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A Study of Nonlinear Fractional Differential Equations of Arbitrary Order with Riemann-Liouville Type Multistrip Boundary Conditions
We develop the existence theory for nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type boundary conditions involving nonintersecting finite many strips of arbitrary length. Our results are based on some standard tools of fixed point theory. For the illustration of the results, some examples are also discussed.
The subject of fractional calculus has recently developed into a hot topic for the researchers in view of its numerous applications in the field of physics, mechanics, chemistry, engineering, and so forth. One can find the systematic progress of the topic in the books ([1–6]). A significant characteristic of a fractional-order differential operator distinguishing it from the integer-order differential operator is that it is nonlocal in nature, that is, the future state of a dynamical system or process involving fractional derivative depends on its current state as well its past states. In fact, this feature of fractional-order operators has contributed towards the popularity of fractional-order models, which are recognized as more realistic and practical than the classical integer-order models. In other words, we can say that the memory and hereditary properties of various materials and processes can be described by differential equations of arbitrary order. There has been a rapid development in the theoretical aspects such as periodicity, asymptotic behavior, and numerical methods for fractional equations. For some recent work on the topic, see ([7–23]) and the references therein. In particular, Ahmad et al.  studied nonlinear fractional differential equations and inclusions of arbitrary order with multistrip boundary conditions.
In this paper, we continue the study initiated in  and consider a boundary value problem of fractional differential equations of arbitrary order , with finite many multistrip Riemann-Liouville type integral boundary conditions: where denotes the Caputo fractional derivative of order , is a given continuous function, is the Riemann-Liouville fractional integral of order , , , and are suitable chosen constants.
Regarding the motivation of the problem, we know that the strip conditions appear in the mathematical modeling of certain real world problems, for instance, see [24, 25]. In , the authors considered the nonlocal strip conditions of the form: In the problem (1), we have introduced Riemann-Liouville type multistrip integral boundary conditions which can be interpreted as the controller at the right-end of the interval under consideration is influenced by a discrete distribution of finite many nonintersecting sensors (strips) of arbitrary length expressed in terms of Riemann-Liouville type integral boundary conditions. For some engineering applications of strip conditions, see ([26–32]).
The main objective of the present study is to develop some existence results for the problem (1) by using standard techniques of fixed point theory. The paper is organized as follows. In Section 2 we discuss a linear variant of the problem (1), which plays a key role in developing the main results presented in Section 3. For the illustration of the theory, we have also included some examples.
2. Preliminary Result
Definition 1. If , then the Caputo derivative of fractional order is defined as where denotes the integer part of the real number . For details, see Theorem 2.1 ([4, page 92]). Here denote the space of real valued functions which have continuous derivatives up to order on such that .
Definition 2. The Riemann-Liouville fractional integral of order is defined as provided the integral exists.
The following result associated with a linear variant of problem (1) plays a pivotal role in establishing the main results.
Lemma 3. For , the fractional boundary value problem has a unique solution given by where
Proof. The general solution of fractional differential equations in (5) can be written as Using the given boundary conditions, it is found that , . Applying the Riemann-Liouville integral operator on (8), we get Using the condition , together with the fact that we obtain which yields where is given by (7). Substituting the values of in (8), we obtain (6). This completes the proof.
3. Main Results
Let denotes the Banach space of all continuous functions defined on endowed with a topology of uniform convergence with the norm .
By Lemma 3, we define an operator as
Observe that the problem (1) has a solution if and only if the associated fixed point problem has a fixed point.
In the first result we prove an existence and uniqueness result by means of Banach’s contraction mapping principle. For the sake of convenience, we set
Proof. With , we define , where and is given by (14). Then we show that . For , by means of the inequality , it can easily be shown that Now, for and for each , we obtain Note that depends only on the parameters involved in the problem. As , therefore is a contraction. Hence, by Banach’s contraction mapping principle, the problem (1) has a unique solution on .
Example 5. Let us consider the following -strip nonlocal boundary value problem: where , , , , , , , , , , , , , , , , , .
With the given values of the parameters involved, we find that Let us choose Clearly as and , where . Therefore all the conditions of Theorem 4 hold and consequently there exists a unique solution for the problem (19) with given by (21).
In the second result we use the Leray-Schauder alternative.
Theorem 6 ((Leray-Schauder alternative) [33, page 4]). Let be a Banach space. Assume that is completely continuous operator and the set is bounded. Then has a fixed point in .
Theorem 7. Assume that there exists a positive constant such that for , . Then the problem (1) has at least one solution.
Proof. First of all, we show that the operator is completely continuous. Note that the operator is continuous in view of the continuity of . Let be a bounded set. By the assumption that , for , we have
which implies that . Further, we find that
Hence, for , we have
This implies that is equicontinuous on . Thus, 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 Theorem 6, the operator has at least one fixed point, which implies that (1) has at least one solution.
Example 8. Consider the boundary value problem of Example 5 with Observe that with . Thus the hypothesis of Theorem 7 is satisfied. Hence by the conclusion of Theorem 7, the problem (19) with given by (29) has at least one solution.
In the next we prove one more existence result for problem (1), based on the following known result.
Theorem 9 (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 .
Proof. Let us define and take such that , that is, . As before, it can be shown that is completely continuous and which in view of the given condition , gives , . Therefore, by Theorem 9, the operator has at least one fixed point, which in turn implies that the problem (1) has at least one solution.
Example 11. Consider the boundary value problem of Example 5 and let us consider For sufficiently small (ignoring and higher powers of ), we have Choosing , all the assumptions of Theorem 10 hold. Therefore, the conclusion of Theorem 10 implies that the problem (19) with given by (32) has at least one solution.
Our final existence result is based on Leray-Schauder nonlinear alternative.
Lemma 12 ((Nonlinear alternative for single valued maps) [33, page 135]). Let be a Banach space, a closed, convex subset of , an open subset of and . Suppose that is a continuous, compact (i.e., 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 13. Assume that there exist a function , and a nondecreasing function such that , for all ; there exists a constant such that
Then the boundary value problem (1) has at least one solution on .
Proof. Consider the operator defined by (13). 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 following the similar computations as before, we find that In consequence, we have Thus, by , 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 12), we deduce that has a fixed point which is a solution of the problem (1). This completes the proof.
Example 14. Consider the boundary value problem of Example 5 with
Then and . Using , we find by the condition that . Thus all the assumptions of Theorem 13 are satisfied. Hence, it follows by Theorem 13 that the problem (19) with defined by (40) has at least one solution.
If we choose an unbounded nonlinearity as follows: Then with and . Using the earlier arguments, with , , we find that , . Hence the problem (19) with given by (41) has at least one solution.
The authors are grateful to the anonymous referees for their valuable comments.
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