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Existence of solutions for fractional q-integro-difference inclusions with fractional q-integral boundary conditions
Advances in Difference Equations volume 2014, Article number: 257 (2014)
In this paper, we investigate the existence of solutions for a nonlocal boundary value problem of fractional q-integro-difference inclusions of two fractional orders with Riemann-Liouville fractional q-integral boundary conditions. A new existence result is obtained by making use of a nonlinear alternative for contractive maps and is well illustrated with the aid of an example.
Nonlocal nonlinear boundary value problems of fractional order have been extensively investigated in recent years. Several results of interest ranging from theoretical analysis to asymptotic behavior and numerical methods for fractional differential equations are available in the literature on the topic. The introduction of fractional derivative in the mathematical modeling of many real world phenomena has played a key role in improving the integer-order mathematical models. One of the important factors accounting for the popularity of the subject is that differential operators of fractional-order help to understand the hereditary phenomena in many materials and processes in a better way than the corresponding integer-order differential operators. For examples and details in physics, chemistry, biology, biophysics, blood flow phenomena, control theory, wave propagation, signal and image processing, viscoelasticity, identification, fitting of experimental data, economics, etc., we refer the reader to the texts [1–4]. Some recent work on fractional differential equations can be found in a series of papers [5–17] and the references cited therein.
Fractional q-difference equations, known as fractional analogue of q-difference equations, have recently been discussed by several researchers. For some recent work on the topic, see [18–32]. In a recent paper , the authors obtained some existence results for the Langevin type q-difference (integral) equation with two fractional orders and four-point nonlocal integral boundary conditions.
Initial and boundary value problems involving multivalued maps have been studied by many researchers. In fact, the multivalued (inclusions) problems appear in the mathematical modeling of a variety of problems in economics, optimal control, etc. and are widely studied by many authors, see [34–36] and the references therein. Recent works on fractional-order multivalued problems [37–43] clearly indicate the interest in the subject. In  the authors studied the existence of solutions for a problem of nonlinear fractional q-difference inclusions with nonlocal Robin (separated) conditions given by
where is the fractional q-derivative of the Caputo type, is a multivalued map, is the family of all subsets of ℝ and (). However, the study of multivalued problems in the setting of fractional q-difference equations is still at an initial stage and needs to be explored further.
In this paper, motivated by , we consider the multivalued analogue of the problem addressed in . Precisely we discuss the existence of solutions for a boundary value problem of fractional q-integro-difference inclusions with fractional q-integral boundary conditions given by
where denotes the Caputo fractional q-difference operator of order β, , , , , are multivalued maps, is the family of all nonempty subsets of ℝ, a, b, A, B, , , , are real constants and
Here we emphasize that the multivalued (inclusion) problem at hand is new in the sense that it involves fractional q-integro-difference inclusions of two fractional orders with four-point nonlocal Riemann-Liouville fractional q-integral boundary conditions (in contrast to the problem considered in ). Also our method of proof is different from the one employed in . The paper is organized as follows. An existence result for problem (1.1), based on a nonlinear alternative for contractive maps, is established in Section 3. The background material for the problem at hand can be found in the related literature. However, for quick reference, we outline it in Section 2.
2 Preliminaries on fractional q-calculus
Let a q-real number denoted by be defined by
The q-analogue of the Pochhammer symbol (q-shifted factorial) is defined as
The q-analogue of the exponent is
The q-gamma function is defined as
where . Observe that . For any , the q-beta function is given by
which, in terms of q-gamma function, can be expressed as
Definition 2.1 ()
Let f be a function defined on . The fractional q-integral of the Riemann-Liouville type of order is and
Observe that in Definition 2.1 yields q-integral
For more details on q-integral and fractional q-integral, see Section 1.3 and Section 4.2 respectively in .
Remark 2.2 The q-fractional integration possesses the semigroup property (Proposition 4.3 )
Further, it has been shown in Lemma 6 of  that
Before giving the definition of fractional q-derivative, we recall the concept of q-derivative.
We know that the q-derivative of a function is defined as
Definition 2.3 ()
The Caputo fractional q-derivative of order is defined by
where is the smallest integer greater than or equal to β.
Next we recall some properties involving Riemann-Liouville q-fractional integral and Caputo fractional q-derivative (Theorem 5.2 ).
To define the solution for problem (1.1), we need the following lemma.
