# Theory of Nonlinear Caputo-Katugampola Fractional Differential Equations

Saleh S. Redhwan
Department of Mathematics
Dr.Babasaheb Ambedkar Marathwada
University, Aurangabad, (M.S),431001, India
<EMAIL_ADDRESS>
&Sadikali L. Shaikh
Department of Mathematics
Maulana Azad College of arts,
Science and Commerce
Aurangabad, (M.S),431001, India

Mohammed S. Abdo1,2
1Department of Mathematics
Dr.Babasaheb Ambedkar Marathwada University
Aurangabad, (M.S),431001, India
2Hodeidah University
Al-Hodeidah, 3114, Yemen
<EMAIL_ADDRESS>
S.S. Redhwan<EMAIL_ADDRESS>
###### Abstract

This manuscript investigates the existence and uniqueness of solutions to the
first order fractional anti-periodic boundary value problem involving Caputo-
Katugampola (CK) derivative. A variety of tools for analysis this paper
through the integral equivalent equation of the given problem, fixed point
theorems of Leray–Schauder, Krasnoselskii’s, and Banach are used. Examples of
the obtained results are also presented.

_K_ eywords Katugampola fractional operator $\cdot$ fractional differential
equations $\cdot$ fixed point theorems

##  1 Introduction

In this manuscript, we investigate the sufficient conditions of the existence
and uniqueness of solutions for the CK fractional differential equation

${}^{c}D_{a^{+}}^{\alpha;\rho}y(t)=f(t,y(t)),\text{ where }0<\alpha<1,\text{
}t\in J=[a,b].$ (1.1)

with the anti-period boundary condition

$y(a)+y(b)=0,$ (1.2)

where ${}^{c}D_{a^{+}}^{\alpha;\rho}$ is CK fractional derivatives of order
$\alpha$, and $f:J\times\mathbb{R}\longrightarrow\mathbb{R}$ is the
appropriate function.

The subject of fractional differential equations has recently evolved as an
interesting and popular field of research; see the interesting books [21]
[23]. In fact, fractional derivatives provide an excellent tool for the
description of memory and hereditary properties of various materials and
processes. More and more researchers have found that fractional differential
equations play important roles in many research areas, such as physics,
chemical technology, population dynamics, biotechnology, and economics. For
examples and recent developments of the topic, we refer to the papers [8] [11]
[12] and the references cited therein.

In [19] the author introduced a new fractional integral, which generalizes the
Riemann-Liouville and the Hadamard integral into a single form. For further
properties such as expansion formulas, variational calculus applications,
control theoretical applications, convexity and integral inequalities and
Hermite-Hadamard type inequalities of this new operator and similar operators,
for example, see [13] [14] [15] [16]. The corresponding fractional derivatives
were introduced in [17] [20] [22], Which so-called katugampola fractional
operators. The existence and uniqueness results for the CK derivative are
given in [18], the author used the Peano theorem to obtain the existence and
uniqueness results of the Cauchy type problem

${}^{c}D_{0^{+}}^{\alpha;\rho}y(t)=f(t,y(t)),\text{ \ \ }t\in[0,b],$ (1.3)
$y^{(k)}(0)=y_{0}^{(k)},\text{ \ }k=0,1,...,m-1,$ (1.4)

where $m=[\alpha].$ In the same context, R. Almeida in [5], proved the
uniqueness result of the problem (1.3)-(1.4) involving
${}^{c}D_{a^{+}}^{\alpha;\rho}$ via Gronwall inequality type.

On the other hand, Oliveira and de Oliveira in [7], considered the initial
value problem for a nonlinear fractional differential equation including
Hilfer-Katugampola derivative of the form

${}^{\rho}D_{a^{+}}^{\alpha,\beta}y(t)=f(t,y(t)),\text{ \ \ }t\in J,$ (1.5)
${}^{\rho}I_{a^{+}}^{1-\gamma}y(a)=c,\text{ \
}\gamma=\alpha+\beta-\alpha\beta.$ (1.6)

They used the generalized Banach fixed point theorem to investigate the
existence and uniqueness results on problems (1.5)-(1.6).

The recent development of fractional differential equations and the
theoretical analysis can be seen in [1, 2, 3, 4, 9, 18]. As such, the CK type
fractional operators have not been studied and investigated in much detail as
yet as compared to the other classical operators. In this manuscript, we
introduce new results of the solution to the problem (1.1)-(1.2) involving CK
fractional operator. Further, we use some fixed point theorems to analyize our
results. To the best of our knowledge, the fractional boundary value problems
involving Caputo-Katugampola type fractional operators have not yet been
investigated and developed till the present day.

The paper is organized as follows. In the next section, we begin by
summarizing the forms of Katugampola fractional integral and CK type
fractional derivative, and we also present the background material related to
our work and prove an important lemma which plays a key role in the sequel.
The third section contains the existence and uniqueness results to the problem
(1.1)-(1.2) by means of fixed point theorems of (Banach, Leray-schauder,
Krasnoselskii’s). In the last section, we present some illustrative examples.

