- Research Article
- Open Access
Fractional q-difference hybrid equations and inclusions with Dirichlet boundary conditions
© Ahmad and Ntouyas; licensee Springer. 2014
- Received: 6 May 2014
- Accepted: 13 June 2014
- Published: 23 July 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 derivative
- boundary conditions
- fixed point
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.
where denotes the Caputo fractional q-derivative of order α, and .
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.
where . Observe that .
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.
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 .
We use the nonlinear alternative of Schaefer’s type due to Dhage .
Lemma 3.2 ()
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:
Then the problem (1.1) has at least one solution on .
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.
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.
for all . This shows that ℬ is uniformly bounded on .
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 .
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. □
and we can choose r such that . Hence the conclusion of Theorem 3.3 applies to the problem (3.7).
Let be the Banach space of measurable functions which are Lebesgue integrable and normed by .
is measurable for each ;
is upper semicontinuous for almost all .
- (iii)there exists a function such that
for all and for a.e. .
The following lemma is used in the sequel.
Lemma 4.2 ()
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 .
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;
where , , .
Then the boundary value problem (1.2) has at least one solution on .
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.
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 , .
and so is uniformly bounded.
which tends to zero independently of as .
Therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.
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.
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. □
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.
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.
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