# Existence Theorem

###### Abstract

In this paper, we initially derive the equivalent fractional integral equation
to $\Psi$-Hilfer hybrid fractional differential equations and through it, we
prove the existence of a solution in the weighted space. The primary objective
of the paper is to obtain estimates on $\Psi$-Hilfer derivative and utilize it
to derive the hybrid fractional differential inequalities involving
$\Psi$-Hilfer derivative. With the assistance of these fractional differential
inequalities, we determine the existence of extremal solutions, comparison
theorems and uniqueness of the solution.

On the Nonlinear $\Psi$-Hilfer Hybrid Fractional Differential Equations

Kishor D. Kucche 1

<EMAIL_ADDRESS>
Ashwini D. Mali 2

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

Key words: $\Psi$-Hilfer fractional derivative; Fractional differential
inequalities; Existence and uniqueness; Extremal solutions; Comparison
theorems.
2010 Mathematics Subject Classification: 34A38, 26A33, 34A12, 34A40.

##  1 Introduction

The classical theory of fractional differential equations (FDEs) relating to
inequalities and comparison results has developed by Lakshmikantham et al.[1,
2, 3] and utilized this theory to demonstrate the qualitative and quantitative
properties of the solution of various classes of nonlinear FDEs. On the other
hand, Dhage and Lakshmikantham[4] inducted the study of integer order hybrid
nonlinear differential equations. They have proved an existence theorem and
built differential inequalities. The obtained inequalities at that point used
to investigate the existence of extremal solutions and a comparison result. On
the lines of [4], Zhao et al.[5] have developed the theory of hybrid FDEs
involving Riemann–Liouville (RL) fractional derivative operator and acquired
fundamental fractional differential inequalities, the existence of extremal
solutions and comparison principle. Other related works on the hybrid FDEs may
be found in [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].

The idea of the fractional derivative with respect to another function is
introduced by Kilbas et al.[17] in the sense of RL fractional derivative. On a
comparable line, Almeida[18] presented $\Psi$-Caputo fractional derivative and
investigated many fascinating properties of this operator. Hilfer[20]
presented fractional derivative operator
$\mathcal{D}_{a^{+}}^{\mu,\nu}(\cdot)$ with two parameters
$\mu\in(n-1,n),n\in\mathbb{N}$ and $\nu~{}(0\leq\nu\leq 1)$. The Hilfer
derivative $\mathcal{D}_{a^{+}}^{\mu,\nu}(\cdot)$ unifies the theory of FDEs
involving RL fractional derivative ($\nu=0$) and Caputo fractional derivative
($\nu=1$). The calculus of Hilfer derivative and the analysis of nonlinear
FDEs involving it very well may be found in [21, 22].

In [23], Sousa and Oliveira presented the Hilfer version of the fractional
derivative with respect to another function which we refer in the present
paper as $\Psi$-Hilfer fractional derivative. The importance of the
$\Psi$-Hilfer fractional derivative lies in the fact that it incorporates
several well recognized fractional derivative operators as it’s particular
cases, for example, RL[17], Caputo[17], $\Psi$-RL[17], Hadmard[17], Riesz[17],
Erd$\acute{e}$ly-Kober[17], $\Psi$-Caputo[18], Katugampola[19], Hilfer[20] and
so forth. In this way various properties of solutions of FDEs involving
different fractional derivative operators recorded in [23] can be analyzed
under one roof with a single fractional derivative operator. Investigation of
initial and boundary value problems for $\Psi$-Hilfer about existence,
uniqueness, data dependence and Ulam-Hyers stabilities may be found in [24,
25, 26, 27, 28, 29, 30, 31, 32, 33].

Motivated by the work of [4, 5], in the present paper, we consider the
following $\Psi$-Hilfer hybrid FDEs of the form

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{y(t)}{f(t,y(t))}\right]=g(t,y(t)),~{}a.e.~{}t\in(0,\,T],~{}$
(1.1)
$\displaystyle\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}y(t)|_{t=0}=y_{0}\in\mbox{\Bbb
R},$ (1.2)

where $0<\mu<1,~{}0\leq\nu\leq 1,~{}\xi=\mu+\nu(1-\mu)$,
${}^{H}\mathcal{D}^{\mu,\nu;\,\Psi}_{0^{+}}(\cdot)$ is the $\Psi$-Hilfer
fractional derivative of order $\mu$ and type $\nu$, $f\in C(J\times\mbox{\Bbb
R}\,,\mbox{\Bbb R}\setminus\\{0\\})$ is bounded, $J=[0,T]$ and
$g\in\mathfrak{C}(J\times\mbox{\Bbb R}\,,\mbox{\Bbb R})=\\{h~{}|~{}\text{the
map}~{}\omega\to h(\tau,\omega)\;\text{is continuous }\;\text{for
each}~{}\tau\;\text{and the map}~{}\tau\to h(\tau,\omega)\\\ \;\text{is
measurable}\;\text{for each}~{}\omega\\}.$

We obtain, the equivalent fractional integral equation to the $\Psi$-Hilfer
hybrid FDEs (1.1)-(1.2) and establish the existence of solution in the
weighted space $C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$. Our main objective here
is to obtain estimates on $\Psi$-Hilfer derivative and utilize it to develop
the hybrid fractional differential inequalities involving $\Psi$-Hilfer
derivative. Using the fractional differential inequalities in the setting of
$\Psi$-Hilfer derivative, we then derive the existence of extremal solutions,
comparison results and uniqueness of the solution.

  * •

For $~{}\nu=0,~{}\Psi(t)=t,~{}y_{0}=0$, the obtained outcomes in the current
paper incorporates the investigation of [5] relating to nonlinear hybrid FDEs
of the form

${}^{RL}\mathcal{D}^{\mu}_{0^{+}}\left[\frac{y(t)}{f(t,y(t))}\right]=g(t,y(t)),~{}a.e.~{}t\in(0,\,T],$
$\displaystyle y(0)=0.$

  * •

For $\mu=1,~{}\nu=1,~{}\Psi(t)=t$, the obtained outcomes in the current paper
incorporates the investigation of [4] pertaining to nonlinear integer order
hybrid differential equations of the form

$\displaystyle\frac{d}{dt}\left[\frac{y(t)}{f(t,y(t))}\right]=g(t,y(t)),~{}a.e.~{}t\in(0,\,T],$
$\displaystyle y(0)=y_{0}\in\mbox{\Bbb R}.$

  * •

For $f=1$, the obtained results are applicable to nonlinear $\Psi$-Hilfer FDEs
of the form

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}y(t)=g(t,y(t)),~{}a.e.~{}t\in(0,\,T],$
$\displaystyle\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}y(t)|_{t=0}=y_{0}\in\mbox{\Bbb
R}.$

  * •

For $f=1,$ $\nu=0$ (in this case $\xi=\mu$), $\Psi(t)=t$, the acquired results
includes the study of nonlinear FDEs involving Riemann–Liouville
derivative[34]

${}^{RL}\mathcal{D}^{\mu}_{0^{+}}y(t)=g(t,y(t)),~{}a.e.~{}t\in(0,\,T],$
$\displaystyle[t^{1-\mu}y(t)]_{t=0}=y_{0}\in\mbox{\Bbb R}.$

  * •

For $f=1,~{}\nu=1$ (in this case $\xi=1$), $\,\Psi(t)=t$, the acquired results
includes the study of nonlinear FDEs involving Caputo derivative[1]

${}^{C}\mathcal{D}^{\mu}_{0^{+}}y(t)=g(t,y(t)),~{}a.e.~{}t\in(0,\,T],$
$\displaystyle y(0)=y_{0}\in\mbox{\Bbb R}.$

We have referenced above just a couple of extraordinary cases. Aside from
these, for various selections of parameters $\mu$, $\nu$ and the function
$\Psi$, the obtained outcomes in the current paper additionally hold for
nonlinear hybrid FDEs with well known fractional derivative operators recorded
in [23], for example, RL, Caputo,$\Psi$-RL, $\Psi$-Caputo, Hadmard,
Katugampola, Riesz, Erd$\acute{e}$ly-Kober, Hilfer and so forth.

The plan of the paper is as follows: In the section 2, we give some
definitions and results which are useful to prove the main results. In section
3, we derive an equivalent integral equation(IE) to the problem (1.1)-(1.2)
and establish existence of solution to it. In section 4, we obtain estimates
on $\Psi$-Hilfer derivative. Section 5 deals with hybrid fractional
differential inequalities involving $\Psi$-Hilfer derivative. Section 6,
contribute the study of maximal and minimal solutions. In section 7, we obtain
comparison theorems. Finally, uniqueness result is proved in the section 8.

