Abstract
This paper investigates the higher order differential equations with nonlocal boundary conditions { u ( n ) ( t ) + f ( t , u ( t ) , u ′ ( t ) , … , u ( n − 2 ) ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 3 ) ( 0 ) = 0 , u ( n − 2 ) ( 0 ) = ∫ 0 1 u ( n − 2 ) ( s ) d A ( s ) , u ( n − 2 ) ( 1 ) = ∫ 0 1 u ( n − 2 ) ( s ) d B ( s ) . The existence results of multiple monotone positive solutions are obtained by means of fixed point index theory for operators in a cone.MSC:34B10, 34B18.
Highlights
This paper investigates the higher order differential equations with nonlocal boundary conditions
1 Introduction In this paper, we are concerned with the existence of multiple monotone positive solutions for the higher order differential equation u(n)(t) + f t, u(t), u (t), . . . , u(n– )(t) =, t ∈ (, ), ( . )
Boundary value problems (BVPs for short) for nonlinear differential equations arise in many areas of applied mathematics and physics
Summary
We are concerned with the existence of multiple monotone positive solutions for the higher order differential equation u(n)(t) + f t, u(t), u (t), . Boundary value problems with integral boundary conditions for ordinary differential equations represent a very interesting and important class of problems and arise in the study of various physical, biological and chemical processes [ – ], such as heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics They include two, three, multi-point and nonlocal BVPs as special cases. In [ ], Feng, Ji and Ge considered the existence and multiplicity of positive solutions for a class of nonlinear boundary value problems of second order differential equations with integral boundary conditions in ordered Banach spaces. Few papers have considered the monotone positive solutions for a higher order differential equation with integral boundary conditions.
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