Analysis of the Sign of the Solution for Certain Second-Order Periodic Boundary Value Problems with Piecewise Constant Arguments

Abstract

We find sufficient conditions for the unique solution of certain second-order boundary value problems to have a constant sign. To this purpose, we use the expression in terms of a Green’s function of the unique solution for impulsive linear periodic boundary value problems associated with second-order differential equations with a functional dependence, which is a piecewise constant function. Our analysis lies in the study of the sign of the Green’s function.

Boundary value problems for nonlinear second‐order functional differential equations with piecewise constant arguments

“…• We started the research work by recalling several preliminary notions about some suitable spaces of piecewise regular functions, together with some results concerning the expressions of the solutions to the linearized versions of the aforementioned problem (3). In these results, extracted from previous works, 25,27 the deviating argument was based on the integer part function. • Then, after mentioning some useful results from Buedo-Fernández et al, 27 we obtained some other new comparison results for boundary value problems associated to linear delay differential equations.…”

“…In these results, extracted from previous works, 25,27 the deviating argument was based on the integer part function. • Then, after mentioning some useful results from Buedo-Fernández et al, 27 we obtained some other new comparison results for boundary value problems associated to linear delay differential equations. These results were useful to determine the adequate relation (in terms of order) between two functions and also between their corresponding derivatives.…”

“…Remark 2 (Remark 7 of Buedo-Fernández et al 27 ). Condition (III) in Theorems 2-3 is satisfied if one of the following conditions holds:…”

“…Furthermore, K and 𝜕 𝜕t K are bounded, since their expressions depend on the functions h 1 , h 2 and function g in Buedo-Fernández et al 27 and their respective derivatives, which are obviously continuous and hence bounded on [0, 1]. We remark that the nonnegative character of functions K(•, 0) and K(•, s) for almost every s ∈ (0, T) is guaranteed by the conditions in Theorem 2 (see Buedo-Fernández et al 27 ).…”

“…More specifically, it was illustrated how the solution to (1) can be obtained by means of a Green's function which is given by the solution of (2) (see also Yang et al 26 ). Further, in Buedo-Fernández et al, 27 conditions were provided in order to prove the existence of solutions to (1) with a constant sign, deducing comparison results which will be useful to prove in this paper the existence of solutions to nonlinear second-order functional differential equations with piecewise constant arguments, for which the nonlinearity depends on the x ′ term and the delay is also introduced in the derivative.…”