##  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
$\Psi^{\prime}(t)\neq 0$, $\forall~{}t\in[a,b]$. We consider the weighted
space

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

endowed with the norm

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

The weighted space $C_{1-\xi\,;\Psi}(\left[a,b\right],\,\mbox{\Bbb R})$ is a
partially ordered Banach space with the norm
$\left\|\cdot\right\|_{C_{1-\xi;\Psi}\left[a,b\right]}$ and the partial
ordering relation $\preceq$ defined by

$h_{1}\preceq h_{2}~{}\text{if and only
if}~{}\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\xi}h_{1}\left(t\right)\leq\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\xi}h_{2}\left(t\right),~{}t\in\left[a,b\right],$

where $h_{1},h_{2}\in C_{1-\xi\,;\Psi}(\left[a,b\right],\,\mbox{\Bbb R})$.
Note that

  1. (i)

$h_{1}\prec h_{2}~{}\text{if and only
if}~{}\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\xi}h_{1}\left(t\right)<\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\xi}h_{2}\left(t\right),~{}t\in\left[a,b\right];$

  2. (ii)

$h_{1}=h_{2}~{}\text{if and only
if}~{}\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\xi}h_{1}\left(t\right)=\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\xi}h_{2}\left(t\right),~{}t\in\left[a,b\right];$

  3. (iii)

$h_{1}\succ h_{2}~{}\text{if and only
if}~{}\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\xi}h_{1}\left(t\right)>\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{1-\xi}h_{2}\left(t\right),~{}t\in\left[a,b\right].$

######  Definition 2.1 ([17])

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

$I_{a^{+}}^{\mu\,;\Psi}h\left(t\right)=\frac{1}{\Gamma\left(\mu\right)}\int_{a}^{t}\Psi^{\prime}\left(s\right)\left(\Psi\left(t\right)-\Psi\left(s\right)\right)^{\mu-1}h\left(s\right)ds.$
(2.2)

######  Definition 2.2 ([23])

The $\Psi$-Hilfer fractional derivative of a function $h$ of order $0<\mu<1$
and type $0\leq\nu\leq 1$, is defined by

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

######  Lemma 2.1 ([17, 23])

Let $\chi,\delta>0$ and $\rho>n$. Then

  1. (i)

$\mathcal{I}_{a^{+}}^{\mu\,;\,\Psi}\mathcal{I}_{a^{+}}^{\chi\,;\,\Psi}h(t)=\mathcal{I}_{a^{+}}^{\mu+\chi\,;\,\Psi}h(t)$.

  2. (ii)

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

  3. (iii)

${}^{H}\mathcal{D}_{a^{+}}^{\mu,\,\nu\,;\,\Psi}\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{\xi-1}=0.$

  4. (iv)

${}^{H}\mathcal{D}_{a+}^{\mu,\,\nu\,;\,\psi}\left(\psi\left(t\right)-\psi\left(a\right)\right)^{\rho-1}=\frac{\Gamma\left(\rho\right)}{\Gamma\left(\rho-\alpha\right)}\left(\psi\left(t\right)-\psi\left(a\right)\right)^{\mu-\rho-1}.$

######  Lemma 2.2 ([23])

If $h\in C^{n}[a,b]$, $n-1<\mu<n$ and $0\leq\nu\leq 1$, then

  1. (i)

$I_{a^{+}}^{\mu\,;\Psi}\text{
}^{H}\mathcal{D}_{a^{+}}^{\mu,\nu\,;\Psi}h\left(t\right)=h\left(t\right)-\overset{n}{\underset{k=1}{\sum}}\frac{\left(\Psi\left(t\right)-\Psi\left(a\right)\right)^{\xi-k}}{\Gamma\left(\xi-k+1\right)}h_{\Psi}^{\left[n-k\right]}I_{a^{+}}^{\left(1-\nu\right)\left(n-\mu\right)\,;\Psi}h\left(a\right)$
where,
$h_{\Psi}^{\left[n-k\right]}h(t)=\left(\frac{1}{\Psi^{\prime}(t)}\frac{d}{dt}\right)^{n-k}h(t)$.

  2. (ii)

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

######  Definition 2.3 ([17, 35])

Let $\eta>0\,(\,\eta\in\mbox{\Bbb R})$. Then the one parameter Mittag-Leffler
function is defined as

$E_{\eta}(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\Gamma(k\eta+1)}.$

######  Lemma 2.3 ([36])

Let $S$ be a non-empty closed, convex and bounded subset of the Banach algebra
$X$ and let $A:X\rightarrow X$ and $B:S\rightarrow X$ be two operators such
that

  * (a)

$A$ is Lipschitzian with a Lipschitz constant $\alpha$;

  * (b)

$B$ is completely continuous;

  * (c)

$y=AyBx\implies y\in S~{}\text{for all}~{}x\in S$ and

  * (d)

$\alpha M<1$ where $M=\sup\left\\{\left\|Bx\right\|:x\in S\right\\}$.

Then the operator equation $y=AyBy$ has a solution in $S$.

##  3 Existence of solution

###### Lemma 3.1

A function $y\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$ is the solution of the
Cauchy problem for hybrid FDEs (1.1)-(1.2) if and only if it is solution of
the following hybrid fractional IE

$\displaystyle y(t)$
$\displaystyle=f(t,y(t))\left\\{\frac{y_{0}}{f(0,y(0))}\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\xi-1}+\mathcal{I}_{0^{+}}^{\mu\,;\,\Psi}g(t,y(t))\right\\},~{}t\in(0,T].$
(3.1)

Proof: Let $y\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$ be the solution of the
Cauchy problem for hybrid FDEs (1.1)-(1.2). Operating $\Psi$-RL fractional
integral $\mathcal{I}_{0^{+}}^{\mu\,;\,\Psi}$ on the both sides of the
equation (1.1) and using Lemma 2.2(i), we obtain

$\frac{y(t)}{f(t,y(t))}-\frac{\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\xi-1}}{\Gamma(\xi)}\left[\mathcal{I}_{0^{+}}^{1-\xi\,;\Psi}\frac{y(t)}{f(t,y(t))}\right]_{t=0}=\mathcal{I}_{0^{+}}^{\mu\,;\Psi}g(t,y(t)),\,t\in(0,T].$

The above equation can be written as

$y(t)=f(t,y(t))\left\\{\frac{C}{\Gamma(\xi)}\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\xi-1}+\mathcal{I}_{0^{+}}^{\mu\,;\Psi}g(t,y(t))\right\\},~{}t\in(0,T],$
(3.2)

where

$C=\left[\mathcal{I}_{0^{+}}^{1-\xi\,;\Psi}\frac{y(t)}{f(t,y(t))}\right]_{t=0}.$

Next, we evaluate the value of $C$ using initial condition. Multiplying
$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}$ on both sides of
the equation (3.2), we get

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}y(t)=\frac{C\,}{\Gamma(\xi)}f(t,y(t))+\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}f(t,y(t))\,\mathcal{I}_{0^{+}}^{\mu\,;\Psi}g(t,y(t)),~{}t\in
J.$

Putting $t=0$ in the above equation and using the initial condition (1.2), we
obtain

$C=\frac{y_{0}\,\Gamma(\xi)}{f(0,y(0))}.$

Putting value of $C$ in the equation (3.2), we get

$y(t)=f(t,y(t))\left[\frac{y_{0}\,}{f(0,y(0))}\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\xi-1}+\mathcal{I}_{0^{+}}^{\mu\,;\Psi}g(t,y(t))\right],~{}t\in(0,T],$

which is required equivalent hybrid fractional IE to the hybrid FDEs
(1.1)-(1.2).

