A Study of Nonlinear Boundary Value Problem

In this chapter, firstly we apply the iterative method to establish the existence of the positive solution for a type of nonlinear singular higher-order fractional differential equation with fractional multi-point boundary conditions. Explicit iterative sequences are given to approximate the solutions and the error estimations are also given. Secondly, we cover the multi-valued case of our problem. We investigate it for nonconvex compact valued multifunctions via a fixed point theorem for multivalued maps due to Covitz and Nadler. Two illustrative examples are presented at the end to illustrate the validity of our results.

The first definition of fractional derivative was introduced at the end of the nineteenth century by Liouville and Riemann, but the concept of non-integer derivative and integral, as a generalization of the traditional integer order differential and integral calculus, was mentioned already in 1695 by Leibniz [1] and L'Hospital [2]. In fact, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. The mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, Bode's analysis of feedback amplifiers, capacitor theory, electrical circuits, electroanalytical chemistry, biology, control theory, fitting of experimental data, involves derivatives of fractional order. In consequence, the subject of fractional differential equations is gaining much importance and attention. For more details we refer the reader to [1][2][3][4][5][6] and the references cited therein.
Boundary value problems for nonlinear differential equations arise in a variety of areas of applied mathematics, physics and variational problems of control theory. A point of central importance in the study of nonlinear boundary value problems is to understand how the properties of nonlinearity in a problem influence the nature of the solutions to the boundary value problems. The multi-point boundary conditions are important in various physical problems of applied science when the controllers at the end points of the interval (under consideration) dissipate or add energy according to the sensors located, at intermediate points, see [7,8] and the references therein. We quote also that realistic problems arising from economics, optimal control, stochastic analysis can be modeled as differential inclusion. The study of fractional differential inclusions was initiated by El-Sayed and Ibrahim [9]. Also, recently, several qualitative results for fractional differential inclusion were obtained in [10][11][12][13] and the references therein.
The techniques of nonlinear analysis, as the main method to deal with the problems of nonlinear differential equations (DEs), nonlinear fractional differential equations (FDEs), nonlinear partial differential equations (PDEs), nonlinear fractional partial differential equations (FPDEs), nonlinear stochastic fractional partial differential equations (SFPDEs), plays an essential role in the research of this field, such as establishing the existence, uniqueness and multiplicity of solutions (or positive solutions) and mild solutions for nonlinear of different kinds of FPDEs, FPDEs, SFPDEs, inclusion differential equations and inclusion fractional differential equations with various boundary conditions, by using different techniques (approaches). For more details, see  and the references therein. For example, iterative method is an important tool for solving linear and nonlinear Boundary Value Problems. It has been used in the research areas of mathematics and several branches of science and other fields. However, Many authors showed the existence of positive solutions for a class of boundary value problem at resonance case. Some recent devolopment for resonant case can be found in [38,39]. Let us cited few papers. In [40], the authors studied the boundary value problems of the fractional order differential equation: and D α 0þ , D β 0þ are the stantard Riemann-Liouville fractional derivative of order α. They obtained the multiple positive solutions by the Leray-Schauder nonlinear alternative and the fixed point theorem on cones.
In 2020 Li et al. [41] consider the existence of a positive solution for the following BVP of nonlinear fractional differential equation with integral boundary conditions: Þare continuous. To obtain the existence results, the authors used the well-known GuoKrasnoselskiis fixed point theorem.
In 2017, Rezapour et al. [42] investigated a Caputo fractional inclusion with integral boundary condition for the following problem is a compact valued multifunction and c D α denotes the Caputo fractional derivative of order α.
In 2018, Bouteraa and Benaicha [10] studied the existence of solutions for the Caputo fractional differential inclusion subject to three-point boundary conditions is a compact valued multifunction and c D α denotes the Caputo fractional derivative of order α.
In 2019, Ahmad et al.
[43] investigated the existence of solutions for the boundary value problem of coupled Caputo (Liouville-Caputo) type fractional differential inclusions: ( subject to the coupled boundary conditions: where C D α , C D β denote the Caputo fractional derivatives of order α and β respectively, F, G : 0, T ½ Â Â  are given multivalued maps, P  ð Þ is the family of all nonempty subsets of , and ν i , Inspired and motivated by the works mentioned above, we focus on the uniqueness of positive solutions for the nonlocal boundary value problem (1) with the iterative method and properties of f t, u ð Þ, explicit iterative sequences are given to approximate the solutions and the error estimations are also given. We also cover the multi-valued case of problem (2) when the right-hand side is nonconvex compact valued multifunctions via a fixed point theorem for multivalued maps due to Covitz and Nadler.
The chapter is organized as follows. In Section 2, we present some notations and lemmas that will be used to prove our main results of problem (1) and we discuss the uniqueness of problem (1). Finally, we give an example to illustrate our result. In Section 3, we introduce some definitions and preliminary results about essential properties of multifunction that will be used in the remainder of the chapter and we present existence results for the problem (2) when the right-hand side is a nonconvex compact multifunction. We shall use the fixed point theorem for contraction multivalued maps due to Covitz and Nadler [44] to prove the uniqueness of solution of problem (2). Finally, we give an example to ascertain the main result.

