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Solvability of a fractional integral equation with the concept of measure of noncompactness.

Abstract

We study the existence of solutions for a new class of integral equations involving a fractional integral with respect to another function. Our techniques are based on the measure of non-compactness concept combined with a generalized version of Darbo's theorem. Some examples are presented to illustrate the obtained results.

1 Introduction

Many problems in several branches of science such as engineering, physics, biology and other disciplines can be modeled as an integral equation (see, for example [10, 23, 32, 34, 35]). The most used techniques to study the existence of solutions to an integral equation are based on fixed point arguments. For example, Banach contraction principle was used by several authors in order to establish existence results for different kinds of integral equations (see, for example [27, 29, 30, 33]). Similarly, many solvability results were derived via Schauder's fixed point theorem (see, for example [14, 25, 26]). On the other hand, in the absence of compacity and the Lipschitz condition, the above mentioned theorems cannot be applied. In such situation, the measure of non-compactness concept can be useful in order to avoid the mentioned problems.

Several existence results for integral equations were established using fixed point theorems involving measures of non-compactness. For more details, we refer the reader to [1,3,4,5,7,8,9,12,13,15,20,21,28] and the references therein. On the other hand, due to the importance of fractional calculus in modelling various problems from many fields of science and engineering, a great attention has been focused recently on the study of fractional integral equations (see, for example [2, 6,11,16,17,18, 24]).

In this paper, we deal with a new class of fractional integral equations involving a fractional integral with respect to another function. Using a fixed point theorem involving measures of non-compactness, an existence result is established. Some numerical examples are also provided.

2 Problem formulation and preliminaries

The main purpose of this paper is to study the following integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

where [alpha] [member of] (0,1), 0 [less than or equal to] a [less than or equal to] T, k : [a, T] x R [right arrow] R, g : [a,T] [right arrow] R and fi : [a,T] x R [right arrow] R (i = 1,2), from the point of view of the theory of existence of solutions. Eq.(2.1) can be written in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the fractional integral of order [alpha] with respect to the function g defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

In the particular case g([tau]) = [tau], Eq.(2.1) models some problems in the queuing theory and biology (see [19]). In such case, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Riemann-Liouville fractional integral defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.2)

In the case [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Erdelyi-Kober fractional integral defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.3)

However, in the case g([tau]) = ln [tau] and a > 0, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Hadamard fractional integral defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

For more details on fractional calculus, we refer the reader to [31].

Using a measure of non-compactness in the set C([a, T];R) of real continuous functions in [a, T] introduced by Banas and Goebel [8], we give sufficient conditions for the existence of at least one solution to Eq.(2.1). To the best of our knowledge, an integral equation of the type (2.1) has not been considered earlier.

Before presenting and proving the main results of this paper, we need to recall some basic concepts and present some preliminaries results that will be useful later.

Let (H, || * ||) be a given Banach space. We denote by BH the family of all nonempty bounded subsets of H. A mapping 77 : BH [right arrow] [0, oo) is said to be a measure of non-compactness (see [8]) if it satisfies the following axioms: (A1) for all B [member of] BH, we have

77(B) = 0 [??] B is precompact;

(A2) for every pair ([B.sub.1],[B.sub.2]) [member of] [B.sub.H] X [B.sub.H], we have

[B.sub.1] [subset.bar] [B.sub.2] [??] [eta]([B.sub.1]) [less than or equal to] [eta]([B.sub.2]);

(A3) for every B [member of] [B.sub.H],

[eta]([bar.B]) = [eta]([bar.B]) = [eta]([bar.co]B),

where [bar.co]B denotes the closed convex hull of B;

(M) for every pair ([B.sub.1], [B.sub.2]) [member of] [B.sub.H] X [B.sub.H] and [lambda] [member of] (0,1), we have

[eta]([lambda][B.sub.1] + (1 - [lambda])[B.sub.2]) [less than or equal to] [lambda][eta]([B.sub.1]) + (1 - [lambda])[eta]([B.sub.2]);

(A5) if {[B.sub.n]} is a sequence of closed and decreasing (w.r.t [subset.bar]) sets in [B.sub.H] such that [eta]([B.sub.n]) [right arrow] 0 as n [right arrow] [infinity], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is nonempty and compact.

A function [phi] : [0,[infinity]) [right arrow] [0,[infinity]) is said to be a comparison function if it satisfies the following properties:

(i) [phi] is non-decreasing;

(ii) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [[phi].sup.(n)] denotes the n-iteration of [phi].

