On the existence of positive solutions for generalized fractional boundary value problems

The existence of positive solutions is established for boundary value problems defined within generalized Riemann–Liouville and Caputo fractional operators. Our approach is based on utilizing the technique of fixed point theorems. For the sake of converting the proposed problems into integral equations, we construct Green functions and study their properties for three different types of boundary value problems. Examples are presented to demonstrate the validity of theoretical findings.


Introduction
The branch of mathematics that deals with the study of noninteger order derivatives and integrals is called fractional calculus (FC). FC is almost 300 years old as the classical calculus. The interesting thing about this subject is that in contrast to the classical derivatives, the fractional derivatives are not a point quantity. Indeed, the fractional derivative of a function of order α at some point is a local property only for α being an integer. On the other hand, when α is not an integer, the derivative does not only depend on the graph of the function very close to the point but it also depends on some history.
FC has not been a famous applied field of interest among scientists and engineers in the previous years. Thus, many researchers have not recognized the rich applications of FC for a long period of time. In recent decades, however, it has been realized that the FC has several potential applications in different areas of engineering and science such as propagation, electrochemistry, finance, and bio engineering. In the literature, one can figure out that there are many definitions of fractional derivatives. For instance, we refer here to the most well-known types such as Caputo derivative, Liouville derivative, Hadamard derivative, Katugampola derivative, and many others. Consequently, this has led to several types of fractional differential equations defined by different fractional operators [11,22]. The best way to deal with such a variety of fractional operators is to accommodate generalized forms of fractional operators that include other operators.
Researchers who are interested in this subject have introduced many generalizations of fractional derivatives such as ψ-Hilfer fractional derivative, Hilfer-Katugampola fractional derivative, and generalized proportional fractional derivative [1, 2, 12, 14-17, 23, 25-28]. In [1], Abdo and Panchal considered a general form of ψ-Hilfer fractional derivative with respect to another function of a fractional integro-differential equation. They presented results on existence, uniqueness, and stability of the solutions. In [16], the authors introduced a new generalized derivative involving exponential functions in their kernels which, upon considering limiting cases, converges to classical derivatives. They solved Cauchy linear fractional type problems within this derivative. In [17], however, the author introduced Katugampola fractional derivative. Indeed, he presented a generalized fractional derivative that generalizes the regular Hadamard and Riemann-Liouville fractional derivatives. In [23], the authors considered a class of nonlinear fractional initial value problems, and they proved the existence and uniqueness of solutions. Following this trend, the existence of positive solutions of the regular fractional boundary value problems (FBVPs) have been discussed in many papers such as [5,6,9,10,21,31,[33][34][35]. For the sake of completeness, we refer afterwards to some relevant papers that study the existence of solutions in the frame of the classical Riemann-Liouville and Caputo derivatives. More precisely, the authors in [8] considered the problem where D α 0+ is the Riemann-Liouville operator. They investigated the existence and multiplicity of positive solutions for the above problem. In [7], the same author proved the existence and uniqueness of positive solutions for the same problem but under different boundary conditions of the form The existence and multiplicity of positive solutions of the following problem were discussed in [32]: where c D α 0+ is the Caputo operator. On the other hand, and to the best of our insight, the existence of positive solutions of FBVPs within ϕ-Riemann-Liouville and ϕ-Caputo operators has not been discussed so far.
The results of this paper are motivated by the recent work of Almeida in [3] who established results for the existence and uniqueness of solutions of FBVP involving a general form of fractional derivative. We shall consider a general form of the derivative, fractional derivative of a function with respect to another function, that includes other definitions of operators for particular choice of a function. The new derivative herein generalizes the classical definitions of derivatives in the sense that the Riemann-Liouville [30], the Erdelyi-Kober [18,29], and the Hadamard [20,24] fractional derivatives are all recovered by choosing particular forms of ϕ(t). Moreover, the results of this paper generalize the work established in the papers [7,8,32].
In this paper, we discuss the existence and multiplicity of positive solutions of FBVPs defined within ϕ-Riemann-Liouville and ϕ-Caputo operators. For our purpose, we convert FBVPs into equivalent integral equations via constructing Green functions for the proposed problems. The technique of fixed point theorems is employed to prove the main results.
The FBVP under consideration has the form and is associated with two different boundary conditions Moreover, we study FBVP of the form The paper is divided into four sections. Section 1 presents a descriptive introduction. Section 2 states some essential definitions and lemmas that we utilize to prove the main results. Section 3 is devoted to proving the main existence results for FBVP (1.

