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Existence to singular boundary value problems with sign changing nonlinearities using an approximation method approach

Haishen Donal O'ReganRavi P. Agarwal — 2007

Applications of Mathematics

This paper studies the existence of solutions to the singular boundary value problem - u ' ' = g ( t , u ) + h ( t , u ) , t ( 0 , 1 ) , u ( 0 ) = 0 = u ( 1 ) , where g ( 0 , 1 ) × ( 0 , ) and h ( 0 , 1 ) × [ 0 , ) [ 0 , ) are continuous. So our nonlinearity may be singular at t = 0 , 1 and u = 0 and, moreover, may change sign. The approach is based on an approximation method together with the theory of upper and lower solutions.

On the existence of multiple periodic solutions for the vector p -Laplacian via critical point theory

Haishen Donal O'ReganRavi P. Agarwal — 2005

Applications of Mathematics

We study the vector p -Laplacian - ( | u ' | p - 2 u ' ) ' = F ( t , u ) a.e. t [ 0 , T ] , u ( 0 ) = u ( T ) , u ' ( 0 ) = u ' ( T ) , 1 < p < . ( * ) We prove that there exists a sequence ( u n ) of solutions of ( * ) such that u n is a critical point of ϕ and another sequence ( u n * ) of solutions of ( * ) such that u n * is a local minimum point of ϕ , where ϕ is a functional defined below.

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