-1}\left|(\Psi(s)-\Psi(0))^{1-\gamma}f(s,\varphi(s))\right|ds$
$\displaystyle\quad+(\Psi(T)-\Psi(0))^{1-\,\gamma}E_{\alpha,\,\alpha}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)$
$\displaystyle\qquad\quad\times\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}(\Psi(s)-\Psi(0))^{\gamma-1}\left|(\Psi(s)-\Psi(0))^{1-\gamma}f(s,\varphi(s))\right|ds$
$\displaystyle\leq\frac{r\,E_{\alpha,\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\,E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}$
$\displaystyle\qquad\quad\times\mathfrak{K}\,\Gamma(\alpha-\gamma+1)\left[\mathcal{I}_{0+}^{\alpha-\,\gamma+1;\,\Psi}(\Psi(t)-\Psi(0))^{\gamma-1}\right]_{t=T}$
$\displaystyle+(\Psi(T)-\Psi(0))^{1-\,\gamma}E_{\alpha,\,\alpha}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\,\mathfrak{K}\,\Gamma(\alpha)\,\mathcal{I}_{0+}^{\alpha;\,\Psi}(\Psi(t)-\Psi(0))^{\gamma-1}$
$\displaystyle\leq\frac{r\,E_{\alpha,\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\,E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}$
$\displaystyle\qquad\quad\times\mathfrak{K}\,\Gamma(\alpha-\gamma+1)\frac{\Gamma(\gamma)}{\Gamma(\alpha+1)}(\Psi(T)-\Psi(0))^{\alpha}$
$\displaystyle+(\Psi(T)-\Psi(0))^{1-\,\gamma}E_{\alpha,\,\alpha}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\,\mathfrak{K}\,\Gamma(\alpha)\frac{\Gamma(\gamma)}{\Gamma(\gamma+\alpha)}(\Psi(T)-\Psi(0))^{\alpha+\gamma-1}$
$\displaystyle=\mathfrak{K}(\Psi(T)-\Psi(0))^{\alpha}$
$\displaystyle\times\quad\left\\{B\left(\alpha-\gamma+1,\gamma\right)\frac{r\,E_{\alpha,\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\,E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}\right.$
$\displaystyle\quad+B\left(\alpha,\gamma\right)E_{\alpha,\,\alpha}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\Big{\\}}$
$\displaystyle:=\omega$

From above inequality it follows that,

$\left\|\mathcal{A}\varphi\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\leq\omega,~{}\varphi\in\mathfrak{D}.$

This implies, $\mathcal{A}(\mathfrak{D})$ is uniformly bounded. Next, we prove
that $\mathcal{A}(\mathfrak{D})$ is equicontinuous. Let any $t_{1},t_{2}\in J$
such that $t_{2}>t_{1}$. Then, for any $\varphi\in\mathfrak{D}$, we obtain

