Positive solutions to second-order singular nonlocal problems: existence and sharp conditions

Abstract

In this paper we consider sharp conditions on ω and f for the existence of C 1 [0, 1] positive solutions to a second-order singular nonlocal problem u (t) + ω(t)f (t, u(t)) = 0, u(0) = u(1) = 1 0 g(t)u(t) dt; it turns out that this case is more difficult to handle than two point boundary value problems and needs some new ingredients in the arguments. On the technical level, we adopt the topological degree method.

Existence of C1‐Positive Solutions for a Class of Second‐Order Impulsive Differential Equations

“…But in their work, a key assumption is that the nonlinearity f(t, x) is nondecreasing with respect to x ≥ 0. Clearly, if f(t, x) � t 2 x 1/3 + t 3 x − 1/3 , the results obtained in [9,10] are not valid. e aim of this study is to extend the results in [10] to more general cases.…”

“…Chu and Nieto in [8], utilizing the nonlinear alternative principle of Leray-Schauder type and Schauder's fixed-point theorem, presented existence results of positive T-periodic solutions for second-order differential equations. In 2019, Ma and Zhang in [9] proved sharp conditions for the existence of positive solutions of secondorder singular differential equation with integral boundary conditions. Recently, Zhang and Tian [10] established sharp conditions for the existence of positive solutions of secondorder impulsive differential equations.…”