Conversely, let $y\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$ is a solution of
the hybrid IE (3.1). Then it can be written as

$\frac{y(t)}{f(t,y(t))}=\frac{y_{0}\,}{f(0,y(0))}\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\xi-1}+\mathcal{I}_{0^{+}}^{\mu\,;\Psi}g(t,y(t)),~{}t\in(0,T].$
(3.3)

Operating the $\Psi$-Hilfer derivative
${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}$ on both sides of the equation
(3.3) and using the Lemma 2.1(iii) and Lemma 2.2(ii), we obtain

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{y(t)}{f(t,y(t))}\right]$
$\displaystyle=\frac{y_{0}\,}{f(0,y(0))}\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\xi-1}+\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\mathcal{I}_{0^{+}}^{\mu\,;\Psi}g(t,y(t))$
$\displaystyle=g(t,y(t)).$

It remains to verify the initial condition (1.2). From the equation (3.3), we
have

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}\frac{y(t)}{f(t,y(t))}=\frac{y_{0}}{f(0,y(0))}+\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}\mathcal{I}_{0^{+}}^{\mu\,;\Psi}g(t,y(t)).$

At $t=0$, above equation reduces to

$\left[\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}\,y(t)\right]_{t=0}=y_{0},$

which gives the initial condition (1.2). This completes the proof. $\Box$

To prove the existence of solution to the hybrid FDEs (1.1)-(1.2), we need the
following hypotheses on $f$ and $g$:

  1. (H1)

The function $f\in C\left(J\times\mbox{\Bbb R},\mbox{\Bbb
R}\setminus\\{0\\}\right)$ is bounded and satisfies the following conditions:

    1. (i)

The mapping $v\rightarrow\frac{v}{f(t,v)}$ is increasing in ℝ a.e. $t\in(0,T]$
;

    2. (ii)

there exists $L>0$ such that

$\left|f(t,x)-f(t,y)\right|\leq L\left|x-y\right|,~{}t\in
J~{}\text{and}\,\,x,y\in\mbox{\Bbb R}.$

  2. (H2)

The function $g\in\mathfrak{C}(J\times\mbox{\Bbb R}\,,\mbox{\Bbb R})$ and
there exists a function $h\in C(J,\,\mbox{\Bbb R})$ such that

$\left|g(t,y)\right|\leq h(t),\,~{}a.e.\,\,t\in
J~{}\text{and}~{}\,y\in\mbox{\Bbb R}.$

###### Theorem 3.2

Assume that the hypotheses (H1)-(H2) hold. Then the hybrid FDEs (1.1)-(1.2)
has a solution $y\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$ provided

$L\left\\{\left|\frac{y_{0}\,}{f(0,y(0))}\right|+\frac{\left\|h\right\|_{\infty}\left(\Psi\left(T\right)-\Psi\left(0\right)\right)^{\mu}}{\Gamma(\mu+1)}\right\\}<1.$
(3.4)

Proof: Let $X:=\left(C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb
R}),\,\left\|\cdot\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\right)$. Then $X$ is a Banach algebra with the product of vectors
defined by $(xy)(t)=x(t)y(t),\,t\in J$. Define,

$S=\\{x\in X:\left\|x\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\leq R\\},$

where

$R=K\left\\{\left|\frac{y_{0}\,}{f(0,y(0))}\right|+\frac{\left\|h\right\|_{\infty}\left(\Psi\left(T\right)-\Psi\left(0\right)\right)^{\mu+1-\xi}}{\Gamma(\mu+1)}\right\\}$

and $K$ is bound on $f$. Clearly, $S$ is closed, convex and bounded subset of
$X$. Define two operators $A:X\rightarrow X$ and $B:S\rightarrow X$ by

$\displaystyle Ay(t)$ $\displaystyle=f(t,y(t)),\,t\in J,$ $\displaystyle
By(t)$
$\displaystyle=\frac{y_{0}\,\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\xi-1}}{f(0,y(0))}+\frac{1}{\Gamma\left(\mu\right)}\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\mu-1}g(s,y(s))\,ds,\,t\in(0,T].$

Then, the hybrid IE (3.1) is transformed into the following operator equation

$y=AyBy,\,y\in X.$

We prove that the operators $A$ and $B$ satisfies all the conditions of Lemma
2.3. The proof is given in the following steps:
Step 1: $A:X\rightarrow X$ is Lipschitz operator.
Using the hypothesis (H1)(ii), we obtain

$\displaystyle\left|\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}\left(Ax(t)-Ay(t)\right)\right|$
$\displaystyle=\left|\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}\left(f(t,x(t))-f(t,y(t))\right)\right|$
$\displaystyle\leq
L\left|\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}\left(x(t)-y(t)\right)\right|$
$\displaystyle\leq L\left\|x-y\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)}.$

This gives,

$\left\|Ax-Ay\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)}\leq
L\left\|x-y\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)}.$ (3.5)

Step 2: $B:S\rightarrow X$ is completely continuous.
(i) $B:S\rightarrow X$ is continuous.
Let $y_{n}\rightarrow y$ in $S$. Then,

$\displaystyle\left\|By_{n}-By\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)}$ $\displaystyle=\underset{t\in
J}{\max}\left|\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}\left(By_{n}(t)-By(t)\right)\right|$
$\displaystyle\leq\underset{t\in
J}{\max}\frac{\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\mu-1}\left|g(s,y_{n}(s))-g(s,y(s))\right|\,ds.$

By continuity of $g$ and Lebesgue dominated convergence theorem, from the
above inequality, we have

$\left\|By_{n}-By\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\rightarrow 0~{}\text{ as}~{}n\rightarrow\infty.$

This proves $B:S\rightarrow X$ is continuous.
(ii) $B(S)=\left\\{By:y\in S\right\\}$ is uniformly bounded.
Using hypothesis (H2), for any $y\in S$ and $t\in J$, we have

$\displaystyle\left|\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}By(t)\right|$
$\displaystyle\leq\left|\frac{y_{0}}{f(0,y(0))}\right|+\frac{\,\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\mu-1}\left|g(s,y(s))\right|\,ds$
$\displaystyle\leq\left|\frac{y_{0}}{f(0,y(0))}\right|+\frac{\,\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\mu-1}h(s)\,ds$
$\displaystyle\leq\left|\frac{y_{0}}{f(0,y(0))}\right|+\left\|h\right\|_{\infty}\,\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}\frac{\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}}{\Gamma\left(\mu+1\right)}$
$\displaystyle\leq\left|\frac{y_{0}}{f(0,y(0))}\right|+\frac{\left\|h\right\|_{\infty}\left(\Psi\left(T\right)-\Psi\left(0\right)\right)^{\mu+1-\xi}}{\Gamma\left(\mu+1\right)}.$

Therefore,

$\left\|By\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\leq\left|\frac{y_{0}}{f(0,y(0))}\right|+\frac{\left\|h\right\|_{\infty}\left(\Psi\left(T\right)-\Psi\left(0\right)\right)^{\mu+1-\xi}}{\Gamma\left(\mu+1\right)}.$
(3.6)

(iii) $B(S)$ is equicontinuous.
Let any $y\in S$ and $t_{1},t_{2}\in J$ with $t_{1}<t_{2}$. Then using
hypothesis (H2), we have

$\displaystyle\left|\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{1-\xi}By(t_{2})-\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}By(t_{1})\right|$
$\displaystyle=\left|\left\\{\frac{y_{0}}{f(0,y(0))}+\frac{\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{2}}\Psi^{\prime}(s)(\Psi(t_{2})-\Psi(s))^{\mu-1}g(s,y(s))\,ds\right\\}\right.$
$\displaystyle\left.-\left\\{\frac{y_{0}}{f(0,y(0))}+\frac{\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{1}}\Psi^{\prime}(s)(\Psi(t_{1})-\Psi(s))^{\mu-1}g(s,y(s))\,ds\right\\}\right|$
$\displaystyle\leq\left|\frac{\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{2}}\Psi^{\prime}(s)(\Psi(t_{2})-\Psi(s))^{\mu-1}\left|g(s,y(s))\right|\,ds\right.$
$\displaystyle\left.\qquad-\frac{\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{1}}\Psi^{\prime}(s)(\Psi(t_{1})-\Psi(s))^{\mu-1}\left|g(s,y(s))\right|\,ds\right|$
$\displaystyle\leq\left|\frac{\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{2}}\Psi^{\prime}(s)(\Psi(t_{2})-\Psi(s))^{\mu-1}h(s)\,ds\right.$
$\displaystyle\left.\qquad-\frac{\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{1}}\Psi^{\prime}(s)(\Psi(t_{1})-\Psi(s))^{\mu-1}h(s)\,ds\right|$
$\displaystyle\leq\left|\frac{\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{1-\xi}\left\|h\right\|_{\infty}}{\Gamma\left(\mu\right)}\int_{0}^{t_{2}}\Psi^{\prime}(s)(\Psi(t_{2})-\Psi(s))^{\mu-1}\,ds\right.$
$\displaystyle\left.\qquad-\frac{\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}\left\|h\right\|_{\infty}}{\Gamma\left(\mu\right)}\int_{0}^{t_{1}}\Psi^{\prime}(s)(\Psi(t_{1})-\Psi(s))^{\mu-1}\,ds\right|$
$\displaystyle=\frac{\left\|h\right\|_{\infty}}{\Gamma\left(\mu+1\right)}\left\\{(\Psi(t_{2})-\Psi(0))^{\mu+1-\xi}-(\Psi(t_{1})-\Psi(0))^{\mu+1-\xi}\right\\}.$

By the continuity of $\Psi$, from the above inequality it follows that

$\text{if}\left|t_{1}-t_{2}\right|\to
0~{}\text{then}\left|\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{1-\xi}By(t_{2})-\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}By(t_{1})\right|\to
0.$

