Positive solutions for Neumann boundary value problems of nonlinear second-order integro-differential equations in ordered Banach spaces

Yang He, Liang, Liang Yue

Description

The paper deals with the existence of positive solutions for Neumann boundary value problems of nonlinear second-order integro-differential equations - u ″ ( t ) + M u ( t ) = f ( t , u ( t ) , ( S u ) ( t ) ) , 0 < t < 1 , u ′ ( 0 ) = u ′ ( 1 ) = θ and u ″ ( t ) + M u ( t ) = f ( t , u ( t ) , ( S u ) ( t ) ) , 0 < t < 1 , u ′ ( 0 ) = u ′ ( 1 ) = θ in an ordered Banach space E with positive cone K , where M > 0 is a constant, f : [0, 1] × K × K → K is continuous, S : C ([0, 1], K ) → C ([0, 1], K ) is a Fredholm integral operator with positive kernel. Under more general order conditions and measure of noncompactness conditions on the nonlinear term f , criteria on existence of positive solutions are obtained. The argument is based on the fixed point index theory of condensing mapping in cones. Mathematics Subject Classification (2000): 34B15; 34G20.

Informations

• Publié par: biomed
• Publié le: 01 janvier 2011
• Nombre de lectures: 1
• Langue: English

Extrait

Yang and LiangJournal of Inequalities and Applications2011,2011:73
http://www.journalofinequalitiesandapplications.com/content/2011/1/73

R E S E A R C H

Open Access

Positive solutions for Neumann boundary value
problems of nonlinear second-order
integrodifferential equations in ordered Banach spaces
1* 2
He Yangand Yue Liang

* Correspondence:
[e-mail]
1
Department of Mathematics,
Northwest Normal University,
Lanzhou 730070, People’s Republic
of China
Full list of author information is
available at the end of the article

Abstract
The paper deals with the existence of positive solutions for Neumann boundary
value problems of nonlinear second-order integro-differential equations
′′ ′′
−u(t) +Mu(t) =f(t,u(t), (Su)(t)), 0<t<1,u(0) =u(1) =θ

and

′′ ′′
u(t) +Mu(t) =f(t,u(t), (Su)(t)), 0<t<1,u(0) =u(1) =θ

in an ordered Banach spaceEwith positive coneK, whereM >0 is a constant,f: [0,
1] ×K×K®Kis continuous,S:C([0, 1],K)®C([0, 1],K) is a Fredholm integral
operator with positive kernel. Under more general order conditions and measure of
noncompactness conditions on the nonlinear termf, criteria on existence of positive
solutions are obtained. The argument is based on the fixed point index theory of
condensing mapping in cones.
Mathematics Subject Classification (2000):34B15; 34G20.
Keywords:nonlinear second-order integro-differential equation, Neumann boundary
value problem, positive solution, condensing mapping, fixed point index theorem

1 Introduction
LetEbe an ordered Banach space, whose positive coneKis normal with a normal
constantN0, that is, ifθ≤x≤y, then ||x||≤N0||y||, whereθis the zero element inE.
We consider the existence of positive solutions for nonlinear second-order
integro-differential equations
′′
−u(t) +Mu(t) =f(t,u(t), (Su)(t)), 0<t<1(1)

and
′′
u(t) +Mu(t) =f(t,u(t), (Su)(t)), 0<t<1

satisfying Neumann boundary conditions
′ ′
u(0) =u(1) =θ,

(2)

(3)

© 2011 Yang and Liang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
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