Examples of comparison functions are

[phi](t) = [lambda]t, [lambda] [member of] [0,1); [phi](t) = arctant; [phi](t) = ln(1 + t); [phi](t) = [t/[1+t]].

Through this paper, we denote by [PHI] the set of comparison functions.

In [5], Aghajani et al. proved the following useful result in order to prove that a function is a comparison function.

Lemma 2.1. Let [phi] : [0,[infinity]) [right arrow] [0,[infinity]) be a non-decreasing and upper semi-continuous function. Then the following conditions are equivalent:

(i) lim [[phi].sup.(n)](t) = 0 for all t > 0.

(ii) [phi](t) < t for all t > 0.

In the same paper, the authors proved the following generalization of Darbo's theorem (see [1]).

Lemma 2.2. Let Cbe a nonempty, bounded, closed and convex subset of a Banach space H. Let U : C [right arrow] Cbe a continuous mapping such that

[eta](UW) [less than or equal to] [phi]([eta](W)), W [subset.bar] C,

where [phi] [member of] [PHI] and [eta] is a measure of non-compactness in H. Then U has at least one fixed point.

For given functions [phi], [psi] : [0,[infinity]) [right arrow] R and a real number [lambda], define the operators [T.sub.max] and T by

[T.sub.max]([phi],[psi])(t) = max{[phi](t),[psi](t)}, t [greater than or equal to] 0

and

T([lambda],[phi])(t)=[lambda][phi](t), t [greater than or equal to] 0.

Denote by C([0,[infinity]); R) the set of real continuous functions in [0,[infinity]) and by [PHI] the set [PHI] [intersection] C[0,[infinity]);R).

The following lemma is interesting for our purpose.

Lemma 2.3. The following properties hold:

(i) [T.sub.max] ([PHI] x [PHI]) [subset.bar] [PHI].

(ii) T([0,1] x [PHI]) [subset.bar] [PHI].

Proof. Let ([phi],[psi]) [member of] [PHI] x [PHI]. We have to prove that [T.sub.max]([phi],[psi]) : [0,[infinity]) [right arrow] [0,[infinity]) belongs to [PHI]. Since the maximum of a finite number of continuous functions is continuous, we have [T.sub.max]([phi],[psi]) [member of] C[0,[infinity]);R). Moreover, since ([phi],[psi]) [member of] [PHI] x [PHI] [subset.bar] [PHI] x [PHI], then [phi] and [psi] are non-decreasing functions, which yields [T.sub.max]([phi],[psi]) is also a non-decreasing function. From Lemma 2.1, we have

[phi](t) < t, [psi](t) < t, t > 0.

Then

[T.sub.m]ax([phi],[psi])(t) < t, t > 0.

Again, by Lemma 2.1, we get

lim [[[T.sub.max]([phi],[psi])].sup.(n)](t) =0, t > 0.

Therefore, [T.sub.max]([phi],[psi]) [member of] [PHI]. As consequence, [T.sub.max]([phi],[psi]) [member of] [PHI] [intersection] C([0,[infinity]);R) = [PHI], which proves (i). A similar argument can be used to prove (ii).

Let H = C([a, T];R) be the set of real continuous functions in [a, T]. Such a set is a Banach space with respect to the norm

||v|| = max{|v(t)| : t [member of] [a,T]}, v [member of] C([a,T];R).

We will use the following notations through this paper. Let B [member of] [B.sub.H]. For v [member of] B and [rho] [greater than or equal to] 0, set

[omega](v,p) = sup{|v(t) - v(s)| : t,s [member of] [a, T], |t - s| [less than or equal to] [rho]}.

We define the mapping [OMEGA] : [B.sub.h] x [0,[infinity]) [right arrow] [0,[infinity]) by

[OMEGA](B,[rho]) = sup{[omega](v,p) : v [member of] B}, (B,[rho]) [member of] [B.sub.H]x [0,[infinity]).

It was proved in [8] that the mapping [eta] : [B.sub.H] [right arrow] [0,[infinity]) defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

is a measure of non-compactness in the Banach space H.

Now, we are ready to present our main results. This is the aim of the next section.

3 Main results

Our first existence result is obtained under the following hypotheses:

(H1) The functions f :[a,T] x R [right arrow] R (i = 1,2) are continuous.

(H2) There exist functions [[phi].sub.1], [[psi].sub.2] : [0,[infinity]) [right arrow] [0,[infinity]) such that

(i) [[phi].sub.i](0) = 0, i = 1,2.

