Advanced Theoretical and Applied Studies of Fractional Differential Equations 2013View this Special Issue
Existence Theory for th Order Nonlocal Integral Boundary Value Problems and Extension to Fractional Case
This paper is devoted to the study of the existence and uniqueness of solutions for th order differential equations with nonlocal integral boundary conditions. Our results are based on a variety of fixed point theorems. Some illustrative examples are discussed. We also discuss the Caputo type fractional analogue of the higher-order problem of ordinary differential equations.
Boundary value problems with nonclassical boundary conditions are often used to take into account some peculiarities of physical, chemical or other processes, which are impossible by applying classical boundary conditions. Nonlocal conditions appear when values of the function on the boundary are connected to values inside the domain. Integral nonlocal boundary conditions can be used when it is impossible to directly determine the values of the sought quantity on the boundary while the total amount or integral average on space domain is known.
Boundary value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint, and nonlocal boundary value problems as special cases. Integral boundary value problems occur in the mathematical modeling of a variety of physics processes and have recently received considerable attention. For some recent work on boundary value problems with integral boundary conditions we refer to [1–23] and the references cited therein.
In this paper, we discuss some existence and uniqueness results for boundary value problems of th order ordinary differential equations. Precisely, in the first part of the paper we consider the following boundary value problem of nonlinear th-order differential equations with multipoint integral boundary conditions where is a given continuous function, and , , , , are real constants to be chosen appropriately. Existence and uniqueness results are proved by using a variety of fixed point theorems such as Schaefer’s fixed point theorem, Leray-Schauder Nonlinear Alternative, Krasnoselskii’s fixed point theorem, Banach’s fixed point theorem, and Boyd and Wang fixed point theorem for nonlinear contractions . The methods used are well known; however, their exposition in the framework of problem (1) is new.
Next, we extend our discussion to the fractional case by considering the problem consisting of the boundary conditions in (1) along with the Caputo type fractional differential equation as follows: Fractional calculus has emerged as an interesting mathematical modelling tool in many branches of basic sciences, engineering, and technical sciences [25–27]. Differential and integral operators of fractional order do share some of the characteristics exhibited by the processes associated with complex systems having long-memory in time. In other words, we can say that a dynamical system or process involving fractional derivatives takes into account its current as well as past states. This feature has contributed significantly to the popularity of the subject and has motivated many researchers to focus on fractional order models. For some recent development of the topic, for instance, see [13, 28–35].
The paper is organized as follows. In Section 2, we recall some preliminary facts that we need in the sequel. Section 3 contains the existence and uniqueness results for the boundary value problem (1). In Section 4, some illustrative examples are presented. In Section 5, we consider the Caputo type fractional analogue of problem (1).
2. An Auxiliary Lemma
Lemma 1. Let . For any , the unique solution of the boundary value problem is given by where
Proof. It is well known that the solution of the differential equation in (3) can be written as
where , , are arbitrary real constants. Using the boundary conditions in (6), we get and applying the boundary condition , we find that
where is defined by (5).
Substituting the values of and in (6), we get (4).
3. Some Existence and Uniqueness Results
Let denote the Banach space of all continuous functions from endowed with the norm defined by . Let be the Banach space of measurable functions which are Lebesgue integrable and normed by .
Now we are in a position to present several existence results for the problem (1). Our first result is based on Schaefer’s fixed point theorem.
Lemma 2 (see ). Let be a Banach space. Assume that is a completely continuous operator and the set is bounded. Then, has a fixed point in .
Theorem 3. Let be a continuous function. Assume that there exists a constant such that for , . Then, the boundary value problem (1) has at least one solution.
Proof. First we show that the operator defined by (8) is completely continuous. Clearly, continuity of the operator follows from the continuity of . Then, it follows by the assumption that
which implies that . Furthermore,
Hence, for , we have Thus, by the foregoing arguments, one can infer that the operator is equicontinuous on . Hence, by the Arzelá-Ascoli theorem, the operator is completely continuous.
Next, we consider the set and show that the set is bounded. Let , then, , . For any , we have
Thus, for any . So, the set is bounded. Thus, by the conclusion of Lemma 2, the operator has at least one fixed point, which implies that the boundary value problem (1) has at least one solution.
Our next existence result is based on Leray-Schauder Nonlinear Alternative .
Lemma 4 (nonlinear alternative for single valued maps). Let be a Banach space, a closed convex subset of , an open subset of , and . Suppose that is a continuous, compact (that is, is a relatively compact subset of ) map. Then, either(i)has a fixed point in , or (ii)there is a (the boundary of in ) and with .
Theorem 5. Let be a continuous function. Assume that there exist a function , and a nondecreasing function such that , ; there exists a constant such that
Then, the boundary value problem (1) has at least one solution on .
Proof. Consider the operator defined by (8). We show that maps bounded sets into bounded sets in . For a positive number , let be a bounded set in . Then,
Next, we show that maps bounded sets into equicontinuous sets of . Let with and , where is a bounded set of . Then, we obtain
Obviously, the right-hand side of the above inequality tends to zero independently of as . As satisfies the above assumptions; therefore, it follows by the Arzelá-Ascoli theorem that is completely continuous.
