The Existence and Uniqueness of Positive Solutions of an Ordinary Differential Equation with a Nonlocal Conditions

Abstract

Abstract Many researchers have studied problems with non-local conditions of the second-order differential equations. In this work we study the ordinary differential equation v″(t) + g(t, v(t)) = 0, t ∈ (0,1), with the nonlocal conditions v’(1) = 0, v(0) = Dαv(t)|t=1,α ∈ (0,1). First, we study the existence of at least one positive continuous solution under some assumptions on the function g. Then we discuss the uniqueness of solution by assume that there exist a constant k > 0 such that |g(t,v)-g(t,ῡ)| ≤ |v-ῡ|, ∀ t ∈ [0,1], ∀v, ῡ C[0,1] for this ordinary differential equation, a clarifying example was given as an application. The main idea in this paper is to study ordinary differential equations with a fractional order condition.