(ii) [[phi].sub.1] is non-decreasing and continuous.

(iii) [[phi].sub.2] [member of] [PHI].

(iv) There exist [delta] [greater than or equal to] 0 and 6 [member of] [PHI] such that

[[phi].sub.1](t) [less than or equal to] [delta][theta](t), t [greater than or equal to] 0.

(v) For all (t, u, v) [member of] [a,T] x R x R, we have

|[f.sub.i](t,u) - [f.sub.i](t,v)| [less than or equal to] [[phi].sub.i](|u - v|), i = 1,2.

(H3) The function k : [a, T] x R [right arrow] R is continuous and it satisfies

|k(t,u)| < [phi](|u|), (t,u) [member of] [a,T] x R,

where [phi] : [0,[infinity]) [right arrow] [0,[infinity]) is a non-decreasing function.

(H4) g [member of] C([a, T];R) [intersection] [C.sup.1]((a, T];R) with g'(t) > 0, for all t [member of] (a, T].

(H5) There exists [r.sub.0] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and, moreover

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[f*.sub.i] = max{|[f.sub.i](t, 0)| : t [member of] [a, T]}, i = 1,2.

Our main result is the following.

Theorem 3.1. Under the assumptions (H1)-(H5), Eq.(2.1) has at least one solution y* [member of] C([a,T];R). Moreover, we have ||y*|| < [r.sub.0].

Proof. Let us consider the operator U defined onH = C([a, T];R) by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.1)

At first, we show that the operator U maps H into itself. Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.2)

From assumption (H1), we have just to prove that [member of] maps H into itself, that is, Gx : [a, T] [right arrow] R is continuous for every x [member of] H. Observe that Gx is well-defined. In fact, for all t [member of] [a, T], from assumption (H3), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3.3)

Let us check the continuity of Gx at the point a. Let {tn} be a sequence in [a, T] such that [t.sub.n] [right arrow] [a.sup.+] as n [right arrow] [infinity]. Using (3.3), for all n [member of] N, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Letting n [right arrow] [infinity] in the above inequality and using the continuity of g, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which proves the continuity of Gx at the point a. Now, let t [member of] (a, T] be fixed and {[t.sub.n]} be a sequence in (a, T] such that [t.sub.n] [right arrow] t as n [right arrow] [infinity]. Without restriction of the generality, we may assume that [t.sub.n] [greater than or equal to] t for n large enough. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

For n large enough, we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Since g is continuous in [a, T], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then Gx is continuous at t. As consequence, Gx [member of] H, for all x [member of] H, and U : H [right arrow] H is well-defined.

On the other hand, for an arbitrarily fixed xeH and t [member of] [a, T], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

From the above inequality, the fact that the functions [[phi].sub.1], [[phi].sub.2], [phi] : [0,[infinity]) [right arrow] [0,[infinity]) are non-decreasing, and assumption (H5), we infer that the operator U maps [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into itself, where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Now, we claim that the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is continuous. We write U in the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and G is defined by (3.2). In order to prove our claim, it is sufficient to show that the operators [F.sub.i] (i = 1,2) and G are continuous on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Firstly, we show that [F.sub.i] are continuous operators on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. To do this, we take a sequence {[x.sub.n]} [subset] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and x [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that ||[x.sub.n] - x|| [right arrow] 0 as n [right arrow] [infinity], and we have to prove that ||[F.sub.i][x.sub.n] - [F.sub.i]x|| [right arrow] 0 as n [right arrow] [infinity]. In fact, for all t [member of] [a, T], we have

|([F.sub.i][x.sub.n])(t) - ([F.sub.i]x)(t)| = |[f.sub.i](t, [x.sub.n](t)) - [f.sub.i](t, x(t))| [less than or equal to] [[phi].sub.i](|[x.sub.n](t) - x(t)|) [less than or equal to] [[phi].sub.i](||[x.sub.n] - x||).

Then

||[F.sub.i][x.sub.n] - [F.sub.i]x|| [less than or equal to] [[phi].sub.i](||[x.sub.n] - x||), n [member of] N.

Letting n [right arrow] [infinity] in the above inequality, using the continuity of the functions [[phi].sub.i] (i = 1,2) and the fact that [[phi].sub.i](0) = 0, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

This proves that [F.sub.i] is a continuous operator on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all i = 1,2. Next, we show that G is a continuous operator on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. To do this, we fix a real number [epsilon] > 0 and we take arbitrary functions x,y [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that ||x - y|| < [epsilon]. For all t [member of] [a, T], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

u([r.sub.0],[epsilon]) = sup{|k([tau],v) - k(x,w)| : t [member of] [a,T], v,w g [-[r.sub.0],[r.sub.0]], |v - w| < [epsilon]}.