$\displaystyle\left|\left(\Psi(t_{2})-\Psi(0)\right)^{1-\,\gamma}\mathcal{A}\varphi(t_{2})-\left(\Psi(t_{1})-\Psi(0)\right)^{1-\,\gamma}\mathcal{A}\varphi(t_{1})\right|$
$\displaystyle=\left|\left\\{\frac{r\,E_{\alpha,\,\gamma}\left(M(\Psi(t_{2})-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}\right.\right.$
$\displaystyle\left.\left.\qquad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)f(s,\varphi(s))ds\right.\right.$
$\displaystyle\left.\left.\qquad+(\Psi(t_{2})-\Psi(0))^{1-\,\gamma}\int_{0}^{t_{2}}\Psi^{\prime}(s)(\Psi(t_{2})-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t_{2})-\Psi(s))^{\alpha}\right)f(s,\varphi(s))ds\right\\}\right.$
$\displaystyle\left.\quad-\left\\{\frac{r\,E_{\alpha,\,\gamma}\left(M(\Psi(t_{1})-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}\right.\right.$
$\displaystyle\left.\left.\qquad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)f(s,\varphi(s))ds\right.\right.$
$\displaystyle\left.\left.\qquad+(\Psi(t_{1})-\Psi(0))^{1-\,\gamma}\int_{0}^{t_{1}}\Psi^{\prime}(s)(\Psi(t_{1})-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t_{1})-\Psi(s))^{\alpha}\right)f(s,\varphi(s))ds\right\\}\right|$
$\displaystyle\leq\frac{r\,E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}\left|E_{\alpha,\,\gamma}\left(M(\Psi(t_{2})-\Psi(0))^{\alpha}\right)-E_{\alpha,\,\gamma}\left(M(\Psi(t_{1})-\Psi(0))^{\alpha}\right)\right|$
$\displaystyle\qquad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}(\Psi(s)-\Psi(0))^{\gamma-1}\left|(\Psi(s)-\Psi(0))^{1-\gamma}f(s,\varphi(s))\right|ds$
$\displaystyle\quad+\left|(\Psi(t_{2})-\Psi(0))^{1-\,\gamma}E_{\alpha,\,\alpha}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\right.$
$\displaystyle\left.\quad\times\int_{0}^{t_{2}}\Psi^{\prime}(s)(\Psi(t_{2})-\Psi(s))^{\alpha-1}(\Psi(s)-\Psi(0))^{\gamma-1}\left|(\Psi(s)-\Psi(0))^{1-\gamma}f(s,\varphi(s))\right|ds\right.$
$\displaystyle\quad-\left.(\Psi(t_{1})-\Psi(0))^{1-\,\gamma}E_{\alpha,\,\alpha}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\right.$
$\displaystyle\left.\quad\times\int_{0}^{t_{1}}\Psi^{\prime}(s)(\Psi(t_{1})-\Psi(s))^{\alpha-1}(\Psi(s)-\Psi(0))^{\gamma-1}\left|(\Psi(s)-\Psi(0))^{1-\gamma}f(s,\varphi(s))\right|ds\right|$
$\displaystyle\leq\frac{r\,E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}\left|E_{\alpha,\,\gamma}\left(M(\Psi(t_{2})-\Psi(0))^{\alpha}\right)-E_{\alpha,\,\gamma}\left(M(\Psi(t_{1})-\Psi(0))^{\alpha}\right)\right|$
$\displaystyle\qquad\times\mathfrak{K}\Gamma(\alpha-\gamma+1)\left[\mathcal{I}_{0+}^{\alpha-\,\gamma+1;\,\Psi}(\Psi(t)-\Psi(0))^{\gamma-1}\right]_{t=T}$
$\displaystyle\quad+\left|\mathfrak{K}(\Psi(t_{2})-\Psi(0))^{1-\,\gamma}E_{\alpha,\,\alpha}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\Gamma(\alpha)\left[\mathcal{I}_{0+}^{\alpha;\,\Psi}(\Psi(t)-\Psi(0))^{\gamma-1}\right]_{t=t_{2}}\right.$
$\displaystyle\quad-\left.\mathfrak{K}(\Psi(t_{1})-\Psi(0))^{1-\,\gamma}E_{\alpha,\,\alpha}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\Gamma(\alpha)\left[\mathcal{I}_{0+}^{\alpha;\,\Psi}(\Psi(t)-\Psi(0))^{\gamma-1}\right]_{t=t_{1}}\right|$
$\displaystyle\leq\frac{\mathfrak{K}\,r\,E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}B(\alpha-\gamma+1,\gamma)(\Psi(T)-\Psi(0))^{\alpha}$
$\displaystyle\qquad\times\left|E_{\alpha,\,\gamma}\left(M(\Psi(t_{2})-\Psi(0))^{\alpha}\right)-E_{\alpha,\,\gamma}\left(M(\Psi(t_{1})-\Psi(0))^{\alpha}\right)\right|$
$\displaystyle\qquad+\mathfrak{K}\,E_{\alpha,\,\alpha}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)B(\alpha,\gamma)\left|(\Psi(t_{2})-\Psi(0))^{\alpha}-(\Psi(t_{1})-\Psi(0))^{\alpha}\right|.$
(5.6)

By Lemma 2.5, two parameter Mittag-Leffler function is uniformly continuous.
Therefore, we have

$\left|E_{\alpha,\,\gamma}\left(M(\Psi(t_{2})-\Psi(0))^{\alpha}\right)-E_{\alpha,\,\gamma}\left(M(\Psi(t_{1})-\Psi(0))^{\alpha}\right)\right|\rightarrow
0~{}\text{as}~{}\left|t_{2}-t_{1}\right|\rightarrow 0.$ (5.7)

Further, using the continuity of $\Psi$, we have

$\left|(\Psi(t_{2})-\Psi(0))^{\alpha}-(\Psi(t_{1})-\Psi(0))^{\alpha}\right|\rightarrow
0~{}\text{as}~{}\left|t_{2}-t_{1}\right|\rightarrow 0.$ (5.8)

Using the conditions (5.7) and (5.8) in the inequality (5), we obtain

$\left|\left(\Psi(t_{2})-\Psi(0)\right)^{1-\,\gamma}\mathcal{A}\varphi(t_{2})-\left(\Psi(t_{1})-\Psi(0)\right)^{1-\,\gamma}\mathcal{A}\varphi(t_{1})\right|\rightarrow
0~{}\text{as}~{}\left|t_{2}-t_{1}\right|\rightarrow 0.$

This proves $\mathcal{A}(\mathfrak{D})$ is equicontinuous set of family of
functions. Therefore, by Arzel$\grave{a}$-Ascoli theorem,
$\mathcal{A}(\mathfrak{D})$ is relatively compact. Note that, the continuity
of operator $\mathcal{A}$ follows from hypothesis (H1)(i). We have proved
$\mathcal{A}:\mathfrak{D}\rightarrow\mathcal{X}$ is completely continuous
operator.
Part 3: In this part, it is proved that
$\mathcal{A}:\mathfrak{D}\rightarrow\mathcal{X}$ is monotonically increasing
operator.
Let any $\delta_{1},\delta_{2}\in\mathfrak{D}$ with
$w_{0}\leq\delta_{1}\leq\delta_{2}\leq z_{0}$. Define
$\mathcal{B}(t)=f(t,\,\delta_{2}(t))-f(t,\,\delta_{1}(t)),\,t\in(0,T]$. Then
from hypothesis (H1)(ii), we have $\mathcal{B}(t)\geq 0,~{}t\in(0,T]$. For any
$t\in(0,T]$,

