Existence results of fractional di ﬀ erential equations with nonlocal double-integral boundary conditions

: This article presents the existence outcomes concerning a family of singular nonlinear differential equations containing Caputo’s fractional derivatives with nonlocal double integral boundary conditions. According to the nature of Caputo’s fractional calculus, the problem is converted into an equivalent integral equation, while two standard ﬁxed theorems are employed to prove its uniqueness and existence results. An example is presented at the end of this paper to illustrate our obtained results


Introduction
Due to its extensive applications in several fields like science and engineering, fractional calculus (FC) has acquired remarkable generality and significance, especially within the last few decades.FC is widely used to describe such practical problems as viscoelastic bodies, continuous media with memory, transformation of temperature, etc.Compared with the traditional integer-order models, the fractional order models can accurately reflect the properties and laws of related phenomena.Recently, there has been a lot of literature on FC.Some of them focus on the basic theory of FC, and the others focus their research on the solvability of initial problems or boundary problems in term of special functions, readers can refer to references [1][2][3][4][5][6][7] for details.Researchers have made great advancement in the study of qualitative and quantitative properties of solutions for fractional differential equations (FDEs), including existence, uniqueness, boundedness, continuous dependence on initial data and so on [8][9][10][11][12][13][14][15].The methods used for analysis include fixed point theorems, the comparison principle, chaos control, nonlinear alternatives of the Leray-Schauder type, upper and lower solutions and numerical calculation.For various studies performed on FC, we refer the reader to more literature [16][17][18][19][20][21][22][23] and the references therein.
In recent years, the issues related to singular FDEs (SFDEs) have been verified.The positive solutions regarding a category of SFDEs were verified in [21] by where f : (0, 1] × [0, +∞) and lim t→0 + f (t, x(t)) = ∞.They employed the fixed-point theorem and the Leray-Schauder type with nonlinear form in a cone to obtain two results for this problem.
FDEs have been investigated in various studies when integral boundary conditions (BCs) are under consideration.This type of problems arose from many research areas such as heat conduction, chemical engineering, underground water flow, population dynamics, and so forth.For further information about FDEs with integral BCs, we refer the reader to the [32][33][34][35][36][37][38][39] and the references therein.For instance, Ahmad and Agarwal [39] investigated both the existence and uniqueness of solutions (EUS) for fractional boundary value problems (FBVPs) with some novel versions regarding slit-strips conditions.One of the problems that they considered is as follows: where c D q stands for a special derivative with order q called the fractional derivative of Caputo type and a continuous mapping expressed by f (t, x(t)) in ([0, 1] × R) is considered.They obtained the EUS conditions for the mentioned problems by applying fixed principles.
Inspired by the mentioned studies, the current study discusses the following singular nonlinear FDE containing nonlocal double integral BCs: where c D δ 0 + is Caputo's differentiation of order δ; δ, η and γ are real numbers satisfying 1 ≤ n − 1 < δ ≤ n < +∞ and 0 < η < γ < 1, and n = [δ] + 1 is an integer number, the nonlinear term f (t, x(t)) ∈ ((0, 1)× R, R) becomes singular when both t = 0 and t = 1, namely, lim For the physical meaning of the integral BCs in (1.1), x(t) can be interpreted as the distribution of heat on a linear body, and the integral condition x(0) = η 0 x(τ)dτ states that the heat absorbed or emitted by
the body at t = 0 is equal to the variable of its heat over [0, η].The other integral condition has a similar explanatory and physical meaning.The current study aims to demonstrate the EUS to the problem (1.1).The generalized Hölder's inequality and fixed-point theories are applied in this paper, while the use of the generalized Hölder's inequality is the highlight of this article.This category of problems discussed in this article and the methods used make a contribution to the existing literature.This paper consists of a total of five parts.In the first part, the related situation of FDEs is introduced.The second part mainly introduces some basic knowledge of FC, such as definitions and related lemmas, which will be employed in the following content.The third part is the core of the manuscript, including the key conclusions and their proofs.The fourth part includes an example, which aims to use the results of this paper to solve the relevant problems.The last part is the summary of this paper.