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since k is uniform continuous on the compact [a, T] x [-[r.sub.0], [r.sub.0]], we have u([r.sub.0], [epsilon]) [right arrow] 0 as [epsilon] [right arrow] [0.sup.+] and, therefore, the last inequality gives us

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then G is continuous on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and U maps continuously the set [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into itself.

Further, let W be a nonempty subset of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [rho] > 0 be fixed, x [member of] W and [t.sub.1], [t.sub.2] [member of] [a , T] be such that |[t.sub.1] - [t.sub.2]| [less than or equal to] [rho]. Without restriction of the generality, we may assume [t.sub.1] [greater than or equal to] [t.sub.2]. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Passing to the limit as [rho] [right arrow] [0.sup.+], we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Taking in consideration assumption (H2)-(iv), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since [lambda] [member of] [0,1] (see (H5)), from Lemma 2.3, T([lambda], [T.sub.max]([theta], [[phi].sub.2])) [member of] [PHI] Finally, by Lemma 2.2, the operator U has at least one fixed point y* [member of] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], which is a solution to Eq.(2.1) satisfying ||y* || [less than or equal to] [r.sub.0]. This completes the proof.

If in Theorem 3.1 we add the following assumptions:

(H6) For i = 1,2, [f.sub.i] ( [a, T] X [0,[infinity])) [subset.bar] [0,[infinity]) and [f.sub.i] is non-decreasing with respect to each variable on [a, T] x [0,[infinity]).

(H7) k([a,T] x [0,[infinity])) [subset.bar] [0,[infinity]) and k is non-decreasing with respect to each variable on [a, T] x [0,[infinity]).

Then we obtain the following result.

Theorem 3.2. Under the assumptions (H1)-(H7), Eq.(2.1) has at least one solution y* [member of] C([a, T];R) which is non-negative and non-decreasing on the interval [a, T].

Proof. Let us consider the operator U defined on H = C([a, T];R) by (3.1). Following the proof of Theorem 3.1, we know that the operator U maps H into itself, and that, for x[member of]H, we have the estimate

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

which yields [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Now, we consider the subset [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In virtue (H6) and (H7), we may infer that the operator U maps [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] into itself. Following the same argument used in the proof of Theorem 3.1, we obtain that U is a continuous operator on [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Next, we take the subset Q of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Obviously, the set Q is nonempty, bounded, closed and convex. Let us prove that the operator U maps Q into itself. To do this, let x [member of] Q be fixed and (t,s) be a pair of elements in [a, T] x [a, T] such that t [greater than or equal to] s. By assumption (H6), we have

[f.sub.i](t,x(t)) [greater than or equal to] [f.sub.i](s,x(s)), i = 1,2.

So we have just to prove that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Using assumption (H7), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then (3.4) holds and Ux is a non-decreasing function. Therefore, u maps Q into itself. Finally, using the same argument of the proof of Theorem 3.1, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

An application of Lemma 2.2 gives us the desired result.

4 Particular cases and examples

In this section, using Theorem 3.1, we give some existence results for some functional equations involving various types of fractional integrals. We present also some illustrative examples.

4.1 A functional equation involving Riemann-Liouville fractional integral

Let us consider the integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.1)

where [alpha] [member of] (0,1), 0 [less than or equal to] a < T,k : [a,T] x R [right arrow] R and [f.sub.i] : [a,T] x R [right arrow] R (i = 1,2). Eq.(4.1) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Riemann-Liouville fractional integral of order [alpha] defined by (2.2). Take

g(t) = t, t [member of] [a, T]

in Theorem 3.1, we obtain the following existence result.

Corollary 4.1. Suppose that all the assumptions (H1)-(H3) are satisfied. Moreover, suppose that there exists [r.sub.0] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then (4.1) has at least one solution y* [member of] C([a, T];R). Moreover, we have ||y*|| [less than or equal to]

We present the following example to illustrate the above result.

Example 4.2. Consider the integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.2)

The above equation can be written in the form (4.1) with a = 0, T = 1, [alpha] = [1/2],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can check easily that all the assumptions (H1)-(H3) are satisfied with [[phi].sub.1] (t) = [theta](t) =[delta] = 0, [[phi].sub.2](t) = [[square root of ([pi])]/2]t and [phi](r) = [[square root of ([pi])]/2]t. Moreover, in this case, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Observe that for [r.sub.0] = 1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Moreover,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then all the required assumptions of Corollary 4.1 are satisfied. As consequence, Eq.(4.2) has at least one solution y* [member of] C([0,1];R) satisfying ||y*|| [less than or equal to] 1.

