On a singular Riemann–Liouville fractional boundary value problem with parameters

Abstract. We investigate the existence of positive solutions for a nonlinear Riemann–Liouville fractional differential equation with a positive parameter subject to nonlocal boundary conditions, which contain fractional derivatives and Riemann–Stieltjes integrals. The nonlinearity of the equation is nonnegative, and it may have singularities at its variables. In the proof of the main results, we use the fixed point index theory and the principal characteristic value of an associated linear operator. A related semipositone problem is also studied by using the Guo–Krasnosel’skii fixed point theorem.

The integrals from the boundary conditions (2) are Riemann-Stieltjes integrals with H i , i = 1, . . . , m, functions of bounded variation, the nonnegative function f (t, u) may have singularity at u = 0, and the nonnegative function h(t) may be singular at t = 0 and/or t = 1. Under some assumptions for the functions h and f , we establish intervals for the parameter λ such that problem (1), (2) has positive solutions (u(t) > 0 for all t ∈ (0, 1]). These intervals for λ are expressed by using the principal characteristic value of an associated linear operator. In the proof of the main theorems, we use the fixed point index theory. In the case in which h ≡ 1 and f is a function which changes sign and has singularities at t = 0 and/or t = 1, we present two existence results for the positive solutions of this problem. In the proof of these results, we apply the Guo-Krasnosel'skii fixed point theorem. The boundary conditions (2) cover various cases, such as multipoint boundary conditions when the functions H i are step functions, or classical integral boundary conditions, or a combination of them.
We present below some papers, which investigate particular cases of our boundary value problem (1), (2) and other problems related to (1), (2). Equation (1) with h(t) ≡ 1 subject to the boundary conditions u(0) = u (0) = · · · = u (n−2) (0) = 0, where ξ i ∈ R, i = 1, . . . , m, 0 < ξ 1 < · · · < ξ m < 1, p, q ∈ R, p ∈ [1, n−2], q ∈ [0, p], was investigated in [11]. In paper [11], the nonlinearity f changes sign, and it is singular only at t = 0 and/or t = 1. The authors of [11] apply the Guo-Krasnosel'skii fixed point theorem to prove the existence of positive solutions when the parameter belongs to various intervals. Equation (1) with λ = 1 and h(t) ≡ 1 supplemented with the boundary conditions (2) with m = 1, where f may change sign and may be singular at the points t = 0, t = 1 and/or u = 0 has been studied in [20]. In the paper [20], the author presents some conditions for f , which contain height functions defined on special bounded sets under which he proves the existence and multiplicity of positive solutions. The existence of multiple positive solutions for equation (1) with λ = 1 and h(t) ≡ 1 subject to the boundary conditions (2) was investigated in the recent paper [1]. The authors use in [1] various height functions of the nonlinearity defined on special bounded sets and two theorems from the fixed point index theory. In the paper [35], the authors prove the existence of at least three positive solutions for equation (1) with λ = 1 and h(t) ≡ 1 with the boundary conditions where is nonnegative and may be singular at t = 0 and t = 1, and the function f is nonnegative and may be singular at the points t = 0, t = 1 and u = 0. Our boundary conditions (2) are more general than the above boundary conditions (3). Indeed, the last relation from (3) http://www.journals.vu.lt/nonlinear-analysis can be written as D β 1]}, and in the right-hand side of the last condition in (2), we have a sum of Riemann-Stieltjes integrals from Riemann-Liouville derivatives of various orders. In the paper [35], the authors use different height functions of the nonlinear term on special bounded sets, the Krasnosel'skii theorem and the Leggett-Williams fixed point index theorem. We also mention the paper [33], where the authors prove the existence of positive solutions of fractional differential equation (1) supplemented with the boundary conditions . The last condition of the boundary conditions (4) can be written as D β . . .}, so this condition is a particular case of our condition from (2). We mention that condition (I3) (see below, in Section 3) used in our results was first introduced in the paper [18], where the authors proved the existence of at least one positive solution for a fourth-order nonlinear singular Sturm-Liouville eigenvalue problem.
The paper is organized as follows. In Section 2, we present the solution of a linear fractional differential equation associated to equation (1) subject to the boundary conditions (2) and the properties of the corresponding Green functions. Some theorems from the fixed point index theory, the Guo-Krasnosel'skii fixed point theorem and an application of the Krein-Rutman theorem in the space C[0, 1] are recalled in Section 2, and they will be used in the next sections. In Section 3, we give and prove the main theorems for the existence of at least one positive solution for problem (1), (2). In Section 4, we present two existence results for the positive solutions of problem (1), (2) with h ≡ 1, where the nonlinearity changes sign, and it is singular at t = 0 and/or t = 1. Section 5 contains some examples, which illustrate the obtained results, and in Section 6, we give the conclusions for our fractional boundary value problems.