##  2 Preliminaries

In this partition, we recall some essential basic definitions, lemmas, and
preliminary facts related to our results throughout the paper. Let $J=[a,b]$
$\left(-\infty<a<b<\infty\right)\ $be a finite interval of $\mathbb{R}$.
Denote $\mathcal{C(}J,\mathbb{R})$ be the Banach space of all continuous
functions $h:J\longrightarrow\mathbb{R}$ endowed with the norm given by

$\left\|h\right\|_{\mathcal{C}}=\sup_{t\in J}\left|h(t)\right|:t\in J\\},$

where $h\in\mathcal{C(}J,\mathbb{R})$.
$\mathcal{C}^{n}\mathcal{(}J,\mathbb{R})$ $(n\in\mathbb{N}_{0})$ denotes the
set of mappings having $n$ times continuously differentiable on $J.$

For $a<b$, $c\in\mathbb{R}$ and $1\leq p<\infty,$ define the function space

$X_{c}^{p}(a,b)=\left\\{h:J\rightarrow\mathbb{R}:\left\|h\right\|_{X_{c}^{p}}=\left(\int_{a}^{b}\left|t^{c}h(t)\right|^{p}\frac{dt}{t}\right)^{\frac{1}{p}}<\infty\right\\}.$

for $p=\infty,$

$\left\|h\right\|_{X_{c}^{p}}=ess\sup_{a\leq t\leq
b}\left[\left|t^{c}h(t)\right|\right].$

Now we state some definitions of the generalized fractional operators were
introduced in [17, 18, 19].

######  Definition 2.1.

Let $t>a$ be two reals, $\alpha>0$, $\rho>0,$ and $h:J\rightarrow\mathbb{R}$
be an integrable function. The left-sided Katugampola fractional integral of
order $\alpha$ and type $\rho$ is defined by

$I_{a^{+}}^{\alpha;\rho}h(t)=\frac{1}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}\left(\frac{t^{\rho}-s^{\rho}}{\rho}\right)^{\alpha-1}h(s)ds,$
(2.1)

where, $\Gamma(.)$ is a gamma function.

######  Definition 2.2.

Let $n-1<\alpha<n$ , $(n=[\alpha]+1),$ $\rho>0,$ and
$h\in\mathcal{C}^{n}\mathcal{(}J,\mathbb{R}).$ The left-sided Katugampola
fractional derivative of order $\alpha$ with dependence on a parameter $\rho$
is defined as

$D_{a^{+}}^{\alpha;\rho}h(t)=\left(t^{1-\rho}\frac{d}{dt}\right)^{n}I_{a^{+}}^{n-\alpha;\rho}h(t)=\frac{\gamma^{n}}{\Gamma(1-\alpha)}\int_{a}^{t}\left(\frac{t^{\rho}-s^{\rho}}{\rho}\right)^{n-\alpha-1}\frac{h(s)}{s^{1-\rho}}ds\text{,
}t>a,$ (2.2)

where $\gamma=\left(t^{1-\rho}\frac{d}{dt}\right).$ In particular, if
$0<\alpha<1,$ $\rho>0,$ and $h\in C^{1}(J,\mathbb{R}),$ we have

$D_{a^{+}}^{\alpha;\rho}h(t)=\left(t^{1-\rho}\frac{d}{dt}\right)I_{a^{+}}^{1-\alpha;\rho}h(t)=\frac{\gamma}{\Gamma(1-\alpha)}\int_{a}^{t}\left(\frac{t^{\rho}-s^{\rho}}{\rho}\right)^{-\alpha}\frac{h(s)}{s^{1-\rho}}ds\text{,
}t>a.$

######  Definition 2.3.

Let $\alpha\geq 0,$ $n=[\alpha]+1.$ If
$h\in\mathcal{C}^{n}\mathcal{(}J,\mathbb{R}),$ we define the left sided CK
fractional derivatives of $h$ of order $\alpha$ with a parameter $\rho>0$ by

${}^{C}D_{a^{+}}^{\alpha;\rho}h(t)=D_{a^{+}}^{\alpha;\rho}\left[h(t)-\sum_{k=0}^{n-1}\frac{h_{\rho}^{(k)}(a)}{k!}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{k}\right],$
(2.3)

where $h_{\rho}^{(k)}(t)=\left(t^{1-\rho}\frac{d}{dt}\right)^{k}h(t).$ In case
$0<\alpha<1$, and $h\in\mathcal{C}^{1}\mathcal{(}J,\mathbb{R})$ we have

${}^{C}D_{a^{+}}^{\alpha;\rho}h(t)=D_{a^{+}}^{\alpha;\rho}\left[h(t)-h(a)\right].$

From last equation and (2.2), one deduces

${}^{C}D_{a^{+}}^{\alpha;\rho}h(t)=\frac{\gamma}{\Gamma(1-\alpha)}\int_{a}^{t}\left(\frac{t^{\rho}-s^{\rho}}{\rho}\right)^{-\alpha}\frac{\left[h(s)-h(a)\right]}{s^{1-\rho}}ds\text{,
}t>a.$

Notice that, if $\alpha\notin\mathbb{N}_{0}$, $h$ is an absolutely continuous
function on $J,$ then the CK fractional derivative exists a.e. Moreover, we
have

${}^{C}D_{a^{+}}^{\alpha;\rho}h(t)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(1-\alpha)}\int_{a}^{t}\left(\frac{t^{\rho}-s^{\rho}}{\rho}\right)^{-\alpha}\frac{h_{\rho}^{(1)}(s)}{s^{1-\rho}}ds\text{,
}t>a,$ $\displaystyle=$ $\displaystyle
I_{a^{+}}^{1-\alpha;\rho}h_{\rho}^{(1)}(t).$

Also, if $\alpha\in\mathbb{N},$
${}^{C}D_{a^{+}}^{\alpha;\rho}h(t)=h_{\rho}^{(n)}(t).$ Particularly,
${}^{C}D_{a^{+}}^{0;\rho}h(t)=h_{\rho}^{(0)}(t)=h(t).$

######  Lemma 2.4.

$I_{a^{+}}^{\alpha;\rho}$ is bounded on the function space $X_{c}^{p}(a,b).$

######  Lemma 2.5.

Let $\alpha>0,$ $\beta>0,$ $h$ $\in X_{c}^{p}(a,b)$ $(1\leq p<\infty),$
$\rho,c\in\mathbb{R},$ $\rho\geq c.$ Then we have

$I_{a^{+}}^{\alpha;\rho}I_{a^{+}}^{\beta;\rho}h(t)=I_{a^{+}}^{\alpha+\beta;\rho}h(t),\qquad^{c}D_{a^{+}}^{\alpha;\rho}\text{
}^{\rho}I_{a^{+}}^{\alpha}h(t)=h(t).$

######  Lemma 2.6.

Let $t>a,$ $\alpha,\delta\in(0,\infty),$ and
$I_{a^{+}}^{\alpha;\rho},D_{a^{+}}^{\alpha;\rho}$ and
${}^{C}D_{a^{+}}^{\alpha;\rho}$are according on (2.1), (2.2) and (2.3)
respectively. Then we have

$I_{a^{+}}^{\alpha;\rho}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{\delta-1}=\frac{\Gamma(\delta)}{\Gamma(\delta+\alpha)}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{\alpha+\delta-1},$
${}^{C}D_{a^{+}}^{\alpha;\rho}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{\delta-1}=\frac{\Gamma(\delta)}{\Gamma(\delta-\alpha)}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{\delta-\alpha-1},$

and

${}^{C}D_{a^{+}}^{\alpha;\rho}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{k}=0,\text{
\ }\alpha\geq 0,\text{ }k=0,1,...,n-1.$

Particularly, ${}^{C}D_{a^{+}}^{\alpha;\rho}\left(1\right)=0.$

######  Theorem 2.7.

[11] (Banach fixed point theorem) Let $(X;d)$ be a nonempty complete metric
space with $T:X\rightarrow X$ is a contraction mapping. Then map $T$ has a
fixed point.

######  Theorem 2.8.

[11] (Krasnoselskii’s fixed point theorem) Let $X$ be a Banach space, let
$\Omega$ be a bounded closed convex subset of $X$ and let $T_{1},T_{2}$ be
mapping from $\Omega$ into $X$ such that $T_{1}x+T_{2}y$ $\in$ $\Omega$ for
every pair $x,y\in\Omega$. If $T_{1}$ is contraction and $T_{2}$ is completely
continuous, then the equation $T_{1}x+T_{2}x$ $=x$ has a solution on $\Omega.$

######  Theorem 2.9.

[11] (Leray-Schauder Nonlinear Alternative). Let $X$ be a Banach space and
$\Omega\subseteq X$ closed and convex. Assume that $K$ is a relatively open
subset of $\Omega$ with $0\in K$ and $T:\overline{K}\longrightarrow\Omega$ is
a compact and continuous mapping. Then ethier

  1. 1.

