Boundary and initial value problems for second-order neutral functional differential equations
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In this paper, we consider the three-point boundary-value problem for the second order neutral functional differential equation u″ + ƒ(t, ut, u′(t)) = 0, 0 ≤ t ≤ 1, with the three-point boundary condition u0 = ϕ, u(1) = u(η). Under suitable assumptions on the function ƒ we prove the existence, uniqueness and continuous dependence of solutions. As an application of the methods used, we study the existence of solutions for the same equation with a "mixed" boundary condition u0 = ϕ, u(1) = α[u′(η) - u′(0)], or with an initial condition u0 = ϕ, u′(0) = 0. For the initial-value problem, the uniqueness and continuous dependence of solutions are also considered. Furthermore, the paper shows that the solution set of the initial-value problem is nonempty, compact and connected. Our approach is based on the fixed point theory.
CitationLe, H. H., & Le, T. P. N. (2006). Boundary and initial value problems for second-order neutral functional differential equations. Electronic Journal of Differential Equations, 2006(62), pp. 1-19.
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