A discussion on the existence of positive solutions of the boundary value problems via ψ-Hilfer fractional derivative on b-metric spaces

In this paper, we investigate the existence of positive solutions for the new class of boundary value problems via ψ-Hilfer fractional differential equations. For our purpose, we use the α−ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha -\psi $\end{document} Geraghty-type contraction in the framework of the b-metric space. We give an example illustrating the validity of the proved results.

In the last few decades, the natural extension of differential equations, fractional differential equations, have been investigated densely in the setting of the standard metric spaces. As it is well known, there are several distinct fractional derivative types, such as Caputo, Hadamard, Grunwald-Letnikov, Hilfer, Riemann-Liouville, Riesz, Atangana-Baleanu, and so on. Among these different types of fractional derivatives, we focus on the Hilfer fractional derivative; see, for example, . By using this definition we will investigate the existence of positive solutions for certain boundary value problems in the context of b-metric spaces.

Preliminaries
In this section, we recall some notations and definitions of the fractional differential equation. Throughout this paper, we assume that all considered sets are nonempty and denote R + = [0, ∞).

Definition 2.3 ([22]
) Let n -1 < ι < n (n ∈ N), and let y, δ ∈ C n [a, T] be two functions such that δ is increasing and δ (t) = 0 for all t ∈ [a, T]. Then the left-sided δ-Hilfer fractional derivative of a function y of order ι and type 0 ≤ β ≤ 1 is defined by In this paper, we consider the case n = 1, because 0 < ι < 1.

Now we introduce the spaces
.

δ [a, T] is continuous on [a, T] and satisfies
provided that I ι,κ a + exists. Note that when κ( ) = , we obtain the well-known classical Riemann-Liouville fractional integral. ([18, 21]) Let ι > 0, let n be the smallest integer greater than or equal to

Definition 2.11
The left-sided κ-Riemann-Liouville fractional differential of h of order ι is given by The left-sided κ-Caputo fractional differential of h of order ι is given by Let be the set of all increasing and continuous functions φ : R + → R + satisfying the property φ(c ) ≤ cφ( ) ≤ c for c > 1 and φ(0) = 0. We denote by F the family of all nondecreasing functions λ : R + → [0, 1 r 2 ) for some r ≥ 1.
if and only if y satisfies the integral equation

Main results
Let M = C Then (M, d) is a complete b-metric space with r = 2.
Then problem (4) has at least one solution.
In [23] the authors investigated the existence, uniqueness, and continuous dependence of global solution to the following singular fractional differential equation involving the left generalized Caputo fractional derivative with respect to another function δ: where 0 < ι ≤ 1, and c D ι;δ 0 + is the δ-Caputo fractional derivative introduced by Almeida [11], f : (0, b] × R → R is given function with lim t→0 + f (t, ·) = ∞, and u 0 is a constant.
Proof By Lemma 3.3, ζ ∈ C(K) is a solution of (8) if and only if it is a solution of the integral equation (9). Define O : C(K) → C(K) by We find a fixed point of O. Now let ζ , w ∈ C(K) be such that τ (ζ (κ), w(κ)) ≥ 0. Using (i),