Existence and Uniqueness of Solutions for Fractional Boundary Value Problems under Mild Lipschitz Condition

This paper deals with the following boundary value problem ( where 3 < α ≤ 4, D α is the Riemann-Liouville fractional derivative, and the nonlinearity f , which could be singular at both t = 0 and t = 1, is required to be continuous on ð 0, 1 Þ × ℝ satisfying a mild Lipschitz assumption. Based on the Banach ﬁ xed point theorem on an appropriate space, we prove that this problem possesses a unique continuous solution u satisfying j u ð t Þj ≤ c ω ð t Þ , for t ∈ ½ 0, 1 (cid:2) and c > 0, where ω ð t Þ ≔ t α − 2 ð 1 − t Þ 2 :


Introduction
Higher order fractional differential equations subject to twopoint boundary value problems occur naturally when modeling various phenomena in the applied sciences (see, for example, [1][2][3][4] and references therein). The study of existence, uniqueness, and qualitative properties of the solutions of such problems subject to various type of boundary conditions become an active area of research (see, for instance, [5][6][7][8][9][10][11][12][13] and references therein).
In [10], the authors considered the following problem where 3 < α ≤ 4,D α is the standard Riemann-Liouville fractional derivative. By reducing problem (1) to an equivalent Fredholm integral equation and using some fixed-point theorems, they have proved the existence, multiplicity, and uniqueness of positive solutions. In their approach, properties of the corresponding Green's function are used.
In [6], the authors proved the existence and uniqueness of positive solutions of the problem where σ ∈ ð−1, 1Þ,3 < α ≤ 4,D α is the standard Riemann-Liouville fractional derivative. Their approach relies on properties of Karamata regular variation functions and the Schauder fixed point theorem.
To simplify our statement, for α ∈ ð3, 4, we fix the following notation: (i) G α ðt, sÞ denotes the Green's function of the operator u → D α u with boundary conditions uð0Þ = uð1Þ = D α−3 uð0Þ = u ′ ð1Þ = 0, which is given (see [6], Lemma 5) for t, s ∈ ½0, 1 by (ii) Let L be the minimum positive constant such that where We will prove that L satisfies where γ ≔ max ððα − 2Þ 2 , α − 1Þ: Our main result is the following.
The paper is organized as follows. In Section 2, we recall basic properties of the Green's function G α ðt, sÞ, and we prove that L satisfies the range estimates (10). In Section 3, we prove our main result. To illustrate our existence result, some examples and approximations are given.

Preliminaries
For the convenient of the reader, we recall the following definition.
Definition 4 ( [2,14]). The Riemann-Liouville fractional derivative of order α > 0 for a measurable function g : ð0,∞Þ → ℝ is defined as provided that the right-hand side is pointwise defined on ð0, ∞Þ, where n = ½α + 1 and ½α denotes the integer part of α: The next key lemma is useful. For the proof, we refer the reader to [6].

Main Results
Assume that (H1) and (H2) hold and L < 1: We prove that problem (5) has a unique solution u in C ω ð½0, 1Þ: In addition, for any u 0 ∈ C ω ð½0, 1Þ, the iterative sequence u k ðtÞ ≔ Ð 1 0 G α ðt, sÞf ðs, u k−1 ðsÞÞds converges to u with respect to the k:k ω -norm, and we have Consider the operator T defined by We claim that T is a contraction operator from ðC ω ð½0, 1Þ, k:k ω Þ into itself:
By using similar arguments as in the previous example, we verify that conditions (H1) and (H2) are fulfilled.

Data Availability
No data were used to support this study.