Three positive solutions to initial-boundary value problems of nonlinear delay differential equations

Yuming Wei,Patricia J Y Wong

doi:10.14232/ejqtde.2009.1.14

Abstract

In this paper, we consider the existence of triple positive solutions to the boundary value problem of nonlinear delay differential equation (�(x 0 (t))) 0 +a(t)f(t,x(t),x 0 (t),xt) = 0, 0 < t < 1, x0 = 0, x(1) = 0, where � : R ! R is an increasing homeomorphism and positive homomorphism with �(0) = 0, and xt is a function in C(( �,0),R) defined by xt(�) = x(t + �) for � � � � 0. By using a fixed-point theorem in a cone introduced by Avery and Peterson, we provide sufficient conditions for the existence of triple positive solutions to the above boundary value problem. An example is also presented to demonstrate our result. The conclusions in this paper essentially extend and improve the known results.

Highlights

Throughout this paper, for any intervals I and J of R, we denote by C(I, J) the set of all continuous functions defined on I with values in J
F (t, u, v, μ) is a nonnegative real-valued continuous function defined on [0, 1] × [0, ∞) × R × C([−τ, 0), R), a(t) is a nonnegative continuous function defined on (0, 1), and φ : R → R is an increasing homeomorphism and positive homomorphism with φ(0) = 0
In the literature, the existence of three positive solutions has never been established for boundary value problems of delay differential equations with increasing homeomorphism and positive homomorphism operators

Summary

Introduction

Throughout this paper, for any intervals I and J of R, we denote by C(I, J) the set of all continuous functions defined on I with values in J. A projection φ : R → R is called an increasing homeomorphism and positive homomorphism if the following conditions are satisfied: (1) for all x, y ∈ R, φ(x) ≤ φ(y) if x ≤ y; (2) φ is a continuous bijection and its inverse mapping φ−1 is continuous; (3) φ(xy) = φ(x)φ(y) for all x, y ∈ [0, +∞). They obtain the existence of one or two positive solutions by using a fixed-point theorem in cones.

Objectives

Results

Conclusion