A Study of Nonlinear Fractional Differential Equations of Arbitrary Order with Riemann-Liouville Type Multistrip Boundary Conditions

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 fractionalorder 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. [22] studied nonlinear fractional differential equations and inclusions of arbitrary order with multistrip boundary conditions. In this paper, we continue the study initiated in [22] and consider a boundary value problem of fractional differential equations of arbitrary order q ∈ (n − 1, n], n ≥ 2 with finite many multistrip Riemann-Liouville type integral boundary conditions:


Introduction
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][2][3][4][5][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 fractionalorder 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][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]) and the references therein.In particular, Ahmad et al. [22] studied nonlinear fractional differential equations and inclusions of arbitrary order with multistrip boundary conditions.
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 [22], 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][27][28][29][30][31][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.

Preliminary Result
Let us begin this section with some basic definitions of fractional calculus [2][3][4].
Definition 1.If () ∈   [, ], then the Caputo derivative of fractional order  is defined as where provided the integral exists.
The following result associated with a linear variant of problem (1) plays a pivotal role in establishing the main results.
By Lemma 3, we define an operator P : C → C as Observe that the problem (1) has a solution if and only if the associated fixed point problem P =  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 Theorem 4. Suppose that  : [0, ] × R → R is a continuous function and satisfies the following assumption: Then the boundary value problem (1) has a unique solution provided where Λ is given by (14).
Note that Λ depends only on the parameters involved in the problem.As Λ < 1, therefore P is a contraction.Hence, by Banach's contraction mapping principle, the problem (1) has a unique solution on [0, ].
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 which implies that ‖(P)‖ ≤  2 .Further, we find that Hence, for  1 ,  2 ∈ [0, ], we have This implies that P is equicontinuous on [0, ].Thus, by the Arzelá-Ascoli theorem, the operator P : C → C is completely continuous.Next, we consider the set and show that the set  is bounded.Let  ∈ , then  = P, 0 <  < 1.For any  ∈ [0, ], we have Thus, ‖‖ ≤  1 for any  ∈ [0,].So, the set  is bounded.Thus, by the conclusion of Theorem 6, the operator P has at least one fixed point, which implies that (1) has at least one solution.
In the next we prove one more existence result for problem (1), based on the following known result.
Proof.Let us define B  = { ∈ C | ‖‖ < } and take  ∈ C such that ‖‖ = , that is,  ∈ B  .As before, it can be shown that P is completely continuous and ‖P‖ ≤ sup which in view of the given condition (Λ ≤ 1), gives ‖P‖ ≤ ‖‖,  ∈ B  .Therefore, by Theorem 9, the operator P has at least one fixed point, which in turn implies that the problem (1) has at least one solution.( For sufficiently small  (ignoring  2 and higher powers of ), we have        ( 5 +  4 ())  (36)
[]denotes the integer part of the real number .
Let B ⊂ C be a bounded set.By the assumption that |(, )| ≤  1 , for  ∈ B, we have ) is bounded.Then  has a fixed point in .Theorem 7. Assume that there exists a positive constant  1 such that |(, )| ≤  1 for  ∈ [0, ],  ∈ R. Then the problem (1) has at least one solution.Proof.First of all, we show that the operator P is completely continuous.Note that the operator P is continuous in view of the continuity of .
Let  be a Banach space,  a closed, convex subset of ,  an open subset of  and 0 ∈ .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  ∈ (0, 1) with  = ().