Positive Solutions of a Fractional Boundary Value Problem with Sequential Derivatives

: We investigate the existence of positive solutions of a Riemann-Liouville fractional differential equation with sequential derivatives, a positive parameter and a nonnegative singular nonlinearity, supplemented with integral-multipoint boundary conditions which contain fractional derivatives of various orders and Riemann-Stieltjes integrals. Our general boundary conditions cover some symmetry cases for the unknown function. In the proof of our main existence result, we use an application of the Krein-Rutman theorem and two theorems from the ﬁxed point index theory.

We present some assumptions on the functions g and r, and intervals for the parameter λ such that problem (1), (2) has at least one positive solution. By a positive solution of (1), (2) we mean a function v ∈ C[0, 1] satisfying (1) and (2) with v(t) > 0 for all t ∈ (0, 1]. In the proof of our main theorem we use some results from the fixed point index theory. Positive solutions for such fractional problems are of great practical importance for describing nonlocal processes with memory, which determines the relevance of the chosen research topic. In what follows we present some recent results connected with our problem (1), (2). In [1], the authors studied the fractional differential equation subject to nonlocal boundary conditions where γ ∈ R, γ ∈ (n − 1, n], n ∈ N, n ≥ 3, , m ∈ N, ζ j ∈ R for all j = 0, . . . , m, . . , m are bounded variation functions, µ is a positive parameter, the function g(t, w) is nonnegative and it may have singularity at w = 0 and the function h(t) is nonnegative and it may be singular at the points t = 0 and/or t = 1. They gave various assumptions for the functions h and g, and established intervals for the parameter µ such that problem (3), (4) has at least one positive solution. The expression for intervals of µ are given by using the principal characteristic value of an associated linear operator. The fixed point index theory was used in the proof of the main theorems. By using the Guo-Krasnosel'skii fixed point theorem, a related semipositone problem is also studied in [1]. In [2], the authors studied the system of fractional differential equations with sequential derivatives with the coupled boundary conditions where γ 1 , γ 2 ∈ (0, 1], δ 1 ∈ (p − 1, p], δ 2 ∈ (q − 1, q], p, q ∈ N, p, q ≥ 3, n, m ∈ N, α j ∈ R for all j = 0, 1, . . . , n, 0 ≤ α 1 < α 2 < · · · < α n ≤ β 0 < δ 2 − 1, β 0 ≥ 1, β j ∈ R for all j = 0, 1, . . . , m, 0 ≤ β 1 < β 2 < · · · < β m ≤ α 0 < δ 1 − 1, α 0 ≥ 1, µ > 0, ν > 0, φ and ψ are continuous functions which change sign and they may be singular at the points t = 0 and/or t = 1, H j , j = 1, . . . , n and K j , j = 1, . . . , m are bounded variation functions. They present various assumptions for the nonsingular/singular functions φ and ψ, and intervals for parameters µ and ν such that problem (5), (6) has at least one or two positive solutions. They applied the nonlinear alternative of Leray-Schauder type and the Guo-Krasnosel'skii fixed point theorem in the proofs of the main existence results. We also mention the papers [3][4][5][6][7], and the books [4,[8][9][10][11][12][13][14][15][16][17][18] with their references for other results obtained in the last years and for applications of the fractional differential equations and systems in various fields. The structure of the paper is as follows. In Section 2 we study a linear fractional boundary value problem associated to our problem (1), (2), and we present the associated Green functions with their properties and bounds. Section 3 is concerned with the main existence theorem for (1), (2), and an example illustrating our result is presented in Section 4.

Existence of Positive Solutions
In this section we present some conditions on the functions g and r, and intervals for λ such that there exists at least one positive solution of (1), (2). We introduce the Banach space X = C[0, 1] with supremum norm v = sup s∈[0,1] |v(s)|, and we introduce the cones We define the operator E : C → C and the linear operator F : X → X by We observe that v is a solution of problem (1), (2) if and only if v is a fixed point of operator We give now the assumptions that we will use in this section.
Proof. By using (H3), we find that there exists a number m 1 ≥ 3 with the property For v ∈ S θ 2 \ S θ 1 , we find that there exists ω 0 ∈ [θ 1 , θ 2 ] such that v = ω 0 , and then and (16), we deduce where J 10 = max s∈[0,1] J 1 (s) > 0. This gives us that the operator E is well defined. Next we show that E : S θ 2 \ S θ 1 → S. Indeed, for any v ∈ S θ 2 \ S θ 1 and t ∈ [0, 1], we have and then By Lemma 2, we also obtain We prove now that E : S θ 2 \ S θ 1 → S is a completely continuous operator. We assume that D ⊂ S θ 2 \ S θ 1 is an (arbitrary) bounded set. By using the first part of the proof, we deduce that E (D) is uniformly bounded. We show next that E (D) is equicontinuous. For > 0 there exists a natural number m 2 ≥ 3 satisfying the condition 1], then for the above > 0 we deduce that there exists ρ > 0 such that for any t 1 , Here Therefore for any v ∈ D, t 1 , So we obtain that E (D) is equicontinuous. By the Arzela-Ascoli theorem, we deduce that E : S θ 2 \ S θ 1 → S is compact.
We prove next that E : S θ 2 \ S θ 1 → S is continuous. We assume that v n , v 0 ∈ S θ 2 \ S θ 1 for all n ≥ 1, and v n − v 0 → 0 as n → ∞. Then θ 1 ≤ v n ≤ θ 2 for all n ≥ 0. By (H3), for > 0 there exists a natural number m 3 ≥ 3 satisfying the condition Because g is uniformly continuous in 1 Therefore by using the Lebesgue dominated convergence theorem, we deduce So, for the above > 0 there exists another natural number m 4 such that for all n > m 4 we obtain By (17) and (18) we conclude that So we find that E : S θ 2 \ S θ 1 → S is a continuous operator. Therefore E is a completely continuous operator.
Under assumptions (H1)-(H3), by using the extension theorem, the operator E has a completely continuous extension (we also denoted it by E ) from S to S. By using the Krein-Rutman theorem in the space C[0, 1] and similar methods as those used in the proof of Lemma 3.2 from [1], we obtain the following lemma.
By using a similar approach as that used in the proof of Lemma 4 for operator E , we deduce that F (S) ⊂ S.

Conclusions
In this paper we study the fractional differential Equation (1) with sequential derivatives and a positive parameter, subject to general nonlocal boundary conditions (2) containing Riemann-Stieltjes integrals and fractional derivatives of various orders. The function g from the equation is a nonnegative continuous function and it may be singular at the second variable in the point 0, and the function r is also nonnegative continuous one and it may be singular at the points t = 0 and t = 1. The general boundary conditions (2) cover some symmetry cases (as periodicity conditions for derivatives) for the unknown function. By using an application of the Krein-Rutman theorem in the space C[0, 1], and some theorems from the fixed point index theory, we prove that problem (1), (2) has at least one positive solution v(t), t ∈ [0, 1]. To illustrate our main existence theorem, we finally present an example.