$\displaystyle\mathcal{A}\delta_{2}(t)-\mathcal{A}\delta_{1}(t)$
$\displaystyle=\left\\{\frac{r\,(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}\right.$
$\displaystyle\left.\qquad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)f(s,\delta_{2}(s))ds\right.$
$\displaystyle\left.\qquad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)f(s,\delta_{2}(s))ds\right\\}$
$\displaystyle-\left\\{\frac{r\,(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}\right.$
$\displaystyle\left.\qquad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)f(s,\delta_{1}(s))ds\right.$
$\displaystyle\left.\qquad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)f(s,\delta_{1}(s))ds\right\\}$
$\displaystyle=\frac{r\,(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}$
$\displaystyle\qquad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(s)\right)^{\alpha}\right)\mathcal{B}(s)ds$
$\displaystyle\qquad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)\mathcal{B}(s)ds.$

Since $\mathcal{B}(t)\geq 0,~{}t\in(0,T]$ and $\Psi$ is increasing function on
$J$, the integrands of both the integrals in the right hand side of the above
inequality is non-negative on $(0,T]$. Therefore, we have
$\mathcal{A}\delta_{2}(t)-\mathcal{A}\delta_{1}(t)\geq 0,~{}t\in(0,T]$. This
gives $\mathcal{A}\delta_{2}\succeq\mathcal{A}\delta_{1}$. Therefore,
$\mathcal{A}:\mathfrak{D}\rightarrow\mathcal{X}$ is monotonically increasing
operator.
Part 4: For each $n(n=1,2,3,\cdots)$ define $w_{n}=\mathcal{A}\,w_{n-1}$ and
$z_{n}=\mathcal{A}\,z_{n-1}$. By Part 1, it follows that

$w_{0}\preceq\mathcal{A}w_{0}=w_{1}~{}\text{and}~{}z_{1}=\mathcal{A}z_{0}\preceq
z_{0}\text{ in}~{}C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right).$

Therefore, by using increasing nature of an operator $\mathcal{A}$, we have

$w_{1}\preceq w_{2}\preceq\cdots\preceq w_{n}\preceq z_{n}\preceq\cdots\preceq
z_{2}\preceq z_{1}\text{ in}~{}C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right).$

This implies $\\{w_{n}\\}_{n=1}^{\infty}$ and $\\{z_{n}\\}_{n=1}^{\infty}$ are
the monotonic sequences in $\mathcal{A}(\mathfrak{D})\subseteq\mathcal{X}$
which are relatively compact also. Therefore, by applying Lemma 2.7, there
exists $w^{*},z^{*}\in C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$
such that

$\lim\limits_{n\rightarrow\infty}\left\|w_{n}-w^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\mbox{\Bbb
R}\right)}=0$ (5.9)

and

$\lim\limits_{n\rightarrow\infty}\left\|z_{n}-z^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\mbox{\Bbb
R}\right)}=0.$ (5.10)

Since for each $n$, $w_{n}=\mathcal{A}\,w_{n-1}$ and
$z_{n}=\mathcal{A}\,z_{n-1}$, by continuity of the operator $\mathcal{A}$ and
using the limits (5.9) and (5.10), we obtain

$w^{*}=\mathcal{A}w^{*}~{}\text{and}~{}z^{*}=\mathcal{A}z^{*}.$

Therefore, $w^{*}$ and $z^{*}$ are the fixed points of an operator
$\mathcal{A}$. Further, we know that, $y\in
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ is the solution of the
nonlinear $\Psi$-Hilfer BVP (1.4)-(1.5) if and only if it is the fixed point
of an operator $\mathcal{A}$. Thus, $w^{*}$ and $z^{*}$ are the solutions of
the nonlinear $\Psi$-Hilfer BVP (1.4)-(1.5).
Part 5: Finally, we prove that $w^{*}$ and $z^{*}$ are the minimal and the
maximal solutions respectively of the nonlinear $\Psi$-Hilfer BVP (1.4)-(1.5).
Let $y\in[w_{0},\,z_{0}]$ be any solution of the $\Psi$-Hilfer BVP
(1.4)-(1.5). Then,

$y=\mathcal{A}y~{}~{}\text{and}~{}~{}w_{0}\preceq y\preceq z_{0}.$

Since $\mathcal{A}$ is an increasing operator, we have

$w_{1}=\mathcal{A}w_{0}\preceq\mathcal{A}y=y\preceq\mathcal{A}z_{0}=z_{1}.$

Again, using the increasing nature of an operator $\mathcal{A}$, from above
inequality, we obtain

$w_{2}\preceq y\preceq z_{2}.$

Continuing in this way, we obtain

$w_{n}\preceq y\preceq z_{n},\,\,n=1,2,3,\cdots.$

Taking the limit as $n\rightarrow\infty$ in the above inequality with respect
to the norm $\left\|\cdot\right\|_{C_{1-\,\gamma;\Psi}\left(J,\mbox{\Bbb
R}\right)}$, we obtain