From the parts (ii) and (iii), it follows that $B(S)$ is uniformly bounded and
equicontinous set in $X$. Then by Arzel$\grave{a}$-Ascoli theorem, $B(S)$ is
relatively compact. Therefore, $B:S\rightarrow X$ is a compact operator. Since
$B:S\rightarrow X$ is the continuous and compact operator, it is completely
continuous.
Step 3: For $y\in X,$ $y=AyBx\implies y\in S~{}\text{for all}~{}x\in S$.
Let any $y\in X$ and $x\in S$ such that $y=AyBx$. Using the hypothesis (H2)
and boundedness of $f$, for any $t\in J$, we have

$\displaystyle\left|\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}y(t)\right|$
$\displaystyle=\left|\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}Ay(t)Bx(t)\right|$
$\displaystyle=\left|\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}f(t,y(t))\left\\{\frac{y_{0}\,\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\xi-1}}{f(0,x(0))}+\frac{1}{\Gamma\left(\mu\right)}\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\mu-1}g(s,x(s))\,ds\right\\}\right|$
$\displaystyle\leq\left|f(t,y(t))\right|\left\\{\left|\frac{y_{0}}{f(0,x(0))}\right|+\frac{\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\mu-1}\left|g(s,x(s))\right|\,ds\right\\}$
$\displaystyle\leq
K\left\\{\left|\frac{y_{0}}{f(0,x(0))}\right|+\frac{\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t}\Psi^{\prime}(s)(\Psi(t)-\Psi(s))^{\mu-1}h(s)\,ds\right\\}$
$\displaystyle\leq
K\left\\{\left|\frac{y_{0}}{f(0,x(0))}\right|+\frac{\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu+1-\xi}\left\|h\right\|_{\infty}}{\Gamma\left(\mu+1\right)}\right\\}.$

This gives

$\left\|y\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)}\leq
K\left\\{\left|\frac{y_{0}}{f(0,x(0))}\right|+\frac{\left(\Psi\left(T\right)-\Psi\left(0\right)\right)^{\mu+1-\xi}\left\|h\right\|_{\infty}}{\Gamma\left(\mu+1\right)}\right\\}=R.$

This implies, $y\in S$.
Step 4: To prove $\alpha M<1$ where
$M=\sup\left\\{\left\|By\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)}:y\in S\right\\}$.
From inequality (3.6), we have

$\displaystyle
M=\sup\left\\{\left\|By\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)}:y\in
S\right\\}\leq\left|\frac{y_{0}}{f(0,y(0))}\right|+\frac{\left\|h\right\|_{\infty}\left(\Psi\left(T\right)-\Psi\left(0\right)\right)^{\mu+1-\xi}}{\Gamma\left(\mu+1\right)}.$

From the inequality (3.5), we have $\alpha=L$. Therefore, using the condition
(3.4), we have

$\alpha M\leq
L\left\\{\left|\frac{y_{0}}{f(0,y(0))}\right|+\frac{\left\|h\right\|_{\infty}\left(\Psi\left(T\right)-\Psi\left(0\right)\right)^{\mu+1-\xi}}{\Gamma\left(\mu+1\right)}\right\\}<1.$

From steps 1 to 4, it follows that all the conditions of Lemma 2.3 are
fulfilled. Consequently, applying Lemma 2.3, the operator equation $y=AyBy$
has a solution in $S$, which acts as a solution of hybrid FDEs (1.1)-(1.2).
$\Box$

##  4 Estimates on $\Psi$-Hilfer derivative

###### Theorem 4.1

Let $m\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$. Let $t_{1}\in(0,T]$ be such
that $m(t_{1})=0$ and $m(t)\leq 0,\,t\in(0,t_{1})$. Then,
${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}m(t_{1})\geq 0$.

Proof: Define

$M_{\Psi}(t)=\int_{0}^{t}\Psi^{\prime}(s)\left(\Psi(t)-\Psi(s)\right)^{-\xi}m(s)ds,~{}t\in
J.$

Let any $h>0$ such that $0<t_{1}-h<t_{1}$. Then

$\displaystyle M_{\Psi}(t_{1})-M_{\Psi}(t_{1}-h)$
$\displaystyle=\int_{0}^{t_{1}}\Psi^{\prime}(s)\left(\Psi(t_{1})-\Psi(s)\right)^{-\xi}m(s)ds-\int_{0}^{t_{1}-h}\Psi^{\prime}(s)\left(\Psi(t_{1}-h)-\Psi(s)\right)^{-\xi}m(s)ds$
$\displaystyle=\int_{0}^{t_{1}-h}\Psi^{\prime}(s)\left\\{\left(\Psi(t_{1})-\Psi(s)\right)^{-\xi}-\left(\Psi(t_{1}-h)-\Psi(s)\right)^{-\xi}\right\\}m(s)ds$
$\displaystyle\quad+\int_{t_{1}-h}^{t_{1}}\Psi^{\prime}(s)\left(\Psi(t_{1})-\Psi(s)\right)^{-\xi}m(s)ds.$

Let

$\displaystyle
I_{1}=\int_{0}^{t_{1}-h}\Psi^{\prime}(s)\left\\{\left(\Psi(t_{1})-\Psi(s)\right)^{-\xi}-\left(\Psi(t_{1}-h)-\Psi(s)\right)^{-\xi}\right\\}m(s)ds$

and

$\displaystyle
I_{2}=\int_{t_{1}-h}^{t_{1}}\Psi^{\prime}(s)\left(\Psi(t_{1})-\Psi(s)\right)^{-\xi}m(s)ds.$

Then

$M_{\Psi}(t_{1})-M_{\Psi}(t_{1}-h)=I_{1}+I_{2}.$ (4.1)

Since $\xi>0$ and $\Psi$ is increasing function, we have

$0<\left(\Psi(t_{1}-h)-\Psi(s)\right)^{\xi}<\left(\Psi(t_{1})-\Psi(s)\right)^{\xi},\,\text{for}~{}0<s<t_{1}-h<t_{1}.$

This gives

$\left(\Psi(t_{1})-\Psi(s)\right)^{-\xi}-\left(\Psi(t_{1}-h)-\Psi(s)\right)^{-\xi}<0,\,\text{for}~{}0<s<t_{1}-h<t_{1}.$

This coupled with hypothesis $m(s)\leq 0,s\in(0,t_{1})$ and the fact
$\Psi^{\prime}(s)>0,\,s\in J,$ we have

$\Psi^{\prime}(s)\left\\{\left(\Psi(t_{1})-\Psi(s)\right)^{-\xi}-\left(\Psi(t_{1}-h)-\Psi(s)\right)^{-\xi}\right\\}m(s)\geq
0,\,\text{for}~{}0<s<t_{1}-h<t_{1}.$

This implies $I_{1}\geq 0$. Therefore, the equation (4.1) reduces to

$M_{\Psi}(t_{1})-M_{\Psi}(t_{1}-h)\geq I_{2}.$ (4.2)

Since $\left(\Psi(t)-\Psi(0)\right)^{1-\xi}m(t)$ is continuous on $J$,
corresponding to $t_{1}$, there exists constant $K(t_{1})>0$ such that for
$0<t_{1}-h<s<t_{1},$

$-K(t_{1})(t_{1}-s)\leq\left(\Psi(t_{1})-\Psi(0)\right)^{1-\xi}m(t_{1})-\left(\Psi(s)-\Psi(0)\right)^{1-\xi}m(s)\leq
K(t_{1})(t_{1}-s).$

But by hypothesis $m(t_{1})=0$. Therefore, for $0<t_{1}-h<s<t_{1}$, we have

$-K(t_{1})(t_{1}-s)\leq-\left(\Psi(s)-\Psi(0)\right)^{1-\xi}m(s)\leq
K(t_{1})(t_{1}-s).$

This gives

$\left(\Psi(s)-\Psi(0)\right)^{1-\xi}m(s)\geq-K(t_{1})(t_{1}-s),~{}0<t_{1}-h<s<t_{1}.$
(4.3)

Since

$-K(t_{1})(t_{1}-s)>-K(t_{1})h.$ (4.4)

From inequalities (4.3) and (4.4), we have

$\left(\Psi(s)-\Psi(0)\right)^{1-\xi}m(s)>-K(t_{1})h,\,\,\,0<t_{1}-h<s<t_{1}.$

This implies

$\Psi^{\prime}(s)\left(\Psi(t_{1})-\Psi(s)\right)^{-\xi}m(s)\geq-
hK(t_{1})\Psi^{\prime}(s)\left(\Psi(t_{1})-\Psi(s)\right)^{-\xi}\left(\Psi(s)-\Psi(0)\right)^{\xi-1},$
(4.5)

for $0<t_{1}-h<s<t_{1}$. By increasing nature of $\Psi$, we have

$\Psi(t_{1}-h)<\Psi(s)<\Psi(t_{1}),\,s\in(t_{1}-h,t_{1}).$

Since $\xi\leq 1,$ from above inequality, we have

$\left(\Psi(t_{1}-h)-\Psi(0)\right)^{\xi-1}>\left(\Psi(s)-\Psi(0)\right)^{\xi-1}>\left(\Psi(t_{1})-\Psi(0)\right)^{\xi-1},~{}s\in(t_{1}-h,t_{1}).$
(4.6)