$T$ has a fixed point in $\overline{K},$ or

  2. 2.

there exists $x\in\partial K$ such that $x=\lambda Tx$ for some
$\lambda\in(0,1)$, where $\partial K$ is boundary of $K.$

##  3 Existence and uniqueness theorems

In this partition, we present the existence and uniqueness results of
fractional boundary value problem (1.1)-(1.2) involving CK fractional
derivatives. To prove the existence of solutions to (1.1)-(1.2), we need the
following auxiliary Lemmas

######  Lemma 3.1.

Let $\alpha,\rho>0\ $and
$y\in\mathcal{C(}J,\mathbb{R})\cap\mathcal{C}^{1}\mathcal{(}J,\mathbb{R}).$
Then

  1. 1.

the CK fractional deferential equation

${}^{c}D_{a^{+}}^{\alpha;\rho}y(t)=0$

has a solutions

$y(t)=c_{0}+c_{1}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)+c_{2}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{2}+....+c_{n-1}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{n-1},$

where $c_{i}\in\mathbb{R}$, $i=0,1,2,...,n-1$ and $n=[\alpha]+1.$

  2. 2.

If
$y,^{C}D_{a^{+}}^{\alpha;\rho}y\in\mathcal{C(}J,\mathbb{R})\cap\mathcal{C}^{1}\mathcal{(}J,\mathbb{R})$.
Then

$I_{a^{+}}^{\alpha;\rho}\text{
}^{C}D_{a^{+}}^{\alpha;\rho}y(t)=y(t)+c_{0}+c_{1}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)+c_{2}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{2}+....+c_{n-1}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{n-1},$
(3.1)

where $c_{i}\in\mathbb{R}$, $i=0,1,2,...,n-1$ and $n=[\alpha]+1.$

###### Proof.

The first part, follows immediately from the fact

${}^{C}D_{a^{+}}^{\alpha;\rho}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{k}=0,\text{
\ }k=0,1,2,...,n-1.$

To prove the second part, we apply the operator
${}^{C}D_{a^{+}}^{\alpha;\rho}$ to $I_{a^{+}}^{\alpha;\rho}$
${}^{C}D_{a^{+}}^{\alpha;\rho}y(t)-y(t),$ and use Lemma 2.5, it follows that

${}^{C}D_{a^{+}}^{\alpha;\rho}\left[I_{a^{+}}^{\alpha;\rho}\text{
}^{C}D_{a^{+}}^{\alpha;\rho}y(t)-y(t)\right]=\text{
}^{C}D_{a^{+}}^{\alpha;\rho}I_{a^{+}}^{\alpha;\rho}\text{
}^{C}D_{a^{+}}^{\alpha;\rho}y(t)-^{C}D_{a^{+}}^{\alpha;\rho}y(t)=0.$

From the first part, we deduce there exist $c_{i}\in\mathbb{R}$
($i=0,1,2,...,n-1$) such that

$I_{a^{+}}^{\alpha;\rho}\text{
}^{C}D_{a^{+}}^{\alpha;\rho}y(t)-y(t)=c_{0}+c_{1}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)+c_{2}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{2}+....+c_{n-1}\left(\frac{t^{\rho}-a^{\rho}}{\rho}\right)^{n-1},$

which implies the law of composition (3.1). The proof is completed. ∎

######  Lemma 3.2.

Let $0<\alpha<1,$ $\rho>0$ and $g\in\mathcal{C}(J,\mathbb{R}).$ Then the
linear anti-preiodic boundary value problem

${}^{c}D_{a^{+}}^{\alpha;\rho}y(t)=g(t),\text{ \ }t\in J,$ (3.2)
$y(a)+y(b)=0,\qquad\qquad$ (3.3)

has a unique solution defined by

$y(t)=-\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}g(s)ds+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}(t^{\rho}-s^{\rho})^{\alpha-1}g(s)ds.$
(3.4)

###### Proof.

Applying the Katugampola fractional integral of order $\alpha$ to both sides
of equation in (3.2), and using Lemma 3.1, we get

$y(t)=c_{0}+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}(t^{\rho}-s^{\rho})^{\alpha-1}g(s)ds\text{,}$
(3.5)

where $c_{0}\in\mathbb{R}$. Using (3.5) in the boundary conditions of (3.3),
we get

$c_{0}=-\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}g(s)ds.$

which, on substituting in (3.5), yields the solution (3.4). The converse
follows by direct calculations. The proof is completed. ∎

######  Lemma 3.3.

Assume that (H1) holds. A function $y(t)$ solves the problem (1.1)-(1.2) if
and only if it is a fixed-point of the operator
$T:\mathcal{C}(J,\mathbb{R})\rightarrow\mathcal{C}(J,\mathbb{R})$ defined by

$\displaystyle Ty(t)$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}f(s,y(s))ds$
(3.6)
$\displaystyle+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}(t^{\rho}-s^{\rho})^{\alpha-1}f(s,y(s))ds.$

Our first result is based on the Banach fixed point theorem to obtain the
existence of a unique solution of problem (1.1)-(1.2).

