Existence of positive solutions of higher-order nonlinear neutral equations

Tuncay Candan

doi:10.1186/1029-242x-2013-573

Abstract

In this work, we consider the existence of positive solutions of higher-order nonlinear neutral differential equations. In the special case, our results include some well-known results. In order to obtain new sufficient conditions for the existence of a positive solution, we use Schauder’s fixed point theorem.

Highlights

The motivation for the present work was the recent work of Culáková et al [ ] in which the second-order neutral nonlinear differential equation of the form r(t) x(t) – P(t)x(t – τ ) + Q(t)f x(t – σ ) =
We show that S satisfies the assumptions of Schauder’s fixed point theorem
Competing interests The author declares that they have no competing interests

Summary

Introduction

Existence of nonoscillatory or positive solutions of higher-order neutral differential equations was investigated in [ – ], but in this work our results contain existence of solutions and behavior of solutions. Theorem (Schauder’s fixed point theorem [ ]) Let A be a closed, convex and nonempty subset of a Banach space. Let S : A → A be a continuous mapping such that SA is a relatively compact subset of. Proof Let be the set of all continuous and bounded functions on [t , ∞) with the sup norm. In order to prove that SA is relatively compact, it suffices to show that the family of functions {Sx : x ∈ A} is uniformly bounded and equicontinuous on [t , ∞). For x ∈ A and T > T ≥ T , we have (Sx)(T ) – (Sx)(T )

Note that

Assume that