Existence Theory for n th Order Nonlocal Integral Boundary Value Problems and Extension to Fractional Case

and Applied Analysis 3 Theorem 3. Let f : [0, 1] × R → R be a continuous function. Assume that there exists a constant 1 > 0 such that |f(t, u(t))| ≤ 1 for t ∈ [0, 1], u ∈ C. Then, the boundary value problem (1) has at least one solution. Proof. First we show that the operator F defined by (8) is completely continuous. Clearly, continuity of the operator F follows from the continuity of f. Then, it follows by the assumption |f(t, u(t)| ≤ 1 that


Introduction
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  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  () () =  (,  ()) ,  ∈ [0, 1] ,  (0) = 0,   (0) = 0,   (0) = 0, . . .,  (−2) (0) = 0, where  : [0, 1] × R → R is a given continuous function, and , ,   ,   , ( = 1, 2, . . ., ) 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 [24].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 Fractional calculus has emerged as an interesting mathematical modelling tool in many branches of basic sciences, engineering, and technical sciences [25][26][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][29][30][31][32][33][34][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).

Some Existence and Uniqueness Results
Let C = ([0, 1], R) denote the Banach space of all continuous functions from [0, 1] → R endowed with the norm defined by ‖‖ = sup{|()|,  ∈ [0, 1]}.Let  1 ([0, 1], R) be the Banach space of measurable functions  : [0, 1] → R which are Lebesgue integrable and normed by In view of Lemma 1, we define an operator F : C → C by where Λ is given by (5).Observe that the problem (1) has solutions only if the operator equation F =  has fixed points.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.
Hence, for  1 ,  2 ∈ [0, 1], we have Thus, by the foregoing arguments, one can infer that the operator F is equicontinuous on [0, 1].Hence, by the Arzelá-Ascoli theorem, the operator F : C → C is completely continuous.
Next, we consider the set and show that the set  is bounded.Let  ∈ , then,  = F, 0 <  < 1.For any  ∈ [0, 1], we have Thus, ‖‖ ≤  1 for any  ∈ [0, 1].So, the set  is bounded.Thus, by the conclusion of Lemma 2, the operator F 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 [37].
Lemma 4 (nonlinear alternative for single valued maps).Let  be a Banach space,  a closed convex subset of ,  an open subset of , and 0 ∈ .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 V in ) and  ∈ (0, 1) with V = (V).
Proof.Consider the operator F : C → C defined by (8).We show that F maps bounded sets into bounded sets in ([0, 1], R).For a positive number , let   = { ∈ ([0, 1], R) : ‖‖ ≤ } be a bounded set in ([0, 1], R).Then, Next, we show that  maps bounded sets into equicontinuous sets of ([0, 1], R).Let  1 ,  2 ∈ [0, 1] with  1 <  2 and  ∈   , where   is a bounded set of ([0, 1], R).Then, we obtain Obviously, the right-hand side of the above inequality tends to zero independently of  ∈   as  2 −  1 → 0. As F satisfies the above assumptions; therefore, it follows by the Arzelá-Ascoli theorem that  : ( Let  be a solution.Then, for  ∈ [0, 1], and using the computations in proving that F is bounded, we have In consequence, we have In view of ( 2 ), there exists  such that ‖‖ ̸ = .Let us set Note that the operator  :  → ([0, 1], R) is continuous and completely continuous.From the choice of , there is no  ∈  such that  = () for some  ∈ (0, 1).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 [36]).Let  be a Banach space.Assume that Ω is an open bounded subset of  with 0 ∈ Ω and let  : Ω →  be a completely continuous operator, such that ‖‖ ≤ ‖‖ , ∀ ∈ Ω. ( Then,  has a fixed point in Ω. which, in view of (22), implies that ‖F‖ ≤ ‖‖,  ∈ Ω.Therefore, by Lemma 6, the operator F 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 [38].
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) +V ∈  whenever , V ∈ ; (ii)  is compact and continuous; (iii)  is a contraction mapping.Then, there exists  ∈  such that  =  + .Theorem 9. Suppose that  : [0, 1] × R → R is a continuous function and satisfies the following assumptions: Then, the boundary value problem (1) and consider   = { ∈ C : ‖‖ ≤ }.We define the operators  and  on   as Now, we prove the compactness of the operator .In view of ( 1 ), we define sup and consequently, for  1 ,  2 ∈ [0, 1],  1 <  2 , 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 [0, 1].
Next, we discuss the uniqueness of solutions for the problem (1).This result relies on Banach's fixed point theorem.

Theorem 10. Assume that
Since F 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  1/ ([0, 1], R + ) the space of 1/-Lebesgue measurable functions from [0, 1] to R + with norm Then, the boundary value problem (1) has a unique solution, provided that Proof.For ,  ∈ C and for each  ∈ [0, 1] together with Hölder's inequality, we have By the given condition (37), it follows that F is a contraction mapping.Hence, the Banach fixed point theorem applies and F 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 Ψ : R + → R + such that Ψ(0) = 0 and Ψ() <  for all  > 0 with the following property: (40) Then, the boundary value problem (1) has a unique solution.
Proof.We consider the operator F : C → C defined by (8).
Let Ψ : R + → R + be the continuous nondecreasing function satisfying Ψ(0) = 0 and Ψ() <  for all  > 0 which is defined by where we have used (40).By the definition of Ψ, it follows that ‖F − F‖ ≤ Ψ(‖ − ‖).This shows that F is a nonlinear contraction.Thus, by Lemma 13, the operator F has a unique fixed point in C, which in turn is a unique solution of the problem (1).We find that

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][26][27].