Positive solutions for a new class of Hadamard fractional differential equations on infinite intervals

Abstract

In the present article, the following nonlinear problem of new Hadamard fractional differential equations on an infinite interval H D ν x(t) + b(t)f (t, x(t)) + c(t) = 0, 1 < ν < 2, t ∈ (1, ∞), x(1) = 0, H D ν-1 x(∞) = m i=1 γ i H I β i x(η), is studied, where H D ν denotes the Hadamard fractional derivative of order ν, H I(•) is the Hadamard fractional integral, β i , γ i ≥ 0 (i = 1, 2,. .. , m), η ∈ (1, ∞) are constants and Γ (ν) > m i=1 γ i Γ (ν) Γ (ν + β i) (log η) ν+β i-1. By making use of a fixed point theorem for generalized concave operators, the existence and uniqueness of positive solutions is established. Moreover, an application of the established results is also presented via an interesting example.

Existence, uniqueness, and multiplicity results on positive solutions for a class of Hadamard‐type fractional boundary value problem on an infinite interval

“…In this article, we aim to obtain the existence, uniqueness, and multiplicity of positive solutions for BVP . To the authors' knowledge, while the Hadamard‐type fractional derivative is commonly considered for fractional BVPs, to date, few studies have considered BVPs of Hadamard‐type fractional differential equations on infinite intervals . Compared with existing papers, the new insights provided in this paper can be summarized as follows: First, we apply several different techniques to obtain our results including Schauder's fixed point theorem, Banach's contraction mapping principle, the monotone iterative method, and the Avery‐Peterson fixed point theorem.…”

“…Li and Zhai used a fixed‐point theorem to establish the existence and uniqueness of positive solutions of the nonlinear Hadamard fractional differential equations with integral boundary conditions: where is the Hadamard‐type fractional derivative of order ν ; is the Hadamard‐type fractional integral of order β i ≥ 0, γ i ≥ 0( i =1,2,…, m ), η >1.…”

“…Several research papers dealing with Hadamard-type fractional differential equations with various boundary conditions have been recently published. [12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28] Refer to Ahmad et al 9 for a comprehensive monograph on Hadamard-type fractional differential equations, inclusions, and inequalities.…”

“…Li and Zhai 24 used a fixed-point theorem to establish the existence and uniqueness of positive solutions of the nonlinear Hadamard fractional differential equations with integral boundary conditions:…”

“…To the authors' knowledge, while the Hadamard-type fractional derivative is commonly considered for fractional BVPs, to date, few studies have considered BVPs of Hadamard-type fractional differential equations on infinite intervals. [22][23][24][25][26][27][28] Compared with existing papers, the new insights provided in this paper can be summarized as follows: First, we apply several different techniques to obtain our results including Schauder's fixed point theorem, Banach's contraction mapping principle, the monotone iterative method, and the Avery-Peterson fixed point theorem. Not only do we discuss the existence and uniqueness of positive solutions for BVP (1.1) but also the existence of two and three positive solutions for BVP (1.1).…”