Nonlinear second order system of Neumann boundary value problems at resonance

Chaitan P. Gupta

International Journal of Stochastic Analysis, 1989, vol. 2, 1-16

Abstract:

Let f : [ 0 , π ] × ℝ N → ℝ N , ( N ≥ 1 ) satisfy Caratheodory conditions, e ( x ) ∈ L 1 ( [ 0 , π ] ; ℝ N ) . This paper studies the system of nonlinear Neumann boundary value problems x ″ ( t ) + f ( t , x ( t ) ) = e ( t ) , 0 < t < π , x ′ ( 0 ) = x ′ ( π ) = 0 . This problem is at resonance since the associated linear boundary value problem x ″ ( t ) = λ x ( t ) , 0 < t < π , x ′ ( 0 ) = x ′ ( π ) = 0 , has λ = 0 as an eigenvalue. Asymptotic conditions on the nonlinearity f ( t , x ( t ) ) are offered to give existence of solutions for the nonlinear systems. The methods apply to the corresponding system of Lienard-type periodic boundary value problems.

Date: 1989
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnijsa:687630

DOI: 10.1155/S1048953389000134

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