$w^{*}\preceq y\preceq z^{*}~{}\text{
in}~{}C_{1-\,\gamma;\Psi}\left(J,\mbox{\Bbb R}\right).$

Therefore, $w^{*}$ and $z^{*}$ are the minimal solution and the maximal
solution respectively in $C_{1-\,\gamma;\Psi}\left(J,\mbox{\Bbb R}\right)$ of
the nonlinear $\Psi$-Hilfer BVP (1.4)-(1.5). $\Box$

###### Theorem 5.2

Suppose that the conditions of Theorem 5.1 hold, and let there exists a
constant
$\tilde{L}\in\left[0,\,\frac{1}{(\Psi(T)-\Psi(0))^{\alpha}}\Omega^{-1}\right),$
where

$\displaystyle\Omega$
$\displaystyle=B(\alpha-\gamma+1,\gamma)\frac{r\,E_{\alpha,\,\gamma}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)E_{\alpha,\,\alpha+1-\,\gamma}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)}{\left[1-r\,E_{\alpha,\,1}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right)\right]}$
$\displaystyle\qquad\qquad+B(\alpha,\gamma)E_{\alpha,\,\alpha}\left(M\left(\Psi(T)-\Psi(0)\right)^{\alpha}\right).$
(5.11)

Further, assume that $f$ satisfies

$f(t,\,x_{2})-f(t,\,x_{1})\leq\tilde{L}(x_{2}-x_{1}),~{}\text{for any
}x_{1},x_{2}\in\mbox{\Bbb R}\,\,\text{and}\,\,x_{1}\leq x_{2}.$ (5.12)

Then, the $\Psi$-Hilfer BVP (1.4)-(1.5) has a unique solution $y^{*}$ in
$[w_{0},\,z_{0}]$. Moreover, for each $y_{0}\in[w_{0},\,z_{0}]$ the iterative
sequence defined by

$\displaystyle y_{n}(t)$
$\displaystyle=\frac{r\,(\Psi(t)-\Psi(0))^{\gamma-1}E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,\,1}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\right]}$
$\displaystyle\quad\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(s))^{\alpha}\right)f(s,\,y_{n-1}(s))ds$
$\displaystyle\qquad+\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)f(s,\,y_{n-1}(s))ds,~{}n=1,2,3,\cdots,$

is such that

$\lim\limits_{n\rightarrow\infty}\left\|y_{n}-y^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}=0$

and

$\left\|y_{n}-y^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\leq\varrho^{n}\left\|z_{0}-w_{0}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)},$

where

$\displaystyle\varrho=$
$\displaystyle\Omega\,(\Psi(T)-\Psi(0))^{\alpha}\,\tilde{L}.$ (5.13)

Proof: Let any $w_{1},z_{1}\in[w_{0},z_{0}]$ with $w_{1}\preceq z_{1}$. Then
using the definition of operator $\mathcal{A}$ defined in (5) and the
condition (5.12), we obtain

$\displaystyle\left(\left(\Psi(t)-\Psi(0)\right)^{1-\,\gamma}\left(\mathcal{A}z_{1}(t)-\mathcal{A}w_{1}(t)\right)\right)$
$\displaystyle=\left\\{\frac{r\,E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,\,1}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\right]}\right.$
$\displaystyle\quad\left.\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(s))^{\alpha}\right)f(s,\,z_{1}(s))ds\right.$
$\displaystyle\qquad\left.+(\Psi(t)-\Psi(0))^{1-\,\gamma}\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)f(s,\,z_{1}(s))ds\right\\}$
$\displaystyle\qquad-\left\\{\frac{r\,E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,\,1}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\right]}\right.$
$\displaystyle\quad\left.\times\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(s))^{\alpha}\right)f(s,\,w_{1}(s))ds\right.$
$\displaystyle\qquad\left.+(\Psi(t)-\Psi(0))^{1-\,\gamma}\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)f(s,\,w_{1}(s))ds\right\\}$
$\displaystyle\leq\frac{\tilde{L}\,r\,E_{\alpha,\,\gamma}\left(M(\Psi(t)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,\,1}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\right]}\int_{0}^{T}\Psi^{\prime}(s)(\Psi(T)-\Psi(s))^{\alpha-\gamma}E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(s))^{\alpha}\right)$
$\displaystyle\qquad\times(\Psi(s)-\Psi(0))^{\gamma-1}(\Psi(s)-\Psi(0))^{1-\,\gamma}\left[z_{1}(s)-w_{1}(s)\right]ds$
$\displaystyle\qquad+\tilde{L}(\Psi(t)-\Psi(0))^{1-\,\gamma}\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\alpha-1}E_{\alpha,\,\alpha}\left(M(\Psi(t)-\Psi(s))^{\alpha}\right)$
$\displaystyle\qquad\times(\Psi(s)-\Psi(0))^{\gamma-1}(\Psi(s)-\Psi(0))^{1-\,\gamma}\left[z_{1}(s)-w_{1}(s)\right]ds$
$\displaystyle\leq\frac{\tilde{L}\,r\,E_{\alpha,\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,\,1}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\right]}E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\left\|z_{1}-w_{1}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}$
$\displaystyle\qquad\times\Gamma(\alpha-\gamma+1)\left[\mathcal{I}_{0+}^{\alpha-\,\gamma+1;\,\Psi}(\Psi(t)-\Psi(0))^{\gamma-1}\right]_{t=T}$
$\displaystyle+\tilde{L}\,(\Psi(T)-\Psi(0))^{1-\,\gamma}E_{\alpha,\,\alpha}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\left\|z_{1}-w_{1}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\Gamma(\alpha)\,\mathcal{I}_{0+}^{\alpha;\,\Psi}(\Psi(t)-\Psi(0))^{\gamma-1}$
$\displaystyle\leq\frac{\tilde{L}\,r\,E_{\alpha,\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)}{\left[1-r\,E_{\alpha,\,1}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\right]}E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\left\|z_{1}-w_{1}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}$
$\displaystyle\qquad\times\Gamma(\alpha-\gamma+1)\frac{\Gamma(\gamma)}{\Gamma(\alpha+1)}(\Psi(T)-\Psi(0))^{\alpha}+\tilde{L}\,(\Psi(T)-\Psi(0))^{1-\,\gamma}E_{\alpha,\,\alpha}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)$
$\displaystyle\qquad\times\left\|z_{1}-w_{1}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\Gamma(\alpha)\frac{\Gamma(\gamma)}{\Gamma(\alpha+\gamma)}(\Psi(T)-\Psi(0))^{\alpha+\gamma-1}$
$\displaystyle=\tilde{L}\,(\Psi(T)-\Psi(0))^{\alpha}\left\|z_{1}-w_{1}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}$
$\displaystyle\qquad\times\left\\{B(\alpha-\gamma+1,\gamma)\frac{r\,E_{\alpha,\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)}{1-r\,E_{\alpha,\,1}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)}E_{\alpha,\,\alpha+1-\,\gamma}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\right.$
$\displaystyle\qquad\left.+B(\alpha,\gamma)E_{\alpha,\,\alpha}\left(M(\Psi(T)-\Psi(0))^{\alpha}\right)\right\\}.$

