Nonzero solutions of perturbed Hammerstein integral equations with deviated arguments and applications

We provide a theory to establish the existence of nonzero solutions of perturbed Hammerstein integral equations with deviated arguments, being our main ingredient the theory of fixed point index. Our approach is fairly general and covers a variety of cases. We apply our results to a periodic boundary value problem with reflections and to a thermostat problem. In the case of reflections we also discuss the optimality of some constants that occur in our theory. Some examples are presented to illustrate the theory.Comment: 3 figures, 23 page

On structure of solutions of 1-dimensional 2-body problem in Wheeler-Feynman electrodynamics

The problem of 1-dimensional ultra-relativistic scattering of 2 identical charged particles in classical electrodynamics with retarded and advanced interactions is investigated.Comment: 16 pages, 14 figure

The Collocation Method in the Numerical Solution of Boundary Value Problems for Neutral Functional Differential Equations. Part I: Convergence Results

We consider the numerical solution of boundary value problems for general neutral functional differential equations by the collocation method. The collocation method can be applied in two versions: the finite element method and the spectral element method. We give convergence results for the collocation method deduced by the convergence theory developed in [S. Maset, Numer. Math., (2015), pp. 1--31] for a general discretization of an abstract reformulation of the problems. Such convergence results are then applied in Part II [S. Maset, SIAM J. Numer. Anal., 53 (2015), pp. 2794--2821] of this paper to boundary values problems for a particular type of neutral functional differential equations, namely, differential equations with deviating arguments

Inclusion theorems for boundary value problems for delay differential equations

In this thesis existence and uniqueness of solutions to certain second and third order boundary value problems for delay differential equations is established. Sequences of upper and lower solutions similar to those used by Kovač and Savčenko are defined by means of an integral operator, and conditions are given under which these sequences converge monotonically from above and below to the unique solution of the problem. Some numerical examples for the second order case are presented. Existence and uniqueness is also proved for the case where the delay is a function of the solution as well as the independent variable --Abstract, page ii

Multiple fixed-sign solutions for a system of generalized right focal problems with deviating arguments

AbstractWe consider the following system of generalized right focal boundary value problemsui‴(t)=fi(t,u1(ϕ1(t)),u2(ϕ2(t)),…,un(ϕn(t))),t∈[a,b],ui(a)=ui′(t∗)=0,ξui(b)+δui″(b)=0,1⩽i⩽n, where 12(a+b)<t∗<b, ξ⩾0, δ>0 and ϕi, 1⩽i⩽n are deviating arguments. By using different fixed point theorems, we develop several criteria for the existence of three solutions of the system which are of fixed sign on the interval [a,b], i.e., for each 1⩽i⩽n, θiui(t)⩾0 for all t∈[a,b] and fixed θi∈{1,−1}. Examples are also included to illustrate the results obtained

On solvability of periodic boundary value problems for second order linear functional differential equations

The periodic boundary value problem for second order linear functional differential equations with pointwise restrictions (instead of integral ones) is considered. Sharp sufficient conditions for the solvability are obtained

Nonlocal boundary value problems for strongly singular higher-order linear functional-differential equations

For strongly singular higher-order differential equations with deviating arguments, under nonlocal boundary conditions, Agarwal-Kiguradze type theorems are established, which guarantee the presence of Fredholm's property for the above mentioned problems. Also we provide easily verifiable conditions that guarantee the existence of a unique solution of the studied problem