######  Theorem 3.4.

Assume that $f:J\times\mathbb{R}\rightarrow\mathbb{R}$ be a continuous
function satisfying the Lipschitz condition:

(H1)

There exists a constant $L_{f}>0$ such that :

$\left|f(t,x)-f(t,y)\right|\leq L\left|x-y\right|,\text{ \ }\forall t\in
J,\text{ }x,y\in\mathbb{R}.$

If $L\mathcal{N}<1,$ where
$\mathcal{N}:=\frac{3}{2}\frac{\rho^{-\alpha}}{\Gamma(\alpha+1)}(b^{\rho}-a^{\rho})^{\alpha},$
then the boundary value problem (1.1)-(1.2) has a unique solution on $J$.

###### Proof.

Now, we first show that the operator
$T:\mathcal{C}(J,\mathbb{R})\rightarrow\mathcal{C}(J,\mathbb{R})$ defined by
(3.6) is well-defined, i.e., we show that
$T\mathcal{B}_{r}\subseteq\mathcal{B}_{r}\ $where

$\mathcal{B}_{r}=\\{y\in\mathcal{C}(J,\mathbb{R}),\left\|y\right\|\leq r\\},$
(3.7)

with choose $r\geq\frac{\mu\mathcal{N}}{1-L\mathcal{N}},$ and $\sup_{t\in
J}\left|f(t,0)\right|=\mu<\infty.$ For any $y\in\mathcal{B}_{r}$, we obtain by
our hypotheses that

$\displaystyle\left|Ty(t)\right|$ $\displaystyle\leq$
$\displaystyle\underset{t\in J}{\sup}\left|Ty(t)\right|$ $\displaystyle\leq$
$\displaystyle\underset{t\in
J}{\sup}\left\\{\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y(s))-f(t,0)\right|+\left|f(t,0)\right|ds\right.$
$\displaystyle\left.+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}(t^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y(s))-f(t,0)\right|+\left|f(t,0)\right|ds\right\\}$
$\displaystyle\leq$
$\displaystyle\frac{3}{2}\frac{\rho^{-\alpha}}{\Gamma(\alpha+1)}(b^{\rho}-a^{\rho})^{\alpha}\left(Lr+\mu\right)$
$\displaystyle<$ $\displaystyle\left(Lr+\mu\right)\mathcal{N}$
$\displaystyle\leq$ $\displaystyle r.$

which implies that $Ty\in\mathcal{B}_{r}.$Moreover, by (3.6), and lammas 2.5,
2.6, we obtain

$\text{ }^{C}D_{a^{+}}^{\alpha;\rho}Ty(t)=\text{
}^{C}D_{a^{+}}^{\alpha;\rho}\text{
}I_{a^{+}}^{\alpha;\rho}f(t,y(t))=f(t,y(t)).$

Since $f(t,y(t))$ is continuous on $J,$ Then
${}^{c}D_{a^{+}}^{\alpha;\rho}Ty(t)$ is continuous on $J,$ that is
$T\mathcal{B}_{r}\subseteq\mathcal{B}_{r}.$

Next, we apply the Banach fixed point theorem to prove that $T$ has a fixed
point. Indeed, it enough to show that $T$ is contraction map. Let
$y_{1},y_{2}\in\mathcal{C}(J,\mathbb{R})$ and for $t\in J,$ we have

$\displaystyle\left|Ty_{1}(t)-Ty_{2}(t)\right|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y_{1}(s))-f(s,y_{2}(s))\right|ds$
$\displaystyle+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}(t^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y_{1}(s))-f(s,y_{2}(s))\right|ds$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}L\left\|y_{1}-y_{2}\right\|ds$
$\displaystyle+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}(t^{\rho}-s^{\rho})^{\alpha-1}L\left\|y_{1}-y_{2}\right\|ds$
$\displaystyle\leq$
$\displaystyle\frac{3}{2}\frac{\rho^{-\alpha}L}{\Gamma(\alpha+1)}(b^{\rho}-s^{\rho})^{\alpha}\left\|y_{1}-y_{2}\right\|$
$\displaystyle=$ $\displaystyle L\mathcal{N}\left\|y_{1}-y_{2}\right\|,$

which gives $\left\|Ty_{1}-Ty_{2}\right\|\leq
L\mathcal{N}\left\|y_{1}-y_{2}\right\|$. The inequality $L\mathcal{N}<1$ shows
that $T$ is contraction mapping. As a consequence of Banach fixed point
theorem. Then the problem (1.1)-(1.2) has a unique solution. This complete the
proof. ∎

Next, we prove an existence result for the problem (1.1)-(1.2) by using
Leray–Schauder nonlinear alternative.

######  Theorem 3.5.

Assume that $f:J\times\mathbb{R}\rightarrow\mathbb{R}$ is continuous on $J.$
In addition, we assume that:

(H2)

There exist two functions $\psi:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ be a
nondecreasing continuous, and $\eta:J\rightarrow\mathbb{R}^{+}$ is a
continuous such that

$\left|f(t,y)\right|\leq\eta(t)\psi(\left\|y\right\|),\text{ \ }\forall t\in
J,\text{ }y\in\mathbb{R}.$

(H3)

There exists a constant $M>0$ such that

$\frac{\mathcal{N}\left\|\eta\right\|\psi\left(M\right)}{M}<1,$

where $\mathcal{N}\ $is defined as in Theorem 3.4. Then the boundary value
problem (1.1)-(1.2) has at least one solution on $J$.

