Solvability of higher-order BVPs in the half-line with unbounded nonlinearities

Feliz Minhós,Hugo Carrasco

doi:10.3934/proc.2015.0841

Abstract

This work presents sufficient conditions for the existence of unbounded solutions of a Sturm-Liouville type boundary value problem on the half-line. One-sided Nagumo condition plays a special role because it allows an asymmetric unbounded behavior on the nonlinearity. The arguments are based on fixed point theory and lower and upper solutions method. An example is given to show the applicability of our results.

Highlights

This paper is concerned with the study of a general SturmLiouville type boundary value problem composed by a third-order differential equation defined on the half line u (t) = f (t, u(t), u (t), u (t)), t ∈ [0, +∞) (1)and u(0) = A, au (0) + bu (0) = B, u (+∞) = C. (2)with f : [0, +∞) × R3 → R a L1− Caratheodory function, a > 0, b < 0, A, B, C ∈ R
Higher order boundary value problems on infinite intervals appear in several real phenomena such as the gas pressure in a semi-infinite porous medium, or theoretical results as, for example, the study of radially symmetric solutions of nonlinear elliptic equations
Third order boundary value problems can describe the evolution of physical phenomena, for example some draining or coating fluid-flow problems

Summary

Introduction

This paper is concerned with the study of a general SturmLiouville type boundary value problem composed by a third-order differential equation defined on the half line u (t) = f (t, u(t), u (t), u (t)) , t ∈ [0, +∞). Higher order boundary value problems on infinite intervals appear in several real phenomena such as the gas pressure in a semi-infinite porous medium, or theoretical results as, for example, the study of radially symmetric solutions of nonlinear elliptic equations. For these and other applications see, for example, [1].

FELIZ MINHO S AND HUGO CARRASCO

Therefore u