Donal O’Regan

A class of impulsive differential variational inequalities in finite dimensional spaces

Abstract In this paper, a new class of impulsive differential variational inequalities (IDVIs) are introduced and studied in finite dimensional Euclidean spaces. Some existence results on the solutions for the IDVIs are presented under suitable conditions and a convergence theorem of the discrete Euler time-dependent procedure for solving the IDVI is proved by using constructed discrete approximation methods for the impulsive differential inclusions (IDIs). The stability results concerned with the solutions of the IDVIs are also considered when the variation of initial data, impulsive perturbation and right-hand sides happens.

Fixed points of generalized contractive mappings of integral type

AbstractThe aim of this paper is to introduce classes of α-admissible generalized contractive type mappings of integral type and to discuss the existence of fixed points for these mappings in complete metric spaces. Our results improve and generalize fixed point results in the literature. MSC:46T99, 54H25, 47H10, 54E50.

Positive solutions for singular nonlocal boundary value problems involving integral conditions with derivative dependence

Abstract In this paper using a fixed point theory on a cone we present some new results on the existence of multiple positive solutions for singular nonlocal boundary value problems involving integral conditions with derivative dependence.

Lyapunov Functions to Caputo Fractional Neural Networks with Time-Varying Delays

One of the main properties of solutions of nonlinear Caputo fractional neural networks is stability and often the direct Lyapunov method is used to study stability properties (usually these Lyapunov functions do not depend on the time variable). In connection with the Lyapunov fractional method we present a brief overview of the most popular fractional order derivatives of Lyapunov functions among Caputo fractional delay differential equations. These derivatives are applied to various types of neural networks with variable coefficients and time-varying delays. We show that quadratic Lyapunov functions and their Caputo fractional derivatives are not applicable in some cases when one studies stability properties. