Solving discrete zero point problems

In this paper we present two theorems on the existence of a discrete zero point of a function from the n-dimensional integer lattice ℤⁿ to the n-dimensional Euclidean space ℝⁿ. The theorems differ in their boundary conditions. For both theorems we give a proof using a combinatorial lemma and present a constructive proof based on a simplicial algorithm that finds a discrete zero point within a finite number of steps. This research was carried out when Gerard van der Laan and Zaifu Yang were visiting the CentER of Tilburg University in the summer of 2004. The visit of Zaifu Yang has been made possible by financial support of CentER and the Netherlands Organization for Scientific Research (NWO). The authors gratefully acknowledge the inspiring and helpful remarks of the referees.     

Positive solvability of systems of nonlinear Hammerstein integral equations

The paper deals with the existence of positive continuous solutions to systems of nonlinear Hammerstein integral equations. The main tool used in the proofs is fixed point index theory in a cone. The results obtained here are essentially different from existing ones in the literature.     

Stability in totally nonlinear delay difference equations

In this paper we use fixed point method to prove asymptotic stability results of the zero solution of a nonlinear delay difference equation. An asymptotic stability theorem with a sufficient condition is proved, which improves and generalizes some results due to Raffoul (2006) \cite{r1}, Yankson (2009) \cite{y1}, Jin and Luo (2009) \cite{jin} and Chen (2013) \cite{c}     

Stability and periodicity in discrete delay equations

In this research we study the stability of the zero solution of delay difference equations and the existence of unique periodic solution by utilizing fixed point theory. Mainly, we are interested in the qualitative analysis of the complete delay difference equation

Δx(t)=−a(t)x(t−τ).     

Numerical verification of solutions for nonlinear hyperbolic equations

We consider a numerical method to verify the solutions for nonlinear hyperbolic problems with guaranteed error bounds in the one-space dimensional case. We present verification procedures and show some numerical examples.     

Global attractivity for nonlinear fractional differential equations

We present some results for the global attractivity of solutions for fractional differential equations involving Riemann-Liouville fractional calculus. The results are obtained by employing Krasnoselskii’s fixed point theorem. Similar results for fractional differential equations involving Caputo fractional derivative are also obtained by using the classical Schauder’s fixed point theorem. Several examples are given to illustrate our main results.     

On the asymptotic integration of nonlinear differential equations

The aim of the present paper is twofold. Firstly, the paper surveys the literature concerning a specific topic in asymptotic integration theory of ordinary differential equations: the class of second order equations with Bihari-like nonlinearity. Secondly, some general existence results are established with regard to a condition that has been found recently to be of significant use in the theory of elliptic partial differential equations.     

Positive circuits and maximal number of fixed points in discrete dynamical systems

We consider a product X of n finite intervals of integers, a map F from X to itself, the asynchronous state transition graph Γ(F) on X that Thomas proposed as a model for the dynamics of a network of n genes, and the interaction graph G(F) that describes the topology of the system in terms of positive and negative interactions between its n components. Then, we establish an upper bound on the number of fixed points for F, and more generally on the number of attractors in Γ(F), which only depends on X and on the topology of the positive circuits of G(F). This result generalizes the following discrete version of Thomas’ conjecture recently proved by Richard and Comet: If G(F) has no positive circuit, then Γ(F) has a unique attractor. This result also generalizes a result on the maximal number of fixed points in Boolean networks obtained by Aracena, Demongeot and Goles. The interest of this work in the context of gene network modeling is briefly discussed.     

Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces

In this paper, first a new fixed point theorem is established, and then, by the use of it, the existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces are investigated. The results obtained in this paper generalize and improve the results corresponding to those obtained by others.     

The existence of nonoscillatory solutions of higher order nonlinear neutral equations

In this work, we consider the existence of nonoscillatory solutions of variable coefficient higher order nonlinear neutral differential equations. Our results include as special cases some well-known results for linear and nonlinear equations of first, second and higher order. We use the Banach contraction principle to obtain new sufficient conditions for the existence of nonoscillatory solutions.     

Admissibility for discrete Volterra equations

In this paper we develop the theory of admissibility for linear discrete Volterra operators and obtain several necessary and sufficient conditions for admissibility in various sequence spaces. Using the results obtained, we study the existence of solutions (such as bounded, exponential or convergent solutions), of linear or nonlinear discrete Volterra summation equations.     

Some computational methods for systems of nonlinear equations and systems of polynomial equations

This paper gives a brief survey and assessment of computational methods for finding solutions to systems of nonlinear equations and systems of polynomial equations. Starting from methods which converge locally and which find one solution, we progress to methods which are globally convergent and find an a priori determinable number of solutions. We will concentrate on simplicial algorithms and homotopy methods. Enhancements of published methods are included and further developments are discussed.     

Existence of solutions to second-order nonlinear discrete elliptic equations

In this paper, we consider a boundary value problem (BVP) for second-order nonlinear partial difference equations on finite lattice domains. Some conditions are established that ensure existence and uniqueness of solutions to the BVP under consideration.     

Three solutions to a perturbed nonlinear discrete Dirichlet problem

Using critical point theory, we study the existence of at least three solutions for a perturbed nonlinear Dirichlet boundary value problem for difference equations depending on two positive parameters.     

Existence theorems for systems of nonlinear integro-differential equations

The existence of solutions for many systems of integro-differential equations discovered and generalized in the process of applying the Galerkin method for some initial-boundary value problems will be investigated in this paper. U. V. Le is currently supported by the Academy of Finland and the Emil Aaltonen Foundation.     

Homoclinic solutions of discrete nonlinear systems via variational method

Homoclinic solutions arise in various discrete models with variational structure, from discrete nonlinear Schr\"{o}dinger equations to discrete Hamiltonian systems. In recent years, a lot of interesting results on the homoclinic solutions of difference equations have been obtained. In this paper, we review some recent progress by using critical point theory to study the existence and multiplicity results of homoclinic solutions in some discrete nonlinear systems with variational structure.