UNIQUE POSITIVE SOLUTION FOR A CLASS OF THE SYSTEM OF THE NONLINEAR SUSPENSION BRIDGE EQUATIONS
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We prove the existence of a unique positive solution for a class of systems of the following nonlinear suspension bridge equa- tions with Dirichlet boundary conditions and periodic conditions ½ utt + uxxxx + 1 4uttxx +av+ = Á00 + ²1h1(x; t) in (¡¼ 2 ; ¼ 2 ) £ R; vtt + vxxxx + 1 4uttxx +bu+ = Á00 + ²2h2(x; t) in (¡¼ 2 ; ¼ 2 ) £ R; where u+ = maxfu; 0g, ²1, ²2 are small numbers and h1(x; t), h2(x; t) are bounded, ¼-periodic in t and even in x and t and kh1k = kh2k = 1. We ¯rst show that the system has a positive solution, and then prove the uniqueness by the contraction mapping principle on a Ba- nach space.
Abstract
We prove the existence of a unique positive solution for a class of systems of the following nonlinear suspension bridge equa- tions with Dirichlet boundary conditions and periodic conditions ½ utt + uxxxx + 1 4uttxx +av+ = Á00 + ²1h1(x; t) in (¡¼ 2 ; ¼ 2 ) £ R; vtt + vxxxx + 1 4uttxx +bu+ = Á00 + ²2h2(x; t) in (¡¼ 2 ; ¼ 2 ) £ R; where u+ = maxfu; 0g, ²1, ²2 are small numbers and h1(x; t), h2(x; t) are bounded, ¼-periodic in t and even in x and t and kh1k = kh2k = 1. We ¯rst show that the system has a positive solution, and then prove the uniqueness by the contraction mapping principle on a Ba- nach space.
참고문헌 (6)
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