Therefore,

$\displaystyle\left\|\mathcal{A}z_{1}-\mathcal{A}w_{1}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}$
$\displaystyle\leq\tilde{L}\,(\Psi(T)-\Psi(0))^{\alpha}\left\|z_{1}-w_{1}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\Omega.$

Using (5.13), above inequality reduces to

$\left\|\mathcal{A}z_{1}-\mathcal{A}w_{1}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\leq\varrho\left\|z_{1}-w_{1}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)},~{}w_{0}\preceq w_{1}\preceq z_{1}\preceq z_{0}.$

Consider the sequences $\\{z_{n}\\}$ and $\\{w_{n}\\}$ defined in Theorem 5.1.
Then $z_{n}=\mathcal{A}z_{n-1}$ and
$w_{n}=\mathcal{A}w_{n-1}(n=1,2,3,\cdots)$. By repeated application of the
above inequality, we obtain

$\displaystyle\left\|z_{n}-w_{n}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}$
$\displaystyle=\left\|\mathcal{A}z_{n-1}-\mathcal{A}w_{n-1}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}$
$\displaystyle\leq\varrho\left\|z_{n-1}-w_{n-1}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}=\varrho\left\|\mathcal{A}z_{n-2}-\mathcal{A}w_{n-2}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}$
$\displaystyle\leq\varrho^{2}\left\|z_{n-2}-w_{n-2}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}=\varrho^{2}\left\|\mathcal{A}z_{n-3}-\mathcal{A}w_{n-3}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}$
$\displaystyle\leq\cdots\leq\varrho^{n}\left\|z_{0}-w_{0}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}.$ (5.14)

Using the condition on $\tilde{L}$, we obtain $0\leq\varrho<1$. This implies
that $\varrho^{n}\rightarrow 0$ as $n\rightarrow\infty$. Therefore, from the
inequality (5), it follows that

$\lim\limits_{n\rightarrow\infty}\left\|z_{n}-w_{n}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\rightarrow 0.$

By applying Theorem 5.1, there exists minimal solution $w^{*}$ and maximal
solution $z^{*}$ in $[w_{0},z_{0}]$ such that

$\mathcal{A}w^{*}=w^{*}~{}~{}\text{and}~{}~{}\mathcal{A}z^{*}=z^{*}.$ (5.15)

Further,

$\lim\limits_{n\rightarrow\infty}\left\|w_{n}-w^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}=0\,~{}\text{and}~{}\lim\limits_{n\rightarrow\infty}\left\|z_{n}-z^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}=0.$ (5.16)

Using the equations in (5.16) and the continuity of norm, we have

$0=\lim\limits_{n\rightarrow\infty}\left\|z_{n}-w_{n}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}=\left\|z^{*}-w^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}.$

This gives

$z^{*}=w^{*}:=y^{*}~{}\text{ in}~{}C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right).$ (5.17)

