The paper deals with the boundary value problem u ′′ + k u = g(u) + e(t), u(0) = u(2 ), u(0) = u(2 ), where k ∈ , g : (0,∞) 7→ is continuous, e ∈ [0, 2 ] and lim x→0+ 1 x g(s) ds = ∞. In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.

Ruyun Ma, Chenghua Gao, and Ruipeng Chen Department of Mathematics, Northwest Normal University, Lanzhou 730070, China Correspondence should be addressed to Ruyun Ma, ruyun [email protected] Received 31 August 2010; Revised 30 October 2010; Accepted 8 November 2010 Academic Editor: Irena Rachůnková Copyright q 2010 Ruyun Ma et al. This is an open access article distributed under the Creative Commons A...

We consider boundary value problems for semilinear evolution inclusions. We establish the existence of extremal solutions. Using that result, we show that the evolution inclusion has periodic extremal trajectories. These results are then applied to closed loop control systems. Finally, an example of a semilinear parabolic distributed parameter control system is worked out in detail.

This article investigates the existence of solutions to boundary value problems (BVPs) involving systems of first-order ordinary differential equations and two-point, periodic boundary conditions. The methods involve novel differential inequalities and fixed-point theory to yield new theorems guaranteeing the existence of at least one solution. AMS 2000 Classification: 34B15

In this work we obtain some new results concerning the existence of solutions to an impulsive first-order, nonlinear ordinary differential equation with periodic boundary conditions. The ideas involve differential inequalities and Schaefer's fixed-point theorem.

This paper is devoted to the study of periodic boundary value problems for nonlinear third order di¤erential equations subjected to impulsive e¤ects. We provide su¢ cient conditions on the nonlinearity and the impulse functions that guarantee the existence of at least one solution. Our approach is based on a priori estimates, the method of upper and lower solutions combined with an iterative te...

Abstract: In this paper we study a periodic boundary value problem of first order nonlinear differential equations with maxima and discuss the existence and approximation of the solutions. The main result relies on the Dhage iteration method embodied in a recent hybrid fixed point theorem of Dhage (2014) in a partially ordered normed linear space. At the end, we give an example to illustrate th...

Upper and lower solution method plays an important role in studying boundary value problems for nonlinear differential equations; see 1 and the references therein. Recently, many authors are devoted to extend its applications to boundary value problems of functional differential equations 2–5 . Suppose α is one upper solution or lower solution of periodic boundary value problems for second-orde...

We deal with the nonlinear impulsive periodic boundary value problem u′′ = f (t,u,u′), u(ti+) = Ji(u(ti)), u′(ti+) = Mi(u′(ti)), i= 1,2, . . . ,m, u(0) = u(T), u′(0) = u′(T). We establish the existence results which rely on the presence of a well-ordered pair (σ1,σ2) of lower/upper functions (σ1 ≤ σ2 on [0,T]) associated with the problem. In contrast to previous papers investigating such proble...