From the inequalities (4.5) and (4.6), we have

$\Psi^{\prime}(s)\left(\Psi(t_{1})-\Psi(s)\right)^{-\xi}m(s)>-hK(t_{1})\Psi^{\prime}(s)\left(\Psi(t_{1})-\Psi(s)\right)^{-\xi}\left(\Psi(t_{1}-h)-\Psi(0)\right)^{\xi-1}$

for $s\in(t_{1}-h,t_{1}).$ Integrating above inequality between $t_{1}-h$ to
$t_{1}$ and using the definition of $I_{2}$, we obtain

$\displaystyle I_{2}$
$\displaystyle>-hK(t_{1})\left(\Psi(t_{1}-h)-\Psi(0)\right)^{\xi-1}\int_{t_{1}-h}^{t_{1}}\Psi^{\prime}(s)\left(\Psi(t_{1})-\Psi(s)\right)^{-\xi}ds$
$\displaystyle=-hK(t_{1})\left(\Psi(t_{1}-h)-\Psi(0)\right)^{\xi-1}\frac{\left(\Psi(t_{1})-\Psi(t_{1}-h)\right)^{1-\xi}}{1-\xi}.$
(4.7)

From the inequalities (4.2) and (4), we have

$\frac{M_{\Psi}(t_{1})-M_{\Psi}(t_{1}-h)}{h}>-K(t_{1})\left(\Psi(t_{1}-h)-\Psi(0)\right)^{\xi-1}\frac{\left(\Psi(t_{1})-\Psi(t_{1}-h)\right)^{1-\xi}}{1-\xi}.$

Taking the limit as $h\rightarrow 0$ in the above inequality and using the
continuity of $\Psi$, we obtain

$\left[\frac{d}{dt}M_{\Psi}(t)\right]_{t=t_{1}}\geq 0.$

Since $\Psi^{\prime}(t_{1})>0$, using the definition of $M_{\Psi}$, we have

$\left[\frac{1}{\Psi^{\prime}(t)}\frac{d}{dt}\left(\frac{1}{\Gamma(1-\xi)}\int_{0}^{t}\Psi^{\prime}(s)\left(\Psi(t)-\Psi(s)\right)^{-\xi}m(s)ds\right)\right]_{t=t_{1}}\geq
0.$

This gives

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}m(t_{1})=\left[I_{0^{+}}^{\nu\left(1-\mu\right)\,;\Psi}\left(\frac{1}{\Psi^{\prime}\left(t\right)}\frac{d}{dt}\right)I_{0^{+}}^{\left(1-\nu\right)\left(1-\mu\right)\,;\Psi}m(t)\right]_{t=t_{1}}\geq
0.$

$\Box$

The dual of the Theorem 4.1 is also hold.

###### Theorem 4.2

Let $m\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$. Let $t_{1}\in(0,T]$ be such
that $m(t_{1})=0$ and $m(t)\geq 0,\,t\in(0,t_{1})$. Then,
${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}m(t_{1})\leq 0$.

Proof: By utilizing the hypothesis on $m$, it follows that $(-m)$ fulfills the
assumptions of Theorem 4.1 and the proof follows by applying it. $\Box$

##  5 Hybrid differential inequalities with $\Psi$-Hilfer derivative

###### Theorem 5.1

Let $f\in C\left(J\times\mbox{\Bbb R},\mbox{\Bbb
R}\setminus\\{0\\}\right),\,g\in\mathfrak{C}\left(J\times\mbox{\Bbb
R},\mbox{\Bbb R}\right)$ and assume that the hypothesis (H1)(i) hold. Let
$y,\,z\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$ are such that

$\displaystyle^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{y(t)}{f(t,y(t))}\right]$
$\displaystyle\leq g(t,y(t)),~{}a.e.~{}t\in(0,\,T],~{}$ (5.1)
$\displaystyle^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{z(t)}{f(t,z(t))}\right]$
$\displaystyle\geq g(t,z(t)),~{}a.e.~{}t\in(0,\,T],$ (5.2)

one of the inequalities being strict. Then

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}y(t)|_{t=0}<\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}z(t)|_{t=0}$

implies that

$y\prec z~{}\text{in}~{}C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R}).$

Proof: Assume that the conclusion of the theorem doesn’t hold. Using the
continuity of $\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}y(t)$
and $\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}z(t)$ on $J$,
there exists $t_{1}\in(0,\,T]$ such that

$\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}y(t_{1})=\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}z(t_{1})$

and

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}y(t)<\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}z(t),~{}\,\,t\in[0,t_{1}).$

This gives

$y(t_{1})=z(t_{1})~{}\text{and}~{}y(t)<z(t),\,~{}\,t\in(0,t_{1}).$ (5.3)

Define,

$Y(t)=\frac{y(t)}{f(t,y(t))}~{}~{}\text{and}~{}~{}Z(t)=\frac{z(t)}{f(t,z(t))},~{}t\in(0,T].$

Then $Y,Z\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$. Using the relations in the
equation (5.3) and the hypothesis (H1)(i), we obtain

$Y(t_{1})=Z(t_{1})~{}\text{and}~{}Y(t)\leq Z(t),~{}\,t\in(0,t_{1}).$

Define $m(t)=Y(t)-Z(t),~{}t\in(0,T]$. Then, $m\in
C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$. Further, $t_{1}\in(0,\,T]$ such that

$m(t_{1})=0~{}\text{and}~{}m(t)\leq 0,~{}t\in(0,\,t_{1}).$

Therefore by Theorem 4.1, we obtain

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}m(t_{1})\geq 0.$

This implies

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}Y(t_{1})\geq\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}Z(t_{1}).$

By inequality (5.1) and assuming inequality (5.2) is strict, we obtain

$g(t_{1},y(t_{1}))\geq\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}Y(t_{1})\geq\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}Z(t_{1})>g(t_{1},z(t_{1})).$

This is contradicts to the fact $y(t_{1})=z(t_{1})$. Therefore, we must have

$y\prec z~{}\text{in}~{}C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R}).$

$\Box$

###### Theorem 5.2

Assume that the conditions of Theorem 5.1 hold with nonstrict inequalities
(5.1) and (5.2). Further, assume that there exists a real number $L>0$ such
that

$g(t,x_{1})-g(t,x_{2})\leq
L\left(\frac{x_{1}}{f(t,x_{1})}-\frac{x_{2}}{f(t,x_{2})}\right),\,~{}a.e.\,\,t\in
J,$ (5.4)

for all $x_{1},x_{2}\in\mbox{\Bbb R}$ with $x_{1}\geq x_{2}$. Then

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}y(t)|_{t=0}\leq\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}z(t)|_{t=0}$

implies that

$y\preceq z~{}\text{in}~{}C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R}).$

Proof: Let any $\epsilon>0$. Set

$\frac{z_{\epsilon}(t)}{f(t,z_{\epsilon}(t))}=\frac{z(t)}{f(t,z(t))}+\epsilon\,E_{\mu}\left(2L\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}\right),~{}t\in(0,T],$
(5.5)

where $E_{\mu}(\cdot)$ is the one parameter Mittag-Leffler function. This
implies that

$\frac{z_{\epsilon}(t)}{f(t,z_{\epsilon}(t))}>\frac{z(t)}{f(t,z(t))},~{}t\in(0,T].$

By using the hypothesis (H1)(i), we have

$z_{\epsilon}(t)>z(t),~{}t\in(0,T].$ (5.6)

Define

$Z_{\epsilon}(t)=\frac{z_{\epsilon}(t)}{f(t,z_{\epsilon}(t))}~{}\text{and}~{}Z(t)=\frac{z(t)}{f(t,z(t))},~{}t\in(0,T].$

Then, the equation (5.5) takes the form

$Z_{\epsilon}(t)=Z(t)+\epsilon
E_{\mu}\left(2L\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}\right),~{}t\in(0,T].$

Operating $\Psi$-Hilfer fractional derivative operator
${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}$ on both sides of the above
equation and using the inequality (5.2), we obtain

$\displaystyle^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}Z_{\epsilon}(t)$
$\displaystyle=\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}Z(t)+\epsilon\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}E_{\mu}\left(2L\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}\right)$
$\displaystyle\geq
g(t,z(t))+\epsilon\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}E_{\mu}\left(2L\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}\right).$
(5.7)

