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Ahmed Alsaedi, Sotiris K. Ntouyas, Bashir Ahmad, "New Existence Results for Fractional Integrodifferential Equations with Nonlocal Integral Boundary Conditions", Abstract and Applied Analysis, vol. 2015, Article ID 205452, 10 pages, 2015. https://doi.org/10.1155/2015/205452
New Existence Results for Fractional Integrodifferential Equations with Nonlocal Integral Boundary Conditions
We consider a boundary value problem of fractional integrodifferential equations with new nonlocal integral boundary conditions of the form: , and . According to these conditions, the value of the unknown function at the left end point is proportional to its value at a nonlocal point while the value at an arbitrary (local) point is proportional to the contribution due to a substrip of arbitrary length . These conditions appear in the mathematical modelling of physical problems when different parts (nonlocal points and substrips of arbitrary length) of the domain are involved in the input data for the process under consideration. We discuss the existence of solutions for the given problem by means of the Sadovski fixed point theorem for condensing maps and a fixed point theorem due to O’Regan. Some illustrative examples are also presented.
We consider a boundary value problem of fractional differential equations with nonlocal integral boundary conditions given bywhere denotes the Caputo fractional derivative of order , is a given continuous function, , , and , , , are real constants.
Here we remark that the boundary conditions introduced in the problem (1) are of nonlocal strip type and describe the situation when the receptors at the end points of the boundary are influenced by the nonlocal contributions due to interior points and strips of the domain for the problem. For practical examples, see [1, 2]. The problem (1) can also be termed as a five-point nonlocal fractional boundary value problem.
In recent years, several aspects of fractional boundary value problems, ranging from theoretical analysis to numerical simulation, have been investigated. The nonlocal nature of fractional order differential operators has significantly contributed to the popularity and development of the subject. As a matter of fact, this characteristic of such operators help to understand the memory and hereditary properties of many useful materials and processes. For details and applications of fractional differential equations in physical and technical sciences such as biology, physics, biophysics, chemistry, statistics, economics, blood flow phenomena, control theory, and signal and image processing, see [3–6]. For some recent works on nonlocal fractional boundary value problems, we refer the reader to the papers [7–11], while the results based on monotone method for such problems can be found in [12, 13]. In , the limit properties of positive solutions of fractional boundary value problems have been discussed. Fractional differential inclusions supplemented with different kinds of boundary conditions have also been studied by several researchers, for instance, see [15–20].
The paper is organized as follows. In Section 2, we recall some basic definitions from fractional calculus and establish an auxiliary lemma which plays a pivotal role in the sequel. Section 3.1 contains an existence result for the problem (1) which is established by applying Sadovskii’s fixed point theorem for condensing maps. In Section 3.2, we show the existence of solutions for the problem (1) by means of a fixed point theorem due to O’Regan.
Definition 1. The Riemann-Liouville fractional integral of order for a continuous function is defined asprovided the integral exists.
Definition 2. For at least -times continuously differentiable function , the Caputo derivative of fractional order is defined aswhere denotes the integer part of the real number .
Lemma 3. For any the unique solution of the linear fractional boundary value problemiswhere
Proof. It is well known that the general solution of the fractional differential equation in (4) can be written aswhere are arbitrary constants.
Applying the given boundary conditions, we obtain the following systemfrom which we getSubstituting the values of in (7), we get (5). This completes the proof.
3. Existence Results
We denote by 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 .
3.1. Existence Results via Sadovskii’s Fixed Point Theorem
Definition 4. Let be a bounded set in metric space ; then Kuratowskii measure of noncompactness, , is defined as covered by finitely many sets such that the diameter of each set .
Definition 5 (see ). Let be a bounded and continuous operator on Banach space . Then is called a condensing map if for all bounded sets , where denotes the Kuratowski measure of noncompactness.
Lemma 6 (see [23, Example 11.7]). The map is a -set contraction with and is thus condensing, if(i) are operators on the Banach space ;(ii) is -contractive; that is,for all and fixed ;(iii) is compact.
Theorem 7 (see ). Let be a convex, bounded, and closed subset of a Banach space and let be a condensing map. Then has a fixed point.
In view of Lemma 3, we define an operator bywhere
Theorem 8. Let be continuous functions satisfying the following conditions.
(H1) satisfies the Lipschitz condition:(H2)there exist a function and a nondecreasing function such thatThen the boundary value problem (1) has at least one solution provided that
Proof. Let be a closed bounded and convex subset of , where will be fixed later. We define a map aswhere and are defined by (12) and (13), respectively. Notice that the problem (1) is equivalent to a fixed point problem .
Step 1 . For that, we set and select , whereUsing , for , we getwhere we have used the following relations:Consequentlywhich implies that .
Step 2 ( is continuous and -contractive). To show the continuity of for , let us consider a sequence converging to . Then, by the assumption , we haveNext, we show that is -contractive. For , we getBy the given assumptionit follows that is -contractive.
Step 3 ( is compact). In Step 1, it has been shown that is uniformly bounded. Now we show that maps bounded sets into equicontinuous sets of . Let with and . Then we obtainObviously 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. Thus is compact on .
Step 4 ( is condensing). Since is continuous, -contractive and is compact, by Lemma 6, with is a condensing map on .
Consequently, by Theorem 7, the map has a fixed point which, in turn, implies that the problem (1) has a solution.
Example 9. Consider a nonlocal integral boundary value problem of fractional integrodifferential equations given bywhere , and .
3.2. Existence Results via O’Regan’s Fixed Point Theorem
Our next existence result relies on a fixed point theorem due to O’Regan in .
Lemma 10. Denote by an open set in a closed, convex set of a Banach space . Assume . Also assume that is bounded and that is given by , in which is continuous and completely continuous and is a nonlinear contraction (i.e., there exists a nonnegative nondecreasing function satisfying for , such that for all ). Then, either(C1) has a fixed point , or(C2)there exist a point and with , where and , respectively, represent the closure and boundary of .
For convenience we set
Letand denote the maximum number by
Theorem 11. Let be continuous functions. Assume that(A1)there exist a nonnegative function and a nondecreasing function such that(A2)there exist a positive constant and a continuous function such that and for all and ;(A3), where .Then the boundary value problem (1) has at least one solution on .
Proof. By the assumption (A3), there exists a number such thatWe will show that the operators and defined by (12) and (13), respectively, satisfy all the conditions of Lemma 10. The proof consists of a series of steps.
Step 1 (the operator is continuous and completely continuous). We first show that is bounded. For any , we haveThus the operator is uniformly bounded. For any , we havewhich is independent of and tends to zero as . Thus, is equicontinuous. Hence, by the Arzelá-Ascoli theorem, is a relatively compact set. Now, let with . Then the limit is uniformly valid on . From the uniform continuity of on the compact set , it follows that is uniformly valid on . Hence as . This shows the continuity of .
Step 2 (the operator is contractive). ConsiderThis, together with (A2), implies thatso is a nonlinear contraction.
Step 3 (the set is bounded). Using the inequalitywe havefor any . This, with the boundedness of the set , implies that the set is bounded.
Step 4 (finally, it will be shown that the case (C2) in Lemma 10 does not hold). On the contrary, we suppose that (C2) holds. Then, we have that there exist and such that . So, we have and