- Open Access
Existence of solutions for fractional q-integro-difference inclusions with fractional q-integral boundary conditions
© Ahmad et al.; licensee Springer 2014
- Received: 15 July 2014
- Accepted: 30 September 2014
- Published: 13 October 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.
- fractional differential inclusions
- nonlocal boundary conditions
- fixed point theorems
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.
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.
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.
Definition 2.1 ()
For more details on q-integral and fractional q-integral, see Section 1.3 and Section 4.2 respectively in .
Before giving the definition of fractional q-derivative, we recall the concept of q-derivative.
Definition 2.3 ()
where is the smallest integer greater than or equal to β.
To define the solution for problem (1.1), we need the following lemma.
Lemma 2.4 ()
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].
is a contraction, and
is upper semi-continuous (u.s.c.) and compact.
ℋ has a fixed point in , or
there is a point and with .
is measurable for each ,
is upper semi-continuous for almost all , and
- (iii)for each real number , there exists a function such that
for all with .
Lemma 2.7 (Lasota and Opial )
is a closed graph operator in .
Before presenting the main results, we define the solutions of the boundary value problem (1.1).
Theorem 3.2 Assume that
(H1) is an -Carathéodory multivalued map;
for all and , where is given by (3.1) and H denotes the Hausdorff metric;
(H3) is an -Carathéodory multivalued map;
for all ;
where , , , are given by (3.1), (3.2) and (3.3) respectively, and .
Then problem (1.1) has a solution on .
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 .
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 .
and hence is bounded.
Claim II: maps bounded sets into equicontinuous sets.
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.
for some .
Hence has a closed graph (and therefore has closed values). In consequence, is compact valued.
where , , , , , , , .
We found , , , , , , , , .
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).
This paper was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge technical and financial support of KAU.
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