But by Lemma 2.1(iv), we obtain

$\displaystyle^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}E_{\mu}\left(2L\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}\right)$
$\displaystyle=\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left\\{\sum_{k=0}^{\infty}\frac{[2L\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}]^{k}}{\Gamma(k\mu+1)}\right\\}$
$\displaystyle=\sum_{k=1}^{\infty}\frac{2^{k}L^{k}}{\Gamma(k\mu+1)}\frac{\Gamma(k\mu+1)}{\Gamma(k\mu+1-\mu)}\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{k\mu-\mu}$
$\displaystyle=\sum_{k=0}^{\infty}\frac{2^{k+1}L^{k+1}}{\Gamma(k\mu+\mu+1-\mu)}\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{k\mu+\mu-\mu}$
$\displaystyle=2LE_{\mu}\left(2L\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}\right).$
(5.8)

Therefore the inequality (5), reduces to

$^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}Z_{\epsilon}(t)\geq
g(t,z(t))+2L\epsilon
E_{\mu}\left(2L\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}\right),t\in(0,T].$
(5.9)

Using the condition on $g$ given in (5.4) and using the equation (5.5), we get

$\displaystyle g(t,z_{\epsilon}(t))-g(t,z(t))$ $\displaystyle\leq
L\left[\frac{z_{\epsilon}(t)}{f(t,z_{\epsilon}(t))}-\frac{z(t)}{f(t,z(t))}\right]$
$\displaystyle\leq
L\epsilon\,E_{\mu}\left(2L\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}\right),~{}t\in
J.$ (5.10)

Utilizing the inequality (5) in the inequality (5.9), we get

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}Z_{\epsilon}(t)$
$\displaystyle\geq g(t,z_{\epsilon}(t))-L\epsilon
E_{\mu}\left(2L\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}\right)+2L\epsilon
E_{\mu}\left(2L\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}\right)$
$\displaystyle\geq g(t,z_{\epsilon}(t))+L\epsilon
E_{\mu}\left(2L\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\mu}\right)$
$\displaystyle>g(t,z_{\epsilon}(t)).$

Therefore,

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{z_{\epsilon}(t)}{f(t,z_{\epsilon}(t))}\right]>g(t,z_{\epsilon}(t)),~{}t\in(0,T].$

Now, from the inequality (5.6), we have

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}z_{\epsilon}(t)|_{t=0}>\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}z(t)|_{t=0}$

and from the hypothesis

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}z(t)|_{t=0}\geq\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}y(t)|_{t=0}.$

Therefore,

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}z_{\epsilon}(t)|_{t=0}>\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}y(t)|_{t=0}.$
(5.11)

By applying Theorem 5.1 with $z=z_{\epsilon}$, the condition (5.11) implies

$y\prec z_{\epsilon}~{}\text{in}~{}C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R}).$
(5.12)

Taking $\epsilon\rightarrow 0$ in the equation (5.5) and using the hypothesis
(H1)(i), we have

$\lim\limits_{\epsilon\rightarrow 0}z_{\epsilon}(t)=z(t).$ (5.13)

Using (5.13)

$\displaystyle\lim\limits_{\epsilon\rightarrow
0}\left\|z_{\epsilon}-z\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)}=\lim\limits_{\epsilon\rightarrow
0}\left\\{\underset{t\in\left[0,T\right]}{\max}\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}\left|z_{\epsilon}(t)-z(t)\right|\right\\}=0.$

This gives $\lim\limits_{\epsilon\rightarrow 0}z_{\epsilon}=z$ in
$C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$. Therefore, the inequality (5.12)
reduces to $y\preceq z$ in $C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$. $\Box$

##  6 Maximal and minimal solutions

In this section, we will demonstrate the existence of maximal and minimal
solutions for the hybrid FDEs (1.1)-(1.2) in the weighted space
$C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$.

###### Definition 6.1

A solution $r$ of the hybrid FDEs (1.1)-(1.2) is said to be maximal solution
if for any other solution $y$ to the hybrid FDEs (1.1)-(1.2), we have
$y\preceq r$ in $C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$.

###### Definition 6.2

A solution $q$ of the hybrid FDEs (1.1)-(1.2) is said to be minimal solution
if for any other solution $y$ to the hybrid FDEs (1.1)-(1.2), we have
$q\preceq y$ in $C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$.

Let any $\epsilon>0$ and consider the following hybrid FDEs,

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{y(t)}{f(t,y(t))}\right]=g(t,y(t))+\epsilon,~{}a.e.~{}t\in(0,\,T],~{}$
(6.1)
$\displaystyle\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}y(t)|_{t=0}=y_{0}+\epsilon\in\mbox{\Bbb
R}.$ (6.2)

###### Theorem 6.1

Assume that the hypotheses (H1)-(H2) and the condition (3.4) hold. Then for
every small number $\epsilon>0,$ the hybrid FDEs (6.1)-(6.2) has a solution in
$C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$.

Proof: By the hypothesis, we have

$L\left\\{\left|\frac{y_{0}\,}{f(0,y(0))}\right|+\frac{\left\|h\right\|_{\infty}\left(\Psi\left(T\right)-\Psi\left(0\right)\right)^{\mu}}{\Gamma(\mu+1)}\right\\}<1,$

then there exists $\epsilon_{0}>0$ such that

$L\left\\{\left|\frac{y_{0}+\epsilon\,}{f(0,y(0))}\right|+\frac{\left(\left\|h\right\|_{\infty}+\epsilon\right)\left(\Psi\left(T\right)-\Psi\left(0\right)\right)^{\mu}}{\Gamma(\mu+1)}\right\\}<1,~{}0<\epsilon\leq\epsilon_{0}.$

Following the similar steps as in the proof of Theorem 3.2 for the existence
of a solution, one can complete the remaining part of the proof. $\Box$

###### Theorem 6.2

Assume that the hypotheses (H1)-(H2) and the condition (3.4) hold. Then for
every $\epsilon>0,$ the hybrid FDEs (1.1)-(1.2) has a maximal solution in
$C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$.

Proof: Let $\left\\{\epsilon_{n}\right\\}_{n=0}^{\infty}$ be a decreasing
sequence of positive real numbers such that
$\lim\limits_{n\rightarrow\infty}\epsilon_{n}=0$ where $\epsilon_{0}$ is a
positive real number satisfying the inequality

$L\left\\{\left|\frac{y_{0}+\epsilon_{0}\,}{f(0,y(0))}\right|+\frac{\left(\left\|h\right\|_{\infty}+\epsilon_{0}\right)\left(\Psi\left(T\right)-\Psi\left(0\right)\right)^{\mu}}{\Gamma(\mu+1)}\right\\}<1.$
(6.3)

The existence of number $\epsilon_{0}$ can be achieved in the view of the
inequality (3.4). Since $\left\\{\epsilon_{n}\right\\}_{n=0}^{\infty}$ is a
decreasing sequence, we have $\epsilon_{n}\leq\epsilon_{0},n\in\mbox{\Bbb
N}\cup\\{0\\}$ and one can verify that

$L\left\\{\left|\frac{y_{0}+\epsilon_{n}\,}{f(0,y(0))}\right|+\frac{\left(\left\|h\right\|_{\infty}+\epsilon_{n}\right)\left(\Psi\left(T\right)-\Psi\left(0\right)\right)^{\mu}}{\Gamma(\mu+1)}\right\\}<1,~{}\text{for
all}~{}n\in\mbox{\Bbb N}\cup\\{0\\}.$

In the view of above condition, Theorem 6.1 guarantee the existence of
solution $r(\cdot,\epsilon_{n})\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$ of the
hybrid FDEs

$\displaystyle~{}\begin{cases}&{}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{y(t)}{f(t,y(t))}\right]=g(t,y(t))+\epsilon_{n},~{}a.e.~{}t\in(0,\,T],\\\
&\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}y(t)|_{t=0}=y_{0}+\epsilon_{n}\in\mbox{\Bbb
R}.\end{cases}$ (6.4)

From (6.4), it follows that

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{r(t,\epsilon_{n})}{f(t,r(t,\epsilon_{n}))}\right]>g(t,r(t,\epsilon_{n})),~{}a.e.~{}t\in(0,\,T].$

Further, any solution $u\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$ of the hybrid
FDEs (1.1)-(1.2) satisfies

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{u(t)}{f(t,u(t))}\right]$
$\displaystyle\leq g(t,u(t)),~{}a.e.~{}t\in(0,\,T].$

By Theorem 5.1, the condition

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}u(t)|_{t=0}=y_{0}<y_{0}+\epsilon_{n}=\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t,\epsilon_{n})|_{t=0}$

implies

$u\prec r(\cdot,\epsilon_{n}),~{}\text{for all}~{}n\in\mbox{\Bbb
N}\cup\\{0\\}~{}\text{in}~{}C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R}).$ (6.5)

