The existence of countably many positive solutions for some nonlinear n th order m -point boundary value problems

The existence of countably many positive solutions for some nonlinear n th order m -point... In this paper, we consider the existence of countably many positive solutions for n th-order m -point boundary value problems consisting of the equation u ( n ) ( t ) + a ( t ) f ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , with one of the following boundary value conditions: u ( 0 ) = ∑ i = 1 m − 2 k i u ( ξ i ) , u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( 1 ) = 0 , and u ( 0 ) = 0 , u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m − 2 k i u ( ξ i ) , where n ≥ 2 , k i > 0 ( i = 1 , 2 , … , m − 2 ) , 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1 , a ( t ) ∈ L p ( 0 , 1 ) for some p ≥ 1 and has countably many singularities in ( 0 , 1 2 ) . The associated Green’s function for the n th order m -point boundary value problem is first given, and we show that there exist countably many positive solutions using Holder’s inequality and Krasnoselskii’s fixed point theorem for operators on a cone. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Computational and Applied Mathematics Elsevier

The existence of countably many positive solutions for some nonlinear n th order m -point boundary value problems

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Publisher
Elsevier
Copyright
Copyright © 2009 Elsevier Ltd
ISSN
0377-0427
eISSN
1879-1778
DOI
10.1016/j.cam.2009.05.023
Publisher site
See Article on Publisher Site

Abstract

In this paper, we consider the existence of countably many positive solutions for n th-order m -point boundary value problems consisting of the equation u ( n ) ( t ) + a ( t ) f ( u ( t ) ) = 0 , t ∈ ( 0 , 1 ) , with one of the following boundary value conditions: u ( 0 ) = ∑ i = 1 m − 2 k i u ( ξ i ) , u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( 1 ) = 0 , and u ( 0 ) = 0 , u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , u ( 1 ) = ∑ i = 1 m − 2 k i u ( ξ i ) , where n ≥ 2 , k i > 0 ( i = 1 , 2 , … , m − 2 ) , 0 < ξ 1 < ξ 2 < ⋯ < ξ m − 2 < 1 , a ( t ) ∈ L p ( 0 , 1 ) for some p ≥ 1 and has countably many singularities in ( 0 , 1 2 ) . The associated Green’s function for the n th order m -point boundary value problem is first given, and we show that there exist countably many positive solutions using Holder’s inequality and Krasnoselskii’s fixed point theorem for operators on a cone.

Journal

Journal of Computational and Applied MathematicsElsevier

Published: Oct 15, 2009

References

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