###### Proof.

Firstly, we will prove that the operator $T$ defined by (3.6), maps bounded
sets into bounded sets in $\mathcal{C}(J,\mathbb{R})$. For a positive number
$r$, let $\mathcal{B}_{r}$ be a bounded ball in $\mathcal{C}(J,\mathbb{R})$
defined by (3.7). Then, for $t\in J$, we have

$\displaystyle\left|Ty(t)\right|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y(s))\right|ds$
$\displaystyle+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}(t^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y(s))\right|ds$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}\eta(s)\psi\left(\left\|y\right\|\right)ds$
$\displaystyle+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}(t^{\rho}-s^{\rho})^{\alpha-1}\eta(s)\psi\left(\left\|y\right\|\right)ds$
$\displaystyle\leq$
$\displaystyle\frac{3}{2}\frac{\rho^{-\alpha}}{\Gamma(\alpha+1)}(b^{\rho}-a^{\rho})^{\alpha}\left\|\eta\right\|\psi\left(r\right)$
$\displaystyle=$
$\displaystyle\mathcal{N}\left\|\eta\right\|\psi\left(r\right).$

In view of (H3), we obtain $\left\|Ty\right\|\leq r,$ i.e.
$\left(T\mathcal{B}_{r}\right)$ is uniformly bounded.

Next, we prove that $T$ maps bounded sets into equicontinuous sets of
$\mathcal{C}(J,\mathbb{R})$, i.e. $\left(T\mathcal{B}_{r}\right)$ is
equicontinuous. Let $t_{1},t_{2}\in J,$ with $t_{1}<t_{2}$ and for any $y$
$\in\mathcal{B}_{r}$, then we have

$\displaystyle\left|Ty(t_{1})-Ty(t_{2})\right|$ $\displaystyle\leq$
$\displaystyle\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t_{1}}s^{\rho-1}\left[(t_{1}^{\rho}-s^{\rho})^{\alpha-1}-(t_{2}^{\rho}-s^{\rho})^{\alpha-1}\right]\left|f(s,y(s)\right|)ds$
$\displaystyle+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{t_{1}}^{t_{2}}s^{\rho-1}(t_{2}^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y(s))\right|ds$
$\displaystyle\leq$
$\displaystyle\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t_{1}}s^{\rho-1}\left[(t_{1}^{\rho}-s^{\rho})^{\alpha-1}-(t_{2}^{\rho}-s^{\rho})^{\alpha-1}\right]\eta(s)\psi\left(\left\|y\right\|\right)ds$
$\displaystyle+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{t_{1}}^{t_{2}}s^{\rho-1}(t_{2}^{\rho}-s^{\rho})^{\alpha-1}\eta(s)\psi\left(\left\|y\right\|\right)ds$
$\displaystyle\leq$
$\displaystyle\left\|\eta\right\|\psi\left(r\right)\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t_{1}}s^{\rho-1}\left[(t_{1}^{\rho}-s^{\rho})^{\alpha-1}-(t_{2}^{\rho}-s^{\rho})^{\alpha-1}\right]ds$
$\displaystyle+\left\|\eta\right\|\psi\left(r\right)\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{t_{1}}^{t_{2}}s^{\rho-1}(t_{2}^{\rho}-s^{\rho})^{\alpha-1}ds$
$\displaystyle\leq$
$\displaystyle\frac{2\rho^{-\alpha}\left\|\eta\right\|\psi\left(r\right)}{\Gamma(\alpha+1)}(t_{2}^{\rho}-t_{1}^{\rho})^{\alpha}.$

As $t_{1}\longrightarrow t_{2}$ the right-hand side of the preceding
inequality is not dependent on $y$ and goes to zero. Consequently,
$T\mathcal{B}_{r}$ is equicontinuous i.e.

$\left|Ty(t_{1})-Ty(t_{2})\right|\rightarrow 0\text{, \ }\forall\text{
}\left|t_{2}-t_{1}\right|\rightarrow 0,\text{ }y\in\mathcal{B}_{r}.$

So, the compactness of $T$ follows by Ascoli Arzela’s theorem, we conclude
that $T$ is completely continuous.

Finally, we show there exists an open set
$\mathcal{U}\subseteq\mathcal{C}(J,\mathbb{R})$ with $y\neq\lambda Ty$ for
$\lambda\in(0,1)$ and $y\in\partial\mathcal{U}$.

