Existence Result for a Superlinear Fractional Navier Boundary Value Problems

In this paper, we study the following fractional Navier boundary value problem $$\begin{aligned} \left\{ \begin{array}{lllc} D^{\beta }(D^{\alpha }u)(x)=u(x)g(u(x)),\quad x\in (0,1), \\ \displaystyle \lim _{x\longrightarrow 0}x^{1-\beta }D^{\alpha }u(x)=-a,\quad \,\,u(1)=b, \end{array} \right. \end{aligned}$$Dβ(Dαu)(x)=u(x)g(u(x)),x∈(0,1),limx⟶0x1-βDαu(x)=-a,u(1)=b,where $$\alpha ,\beta \in (0,1]$$α,β∈(0,1] such that $$\alpha +\beta >1$$α+β>1, $$D^{\beta }$$Dβ and $$D^{\alpha }$$Dα stand for the standard Riemann–Liouville fractional derivatives and a, b are nonnegative constants such that $$a+b>0$$a+b>0. The function g is a nonnegative continuous function in $$[0,\infty )$$[0,∞) that is required to satisfy some suitable integrability condition. Using estimates on the Green’s function and a perturbation argument, we prove the existence of a unique positive continuous solution, which behaves like the unique solution of the homogeneous problem.


Introduction
Recently, many papers on fractional differential equations have been studied extensively by many researches. The motivation for those works stems from the fact that fractional equations serve as an excellent tool to describe many phenomena in various fields of science and engineering such as viscoelasticity, electrochemistry, control theory, porous media, electromagnetism and other fields. Also, it provides an excellent tool to describe the hereditary properties of various materials and processes. Concerning the development of theory methods and applications of fractional calculus, we refer to [5,[8][9][10][11][12]14,16,23,24,26,28] and the references therein for discussions of various applications.
In [18], Mâagli et al. studied the following initial value problem: where α ∈ (0, 1), σ < 1 and p is a nonnegative continuous function in (0, 1) satisfying some appropriate conditions related to the Karamata class K ( see Definition 4 below ). By a potential theory approach associated with D α and some technical tools relying to Karamata regular variation theory, the authors proved the existence, uniqueness and asymptotic behavior of a positive solution to problem (1.1) in the weighted space of continuous functions Later, in [19], Mâagli and Dhifli studied the following sublinear fractional Navier boundary value problem: where σ ∈ (−1, 1), α, β ∈ (0, 1] such that α + β > 1 and p is a nonnegative continuous function in (0, 1). Under some appropriate conditions on the function p and using the Schäder fixed point theorem, the authors proved the existence of a unique positive solution to problem (1.2). Further, based on the asymptotic behavior for the Green function and some technical tools relying on Karamata regular variation theory, the authors gave a global asymptotic behavior of such solutions to problem (1.2). Inspired by the above-mentioned papers, we aim at studying in this paper the following superlinear fractional Navier boundary value problem: where α, β ∈ (0, 1] such that α + β > 1 and a and b are nonnegative constants such that a + b > 0. The nonlinear term g(t) is required to be a nonnegative continuous function on [0, ∞) satisfying some appropriate conditions related to the class of functions K α,β defined as follows. Definition 1. Let α, β ∈ (0, 1]. A Borel measurable function q in (0, 1) belongs to the class K α,β if q satisfies the following: We use the properties of this class to investigate an existence result for the fractional Navier boundary value problem (1.3). To state our main result in this paper, we need to introduce some convenient notations. Throughout this paper, we denote B((0, 1)) the set of Borel measurable functions in (0, 1) and B + ((0, 1)) the set of nonnegatives ones. We use C r ([0, 1]) to denote the where H(x, t) is the Green function of the operator u → −D β (D α u) in (0, 1), with boundary conditions lim x−→0 + x 1−β D α u(x) = 0 and u(1) = 0. We will prove that if q ∈ K α,β , then κ q < ∞. We denote by ω the unique solution of the following homogenous problem: We can easily verify that for x ∈ (0, 1) Finally, a combination of the following hypotheses on the term g is required: There exists a nonnegative function q ∈ K α,β ∩ C((0, 1)) satisfying: As a typical example of the function satisfying (H 1 )-(H 3 ), we quote Our main result is the following.

Moreover, this solution is unique if hypothesis (H 3 ) is also satisfied.
Observe that in Theorem 1, we obtain a positive solution u ∈ C 1−α ([0, 1]) to problem (1.3) whose behavior is not affected by the perturbed term. That is, it behaves like the solution ω of the homogeneous problem (1.5).
Our paper is organized as follows. In Sect. 2, we give some estimates on H(x, t). In Sect. 3, for a given function q ∈ K α,β with κ q ≤ 1 2 , we construct the Green function H( derive some of its properties. Exploiting these results, we prove our main result in Sect. 4.

Fractional Calculus
For the convenience of the reader, we recall in the following some basic definitions and elementary properties of fractional calculus (For more details, see [9,23,26]).

Definition 2.
The Riemann-Liouville fractional integral of order α > 0 of a function f : (0, 1) −→ R is given by provided that the right-hand side is pointwise defined on (0, 1).
where n = [α] + 1 and [α] mean the integer part of the number α, provided that the right-hand side is pointwise defined on (0, 1).

Karamata Class K
In this subsection, we introduce the Karamata class K and we recall some fundamental properties of functions belonging to this class.

Definition 4.
The class K is the set of Karamata functions L defined on (0, η] by As a typical example of function belonging to the class K, we quote where ξ j are real numbers, log j x = log • log . . . log x ( j times ) and w is a sufficiently large positive real number such that L is defined and positive on (0, η] for some η > 1.
In particular, Next, we establish the following property of H(x, t).

Proposition 1.
We have, for x, r, t ∈ (0, 1), Proof. Using Lemma 4 (i), we have for each x, r, t ∈ (0, 1), We claim that Indeed, by symmetry, we may assume that x ≤ t. Then we deduce that Now, by using (2.5), we obtain the required result.
Then we have