Svatoslav Stanek

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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. Here L : C1([0, T ];R) → C0([0, T ];R) and F : C1([0, T ];R) → L1([0, T ];R) are continuous operators. Existence results are proved by the Leray-Schauder degree and the Borsuk antipodal theorem for(More)
In many applications, adaptive mesh-refinement is observed to be an efficient tool for the numerical solution of partial differential equations and integral equations. Convergence of adaptive schemes to the correct solution, however, is so far only understood for certain kind of differential equations. In general, it cannot be excluded that the adaptive(More)
In this text we investigate solvability of various nonlinear singular boundary value problems for ordinary differential equations on the compact interval [0, T ]. The nonlinearities in differential equations may be singular both in the time and space variables. Location of all singular points in [0, T ] need not be known. The work is divided into 6(More)
We discuss the properties of the differential equation u′′ t a/t u′ t f t, u t , u′ t , a.e. on 0, T , where a ∈ R\{0}, and f satisfies the Lp-Carathéodory conditions on 0, T × R2 for some p > 1. A full description of the asymptotic behavior for t → 0 of functions u satisfying the equation a.e. on 0, T is given. We also describe the structure of boundary(More)
A computationally efficient a posteriori error estimator is introduced and analyzed for collocation solutions to linear index-1 DAEs with properly stated leading term. The procedure is based on a modified defect correction principle, extending an established technique from the ODE context to the DAE case. We prove that the resulting error estimate is(More)