Let $y\in\mathcal{B}_{r}$ be any solution of

$y=\lambda Ty,\text{ }\lambda\in(0,1).$ (3.8)

Then

$\displaystyle\left|y(t)\right|$ $\displaystyle=$
$\displaystyle\lambda\left|Ty(t)\right|$ (3.9) $\displaystyle<$
$\displaystyle\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y(s))\right|ds$
$\displaystyle+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}(t^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y(s))\right|ds$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}\eta(s)\psi\left(\left\|y\right\|\right)ds$
$\displaystyle+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}(t^{\rho}-s^{\rho})^{\alpha-1}\eta(s)\psi\left(\left\|y\right\|\right)ds$
$\displaystyle\leq$
$\displaystyle\frac{3}{2}\left\|\eta\right\|\psi\left(\left\|y\right\|\right)\frac{\rho^{-\alpha}}{\Gamma(\alpha+1)}(b^{\rho}-a^{\rho})^{\alpha},$

which leads to

$\frac{\left\|y\right\|}{\mathcal{N}\left\|\eta\right\|\psi\left(\left\|y\right\|\right)}<1.$

In view of (H2), there exists $M$ such that $\left\|y\right\|\neq M$. This
means that, any solution $y$ of equation (3.8) satisfies $\left\|y\right\|\neq
M,$ let

$\mathcal{U=\\{}z\in\mathbb{K};\left\|y\right\|<M\\}.$

Thus, the Leray–Schauder nonlinear alternative guarantees that (3.8) has a
fixed point in $\partial\mathcal{U}$, which is a solution of the boundary
value problem (1.1)-(1.2), The proof is over. ∎

We will study the next existence result by using
Krasnoselskii${}^{{}^{\prime}}$s fixed point theorem. To this end, we change
hypothesis (H2) into the following one:

(H4)

There exists a function $q(t)\in\mathcal{C}(J,\mathbb{R})$ suth that

$\left|f(t,y)\right|\leq q(t),\text{ \ }\forall t\in J,\text{
}y\in\mathbb{R}.$

######  Theorem 3.6.

Assume that $(H_{4})$ holds. In addition, we assume that:

$($H${}_{5})$

There exists a function $\delta(t)\in\mathcal{C}(J,\mathbb{R})$ suth that

$\left|f(t,x)-f(t,y)\right|\leq\delta(t)\left|x-y\right|,\text{ }\forall t\in
J,\text{ }x,y\in\mathbb{R}.$

If

$\Lambda:=\frac{1}{2}\frac{\rho^{-\alpha}\left\|\delta\right\|}{\Gamma(\alpha+1)}(b^{\rho}-a^{\rho})^{\alpha}<1,$
(3.10)

then the boundary value problem (1.1)-(1.2) has at least one solution on $J$.

###### Proof.

Consider the operator
$T:\mathcal{C}(J,\mathbb{R})\longrightarrow\mathcal{C}(J,\mathbb{R})$ defined
by (3.6). We define
$\mathcal{B}_{r_{0}}:=\\{y\in\mathcal{C}(J,\mathbb{R}):\left\|y\right\|\leq
r_{0}\\}$, $\sup_{t\in J}\left|q(t)\right|=\left\|q\right\|,$ and select a
suitable constant $r_{0}$ such that

$r_{0}=\mathcal{N}\left\|q\right\|+1,$

where $\mathcal{N}\ $is defined as in Theorem 3.4. Furthermore, we need to
analyze the operator $T$ into sum two operators $T_{1}+T_{2}$, as follows

$T_{1}y(t)=-\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}f(s,y(s))ds,$

and

$T_{2}y(t)=\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}(t^{\rho}-s^{\rho})^{\alpha-1}f(s,y(s))ds.$

Taking into account that $T_{1}$ and $T_{2}$ are defined on
$\mathcal{B}_{r_{0}}.$ The proof will be given in several steps.

Step 1: $T_{1}y_{1}+T_{2}y_{2}$ $\in\mathcal{B}_{r_{0}}$ for every
$y_{1},y_{2}\in\mathcal{B}_{r_{0}}$.

For $y_{1},y_{2}\in\mathcal{B}_{r_{0}}$, we have

$\displaystyle\left|T_{1}y_{1}(t)+T_{2}y_{2}(t)\right|$ $\displaystyle\leq$
$\displaystyle\left|T_{1}y_{1}(t)\right|+\left|T_{2}y(t)\right|$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y_{1}(s))\right|ds$
$\displaystyle+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t}s^{\rho-1}(t^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y_{2}(s))\right|ds$
$\displaystyle\leq$
$\displaystyle\frac{3}{2}\frac{\rho^{-\alpha}}{\Gamma(\alpha+1)}(b^{\rho}-a^{\rho})^{\alpha}\left\|q\right\|,$

which gives

$\left\|T_{1}y_{1}+T_{2}y_{2}\right\|\leq\mathcal{N}\left\|q\right\|\leq
r_{0}.$ (3.11)

This proves that $T_{1}y_{1}+T_{2}y_{2}$ $\in\mathcal{B}_{r_{0}}$ for every
$y_{1},y_{2}\in\mathcal{B}_{r_{0}}.$

Step 2: $T_{1}$ is a contration mapping on $\mathcal{B}_{r_{0}}$.

