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Systems of second-order functional differential equations (x (t)+L(x)(t)) = F (x)(t) together with nonlinear functional boundary conditions are considered. n) are continuous operators. Existence results are proved by the Leray-Schauder degree and the Borsuk antipodal theorem for α-condensing operators. Examples demonstrate the optimality of conditions.
In this paper we study the existence and uniqueness of solutions to a nonlinear Neumann problem for a scalar second order ordinary differential equation u = a t u + f (t, u, u), where a < 0, and f (t, x, y) satisfies the local Carathéodory conditions on [0, T ] × R × R. 1 Motivation The aim of this work is to show the existence and uniqueness of solutions… (More)
Optimized Imex Runge-Kutta methods for simulations in astrophysics: A detailed study 13/2012 H. Woracek Asymptotics of eigenvalues for a class of singular Krein strings 12/2012 Abstract: The paper discusses the solvability of the singular Dirichlet boundary value problem u ′′ (t) + a t u ′ (t) − a t 2 u(t) = f (t, u(t), u ′ (t)), u(0) = 0, u(T) = 0. Here a… (More)
In this paper we show the existence of solutions to a nonlinear singular second order ordinary differential equation, u ′′ (t) = a t u ′ (t) + λf (t, u(t), u ′ (t)), subject to periodic boundary conditions, where a > 0 is a given constant, λ > 0 is a parameter, and the nonlinearity f (t, x, y) satisfies the local Carathéodory conditions on [0, T ] × R × R.… (More)