Lemma 2.4 ()
For a given , the integral solution of the boundary value problem
is given by
Let denote the Banach space of all continuous functions from endowed with the norm defined by . Also by we denote the Banach space of measurable functions which are Lebesgue integrable and normed by .
In order to prove our main existence result, we make use of the following form of the nonlinear alternative for contractive maps [, Corollary 3.8].
Theorem 2.5 Let X be a Banach space, and D be a bounded neighborhood of . Let (here denotes the family of all nonempty, compact and convex subsets of X) and be two multi-valued operators such that
is a contraction, and
is upper semi-continuous (u.s.c.) and compact.
Then, if , either
ℋ has a fixed point in , or
there is a point and with .
Definition 2.6 A multivalued map is said to be -Carathéodory if
is measurable for each ,
is upper semi-continuous for almost all , and
for each real number , there exists a function such that
for all with .
Lemma 2.7 (Lasota and Opial )
Let X be a Banach space. Let be an -Carathéodory multivalued map, and let Θ be a linear continuous mapping from to . Then the operator
is a closed graph operator in .
3 Existence result
Before presenting the main results, we define the solutions of the boundary value problem (1.1).
Definition 3.1 A function is said to be a solution of problem (1.1) if , , and there exist functions , such that
In the sequel, we set
Theorem 3.2 Assume that
(H1) is an -Carathéodory multivalued map;
(H2) there exists a function such that
for all and , where is given by (3.1) and H denotes the Hausdorff metric;
(H3) is an -Carathéodory multivalued map;
(H4) there exist functions and a nondecreasing function such that
for all ;
(H5) there exists a number such that
where , , , are given by (3.1), (3.2) and (3.3) respectively, and .
Then problem (1.1) has a solution on .
Proof To transform problem (1.1) to a fixed point problem, let us introduce an operator as
We shall show that the operators and satisfy all the conditions of Theorem 2.5 on . For the sake of clarity, we split the proof into a sequence of steps and claims.
Step 1. We show that is a multivalued contraction on .
Let and . Then and
for some . Since , there exists such that . Thus the multivalued operator U is defined by , where
It follows that and
Taking the supremum over the interval , we obtain
Combining the previous inequality with the corresponding one obtained by interchanging the roles of x and y, we find that
for all . This shows that is a multivalued contraction as
Step 2. We shall show that the operator is u.s.c. and compact. It is well known [, Proposition 1.2] that a completely continuous operator having a closed graph is u.s.c. Therefore we will prove that is completely continuous and has a closed graph. This step involves several claims.
Claim I: maps bounded sets into bounded sets in .
Let be a bounded set in and for some . Then we have
and hence is bounded.
Claim II: maps bounded sets into equicontinuous sets.
As in the proof of Claim I, let be a bounded set and for some . Let with . Then we have
Obviously the right-hand side of the above inequality tends to zero independently of as . Therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.
Claim III: Next we prove that has a closed graph.
Let , and . Then we need to show that . Associated with , there exists such that for each ,
Thus it suffices to show that there exists such that for each ,
Let us consider the linear operator given by
as . Thus, it follows by Lemma 2.7 that is a closed graph operator. Further, we have . Since , therefore, we have
for some .
Hence has a closed graph (and therefore has closed values). In consequence, is compact valued.
Therefore the operators and satisfy all the conditions of Theorem 2.5. So the conclusion of Theorem 2.5 applies and either condition (i) or condition (ii) holds. We show that conclusion (ii) is not possible. If for , then there exist and such that
By hypothesis (H2), for all , we have
Hence, for any ,
for all . Then, by using the computations from Step 1 and Step 2, Claim I, we have
Now, if condition (ii) of Theorem 2.5 holds, then there exist and such that . This implies that x is a solution with and consequently, inequality (3.5) yields
which contradicts (3.4). Hence, has a fixed point in by Theorem 2.5, which in fact is a solution of problem (1.1). This completes the proof. □
Example 3.3 Consider a nonlocal integral boundary value problem of fractional integro-differential equations given by
where , , , , , , , .
We found , , , , , , , , .
Then we have
with . Clearly, , , , , , , and . By the condition
it is found that , where . Thus, all the assumptions of Theorem 3.2 are satisfied. Hence, the conclusion of Theorem 3.2 applies to problem (3.6).
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