From equations (5.15) and (5.17), we have

$\mathcal{A}y^{*}=y^{*}.$

Thus, we have a unique $y^{*}\in\left[w_{0},\,z_{0}\right]$ such that

$\lim\limits_{n\rightarrow\infty}\left\|w_{n}-y^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}=0\,~{}\text{and}~{}\lim\limits_{n\rightarrow\infty}\left\|z_{n}-y^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}=0.$ (5.18)

Since $\\{w_{n}\\}_{n=1}^{\infty}\subseteq
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ is increasing bounded
sequence and $\\{z_{n}\\}_{n=1}^{\infty}\subseteq
C_{1-\,\gamma;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$ is decreasing bounded
sequence, from (5.18) it follows that

$w_{n}\preceq y^{*}\preceq z_{n}~{}\text{
in}~{}C_{1-\,\gamma;\Psi}\left(J,\mbox{\Bbb R}\right).$ (5.19)

For each $y_{0}\in\left[w_{0},\,z_{0}\right]$, consider the iterative sequence
$y_{n}=\mathcal{A}y_{n-1}$. Then using the increasing nature of an operator
$\mathcal{A}$ and the definitions of $w_{n}$ and $z_{n}$, we obtain

$w_{n}\preceq y_{n}\preceq z_{n}~{}\text{
in}~{}C_{1-\,\gamma;\Psi}\left(J,\mbox{\Bbb R}\right).$ (5.20)

Using the inequalities (5), (5.19) and (5.20), for each $t\in J$, we have

$\displaystyle\left(\left(\Psi(t)-\Psi(0)\right)^{1-\,\gamma}\left(y_{n}(t)-y^{*}(t)\right)\right)$
$\displaystyle\leq\left(\left(\Psi(t)-\Psi(0)\right)^{1-\,\gamma}\left(z_{n}(t)-w_{n}(t)\right)\right)$
$\displaystyle\leq\left|\left(\left(\Psi(t)-\Psi(0)\right)^{1-\,\gamma}\left(z_{n}(t)-w_{n}(t)\right)\right)\right|$
$\displaystyle\leq\left\|z_{n}-w_{n}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}$
$\displaystyle\leq\varrho^{n}\left\|z_{0}-w_{0}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}.$

Therefore,

$\left\|y_{n}-y^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\leq\varrho^{n}\left\|z_{0}-w_{0}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}.$ (5.21)

From the above inequality it follows that

$\lim\limits_{n\rightarrow\infty}\left\|y_{n}-y^{*}\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb
R}\right)}=0.$

Observe that the inequality (5.21) gives the error bound with respect to the
$\left\|\cdot\right\|_{C_{1-\,\gamma;\Psi}\left(J,\,\mbox{\Bbb R}\right)}$
between approximation $y_{n}$ and the exact solution $y^{*}$ of the nonlinear
$\Psi$-Hilfer BVP (1.4)-(1.5). $\Box$

##  6 Example

We consider a specific case of the problem (1.4)-(1.5) to illustrate the main
results that we acquired.

###### Example 6.1

Consider the following BVP for the nonlinear Caputo FDEs

${}^{C}\mathcal{D}^{\frac{1}{2}}_{0+}y(t)=\frac{\sqrt{\pi}}{10}-\frac{\sqrt{t}+1}{25}+\frac{1}{25}\sin\left(\frac{\sqrt{t}+1}{5}\right)+\frac{1}{25}\left(5y(t)-\sin\left(y(t)\right)\right),~{}t\in(0,\,1],~{}$
(6.1) $\displaystyle y(0)=\frac{1}{2}\,y(1).$ (6.2)

One can verify that $y^{*}(t)=\frac{\sqrt{t}+1}{5},~{}t\in[0,\,1]$ is an exact
solution of the problem (6.1)-(6.2). Comparing the above problem with the
nonlinear $\Psi$-Hilfer BVP (1.4)-(1.5), we obtain

$\alpha=\frac{1}{2},\beta=1,\gamma=\alpha+\beta(1-\alpha)=1,M=0,r=\frac{1}{2},T=1~{}\text{and}~{}\Psi(t)=t,t\in[0,\,1].$
(6.3)

In this case the weighted space $C_{1-\,\gamma;\Psi}\left([0,\,1],\,\mbox{\Bbb
R}\right)$ reduces to the space of continuous functions
$C\left([0,\,1],\,\mbox{\Bbb R}\right)$ endowed with the supremum norm.

Define the function $f:[0,\,1]\times\mbox{\Bbb R}\rightarrow\mbox{\Bbb R}$ by

$f(t,y)=\frac{\sqrt{\pi}}{10}-\frac{\sqrt{t}+1}{25}+\frac{1}{25}\sin\left(\frac{\sqrt{t}+1}{5}\right)+\frac{1}{25}\left(5y-\sin
y\right).$ (6.4)

(1) For any $k\in\mbox{\Bbb R}$ consider the function $\tilde{f}:\mbox{\Bbb
R}\rightarrow\mbox{\Bbb R}$ defined by

