Nonexistence of solutions for second-order initial value problems

Abstract

We consider nonexistence of solutions for second-order initial value problems. Two results are given: one in which the problems are singular in the time variable, and one in which the problems are singular in both the time and state variables. We consider nonexistence of solutions to singular second-order initial value problems. The results and proofs were originally motivated by Proposition 3.2 in [6]. Existence of solutions to singular differential equations has received a great deal of attention – see, for example, the monograph [1]. For more recent results regarding second-order problems, see [2], [4], [7], [9], [10], [12], [13], [16] and [17]. On the other hand, sometimes nonexistence can be trivial: For example, if f is not Lebesgue integrable in a neighborhood of 0, then clearly x′′(t) = f (t) , x(0) = x0 , x′(0) = x1 has no Carathéodory solution. Results in the literature for nonexistence for singular second-order differential equations typically involve boundary conditions, see for example, [3], [5], [11], [14] and [15]. In [8], existence and nonexistence of positive solutions are studied for the problem x′′ = f (t,x,x′) , x(0) = 0, x′(0) = 0. We begin with the following definition. DEFINITION 1. u is a solution to the initial value problem p(t)u′′(t) = g(t,u(t),u′(t)) u(0) = α, u′(0) = β if there exists a T > 0 such that all of the following are satisfied: i) u , u′ are absolutely continuous on [0,T ] , ii) p(t)u′′(t) = g(t,u(t),u′(t)) a.e. on [0,T ] , iii) u(0) = α , u′(0) = β . We define solution for the problem in Theorem 2 below similarly. Throughout the paper, we assume a,b, f , p,q and u are real-valued. Our first result is the following: Mathematics subject classification (2010): 34A12, 34A34, 34A36.