- Research Article
- Open Access
Fractional q-difference hybrid equations and inclusions with Dirichlet boundary conditions
Advances in Difference Equations volume 2014, Article number: 199 (2014)
We study the existence of solutions for boundary value problems of nonlinear fractional q-difference hybrid equations and inclusions by means of fixed point theorems for single- and multi-valued maps. The main results are illustrated with the aid of examples.
MSC:34A60, 34A08, 34B18.
Fractional calculus has evolved into an interesting and popular field of research due to its theoretical development and extensive applications in the mathematical modeling of many real world phenomena occurring in several branches of the physical, biological, and technical sciences [1–8].
Fractional q-difference (q-fractional) equations are regarded as fractional analogs of q-difference equations and have been investigated by many researchers [9–19]. For some earlier work on the topic, we refer to [20, 21], whereas the preliminary concepts on q-fractional calculus can be found in a recent text .
Fractional hybrid differential equations have also received a considerable attention; for instance, see [23–25] and the references cited therein. In , the authors studied the existence of solutions for a boundary value problem of Riemann-Liouville fractional hybrid differential equations.
In this paper, motivated by , we study the existence of solutions for Dirichlet boundary value problems of fractional q-difference hybrid equations and inclusions. As a first problem, we consider
where denotes the Caputo fractional q-derivative of order α, and .
Next, we study a boundary value problem of fractional q-difference hybrid inclusions given by
where is a multi-valued map, is the family of all nonempty subsets of ℝ.
The paper is organized as follows: in Section 2 we recall some preliminary facts. The existence of solutions for the problem (1.1) is shown in Section 3 while the multi-valued problem (1.2) is investigated in Section 4. The main tool of our study are fixed point theorems due to Dhage for single-valued  and multi-valued  maps.
For a real parameter , a q-real number denoted by is defined by
The q-analog of the Pochhammer symbol (q-shifted factorial) is defined as
The q-analog of the exponent is
The q-gamma function is defined as
where . Observe that .
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 the 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.
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).
3 An existence result for the single-valued problem
This section begins with a basic result, which plays a pivotal role in the forthcoming analysis. Let denote a Banach space of continuous functions from into ℝ with the norm .
Lemma 3.1 For , the unique solution of the problem
is given by
Proof It is well known that the general solution of the q-fractional differential equation in (3.1) can be written as
where are arbitrary unknown constants. Using the boundary conditions given in (3.2), we have
Substituting the values of and in (3.2), we get
We use the nonlinear alternative of Schaefer’s type due to Dhage .
Lemma 3.2 ()
Let and respectively denote an open and closed ball in a Banach algebra X centered at origin of radius r, for some real number . Let and be two operators such that:
A is Lipschitzian with a Lipschitz constant k,
B is completely continuous,
, where .
the equation has a solution in , or
there exists an with such that for some .
Now we are in a position to present the first main result of our paper.
Theorem 3.3 Assume that:
(H1) the function is continuous and there exists a bounded function ϕ, with bound , such that for and
(H2) there exist a continuous nondecreasing function and a function such that
(H3) there exists a real number such that
Then the problem (1.1) has at least one solution on .
Proof Let us set and consider a closed ball in X, where r satisfies the inequality (3.3). By Lemma 3.1, the problem (1.1) is equivalent to the integral equation
Define two operators by
Observe that . We shall show that the operators and ℬ satisfy all the conditions of Lemma 3.2. The proof is constructed in several steps.
Step 1. is Lipschitz on X, that is, the assumption (a) of Lemma 3.2 holds.
Let . Then by (H1), we have
for all . Taking the supremum over the interval , we get
for all . So is Lipschitz on with Lipschitz constant .
Step 2. The operator ℬ is completely continuous on , that is, (b) of Lemma 3.2 holds.
First we show that ℬ is continuous on . Let be a sequence in converging to a point . Then, by Lebesgue’s dominated convergence theorem, we have
for all . This shows that ℬ is continuous on . It is enough to show that is a uniformly bounded and equicontinuous set in X. First we note that
for all . Taking supremum over the interval yields
for all . This shows that ℬ is uniformly bounded on .
Next we show that ℬ is an equicontinuous set in X. 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 ℬ is a completely continuous operator on .
Step 3. , that is, (c) of Lemma 3.2 holds.
This is obvious by (H3) since we have and .
Thus the conditions (a), (b), and (c) of Lemma 3.2 are satisfied. Hence, either the conclusion (i) or the conclusion (ii) of Lemma 3.2 holds. We show that the conclusion (ii) is not possible. Let x be a solution of the operator equation with for some λ, . Then we have
which is a contradiction to (3.3), and hence the conclusion (ii) is not valid. Consequently, the conclusion (i) is valid, and hence the problem (1.1) has a solution on . This completes the proof. □
Example 3.4 Consider the boundary value problem
Here , . Observe that (H1) and (H2) hold with and , , respectively. With the given data,
and we can choose r such that . Hence the conclusion of Theorem 3.3 applies to the problem (3.7).