$\tilde{f}(y)=k+\frac{1}{25}\left(5y-\sin y\right),\,y\in\mbox{\Bbb R}.$

Then, $\tilde{f}^{\prime}(y)=\frac{1}{25}\left(5-\cos y\right)\geq
0,~{}\text{for all }~{}y\in\mbox{\Bbb R}.$ Therefore the function $\tilde{f}$
is increasing on ℝ for any real $k$. This implies the function
$f:[0,\,1]\times\mbox{\Bbb R}\rightarrow\mbox{\Bbb R}$ defined in (6.4) is
increasing in $y\in\mbox{\Bbb R}$ for each $t\in[0,1]$.
(2) Let any $y_{1},y_{2}\in\mbox{\Bbb R}$ with $y_{1}\leq y_{2}$. Then

$\displaystyle f(t,y_{2})-f(t,y_{1})$
$\displaystyle=\frac{1}{25}\left\\{5(y_{2}-y_{1})-\left(\sin y_{2}-\sin
y_{1}\right)\right\\}$
$\displaystyle\leq\frac{1}{25}\left\\{5|y_{2}-y_{1}|+|\sin y_{2}-\sin
y_{1}|\right\\}.$ (6.5)

Since $\sin y$ is continuous and differentiable on the interval
$[y_{1},y_{2}]$, applying mean value theorem, there exists
$\tilde{y}\in[y_{1},y_{2}]$ such that

$\frac{\sin y_{2}-\sin y_{1}}{y_{2}-y_{1}}=\cos\tilde{y}.$

This implies $|\sin y_{2}-\sin y_{1}|\leq|y_{2}-y_{1}|$. Therefore, the
inequality (6) reduces to

$\displaystyle f(t,y_{2})-f(t,y_{1})$
$\displaystyle\leq\frac{1}{25}\left\\{5|y_{2}-y_{1}|+|y_{2}-y_{1}|\right\\}$
$\displaystyle\leq\frac{6}{25}\left(y_{2}-y_{1}\right),~{}\text{for
any}~{}y_{1},y_{2}\in\mbox{\Bbb R}~{}\text{with}~{}y_{1}\leq y_{2}.$ (6.6)

Comparing the above inequality with (5.12), we have $\tilde{L}=\frac{6}{25}$.
(3) Next, we prove that $\tilde{L}\in\left[0,\,\Omega^{-1}\right)$ where
$\Omega$ is defied in (5.2). Using the values given in (6.3), the equation
(5.2) reduces to

$\Omega=B\left(\frac{1}{2}-1+1,1\right)\frac{\frac{1}{2}\,E_{\frac{1}{2},\,1}\left(0\right)E_{\frac{1}{2},\,\frac{1}{2}+1-\,1}\left(0\right)}{\left[1-\frac{1}{2}\,E_{\frac{1}{2},\,1}\left(0\right)\right]}+B\left(\frac{1}{2},1\right)E_{\frac{1}{2},\,\frac{1}{2}}\left(0\right).$

Using Lemma 2.5, we obtain

$E_{n_{1},\,n_{2}}(0)=\frac{1}{\Gamma\left(n_{2}\right)},~{}n_{1},~{}n_{2}>0.$
(6.7)

Therefore

$\Omega=\frac{\Gamma\left(\frac{1}{2}\right)\Gamma\left(1\right)}{\Gamma\left(\frac{1}{2}+1\right)}\frac{\frac{1}{2}\,\frac{1}{\Gamma\left(1\right)}\frac{1}{\Gamma\left(\frac{1}{2}\right)}}{\left[1-\frac{1}{2}\,\frac{1}{\Gamma\left(1\right)}\right]}+\frac{\Gamma\left(\frac{1}{2}\right)\Gamma\left(1\right)}{\Gamma\left(\frac{1}{2}+1\right)}\frac{1}{\Gamma\left(\frac{1}{2}\right)}=\frac{4}{\sqrt{\pi}}.$

Note that, for
$\tilde{L}=\frac{6}{25}~{}\mbox{and}~{}\Omega=\frac{4}{\sqrt{\pi}},$ we have
$\tilde{L}\in\left[0,\,\Omega^{-1}\right)$.
(4) Define $z_{0}(t)=\sqrt{t}+1,\,t\in[0,1]$ and
$w_{0}(t)=\frac{-(\sqrt{t}+1)}{6},\,t\in[0,1]$. Then,

$\displaystyle h_{1}(t):$
$\displaystyle=\,^{C}\mathcal{D}^{\frac{1}{2}}_{0+}z_{0}(t)=\frac{\sqrt{\pi}}{2},$
$\displaystyle h_{2}(t):$ $\displaystyle=f(t,z_{0}(t))$
$\displaystyle=\frac{\sqrt{\pi}}{10}-\frac{\sqrt{t}+1}{25}+\frac{1}{25}\sin\left(\frac{\sqrt{t}+1}{5}\right)+\frac{1}{25}\left(5\left(\sqrt{t}+1\right)-\sin\left(\sqrt{t}+1\right)\right),$
$\displaystyle h_{3}(t):$
$\displaystyle=\,^{C}\mathcal{D}^{\frac{1}{2}}_{0+}w_{0}(t)=\frac{-\sqrt{\pi}}{12},$
$\displaystyle h_{4}(t):$ $\displaystyle=f(t,w_{0}(t))$
$\displaystyle=\frac{\sqrt{\pi}}{10}-\frac{\sqrt{t}+1}{25}+\frac{1}{25}\sin\left(\frac{\sqrt{t}+1}{5}\right)+\frac{1}{25}\left(5\left(\frac{-(\sqrt{t}+1)}{6}\right)-\sin\left(\frac{-(\sqrt{t}+1)}{6}\right)\right).$