Preliminaries
In this part of the paper, we assemble some essential definitions and fixed point theorems that will be used throughout the remaining part of the paper. Besides, some auxiliary lemmas are proved prior to proceeding to the main results of this paper.
In what follows, we convert the FBVPs into integral equations via Green functions.
is equivalent to Proof The general solution of FBVP (2.2) is given by i.e., By using the given conditions z(0) = z(1) = 0, we obtain where G is defined as in equation (2.3).
We prove (ii): Let us denote Observing that G(x, ν) is a decreasing function for ν ≤ x, and it is an increasing function for x ≤ ν, we deduce that where 1/4 < k < 3/4 is a unique solution of the equation Then the conclusion follows by setting For the sake of convenience, we denote has a solution Then we have Thus, we obtain By adding and subtracting ϕ(0) in the first term of the above equation, we get It follows that Adding and subtracting ϕ(0), we have Therefore, we obtain Since μ > 0, we obtain Hence H(x, ν) > 0 for x, ν ∈ (0, 1).

Definition 2.10
Let P be a cone of a real Banach space B and θ : P → [0, ∞) be a continuous map such that for all z, w ∈ P and 0 ≤ λ ≤ 1. Then θ is said to be a nonnegative continuous concave functional on P.
We shall rely on the following fixed point theorems to prove the main results.

Lemma 2.12 ([19]) Suppose that B is a real Banach space, P ⊂ B is a cone, P c = {z ∈ P :
z ≤ c}. Let θ be a nonnegative continuous concave functional on P such that θ (z) ≤ z for all z ∈ P c , and P(θ , b, d) = {z ∈ P : b ≤ θ (z), z ≤ d}. Assume that F : P c → P c is completely continuous and there exist constants Then F has at least three fixed points z 1 , z 2 , and z 3 with z 1 < a, b < θ (z 2 ), a < z 3 with θ (z 3 ) < b.

Existence results
The following section is devoted to stating and proving the existence results for problems Let E = C[0, 1] be a Banach space equipped with the norm z = max x∈[0,1] |z(x)|, and let P, R ⊂ E defined by P = z ∈ E : z(x) ≥ 0 and R = z ∈ E : z(x) ≥ υ(ν) z be the cones, where υ(ν) is defined later.

Lemma 3.1 Let the operator T : P → E be defined by
then T : P → P is a completely continuous operator.

Lemma 3.2 Let the operator F : P → P be defined by
Then F : P → P is completely continuous.
Proof Since f , G, and ϕ are nonnegative and continuous, the operator F : P → P is continuous. We also assume that Ω ⊂ P is bounded, that is, for all z ∈ Ω, z ≤ r 2 , for some r 2 > 0, a positive constant.

Lemma 3.3 Let the operator Q : R → R be defined by
Then Q : R → R is completely continuous.
The proof of the above statement is straightforward and hence is omitted. In the sequel, we make use of the following notations: Proof By Lemma 3.1, we have T : P → P is completely continuous. Let Ω 1 = {z ∈ P : u < ρ 1 }. For z ∈ ∂Ω 1 , we have 0 ≤ z(x) ≤ ρ 1 for all x ∈ [0, 1] such that assumption (A 2 ) holds. For x ∈ [1/4, 3/4], we find that Thus, for z ∈ ∂Ω 1 , we have On the other hand, let Ω 2 = {z ∈ P : z < ρ 2 }. For z ∈ ∂Ω 2 , we have that 0 ≤ z(x) ≤ ρ 2 for all x ∈ [0, 1] such that assumption (A 1 ) holds. For x ∈ [0, 1], we find that Hence by (ii) of Lemma 2.11, it follows that problem (1.1)-(1.2) has a positive solution Theorem 3.5 Let f (x, z) be continuous and a, b, c be positive constants with a < b < c such that the following assumptions hold: Then there exist at least three positive solutions z 1 , Proof For z ∈ P c and z ≤ c, let assumption (C 3 ) hold. Then we have Therefore, T : P c → P c . Similarly, we can show that, for z ∈ P a , condition (H 2 ) of Lemma 2.12 is fulfilled.
Hence there exist at least three positive solutions z 1 , z 2 , z 3 of problem (1.  Proof We want to prove that the operator F n is a contraction for sufficiently large n. For z, w ∈ P, we have and by induction, we obtain Using condition (3.2) and letting LK n-1 (Γ (α)μ) n < 1 2 for sufficiently large n, we have This completes the proof.
Next we prove the existence results for FBVP (1.4). To complete this, we assume a particular form for f . That is, where κ is a positive constant. Besides, assume Theorem 3.7 Suppose that (P 1 ) holds. Proof From Lemma 3.3, we have that the operator Q : R → R is completely continuous.
Similarly one can state and prove the following theorems.

Illustrative examples
Corresponding to the proposed problems, we provide the following examples that demonstrate consistency to the main theorems.