Preliminaries
The characteristics of FC, the lemmas to be used, and pertinent principles are presented in the current subsection.
respectively.In the above relations, Γ(•) stands for the gamma function.
respectively, where D describes the derivative operator and n = Especially, for a = 0, this result can be presented as where Now, BVP (2.1) possesses the following unique solution for a certain function h(t) ∈ C[0, 1] where c D δ 0 + stands for the Caputo's differentiation of order δ; δ, η, γ and n are defined as in problem (1.1), and Proof According to Lemma 2.1, one can gain By differentiating x(t) based on the expression in (2.3), the following relations are obtained and Both sides' integration of (2.7) regarding the lower and upper bounds of 0 and η, respectively is denoted by By transposing and rearranging, we can get the following from the above formula Both sides' integration of (2.7) by using the lower and upper bounds γ and 1, respectively is represented by By transposing and rearranging, we can get the following from the above formula Equations (2.8) and (2.9) constitute a system with η 0 x(τ)dτ and 1 γ x(τ)dτ as the unknown elements, and the coefficients of this system are represented by So, using the Cramer's rule, we can get where The result can be derived after substituting Eqs (2.10) and (2.11) into Eq (2.7).This finishes the proof.
Banach's fixed point theorem and its subsequent theorem help to attain the main outcomes of the current article.This part ends with showing some fundamental understanding of the L p space and introducing an inequality and its corresponding extended format called the Hölder's inequality [46].
Consider that an open (or measurable) set is denoted by V ⊂ R n and a measurable mapping of real numbers defined on V is denoted by g(x).|g(x)| p turns out to be measurable on V for 1 ≤ p < ∞ and V |g(x)| p dx is meaningful.Now, we introduce a function space L p (V) as follows: For g ∈ L p (V), the upcoming norm is defined The mentioned result is extended as where g i (x) ∈ L p i (V) and n i=1 1 p i = 1.The above expression is just called the generalized Hölder's inequality.

Main results
Suppose that E = C([0, 1], R) encompasses continuous function space on interval [0, 1].Now, a Banach space is denoted by X = (E, • ), where • is the maximum norm Define an operator φ : X → X as There exists equality between the solutions of the problem (1.1) and the fixed points regarding the operator φ.This paper presents the following assumptions that are put on f (t, x(t)) that appears in (1.1) in the sequel.
(H1) Both t = 0 and t = 1 lead to a singular f (t, x(t)) which satisfies Besides, there are two constants σ 1 > 0 and σ 2 > 0, where t By the assumption of (H1), it can be deduced that a number N 0 exists and meets where t ∈ [0, 1] and x(t) ∈ E. Throughout the rest of this article, we always employ s, s 1 and s 2 to represent any set of real numbers that meet the following conditions (H2 Accordingly, avoiding excessive conjugate exponent notations is possible while using the generalized Hölder's inequality in different contexts.Lemma 3.1 Assume that 1 ≤ n − 1 < δ ≤ n, and s, s 1 , s 2 , σ 1 and σ 2 are positive constants satisfying (H2).Define an operator K l (t) for some real number l ≥ 1 as Then, the following results are valid: Proof (1) Recall Lemma 3.2 in [44].
(2) According to the generalized Hölder's inequality, one obtains (3) Deriving the function K l (t) and using the generalized Hölder's inequality, one can obtain By the mean value theorem, we have where ξ is a number between t 1 and t 2 .Lemma 3.2 Assume that 1 ≤ n − 1 < δ ≤ n and a function h(t) : (0, 1) → R is continuous and satisfying lim t→0 + h(t) = ∞ and lim t→1 − h(t) = ∞.A new function H(t) is defined as Then the continuity of t According to the definition of H(t), we have For any t ∈ [0, 1], H(t) is continuous and t will be proven.(I) For t = 0 and t ∈ [0, 1], the following equality is attained.
Proof For every x, y ∈ X = (C[0, 1], R), the second assertion of Lemma 3.1 and the generalized Hölder's inequality can be deduced by (H3) The condition (3.3) ensures that the operator φ is a contractive mapping.Accordingly, Banach's fixed-point theorem indicates that φ possesses a unique fixed-point that is equal to the problem (1.1) unique solution.
Proof Take a constant L satisfying The number N 0 is defined in (3.For any x, y ∈ B L , the following relation can be obtained by taking a process similar to Theorem 3.1: The fractional integrals denoted by I α a + f and I α b − f of order α ∈ C( (α) > 0) called the Riemann-Liouville type can be represented by

Definition 2 . 2 (
[3]) Consider y(x) ∈ AC n [a, b].Now, the derivatives ( c D α a + y)(x) and ( c D α b − y)(x), called the Caputo's, can subsist nearly on the whole interval [a, b].(1) If α N 0 , ( c D α a + y)(x) and ( c D α b − y)(x) are defined as follows: Lemma 2.3([45]) (The fixed point theorem by Krasnoselskii) Suppose that M is defined as a non-empty subset of a Banach space X with properties of closedness, boundedness and convexity.Moreover, consider that A and B stand for the operators meeting the subsequent requirements (a) Ax + By ∈ M, for x, y ∈ M; (b) both compactness and continuity of A exist; (c) a contraction mapping is represented by B. Now, z ∈ M exists such that z = Az + Bz.