4.2 A functional equation involving Erdelyi-Kober fractional integral

Let us consider the integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.3)

where a [member of] (0,1), [beta] > 0, 0 [less than or equal to] a < T, k : [a,T] x R [right arrow] R and [f.sub.i] : [a,T] x R [right arrow] R (i = 1,2). Eq.(4.3) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Erdelyi-Kober fractional integral defined by (2.3).

Take

g(t) = [t.sup.[beta]], t [member of] [a, T]

in Theorem 3.1, we obtain the following existence result.

Corollary 4.3. Suppose that all the assumptions (H1)-(H3) are satisfied. Moreover, suppose that there exists [r.sub.0] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then (4.3) has at least one solution y* [member of] C([a,T];R). Moreover, we have ||y*|| [less than or equal to] [r.sub.0].

We present the following example to illustrate the above result.

Example 4.4. We consider the integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.4)

where [alpha] [member of] (0,1) and B > 0. The above equation can be written in the form (4.3) with a = 0, T = 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can check easily that all the assumptions (H1)-(H3) are satisfied with [[phi].sub.1] (t) = [theta](t) =[delta] = 0, [[phi].sub.2](t) = [t/2] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Moreover, in this case, we have [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [f*.sub.2] = 1. On the other hand, we have [22]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then for [r.sub.0] = [1/2], we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Next, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then all the required assumptions of Corollary 4.3 are satisfied. As consequence, Eq.(4.4) has at least one solution y* [member of] C([0,1];R) satisfying ||y*|| [less than or equal to] [1/2].

4.3 A functional equation involving Hadamard fractional integral

We consider the integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.5)

where 0 < a < T, [alpha] [member of] (0,1), k: [a,T] x R [right arrow] R and [f.sub.i] : [a,T] x R [right arrow] R (i = 1,2). Eq.(4.5) can be written as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the Hadamard fractional integral of order [alpha] defined by (2.4).

Take

g(t) = lnt, t [member of] [a,T]

in Theorem 3.1, we obtain the following existence result.

Corollary 4.5. Suppose that all the assumptions (H1)-(H3) are satisfied. Moreover, suppose that there exists [r.sub.0] > 0 such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.6)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.7)

Then (4.5) has at least one solution y* [member of] C([a, T];R). Moreover, we have ||y*|| [less than or equal to]

We end the paper with the following example that illustrates the above result.

Example 4.6. Let us consider the integral equation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4.8)

where [alpha] [member of] (0,1). Eq.(4.8) can be written in the form (4.5) with a = 1, T = [e.sup.1],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We can check easily that all the assumptions (H1)-(H3) are satisfied with [[phi].sub.1] (t) = [theta](t) =[delta] = 0, [[phi].sub.2](t) = [t/2] and [phi](r) = 4[e.sup.-2] [GAMMA]([alpha] + 1)r. In this case, we have [f*.sub.1] = [e.sup.-1] and [f*.sub.2] = 0. Moreover, for [r.sub.0] = [[e.sup.2]/4], the inequalities (4.6) and (4.7) are satisfied. Therefore, by Corollary 4.5, Eq.(4.8) has at least one solution y* [member of] C([1,[e.sup.1]];R) such that ||y*|| [less than or equal to] [[e.sup.2]/4].

Acknowledgements. The first author was partially supported by the project MTM-2013-44357-P. The second author would like to extend his sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding of this research through the International Research Group Project No. IRG14-04.

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Kishin Sadarangani

Bessem Samet

Received by the editors in May 2016 - In revised form in August 2016.

Communicated by E. Colebunders.

2010 Mathematics Subject Classification : 45G05, 26A33, 74H20.

Key words and phrases : Integral equation; fractional integral with respect to another function; Riemann-Liouville fractional integral; Erdelyi-Kober fractional integral; Hadamard fractional integral; measure of non-compactness; Darbo's theorem.

Departamento de Matema`ticas, Universidad de Las Palmas de Gran Canaria, Campus de Tafira Baja, 35017 Las Palmas de Gran Canaria, Spain.

E-mail: ksadaran@dma.ulpgc.es

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia.

E-mail: bsamet@ksu.edu.sa
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