Abstract
We study the general semilinear second-order ODE 𝑢 + 𝑔 ( 𝑡 , 𝑢 , 𝑢 ) = 0 under different two-point boundary conditions. Using the method of upper and lower solutions, we obtain an existence result. Moreover, under a growth condition on 𝑔 , we prove that the set of solutions of 𝑢 + 𝑔 ( 𝑡 , 𝑢 , 𝑢 ) = 0 is homeomorphic to the two-dimensional real space.
Highlights
The Dirichlet problem for the semilinear second-order ODE u g t, u, u 01.1 has been studied by many authors from the pioneering work of Picard 1, who proved the existence of a solution by an application of the well-known method of successive approximations under a Lipschitz condition on g and a smallness condition on T
We study the general semilinear second-order ODE u g t, u, u
Under a growth condition on g, we prove that the set of solutions of u g t, u, u 0 is homeomorphic to the two-dimensional real space
Summary
1.1 has been studied by many authors from the pioneering work of Picard 1 , who proved the existence of a solution by an application of the well-known method of successive approximations under a Lipschitz condition on g and a smallness condition on T. We will study the existence of solutions of 1.1 under Dirichlet, periodic, and nonlinear boundary conditions of the type u 0 f1 u 0 , u T f2 u T , 1.3 where f1 and f2 are given continuous functions. We construct solutions of the aforementioned problems by an iterative method based on the existence of an ordered couple α, β of a lower and an upper solution. This method has been successfully applied to different boundary value problems when g does not depend on u. We will assume instead a Lipschitz condition with respect to u and construct in each case a nonincreasing resp., nondecreasing sequence of upper lower solutions that converges to a solution of the problem
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