Let $r(t,\epsilon_{1})$ and $r(t,\epsilon_{2})$ be the solutions of the hybrid
FDEs (6.4). Then, we have

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{r(t,\epsilon_{1})}{f(t,r(t,\epsilon_{1}))}\right]$
$\displaystyle=g(t,r(t,\epsilon_{1}))+\epsilon_{1}$
$\displaystyle>g(t,r(t,\epsilon_{1}))+\epsilon_{2},$
${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{r(t,\epsilon_{2})}{f(t,r(t,\epsilon_{2}))}\right]\leq
g(t,r(t,\epsilon_{2}))+\epsilon_{2}$

and

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t,\epsilon_{1})|_{t=0}=y_{0}+\epsilon_{1}>y_{0}+\epsilon_{2}=\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t,\epsilon_{2})|_{t=0}.$

Again by applying Theorem 5.1, we have

$r(\cdot,\epsilon_{1})\succ
r(\cdot,\epsilon_{2})~{}\text{in}~{}C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R}).$

Proceeding in this way, we obtain, $r(\cdot,\epsilon_{n})$ is a bounded below
decreasing sequence in $C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$. Therefore, it is
convergent. Let $r\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$ such that

$\left\|r(\cdot,\epsilon_{n})-r\right\|_{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)}\rightarrow 0~{}\text{ as}~{}n\rightarrow\infty.$

Then, we have

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t)=\lim\limits_{n\rightarrow\infty}\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t,\epsilon_{n}),~{}t\in
J.$ (6.6)

We show that the convergence in (6.6) is uniform on $J$. For this it is enough
to prove that the sequence
$\\{\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t,\epsilon_{n})\\}$
is equicontinious on $J$. Let $t_{1},t_{2}\in J$ with $t_{1}>t_{2}$ be
arbitrary. Then,

$\displaystyle\left|\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}r(t_{1},\epsilon_{n})-\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{1-\xi}r(t_{2},\epsilon_{n})\right|$
$\displaystyle=\left|f(t_{1},r(t_{1},\epsilon_{n}))\left\\{\frac{y_{0}+\epsilon_{n}}{f(0,r(0,\epsilon_{n}))}+\frac{\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{1}}\Psi^{\prime}(s)(\Psi(t_{1})-\Psi(s))^{\mu-1}\left(g(s,r(s,\epsilon_{n}))+\epsilon_{n}\right)\,ds\right\\}\right.$
$\displaystyle\left.-f(t_{2},r(t_{2},\epsilon_{n}))\left\\{\frac{y_{0}+\epsilon_{n}}{f(0,r(0,\epsilon_{n}))}+\frac{\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{2}}\Psi^{\prime}(s)(\Psi(t_{2})-\Psi(s))^{\mu-1}\left(g(s,r(s,\epsilon_{n}))+\epsilon_{n}\right)\,ds\right\\}\right|$
$\displaystyle=\left|f(t_{1},r(t_{1},\epsilon_{n}))\left\\{\frac{y_{0}+\epsilon_{n}}{f(0,r(0,\epsilon_{n}))}+\frac{\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{1}}\Psi^{\prime}(s)(\Psi(t_{1})-\Psi(s))^{\mu-1}\left(g(s,r(s,\epsilon_{n}))+\epsilon_{n}\right)\,ds\right\\}\right.$
$\displaystyle\left.-f(t_{2},r(t_{2},\epsilon_{n}))\left\\{\frac{y_{0}+\epsilon_{n}}{f(0,r(0,\epsilon_{n}))}+\frac{\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{1}}\Psi^{\prime}(s)(\Psi(t_{1})-\Psi(s))^{\mu-1}\left(g(s,r(s,\epsilon_{n}))+\epsilon_{n}\right)\,ds\right\\}\right.$
$\displaystyle\left.+f(t_{2},r(t_{2},\epsilon_{n}))\left\\{\frac{y_{0}+\epsilon_{n}}{f(0,r(0,\epsilon_{n}))}+\frac{\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{1}}\Psi^{\prime}(s)(\Psi(t_{1})-\Psi(s))^{\mu-1}\left(g(s,r(s,\epsilon_{n}))+\epsilon_{n}\right)\,ds\right\\}\right.$
$\displaystyle\left.-f(t_{2},r(t_{2},\epsilon_{n}))\left\\{\frac{y_{0}+\epsilon_{n}}{f(0,r(0,\epsilon_{n}))}+\frac{\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{2}}\Psi^{\prime}(s)(\Psi(t_{2})-\Psi(s))^{\mu-1}\left(g(s,r(s,\epsilon_{n}))+\epsilon_{n}\right)\,ds\right\\}\right|$
$\displaystyle\leq\left|f(t_{1},r(t_{1},\epsilon_{n}))-f(t_{2},r(t_{2},\epsilon_{n}))\right|\times$
$\displaystyle\qquad\left\\{\left|\frac{y_{0}+\epsilon_{n}}{f(0,r(0,\epsilon_{n}))}\right|+\frac{\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{1}}\Psi^{\prime}(s)(\Psi(t_{1})-\Psi(s))^{\mu-1}\left(\left|g(s,r(s,\epsilon_{n}))\right|+\epsilon_{n}\right)\,ds\right\\}$
$\displaystyle+\left|f(t_{2},r(t_{2},\epsilon_{n}))\right|\left\\{\frac{\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{1}}\Psi^{\prime}(s)(\Psi(t_{1})-\Psi(s))^{\mu-1}\left(\left|g(s,r(s,\epsilon_{n}))\right|+\epsilon_{n}\right)\,ds\right.$
$\displaystyle\left.\qquad-\frac{\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{1-\xi}}{\Gamma\left(\mu\right)}\int_{0}^{t_{2}}\Psi^{\prime}(s)(\Psi(t_{2})-\Psi(s))^{\mu-1}\left(\left|g(s,r(s,\epsilon_{n}))\right|+\epsilon_{n}\right)\,ds\right\\}.$

Using the hypothesis (H2), we have
$\left|g(t,r(t,\epsilon_{n}))\right|+\epsilon_{n}\leq\left\|h\right\|_{\infty}+\epsilon_{n},\,t\in
J$. Therefore, from the above inequality, we obtain

$\displaystyle\left|\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}r(t_{1},\epsilon_{n})-\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{1-\xi}r(t_{2},\epsilon_{n})\right|$
$\displaystyle\leq\left|f(t_{1},r(t_{1},\epsilon_{n}))-f(t_{2},r(t_{2},\epsilon_{n}))\right|\left\\{\left|\frac{y_{0}+\epsilon_{n}}{f(0,r(0,\epsilon_{n}))}\right|+\frac{\left(\left\|h\right\|_{\infty}+\epsilon_{n}\right)\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{\mu+1-\xi}}{\Gamma\left(\mu+1\right)}\right\\}$
$\displaystyle\qquad+K^{*}\left(\left\|h\right\|_{\infty}+\epsilon_{n}\right)\left\\{\frac{\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{\mu+1-\xi}}{\Gamma\left(\mu+1\right)}-\frac{\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{\mu+1-\xi}}{\Gamma\left(\mu+1\right)}\right\\},$
(6.7)

where $K^{*}=\underset{(t,y)\in
J\times[-R,\,R]}{\sup}\left|f(t,r(t,\epsilon_{n}))\right|$. Since, $f$ is
continuous on compact set $J\times[-R,\,R]$, we have

$\left|f(t_{1},r(t_{1},\epsilon_{n}))-f(t_{2},r(t_{2},\epsilon_{n}))\right|\rightarrow
0~{}\text{ as}~{}\left|t_{1}-t_{2}\right|\rightarrow 0$ (6.8)

uniformly for all $n\in\mbox{\Bbb N}\cup\\{0\\}$. Using the condition (6.8)
and continuity of $\Psi$ function in the inequality (6), we obtain

$\left|\left(\Psi\left(t_{1}\right)-\Psi\left(0\right)\right)^{1-\xi}r(t_{1},\epsilon_{n})-\left(\Psi\left(t_{2}\right)-\Psi\left(0\right)\right)^{1-\xi}r(t_{2},\epsilon_{n})\right|\rightarrow
0~{}\text{ as}~{}\left|t_{1}-t_{2}\right|\rightarrow 0.$

This proves
$\\{\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t,\epsilon_{n})\\}$
is equicontinuous and hence
$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t,\epsilon_{n})\rightarrow\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t)$
converges uniformly on $J$. Next, we prove that
$r\in{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)}$ is a solution of the
hybrid FDEs (1.1)-(1.2). Since
$r(\cdot,\epsilon_{n})\in{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)}$ is
a solution of the hybrid FDEs (6.4), we have

$r(t,\epsilon_{n})=f(t,r(t,\epsilon_{n}))\left\\{\frac{y_{0}+\epsilon_{n}}{f(0,r(0,\,\epsilon_{n}))}\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\xi-1}+\mathcal{I}_{0^{+}}^{\mu\,;\,\Psi}\left(g(t,r(t,\epsilon_{n}))+\epsilon_{n}\right)\right\\},~{}t\in(0,T].$

Using the continuity of functions $f$ and $g$, taking the limit as
$n\rightarrow\infty$ in the above equation, we get

$r(t)=f(t,r(t))\left\\{\frac{y_{0}}{f(0,r(0))}\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{\xi-1}+\mathcal{I}_{0^{+}}^{\mu\,;\,\Psi}g(t,r(t))\right\\},~{}t\in(0,T].$

Thus, $r(t)$ is a solution of the hybrid FDEs (1.1)-(1.2). From the inequality
(6.5) it follows that
$u\preceq\lim\limits_{n\rightarrow\infty}r(\cdot,\epsilon_{n})~{}\text{in}~{}{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right)}$. Therefore $u\preceq
r~{}\text{in}~{}{C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)}$. This proves
$r$ is a maximal solution the hybrid FDEs (1.1)-(1.2) in
$C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$. $\Box$

###### Remark 6.3

The confirmation of the existence of minimal solution for the hybrid FDEs
(1.1)-(1.2) one can finish on comparable lines.