4 Multi-valued case
Let be the Banach space of measurable functions which are Lebesgue integrable and normed by .
For a normed space , let , , , and . A multi-valued map is convex (closed) valued if is convex (closed) for all . The map G is bounded on bounded sets if is bounded in X for all (i.e. ). G is called upper semicontinuous (u.s.c.) on X if for each , the set is a nonempty closed subset of X, and if for each open set N of X containing , there exists an open neighborhood of such that . G is said to be completely continuous if is relatively compact for every . If the multi-valued map G is completely continuous with nonempty compact values, then G is u.s.c. if and only if G has a closed graph, i.e., , , imply . G has a fixed point if there is such that . The fixed point set of the multi-valued operator G will be denoted by FixG. A multi-valued map is said to be measurable if for every , the function
Definition 4.1 A multi-valued map is said to be Carathéodory if
is measurable for each ;
is upper semicontinuous for almost all .
Further a Carathéodory function F is called -Carathéodory if
there exists a function such that
for all and for a.e. .
For each , define the set of selections of F by
The following lemma is used in the sequel.
Lemma 4.2 ()
Let X be a Banach space. Let be an -Carathéodory multi-valued map and let Θ be a linear continuous mapping from to . Then the operator
is a closed graph operator in .
Our second main result for the multi-valued problem (1.2) is based on the following fixed point theorem due to Dhage .
Lemma 4.3 Let X be a Banach algebra and let be a single-valued and be a multi-valued operator satisfying:
A is a single-valued Lipschitz operator with a Lipschitz constant k,
B is compact and upper semicontinuous,
, where .
the operator inclusion has a solution, or
the set is unbounded.
Definition 4.4 A function is called a solution of the problem (1.2) if there exists a function with , a.e. on such that , a.e. on and .
Theorem 4.5 Assume that (H1) holds. In addition we suppose that:
(A1) is -Carathéodory and has nonempty compact and convex values;
(A2) there exists a positive real number such that
where , , .
Then the boundary value problem (1.2) has at least one solution on .
Proof To transform the problem (1.2) into a fixed point problem, we define an operator as
where . Next we introduce two operators by
Obviously . We shall show that the operators and satisfy all the conditions of Lemma 4.3. The proof is structured into a sequence of steps.
Step 1. We first show that is Lipschitz on X, i.e., (a) of Lemma 4.3 holds.
The proof is similar to the one for the operator in Step 1 of Theorem 3.3.
Step 2. Now we show that the multi-valued operator is compact and upper semicontinuous on X, i.e. (b) of Lemma 4.3 holds.
Let us first show that has convex values.
Let . Then there are such that
For any , we have
where for all . Hence and consequently is convex for each . As a result defines a multi-valued operator .
Next we show that maps bounded sets into bounded sets in X. To do this, let Q be a bounded set in X. Then there exists a real number such that , .
Now for each , there exists a such that
Then for each ,
This further implies that
and so is uniformly bounded.
Next we show that maps bounded sets into equicontinuous sets. Let Q be, as above, a bounded set and for some . Then there exists a such that
Then for any with we have
which tends to zero independently of as .
Therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.
In our next step, we show 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
Thus, it follows by Lemma 4.2 that is a closed graph operator. Further, we have . Since , therefore, we have
for some .
As a result we find that the operator is a compact and upper semicontinuous operator on X.
Step 3. Now we show that , i.e. (c) of Lemma 4.3 holds.
It is obvious in view of (H3) with and .
Thus all the conditions of Lemma 4.3 are satisfied and, in consequence, it follows that either the conclusion (i) or the conclusion (ii) holds. We show that the conclusion (ii) is not possible.
Let be arbitrary. Then we have for and there exists such that, for any ,
for all . Thus we have
Consequently, with , we have
Thus the condition (ii) of Theorem 4.3 does not hold in view of the condition (4.1). Therefore, the operator inclusion has a solution, and, in turn, the problem (1.2) has a solution on . This completes the proof. □
Example 4.6 Consider the problem
where is a multi-valued map given by
For we have
Further, , , and
Thus all the conditions of Theorem 4.5 are satisfied and consequently, the problem (4.4) has a solution on .
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
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This paper was funded by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. Therefore, the authors acknowledge with thanks DSR for the financial support.
The authors declare that they have no competing interests.
Each of the authors, BA and SKN, contributed to each part of this work equally and read and approved the final version of the manuscript.