Existence and stability analysis of solutions for a new kind of boundary value problems of nonlinear fractional differential equations

. This research work is dedicated to an investigation for a new kind of boundary value problem of nonlinear fractional differential equation supplemented with general boundary condition. A full analysis of existence and uniqueness of positive solutions is respectively proved by Leray–Schauder nonlinear alternative theorem and Boyd–Wong’s contraction principles. Furthermore, we prove the Hyers–Ulam (HU) stability and Hyers–Ulam–Rassias (HUR) stability of solutions. An example illustrating the validity of the existence result is also discussed.


Introduction
Fractional calculus (FC) has a history of more than 300 years, there are some applications of FC within various fields of mathematics itself. During the last few decades, FC has obtained vigorous development in the applied sciences and gained considerable popularity. Compared with classical integer-order models, fractional derivatives and integrals are more suitable to describe the memory and hereditary properties of various materials, fractional derivatives are more advantageous in simulating mechanical and electrical properties of real materials and describing rheological properties of rocks and many other fields. Based on the description of their properties in terms of fractional derivatives, fractional differential equations (FDEs) are generated naturally, and how to solve these equations is also very necessary. For example, some new models that involve FDEs have been applied successfully, e.g., in mechanics (theory of viscoelasticity and viscoplasticity [6,24]), (bio-)chemistry (modelling of polymers and proteins [11]), electrical engineering (transmission of ultrasound waves [3,28]), medicine (modelling of human tissue under mechanical load [31]). . . Accordingly, more and more researchers and scholars devote themselves to the study of various problem of FDEs. In particular, study of boundary value problems for nonlinear FDEs is particularly concerned among these problems.
By Schauder fixed point theorem and the Banach contraction principle, Rehman et al. [26] investigated existence and uniqueness of solutions for a class of nonlinear multipoint boundary value problems for fractional differential equation In [15], the author studied the nonlinear fractional differential equation subjected with the boundary conditions u(0) = u(1) = 0. Under Carathéodory conditions, using the Leray-Schauder continuation principle, the existence of at least one solution was obtained. KyuNam et al. [16] discussed the existence and uniqueness of solutions for a class of integral boundary value problems of nonlinear multiterm fractional differential equation The existence results are established by the Banach fixed point theorem, and approximate solutions are determined by the Daftardar-Gejji and Jafari iterative method (DJIM) and the Adomian decomposition method (ADM).
In this article, we will deal with singular nonlinear fractional differential equation with a new boundary condition, which is a generalization of many previous researches. To the best of our knowledge, when λ 2 = 0, neither λ 1 = 0 nor λ 1 = 0 was studied like this type of boundary condition. Furthermore, the nonlinear term contained not only lower derivative of order β, but also another lower derivative of order γ. In comparison with the above literature, our results about the difference include both α − β 1 and 0 < α − γ < 1, this has never been seen before.
The stability of differential equations has grown to be one of the considerable areas in the field of mathematical analysis, and we find many different types of stability, such as exponential [7,17,23], Mittag-Leffler [20,27], Hyers-Ulam (HU) stability and other types of stability [13,18,19,21,22]. Among these kinds of stability, Hyers-Ulam stability and its various types provide a bridge between the exact and numerical solutions, so researchers devoted their work to the study of different kinds of HU stability for nonlinear fractional differential equation; see [5,10,12,29].
The paper is organized as follows. In Section 2, we will present some useful lemmas and give some valuable preliminary results. In Section 3, we prove the existence and uniqueness of positive solution to problem (1), (2) by using Leray-Schauder nonlinear alternative theorem and Boyd-Wong's contraction principles. In Section 4, we investigate various kinds of HU stability of solutions.
First, we introduce some fundamental facts of the fractional calculus theory, which are used in this paper; see [14,25].
k=1 is a uniformly convergent sequence of continuous functions on [a, b] and that D α a+ f k exists for every k. Moreover, assume that {D α a+ f k } ∞ k=1 converges uniformly on [a + ε, b] for every ε > 0. Then for every x ∈ (a, b], we have Next, for the sake of readers' convenience, we present some necessary lemmas, which will be used in the main results that follow.
Then the linear fractional boundary value problem has a unique solution given by ϑ( and and , Proof. It is very well known that the equation D α 0+ ϑ(x) + φ(x) = 0 is equivalent to the following integral equation: The boundary condition ϑ(0) = 0 implies that c 2 = 0. Using the property of Riemann-Liouville fractional derivative, we know Combining the boundary condition in (3), it follows that https://www.journals.vu.lt/nonlinear-analysis Therefore the unique solution of problem (3) is given by Lemma 2. Let G be the Green function related to problem (3), which is given by expression (4). Then for 0 β α − 1 1 < γ < α 2, > 1/Γ(α − β) − 1/Γ(α − γ), G, K, K 0 have the following properties: In addition, we also have a relation In the same way, we get K 0 is nonnegative, then we have G 0. It is obvious that (ii) The conclusion is obvious, we omit it. (iii) First, we introduce an inequality. For λ, µ ∈ (0, ∞) and a, x ∈ [0, 1], we have When s x, using inequality (5), we obtain When x s, In addition, Again, we can get the property of K 0 , that is, https://www.journals.vu.lt/nonlinear-analysis Therefore, (iv) By substituting the two inequalities (iii) into formula (4) we naturally come to the conclusion.
Consider the problem ϑ = Gϑ, where operator G is defined by where g ϑ (s) = g(s, ϑ(s), D β 0+ ϑ(s), D γ 0+ ϑ(s)), G is defined in (4). In order to prove that problem (1), (2) has a solution, we just have to show that operator G has a fixed point. Take the fractional derivative of order β for Gϑ, we have where Similarly, we have Then By simple deduction we can get the following properties. (8), (9) and (10) have the following properties: In order to show that operator I α 0+ is continuous, we have to prove I α According to the inequality above, we have I α where From this inequality we know F is continuous on t = 0 if one supplies the definition of F on t = 0: Now, we will evaluate these formulae (13)-(15), respectively.

