Periodic boundary value problems for second order functional differential equations

Abstract

This paper deals with a periodic boundary value problem for a second order functional differential equation. We obtain the existence of extreme solutions under new concept of upper and lower solutions. Also, a mistake in a recent paper (Ding et al. in J. Math. Anal. Appl. 298:341-351, 2004) is corrected.

A monotone iterative method for second order nonlinear problems with boundary conditions driven by maximal monotone multivalued operators

“…Several authors have used it in this sense. For this purpose, see [1,3,4,7,10,11,13] and references therein. The problems studied in [4,10] and (1.1) encompasses the classical problems of Dirichlet, Neumann, Neumann-Steklov, Sturm-Liouville, discussed in [1,7,13].…”

Convergence of solutions for functional integro-differential equations with nonlinear boundary conditions

“…Recently, various classes of differential/difference equations with nonlinear boundary conditions have attracted extensive attention of researchers. For instance, Franco et al [9] discussed the existence conditions of solutions for first-order differential equations with nonlinear boundary conditions; Jankowski [19] obtained the existence conditions of first-order advanced differential equations with nonlinear boundary conditions; Mahdavi [29] investigated the nonlinear boundary value problems involving abstract Volterra operators; Wang et al [36] presented the existence conditions of extreme solutions for first-order functional difference equations with nonlinear boundary conditions; Wang et al [37] proved uniform convergence approximate solutions for second-order functional differential equations with periodic boundary conditions; Wang [38] and Wang and Tian [39] established the existence conditions of extreme solutions for causal differential equations and impulsive differential equations with causal operators, respectively. However, we noticed that the previous studies mostly focused on the existence of solutions and extremal solutions as well as the uniform convergence approximate solutions via the method of upper and lower solutions coupled with the monotone iterative technique; see [22,31].…”

Periodic Solutions Generated by Impulses for State-Dependent Impulsive Differential Equation

“…For example, Chen et al [7] obtained some new results concerning the existence of solutions to an impulsive firstorder, nonlinear ordinary differential equation with periodic boundary conditions via differential inequalities and Schaefer's fixed-point theorem. Wang et al [8] got the existence of extreme solutions of a periodic boundary value problem for a second-order functional differential equation by using upper and lower solutions. Based on a nonlinear alternative principle of Leray-Schauder, together with a truncation technique, Chu and Nieto [9] studied the impulsive periodic solutions of first-order singular ordinary differential equations.…”