##  7 Comparison Theorems

###### Theorem 7.1

Assume that the hypotheses (H1)-(H2) and the condition (3.4) hold. If there
exists a function $u\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$ such that

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{u(t)}{f(t,u(t))}\right]\leq
g(t,u(t)),~{}a.e.~{}t\in(0,\,T],~{}$ (7.1)
$\displaystyle\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}u(t)|_{t=0}\leq
y_{0},$ (7.2)

then

$u\preceq r~{}\text{in}~{}C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right),$

where $r$ is a maximal solution of the hybrid FDEs (1.1)-(1.2).

Proof: Let $\epsilon>0$ be arbitrary small. By Theorem 6.2, the hybrid FDEs
(6.1)-(6.2) has a maximal solution $r(\cdot,\epsilon)\in
C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$. Further, the limit

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t)=\lim\limits_{\epsilon\rightarrow
0}\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t,\epsilon)$
(7.3)

is uniform on $J$, where $r\in C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$
is a maximal solution of the hybrid FDEs (1.1)-(1.2). As $r(t,\epsilon)$ is a
maximal solution of the hybrid FDEs (6.1)-(6.2). Therefore,

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{r(t,\epsilon)}{f(t,r(t,\epsilon))}\right]=g(t,r(t,\epsilon))+\epsilon,~{}a.e.~{}t\in(0,\,T],~{}$
(7.4)
$\displaystyle\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t,\epsilon)|_{t=0}=y_{0}+\epsilon\in\mbox{\Bbb
R}.$ (7.5)

From the above equations, we have

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{r(t,\epsilon)}{f(t,r(t,\epsilon))}\right]>g(t,r(t,\epsilon)),~{}a.e.~{}t\in(0,\,T],~{}$
(7.6)
$\displaystyle\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t,\epsilon)|_{t=0}>y_{0}\in\mbox{\Bbb
R}.$ (7.7)

From equations (7.2) and (7.7), we have

$\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}r(t,\epsilon)|_{t=0}>\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}u(t)|_{t=0}.$
(7.8)

Applying Theorem 5.1, from the inequalities (7.1), (7.6) and (7.8), we obtain

$u\prec r(\cdot,\epsilon)~{}\text{in}~{}C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb
R}\right).$

Taking the limit $\epsilon\rightarrow 0$ of above inequality and using the
equation (7.3), we obtain

$u\preceq r~{}\text{in}~{}C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right).$

$\Box$

The proof of the following theorem relating to the comparison of minimal and
upper solution can be finished on the comparable lines of the proof of Theorem
7.1.

###### Theorem 7.2

Assume that the hypotheses (H1)-(H2) and the condition (3.4) hold. If there
exist a function $v\in C_{1-\xi;\,\Psi}(J,\,\mbox{\Bbb R})$ such that

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{v(t)}{f(t,v(t))}\right]\geq
g(t,v(t)),~{}a.e.~{}t\in(0,\,T],$
$\displaystyle\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}v(t)|_{t=0}\geq
y_{0},$

then

$q\preceq v~{}\text{in}~{}C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right),$

where $q$ is a minimal solution of the hybrid FDEs (1.1)-(1.2).

##  8 Uniqueness of solution

In the accompanying theorem, we demonstrate the uniqueness of the solution to
the hybrid FDEs (1.1)-(1.2) through the Theorem 7.1.

###### Theorem 8.1

Assume that the hypotheses (H1)-(H2) and the condition (3.4) hold. Also if
there exists a function $G:J\times\mbox{\Bbb R}\rightarrow\mbox{\Bbb R}$ such
that

$g(t,y_{1})-g(t,y_{2})\leq
G\left(t,\,\frac{y_{1}}{f(t,y_{1})}-\frac{y_{2}}{f(t,y_{2})}\right),~{}a.e.~{}t\in(0,\,T],$
(8.1)

for all $y_{1},y_{2}\in\mbox{\Bbb R}$ with $y_{1}>y_{2}$. If the identically
zero function is the only solution of the $\Psi$-Hilfer FDEs

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}m(t)=G(t,m(t)),~{}t\in(0,\,T],$
$\displaystyle\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}m(t)|_{t=0}=0,$

then, the hybrid FDEs (1.1)-(1.2) has a unique solution.

Proof: By Theorem 3.2, the hybrid FDEs (1.1)-(1.2) has a solution in
$C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$. Suppose that $u_{1}$ and
$u_{2}$ are two solutions of the hybrid FDEs (1.1)-(1.2) with $u_{1}\succ
u_{2}$ in $C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$. Define a function
$m:(0,T]\rightarrow\mbox{\Bbb R}$ by

$m(t)=\frac{u_{1}(t)}{f(t,u_{1}(t))}-\frac{u_{2}(t)}{f(t,u_{2}(t))},~{}t\in(0,T].$

In the view of hypothesis (H1)(i), we obtain

$m\succ 0~{}\mbox{in}~{}C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right).$ (8.2)

Using the inequality (8.1), we get

${}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}m(t)$
$\displaystyle=\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{u_{1}(t)}{f(t,u_{1}(t))}\right]-\,^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}\left[\frac{u_{2}(t)}{f(t,u_{2}(t))}\right]$
$\displaystyle=g(t,u_{1}(t))-g(t,u_{2}(t))$ $\displaystyle\leq
G\left(t,\frac{u_{1}(t)}{f(t,u_{1}(t))}-\frac{u_{2}(t)}{f(t,u_{2}(t))}\right)$
$\displaystyle=G(t,m(t)).$

Further,

$\displaystyle\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}m(t)|_{t=0}$
$\displaystyle=\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}u_{1}(t)|_{t=0}-\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}u_{2}(t)|_{t=0}=0.$

Therefore,

$\displaystyle\begin{cases}&{}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}m(t)\leq
G(t,m(t)),t\in(0,\,T],\\\
&\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}m(t)|_{t=0}=0.\end{cases}$
(8.3)

By assumption the identically zero function is the maximal solution of

$\displaystyle\begin{cases}&{}^{H}\mathcal{D}^{\mu,\,\nu\,;\,\Psi}_{0^{+}}m(t)=G(t,m(t)),~{}t\in(0,\,T],\\\
&\left(\Psi\left(t\right)-\Psi\left(0\right)\right)^{1-\xi}m(t)|_{t=0}=0.\end{cases}$
(8.4)

Applying Theorem 7.1 to the problems (8.3) and (8.4) with $f(t,\,x)=1$, $g=G$
and $y_{0}=0$, we obtain

$m\preceq 0~{}\text{in}~{}C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$
(8.5)

The equation (8.5) contradicts to the equation (8.2). Thus, we must have
$u_{1}=u_{2}$ in $C_{1-\xi;\,\Psi}\left(J,\,\mbox{\Bbb R}\right)$. $\Box$

## Conclusion

Existence and uniqueness of solution, fractional differential inequalities and
comparison results acquired in the present paper for $\Psi$-Hilfer hybrid FDEs
includes the study of [1, 4, 5, 34]. Since the $\Psi$-Hilfer fractional
derivative gives distinctive fractional derivative operators for different
values of the parameters $\mu$, $\nu$ and the function $\Psi$, the outcomes of
the present paper are also valid for the derivative operators listed in [23]
as its special cases. Further, the fractional integral inequalities and
comparison results obtained in the setting of $\Psi$-Hilfer derivative can be
utilized to analyze the various qualitative and quantitative properties of
solutions for a different classes of $\Psi$-Hilfer hybrid FDEs.

## Acknowledgment

The first 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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