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

Jung, Tack-Sun;Choi, Q-Heung

  • Received : 2008.01.03
  • Accepted : 2008.02.14
  • Published : 2008.03.25

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 $U_{tt}-U_{xx}+av^+=s{\phi}_{00}+f$ in $(-{\frac{\pi}{2},{\frac{\pi}{2}}){\times}R$, ${\upsilon}_{tt}-{\upsilon}_{xx}+bu^+=t{\phi}_{00}+g$ in $(-{\frac{\pi}{2},{\frac{\pi}{2}}){\times}R$, where $u^+$ = max{u, 0}, s, t ${\in}$ R, ${\phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${\lambda}_{00}$ 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

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