Nodal solutions of second-order two-point boundary value problems

Ma Ruyun, Yang Bianxia, Dai, Dai Guowei

Description

We shall study the existence and multiplicity of nodal solutions of the nonlinear second-order two-point boundary value problems, u ″ + f ( t , u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 . The proof of our main results is based upon bifurcation techniques. Mathematics Subject Classifications : 34B07; 34C10; 34C23. We shall study the existence and multiplicity of nodal solutions of the nonlinear second-order two-point boundary value problems, u ″ + f ( t , u ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ( 1 ) = 0 . The proof of our main results is based upon bifurcation techniques. Mathematics Subject Classifications : 34B07; 34C10; 34C23.

Sujets

• Bifurcation

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 8
• Langue: English

Extrait

Maet al.Boundary Value Problems2012,2012:13
http://www.boundaryvalueproblems.com/content/2012/1/13

R E S E A R C H

Nodal solutions of second-order
boundary value problems
*
Ruyun Ma, Bianxia Yang and Guowei Dai

* Correspondence:
[e-mail]
Department of Mathematics,
Northwest Normal University,
Lanzhou 730070, P. R. China

Open Access

two-point

Abstract
We shall study the existence and multiplicity of nodal solutions of the nonlinear
second-order two-point boundary value problems,
′′
u+f(t,u) = 0,t∈(0, 1),u(0) =u(1) = 0.

The proof of our main results is based upon bifurcation techniques.
Mathematics Subject Classifications: 34B07; 34C10; 34C23.
Keywords:nodal solutions, bifurcation

1 Introduction
In [1], Ma and Thompson were considered with determining interval ofμ, in which
there exist nodal solutions for the boundary value problem (BVP)

′′
u(t) +µw(t)f(u) = 0,

t∈(0, 1),

u(0) =u(1) = 0

(1:1)

under the assumptions:
(C1)w(∙)ÎC([0, 1], [0,∞)) and does not vanish identically on any subinterval of [0, 1];
(C2)fÎC(ℝ,ℝ) withsf(s) > 0 fors≠0;
(C3) there existf0,f∞Î(0,∞) such that
f(s)f(s)
f0,= limf∞= lim.
s s
|s|→0|s|→∞

It is well known that under (C1) assumption, the eigenvalue problem

′′
ϕ(t) +µw(t)ϕ(t) = 0,

t∈(0, 1),

ϕ(0) =ϕ(1) = 0

has a countable number of simple eigenvaluesμk,k= 1, 2,..., which satisfy
0< µ1< µ2<· · ·< µk<· · ·, andlimµk=∞,
k→∞

(1:2)

and letμkbe thekth eigenvalue of (1.2) andkbe an eigenfunction corresponding to
μk, thenkhas exactlyk–1 simple zeros in (0,1) (see, e.g., [2]).
Using Rabinowitz bifurcation theorem, they established the following interesting
results:
Theorem A(Ma and Thompson [[1], Theorem 1.1]).Let (C1)-(C3) hold. Assume
µkµkµkuk
< µ <or< µ <as two
that for some kÎN,either.Then BVP (1.1) h
f∞f0f0f∞

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