Atlas: A Third Order Dirichelet Type m-Point Boundary Value Problem involving a p-Laplacian Type Operator. by Chaitan Gupta

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Fifth International Conference on Dynamic Systems and Applications
May 30 - June 2, 2007
Morehouse College
Atlanta, Georgia, USA

Organizers
M. Sambandham, Morehouse College, IFNA

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A Third Order Dirichelet Type m-Point Boundary Value Problem involving a p-Laplacian Type Operator.
by
Chaitan Gupta
University of Nevada, Reno

Abstract

Let f, be an odd increasing homeomorphisms from R onto R satisfying f(0)=0, and let f:[0, 1]×R×R→ R be a function satisfying Caratheodory's conditions. Let a_i ∈ R, x_i ∈ (0, 1), i=1, ..., m-2, 0 < x₁ < x₂ < ... < x_m-2 < 1 be given. We are interested in the problem of existence of solutions for the m-point boundary value problem:

(f(u^''))^'

f(t, u, u^', u^''), t ∈ (0, 1),

u(0)

0, u(1)=

m-2
å
i=1

a_iu(x_i), u^''(0)=0,

in the resonance and non-resonance case. We say that this problem is at resonance if the associated problem

(f(u^''))^'

0, t ∈ (0, 1),

u(0)

0, u(1)=

m-2
å
i=1

a_iu(x_i), u^''(0)=0.

has a non-trivial solution. This is the case if and only if ∑_i=1^m-2a_ix_i=1. Our results use topological degree methods. Interestingly enough in the non-resonance case, i.e., when ∑_i=1^m-2a_ix_i ≠ 1 the sign of degree for the relevant operator depends on whether ∑_i=1^m-2a_ix_i > 1 or ∑_i=1^m-2a_ix_i < 1.

Date received: November 14, 2006