Let be a solution. Then, for , and using the computations in proving that is bounded, we have
In consequence, we have
In view of , there exists such that . Let us set
Note that the operator is continuous and completely continuous. From the choice of , there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder-type (Lemma 4), we deduce that has a fixed point which is a solution of the problem (1). This completes the proof.
To prove the next existence result, we need the following fixed point theorem.
Lemma 6 (see ). Let be a Banach space. Assume that is an open bounded subset of with and let be a completely continuous operator, such that Then, has a fixed point in .
Theorem 7. Let be continuous and there exists with , and Then, the boundary value problem (1) has at least one solution.
Proof. Define and take such that ; that is, . As before, it can be shown that is completely continuous and which, in view of (22), implies that , . Therefore, by Lemma 6, the operator has at least one fixed point, which corresponds to at least one solution of the boundary value problem (1).
Our next existence result is based on Krasnoselskii’s fixed point theorem .
Theorem 8 (Krasnoselskii’s fixed point theorem). Let be a closed, bounded, convex, and nonempty subset of a Banach space . Let and be the operators such that (i) whenever ; (ii) is compact and continuous; (iii) is a contraction mapping. Then, there exists such that .
Theorem 9. Suppose that is a continuous function and satisfies the following assumptions: , , , . , , and .
Then, the boundary value problem (1) has at least one solution on if
Proof. Letting , we choose a real number satisfying the inequality
and consider . We define the operators and on as
For , we find that Thus, . In view of and (24), is a contraction mapping. Continuity of implies that the operator is continuous. Also, is uniformly bounded on as
Now, we prove the compactness of the operator . In view of , we define and consequently, for , , we have which is independent of . Thus, is relatively compact on . Hence, by the Arzelá-Ascoli theorem, is compact on . Thus, all the assumptions of Theorem 8 are satisfied. So, the conclusion of Theorem 8 implies that the boundary value problem (1) has at least one solution on .
Next, we discuss the uniqueness of solutions for the problem (1). This result relies on Banach’s fixed point theorem.
Theorem 10. Assume that is a continuous function satisfying the condition .
If then, the boundary value problem (1) has a unique solution.
we show that , where . For , we have for ,
Thus, we get . Now, for and for each , we obtain
Since is a contraction; therefore, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem).
We give another uniqueness result for the problem (1) by using Banach’s fixed point theorem and Hölder’s inequality. In the following, we denote by the space of -Lebesgue measurable functions from to with norm .
Theorem 11. Let be a continuous function satisfying the following Lipschitz condition:, for all , where , .
Then, the boundary value problem (1) has a unique solution, provided that
Proof. For and for each together with Hölder’s inequality, we have
By the given condition (37), it follows that is a contraction mapping. Hence, the Banach fixed point theorem applies and has a fixed point which is the unique solution of the problem (1). This completes the proof.
Finally, we discuss the uniqueness of solutions for the problem (1) by using a fixed point theorem for nonlinear contractions due to Boyd and Wong.
Definition 12. Let be a Banach space and let be a mapping. is said to be a nonlinear contraction if there exists a continuous nondecreasing function such that and for all with the following property:
Lemma 13 (see Boyd and Wong ). Let be a Banach space and let be a nonlinear contraction. Then, has a unique fixed point in .
Theorem 14. Assume that, , , where is continuous, where
Then, the boundary value problem (1) has a unique solution.
Proof. We consider the operator defined by (8).
Let be the continuous nondecreasing function satisfying and for all which is defined by
For , , we have and so where we have used (40). By the definition of , it follows that . This shows that is a nonlinear contraction. Thus, by Lemma 13, the operator has a unique fixed point in , which in turn is a unique solution of the problem (1).
Example 1. Consider the boundary value problem where , , , , , , , , and .
We find that (a) Let Since , then, is satisfied with . Since therefore, by Theorem 10, the problem (44) with given by (46) has a unique solution. (b) Let Choose . Since then, is satisfied with . We can show that Thus, by Theorem 11, the problem (44) with defined by (48) has a unique solution. (c) Let Clearly, Choosing , , we obtain which implies that . Hence, by Theorem 5, the boundary value problem (44) with defined by (51) has at least one solution on . (d) Let We choose and find that Clearly, Hence, by Theorem 14, the boundary value problem (44) with defined by (54) has a unique solution on .
5. Fractional Case
In this section, we consider a Caputo type fractional analogue of problem (1) given by where denotes the Caputo fractional derivative of order . Before proceeding further, we recall some basic definitions of fractional calculus [25–27].
Definition 15. For an at least -times continuously differentiable function , the Caputo derivative of fractional order is defined as where denotes the integer part of the real number .
Definition 16. The Riemann-Liouville fractional integral of order is defined as
provided that the integral exists.
It is well known  that the general solution of the fractional differential equation with can be written as where are arbitrary constants. Using the boundary conditions for the problem (57), we find that and where . Substituting these values in (61) yields
Integrating the second term in (63) with respect to after interchanging the order of integration, we obtain
Replacing with in (64), the solution of the problem (57) is given by
In relation to the problem (57), we define an operator by
By taking in (66), the resulting operator reduces to the one given by (8) for a th order classical boundary value problem. Thus, all the results for the fractional problem (57), analogous to the classical problem (1), can be obtained with the aid of the operator given by (66). For example, Theorem 10 has the following fractional analogue.
Theorem 17. Assume that