Figure 1: $z_{0}$ is an upper solution

Figure 2: $w_{0}$ is a lower solution

From Figure 1, it follows that

${}^{C}\mathcal{D}^{\frac{1}{2}}_{0+}z_{0}(t)\geq f(t,z_{0}(t)),\,t\in[0,1].$

Further, using the definition of $z_{0}$, we have
$z_{0}(0)=\frac{1}{2}z_{0}(1)$ and hence $b_{z_{0}}(t)=0$. This proves that
$z_{0}(t)=\sqrt{t}+1,\,t\in[0,1]$ is an upper solution of the problem
(6.1)-(6.2). Again, from Figure 2, it follows that

${}^{C}\mathcal{D}^{\frac{1}{2}}_{0+}w_{0}(t)\leq f(t,w_{0}(t)),\,t\in[0,1].$

Further, by definition of $w_{0}$, we have $w_{0}(0)=\frac{1}{2}w_{0}(1)$ and
$a_{w_{0}}(t)=0$. This implies $w_{0}(t)=-\frac{\sqrt{t}+1}{6},\,t\in[0,1]$ is
a lower solution of the problem (6.1)-(6.2). Since all the assumptions of
Theorem 5.2 are satisfied, it guarantee the existence of a unique solution
$y^{*}$ in $[w_{0},z_{0}]$ of the problem (6.1)-(6.2). Indeed,

$-\frac{\sqrt{t}+1}{6}\leq\frac{\sqrt{t}+1}{5}\leq\sqrt{t}+1,~{}\text{for all
}~{}t\in[0,1],$

implies

$w_{0}(t)\leq y^{*}(t)\leq z_{0}(t),~{}t\in[0,1].$

This implies $y^{*}\in[w_{0},z_{0}]$, where $y^{*},~{}w_{0}$ and $z_{0}$
defined above, respectively are the exact, lower and upper solutions of the
problem (6.1)-(6.2).
(5) Using (6.3) and (6.7), the sequence $\\{y_{n}\\}_{n=1}^{\infty}$ defined
in the Theorem 5.2 reduces to

$y_{n}(t)=\frac{1}{\sqrt{\pi}}\int_{0}^{1}\left(1-s\right)^{\frac{-1}{2}}f(s,y_{n-1}(s))ds+\frac{1}{\sqrt{\pi}}\int_{0}^{t}\left(t-s\right)^{\frac{-1}{2}}f(s,y_{n-1}(s))ds$

where $y_{0}\in[w_{0},z_{0}]$. By Theorem 5.2,

$\left\|y_{n}-y^{*}\right\|_{C\left([0,1],\,\mbox{\Bbb
R}\right)}\leq\varrho^{n}\left\|z_{0}-w_{0}\right\|_{C\left([0,1],\,\mbox{\Bbb
R}\right)}.$ (6.8)

Using (6.3) and the values of $\Omega$ and $\tilde{L}$ determined above, from
(5.13), we obtain

$\rho=\Omega\tilde{L}=\frac{4}{\sqrt{\pi}}\frac{6}{25}=\frac{24}{25\sqrt{\pi}}.$

Thus, from the inequality (6.8), we have

$\displaystyle\left\|y_{n}-y^{*}\right\|_{C\left([0,1],\,\mbox{\Bbb
R}\right)}$
$\displaystyle\leq\left(\frac{24}{25\sqrt{\pi}}\right)^{n}\underset{t\in[0,1]}{\sup}\left|z_{0}(t)-w_{0}(t)\right|$
$\displaystyle=\left(\frac{24}{25\sqrt{\pi}}\right)^{n}\underset{t\in[0,1]}{\sup}\left|\sqrt{t}+1-\left(-\frac{\sqrt{t}+1}{6}\right)\right|$
$\displaystyle=\left(\frac{24}{25\sqrt{\pi}}\right)^{n}\frac{7}{6}\,\underset{t\in[0,1]}{\sup}\left|\sqrt{t}+1\right|.$

Therefore,

$\left\|y_{n}-y^{*}\right\|_{C\left([0,1],\,\mbox{\Bbb
R}\right)}\leq\frac{7}{3}\left(\frac{24}{25\sqrt{\pi}}\right)^{n}.$ (6.9)

The inequality (6.9) gives the error between $n^{th}$ approximation $y_{n}$
and exact solution $y^{*}$ of the problem (6.1)-(6.2). Since
$\frac{24}{25\sqrt{\pi}}<1$, it follows that $y_{n}\rightarrow y^{*}$ in
$C\left([0,1],\,\mbox{\Bbb R}\right)$ as $n\rightarrow\infty$.

## Acknowledgement

The second author acknowledges the Science and Engineering Research Board
(SERB), New Delhi, India for the Research Grant (Ref: File no.
EEQ/2018/000407).

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