CONTRACTION MAPPING PRINCIPLE AND ITS APPLICATION TO UNIQUENESS RESULTS FOR THE SYSTEM OF THE WAVE EQUATIONS

• Journal title : Honam Mathematical Journal
• Volume 30, Issue 1,  2008, pp.197-203
• Publisher : The Honam Mathematical Society
• DOI : 10.5831/HMJ.2008.30.1.197
Title & Authors
CONTRACTION MAPPING PRINCIPLE AND ITS APPLICATION TO UNIQUENESS RESULTS FOR THE SYSTEM OF THE WAVE EQUATIONS
Jung, Tack-Sun; Choi, Q-Heung;

Abstract
We show the existence of the unique solution of the following system of the nonlinear wave equations with Dirichlet boundary conditions and periodic conditions under some conditions $\small{U_{tt}-U_{xx}+av^+=s{\phi}_{00}+f}$ in $\small{(-{\frac{\pi}{2},{\frac{\pi}{2}}){\times}R}$, $\small{{\upsilon}_{tt}-{\upsilon}_{xx}+bu^+=t{\phi}_{00}+g}$ in $\small{(-{\frac{\pi}{2},{\frac{\pi}{2}}){\times}R}$, where $\small{u^+}$ = max{u, 0}, s, t $\small{{\in}}$ R, $\small{{\phi}_{00}}$ is the eigenfunction corresponding to the positive eigenvalue $\small{{\lambda}_{00}}$ of the wave operator. We first show that the system has a positive solution or a negative solution depending on the sand t, and then prove the uniqueness theorem by the contraction mapping principle on the Banach space.
Keywords
System of wave equations;jumping nonlinearity;eigenvalues of the matrix;contraction mapping principle;Dirichlet boundary condition;
Language
English
Cited by
References
1.
K. C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhauser, (1993).

2.
Q. H. Choi and T. Jung, An application of a variational reduction method to a nonlinear wave equation, J. Differential Equations, 117, 390-410 (1995).

3.
T. Jung and Q. H. Choi, The existence of a positive solution of the system of the nonlinear wave equations with jumping nonlinearities, Nonlinear Analysis, TMA., to be appeared.

4.
Q. H. Choi and T. Jung, Multiple periodic solutions of a semilinear wave equation at double external resonances, Communications in Applied Analysis 3, No. 1, 73-84 (1999).

5.
Q. H. Choi and T. Jung, Multiplicity results for nonlinear wave equations with nonlinearities crossing eigenvalues, Hokkaido Mathematical Journal Vol.24, No. 1, 53-62 (1995).

6.
P. J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Archive for Rational Mechanics and Analysis Vol. 98, No.2, 167-177 (1987).

7.
J. T. Schwartz, Nonlinear functional analysis, Gordon and Breach, New York, (1969).