Nonlinear Systems of Second-Order ODEs

Patricio Cerda,Pedro Ubilla

doi:10.1155/2008/236386

Abstract

We study existence of positive solutions of the nonlinear system in ; in ; , where and . Here, it is assumed that , are nonnegative continuous functions, , are positive continuous functions, , , and that the nonlinearities satisfy superlinear hypotheses at zero and . The existence of solutions will be obtained using a combination among the method of truncation, a priori bounded and Krasnosel'skii well-known result on fixed point indices in cones. The main contribution here is that we provide a treatment to the above system considering differential operators with nonlinear coefficients. Observe that these coefficients may not necessarily be bounded from below by a positive bound which is independent of and .

Highlights

We study existence of positive solutions for the following nonlinear system of second-order ordinary differential equations:− a1 t u c1g1 u, v h1 t f1 t, u, v in 0, 1,− a2 t v c2g2 u, v h2 t f2 t, u, v in 0, 1, 1.1 u 0 u 1 v 0 v 1 0, where c1, c2 are nonnegatives constants, the functions a1, a2 : 0, 1 → 0, ∞ are continuous, the functions f1, f2 : 0, 1 × 0, ∞ 2→ 0, ∞ are continuous, and h1, h2 ∈ L1 0, 1
In order to overcome these difficulties, we introduce a truncation of system 1.1 depending on n so that the new coefficient of the truncation system becomes bounded from below by a uniformly positive constant
V a2 t c2g2,n u, v h2 t f2 t, u, v in 0, 1, 2.3 u 0 u 1 v 0 v 1 0. For this purpose we need to establish a priori bounds for solutions of a family of systems parameterized by λ ≥ 0

Summary

Introduction

We study existence of positive solutions for the following nonlinear system of second-order ordinary differential equations:. 7. Our main goal is to study systems of type 1.1 by considering local superlinear assumptions at ∞ and global superlinear at zero. In order to overcome these difficulties, we introduce a truncation of system 1.1 depending on n so that the new coefficient of the truncation system becomes bounded from below by a uniformly positive constant.

A priori bounds for a truncation system

A nonexistence result