In view of hypothesis (H5), then for each $y_{1},y_{2}\in\mathcal{B}_{r_{0}}$
and $t\in$ $J$; we have

$\displaystyle\left|T_{1}y_{1}(t)-T_{1}y_{2}(t)\right|$ $\displaystyle\leq$
$\displaystyle\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y_{1}(s))-f(s,y_{2}(s))\right|ds$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{b}s^{\rho-1}(b^{\rho}-s^{\rho})^{\alpha-1}\left\|\delta\right\|\left\|y_{1}-y_{2}\right\|ds$
$\displaystyle\leq$
$\displaystyle\frac{1}{2}\frac{\rho^{-\alpha}\left\|\delta\right\|}{\Gamma(\alpha+1)}(b^{\rho}-a^{\rho})^{\alpha}\left\|y_{1}-y_{2}\right\|,$

which implies

$\left\|T_{1}y_{1}-T_{1}y_{2}\right\|\leq\Lambda\left\|y_{1}-y_{2}\right\|,$

It follows from the inequality (3.10) that $T_{1}$ is contraction mapping.

Step 3: The operator $T_{2}$ is completely continuous on
$\mathcal{B}_{r_{0}}$.

First, from the continuity of $f$, we conclude that the operator $T_{2}$ is
continuous.

Next, It is easy to verify that

$\left\|T_{2}y\right\|\leq\frac{\rho^{-\alpha}\left\|q\right\|}{\Gamma(\alpha+1)}(b^{\rho}-a^{\rho})^{\alpha}=\frac{2}{3}\mathcal{N}\left\|q\right\|=\frac{2}{3}r_{0}-1<r_{0}.$

This proves that $T_{2}$ is uniformly bounded on $\mathcal{B}_{r_{0}}$.

Finally, we show that $T_{2}$ is equicontinuous on $\mathcal{B}_{r_{0}}$. Let
us set $sup_{(t,y)\in J\times\mathcal{B}_{r_{0}}}$
$\left|f(t,y\right|=f_{0}<\infty,$ and let $y\in\mathcal{B}_{r_{0}}$ and
$t_{1},t_{2}\in$ $J$ with $t_{1}<t_{2}$. Then we have

$\displaystyle\left|T_{2}y(t_{1})-T_{2}y(t_{2})\right|$ $\displaystyle\leq$
$\displaystyle\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t_{1}}s^{\rho-1}\left[(t_{1}^{\rho}-s^{\rho})^{\alpha-1}-(t_{2}^{\rho}-s^{\rho})^{\alpha-1}\right]\left|f(s,y(s)\right|ds$
$\displaystyle+\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{t_{1}}^{t_{2}}s^{\rho-1}(t_{2}^{\rho}-s^{\rho})^{\alpha-1}\left|f(s,y(s))\right|ds$
$\displaystyle\leq$ $\displaystyle
f_{0}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{a}^{t_{1}}s^{\rho-1}\left[(t_{1}^{\rho}-s^{\rho})^{\alpha-1}-(t_{2}^{\rho}-s^{\rho})^{\alpha-1}\right]ds$
$\displaystyle+f_{0}\frac{\rho^{1-\alpha}}{\Gamma(\alpha)}\int_{t_{1}}^{t_{2}}s^{\rho-1}(t_{2}^{\rho}-s^{\rho})^{\alpha-1}ds$
$\displaystyle\leq$
$\displaystyle\frac{\rho^{-\alpha}f_{0}}{\Gamma(\alpha+1)}(t_{2}^{\rho}-t_{1}^{\rho})^{\alpha}.$

As $t_{1}\longrightarrow t_{2}$ the right-hand side of the last inequality is
not dependent on $y$ and goes to zero. Consequently,

$\left|T_{2}y(t_{1})-T_{2}y(t_{2})\right|\rightarrow 0\text{, \ }\forall\text{
}\left|t_{2}-t_{1}\right|\rightarrow 0,\text{ }y\in\mathcal{B}_{r_{0}}.$

This proves that $T_{2}$ is equicontinuous on $\mathcal{B}_{r_{0}}$. In view
of Arzela-Ascoli Theorem, it follows that $T_{2}$ is relatively compact on
$\mathcal{B}_{r_{0}}$. Thus, all the assumptions of Krasnosel’skii fixed point
theorem are satisfied. Therefore, we conclude that the boundary value problem
(1.1)-(1.2) has at least one solution on $J.$ ∎

##  4 Examples

Will be provided in the revised submission.

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