Solutions of bounded variation to linear integro-differential equations

The two-point nonhomogeneous boundary value problems generated by an integro-differential expression in the space of normalized bounded variations are replaced by simpler problems generated by differential systems. The boundary conditions considered are most general up to a finite dimension and the results are formulated in such a way that no matrix equations are involved and no intermediate adjoint operators or intermediate subspaces are referred to.     

Nonlinear integral equations with respect to functions having bounded p-variation*

We prove an inequality for the p-variation, $ 1\leqslant p<1 $ , of Picard iterates associated to a nonlinear refinement Riemann–Stieltjes integral equation with respect to a possibly discontinuous function of bounded p-variation.     

Dissipativity of delay functional differential equations with bounded lag

Delay functional differential equations are essentially different from ordinary differential equations because their phase space is infinite dimensional. We first establish a sufficient condition for delay functional differential equations with bounded lag to be dissipative. Then we construct a one-leg θ-method to solve such dissipative equations and prove that it is dissipative if θ=1. One numerical example is given to confirm our theoretical result.     

Solutions of stochastic differential-functional equations via bounded stochastic integral contractors

In this paper we shall use the bounded stochastic integral contractors to investigate the existence and uniqueness of the solution of a general stochastic differential-functional equation $$d\varphi (t) = F(D(\varphi _t ),dt)$$ where, ? _t ={?(t?s): ??s?0}, D:C([?τ,0],R ^d )→R ^m ,F(x,t) is ad-dimensional continuousC-semimartingale with spatial parameterx∈R ^m, and the integral involved here is a nonlinear stochastic integral.     

Solutions of some functional equation bounded on nonzero Christensen measurable sets

Let n be a positive integer. We characterize solutions f: X → ? of the equation f (x + f(x) ⁿ y = f(x)f(y) mapping a real separable F-space X into ?, which are bounded on nonzero Christensen measurable sets.     

Optimal control of stochastic functional differential equations with a bounded memory

This paper treats a finite time horizon optimal control problem in which the controlled state dynamics are governed by a general system of stochastic functional differential equations with a bounded memory. An infinite dimensional Hamilton–Jacobi–Bellman (HJB) equation is derived using a Bellman-type dynamic programming principle. It is shown that the value function is the unique viscosity solution of the HJB equation.     

Oscillation and asymptotic behavior of bounded solutions of functional differential equations with delay

Conditions on a(t), g(t), and f(t) have been found under which the bounded nonoscillatory solutions of the equation y⁽ⁿ⁾(t) ? a(t) y(g(t)) = f(t) approach zero. For the even order equation y⁽²ⁿ⁾(t) ? a(t) y(g(t)) = f(t) the delay is shown to be causing the oscillatory behavior.     

Asymptotic behavior of bounded solutions for a system of neutral functional differential equations

Consider the system of neutral functional differential equations

     

Solutions to two functional equations arising in dynamic programming

This paper deals with the existence, uniqueness and iterative approximation of solutions for two functional equations arising in dynamic programming of multistage decision processes. The results presented in this paper extend, improve and unify the results due to Bellman, Bhakta and Choudhury, Bhakta and Mitra, Liu, Liu and Ume and others.     

Behavior of bounded solutions for some almost periodic neutral partial functional differential equations

We establish a Bochner type characterization for Stepanov almost periodic functions, and we prove a new result about the integration of almost periodic functions. This result is then used together with a reduction principle to investigate the nature of bounded solutions of some almost periodic partial neutral functional differential equations. More specifically, we prove that all bounded solutions on are almost periodic. Copyright © 2016 John Wiley & Sons, Ltd.     

Solutions to the systems of functional equations of multivalued logic

General issues concerning solutions to the systems of functional equations of multivalued logic—the dependence of solutions on functional constants and the possibility of constructing the systems of equations having a given unique solution or a given set of solutions—are considered.