Existence and uniqueness results
This section deals with existence and uniqueness of solutions for problem (1), (2). The nonlinear term g satisfies the following assumptions: (C1) g : (0, 1] × R + × R + × R → R + is continuous, g(x, 0, 0, 0) does not vanish on any compact interval of (0, 1]. Furthermore, there exist nonnegative functions σ i ∈ L p α−γ [0, 1] (i = 1, 2, 3) and continuous and nondecreasing functions ϑ i : (C2) There exists a positive number R such that are upper semicontinuous from the right and nondecreasing such that for any it satisfies Φ(x) < x for all x > 0.
Suppose that ϑ n → ϑ 0 (n → ∞) in cone K, then there exists a constant M > 0 such that ϑ n M (n = 0, 1, . . . ). In order to get the conclusion that operator G is continuous, let us start with the fact that Moreover, on the basis of the continuity of g, we deduce that g ϑn (x) → g ϑ0 (x), n → ∞, for all x ∈ (0, 1]. Taking advantage of Lebesgue dominated convergence theorem, we know Thereupon, by Lemma 4, we have I α 0+ g ϑn convergence to I α 0+ g ϑ0 in C[0, 1]. In addition, by Hölder inequality, we can also get In the end, Ascoli-Arzela theorem guarantees operator G : K → K is compact. That is to say, we can deduce that G(B) is bounded and equicontinuous for any bounded subset B ⊆ K. The proof can be obtained by the conventional procedure, so we omit this step.
From the above we conclude that the operator G is completely continuous.
Theorem 1. (See [1].) Let E be a Banach space with C ⊆ E closed and convex. Assume that U is relatively open subset of C with 0 ∈ U and A : U → C is a continuous compact map. Then either (i) A has a fixed point in U or (ii) There exists u ∈ ∂U and λ ∈ (0, 1) with u = λAu.
Proof. By applying nonlinear alternative of Leray-Schauder-type fixed point theorem (Theorem 1), we will prove that G has a fixed point. Let B R = {u ∈ K | u < R}, R is given in condition (C2). Consider the following integral equation: where λ ∈ (0, 1). We claim that any solution of (16) for any λ ∈ (0, 1) must satisfies ϑ = R. Otherwise, assume that ϑ is a solution of (16) for some λ ∈ (0, 1) such that ϑ = R. Hence, from condition (C1) and Lemma 2(iv), for any x ∈ [0, 1], we have Similarly, in view of Lemma 3 and (7), we have According to (11), we have https://www.journals.vu.lt/nonlinear-analysis Therefore, This is a contradiction and the claim is proved. Leray-Schauder nonlinear alternative theorem guarantees that operator G has a fixed point ϑ ∈ B R . Since g(x, 0, 0, 0) does not vanish on any compact interval of (0, 1), we know ϑ must be positive.
Theorem 3. Let X be a complete metric space and suppose T : X → X satisfies is upper semicontinuous function from the right (i.e., r j ↓ r 0 ⇒ lim sup j→∞ Φ(r j ) Φ(r), and for x > 0, 0 Φ(x) < x for x > 0). Then T has a unique fixed point x ∈ X.
Proof. For any ϑ, y ∈ K and x ∈ [0, 1], by using Lemma 2(iv) and condition (C3), we get Similarly, Synthesizing the above three inequalities and combining with condition (C4), we get Then Boyd-Wong's contraction principle can be applied and G has a unique fixed point which is the unique solution of problem (1), (2). Example.

Stability analysis
In this section, we consider the Banach space Let us introduce some definitions related to Ulam stability.
Similarly, we can get .

Remark 2.
Under the condition of Theorem 7, imitating the process, we can prove that BVP (1), (2) is generalized HUR stable.