Preliminaries
In this section, we recall some definitions and facts which will be used in the later analysis. These details can be found in the recent literature; see [2,4,6,[45][46][47] and the references therein. Let Þdenote the space of i À times differentiable functions u : 0, 1 ½ !  whose i À th derivative u i ð Þ is absolutely continuous and α ½ donotes the integer part of number α.
Þ . The Caputo derivative of fractional order α for the function u : 0, þ∞ ½ Þ! is defined by The Riemann-Liouville fractional derivative order α for the function u : provided that the right hand side is pointwise defined in 0, ∞ ð Þand the function is called Euler's gamma function.
Definition 2.2. The Riemann-Liouville fractional integral of order α > 0 of a function u : 0, ∞ ð Þ! is given by provided that the right hand side is pointwise defined in 0, ∞ ð Þ. We recall in the following lemma some properties involving Riemann-Liouville fractional integral and Riemann-Liouville fractional derivative or Caputo fractional derivative which are need in Lemma where D α and D β represents Riemann-Liouville's or Caputo's fractional derivative of order α and β respectively. Lemma 2.2 [47]. Let α > 0 and y ∈ L 1 0, 1 ð Þ. Then, the general solution of the fractional differential equation where c 0 , c 1 , … , c nÀ1 are real constants and n ¼ α ½ þ 1. Based on the previous Lemma 2:2, we will define the integral solution of our problem 1 ð Þ.
Then the solution of the fractional boundary value problem is given by where d ¼ 1 À P p j¼1 a j η αÀβÀ1 j . Proof. By using Lemma 2:2, the solution of the equation where c 1 , c 2 … , c n are arbitrary real constants. From the boundary condition in (1), one can By the last above equation and Lemma 2:1 i ð Þ, we get this and by D β 0þ u 1 ð Þ ¼ P p j¼1 a j D β 0þ u η j , we have Then, the unique solution of the problem (1) is given by The proof is completed. □ Lemma 2.4. Let P p j¼1 a j η αÀβÀ1 j ∈ 0, 1 ½ Þ, α ∈ n À 1, n ð , β ∈ 1, n À 2 ½ , n ≥ 3. Then, the functions g t, s ð Þ and h t, s ð Þ defined by (6) and (7)  iii. g t, s From the above properties, we deduce the following properties: iv. The function G t, s ð Þ≥ 0 is continuous on 0, 1 ½ Â 0, 1 ½ and G t, s ð Þ> 0 for all t, s ∈ 0, 1 ð Þ.

Existence results
First, for the uniqueness results of problem (1), we need the following assumptions. where In view of Lemma 2:3, we define an operator T as where G t, s ð Þ is given by (5). By A 1 ð Þ it is easy to see that the operator T : D ! C þ 0, 1 ½ is increasing. Observe that the BVP (1) has a solution if and only if the operator T has a fixed point.
Obviously, from A 1 ð Þ we obtain In what follows, we first prove T : D ! D. In fact, for any u ∈ D, there exist a positive constants 0 < m u < 1 < M u such that m u s αÀ1 ≤ u s ð Þ ≤ M u s αÀ1 , s ∈ 0, 1 ½ : Then, from A 1 ð Þ, f t, u ð Þ non-decreasing respect to u and A 2 ð Þ, we can imply that for s ∈ 0, 1 ð Þ, q ∈ 0, 1 ð Þ From (11) and Lemma 2:4, we obtain and Tu t ð Þ ¼ Eqs. (12) and (13) and assumption A 3 ð Þ imply that T : D ! D. Now, we are in the position to give the first main result of this chapter. Theorem 1.1 Suppose A 1 ð Þ À A 3 ð Þ hold. Then problem (1) has a unique, nondecreasing solution u * ∈ D, moreover, constructing successively the sequence of functions for any initial function h 0 t ð Þ ∈ D, then h n t ð Þ f gmust converge to u * t ð Þ uniformly on 0, 1 ½ and the rate of convergence is where 0 < θ < 1, which depends on the initial function h 0 t ð Þ. Proof. For any h 0 ∈ D, we let and Since the operator T is increasing, A 1 ð Þ, A 2 ð Þ and (16)- (20) imply that there exist iterative sequences u n f g, v n f g satisfying In fact, from (19) and (20), we have and Then, by (22)- (24) and induction, the iterative sequences u n f g, v n f g satisfy (10), (19) and (20), it can obtained by induction that where θ ¼ m M .
From (19)- (21) and A 1 ð Þ, we obtain It follows from (26)-(29) that Hence, (15) holds. Since h 0 t ð Þ is arbitrary in D we know that u * t ð Þ is the unique solution of the boundary value problem (1)  We construct an example to illustrate the applicability of the result presented. Example 2.1. Consider the following boundary value problem

Existence result for inclusion problem (2)
We provide another result about the existence of solutions for the problem (2) by using the assumption of nonconvex compact values for multifunction. Our strategy to deal with this problem is based on the Covitz-Nadler theorem for the contraction multivalued maps [44] for lower semi-continuous maps with decomposable values.