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Second order evolution differential functional equations with infinite delay

Człapiński T., Karpowicz A.

Commentationes Mathematicae

2018

Vol. 58, No. 1/2

93--104

We consider a second order semilinear functional evolution equation with infinite delay in a Banach space. We prove the existence of mild solutions for this equation using the measure of noncompactness technique and the Schauder fixed point theorem.

Global convergence of successive approximations of the Darboux problem for partial functional differential equations with infinite delay

Człapiński T.

Opuscula Mathematica

2014

Vol. 34, no. 2

327--338

We consider the Darboux problem for the hyperbolic partial functional differential equation with infinite delay. We deal with generalized (in the "almost everywhere" sense) solutions of this problem. We prove a theorem on the global convergence of successive approximations to a unique solution of the Darboux problem.

On the nonlocal boundary value problem for first order quasilinear partial functional differential equations

Człapiński T.

Annales Societatis Mathematicae Polonae. Seria 1: Commentationes Mathematicae

1999

[Z] 39

57-73

Quasilinear partial functional differential equations of the first order with nonlocal boundary condi-tions are considered. A theorem on the local existence and uniqueness of classical solutions is given. The method of bicharacteristics is used to transform the nonlocal boundary value problem into a functional integral equation and the Banach fixed-point principle to find its solution.