Auxiliary results
In this section, we present some auxiliary results from [1] that we will use in the proof of the main theorems. We consider the fractional differential equation with the boundary conditions (2), where x ∈ C(0, 1) ∩ L 1 (0, 1). We denote is given by where and Based on some properties of functions g 1 and g 2i , i = 1, . . . , m, given by (8) (see [11]), we have the following lemma.
Lemma 2. (See [1].) We suppose that ∆ > 0. Then the Green function G given by (7) is a continuous function on [0, 1] × [0, 1] and satisfies the inequalities: http://www.journals.vu.lt/nonlinear-analysis Lemma 3. (See [1].) We suppose that ∆ > 0, x ∈ C(0, 1) ∩ L 1 (0, 1) and x(t) 0 for all t ∈ (0, 1). Then the solution u of problem (5), (2) given by (6) satisfies the inequality We recall now some theorems concerning the fixed point index theory and the Guo-Krasnosel'skii fixed point theorem. Let X be a real Banach space with the norm · , C ⊂ X a cone, " " the partial ordering defined by C and θ the zero element in X. For > 0, let B = {u ∈ X: u < } be the open ball of radius centered at θ, its closure B = {u ∈ X: u } and its boundary ∂B = {u ∈ X: u = }. The proofs of our results are based on the following fixed point index theorems.
Theorem 3. (See [9].) Let X be a Banach space, and let C ⊂ X be a cone in X.

Main results
In this section, we present intervals for the parameter λ such that our problem (1), (2) has at least one positive solution. We consider the Banach space X = C[0, 1] with the supremum norm u = sup t∈[0,1] |u(t)|, and we define the cones We define the operator A : P → P and the linear operator T : X → X by We see that u is a solution of problem (1), (2) if and only if u is a fixed point of operator A.
We introduce now the assumptions that we will use in what follows.
, and for any 0 < r < R, we have Lemma 4. Assume that assumptions (I1)-(I3) hold. Then, for any 0 < r < R, the operator A : Q R \ Q r → Q is completely continuous.
Proof. By (I3) we deduce that there exists a natural number n 1 3 such that For u ∈ Q R \ Q r , there exists r 1 ∈ [r, R] such that u = r 1 , and then Let  so Au ∈ Q. Therefore A(Q R \ Q r ) ⊂ Q.

This gives us that A(E) is equicontinuous. By the Arzelà-Ascoli theorem we conclude that
Finally, we prove that A : Q R \ Q r → Q is continuous. We suppose that u n , u 0 ∈ Q R \ Q r for all n 1 and u n − u 0 → 0 as n → ∞. Then r u n R for all n 0. By (I3), for ε > 0, there exists a natural number n 3 3 such that h(s) f s, u n (s) − f s, u 0 (s) ds → 0 as n → ∞.
Thus, for the above ε > 0, there exists a natural number N such that, for n > N , we have http://www.journals.vu.lt/nonlinear-analysis By (9) and (10) we conclude that This implies that A : Q R \Q r → Q is continuous. Hence A : Q R \Q r → Q is completely continuous.
Under assumptions (I1)-(I3), by the extension theorem the operator A has a completely continuous extension (also denoted by A) from Q to Q.
Using a similar argument as that used in the proof of Lemma 4 for operator A, we obtain that T (Q) ⊂ Q.
By Theorem 2 we conclude that By (12), (15) and the additivity property of the fixed point index we deduce that So operator A has at least one fixed point on Q R \ Q r1 , which is a positive solution of problem (1), (2).
By using a similar approach as that used in the proof of Theorem 5, we obtain the following result.
In the proof of Theorem 7, we consider R 1 > σ 1 0 p(t) dt > 0, and we define Then there exists λ * > 0 such that, for any λ λ * , the boundary value problem (16), (2) has at least one positive solution.

Conclusion
In this paper, we study the existence of positive solutions for the nonlinear Riemann-Liouville fractional boundary value problem (1), (2), where λ is a positive parameter. The function f is nonnegative, and it may be singular at the second variable, and the function h is also nonnegative, and it may have singularities at t = 0 and/or t = 1. We present conditions for f and h and intervals for λ, which are expressed in term of the principal characteristic value of an associated linear operator. In the proof of the existence theorems, we use two results from the fixed point index theory. We also investigate a related semipositone problem, namely, equation (1) with h ≡ 1 and f a sign-changing function with singularities at t = 0 and/or t = 1 subject to the nonlocal boundary conditions (2). For this problem, we give two existence results for the positive solutions when λ belongs to various intervals. Three examples, which illustrate the obtained existence theorems, are finally presented.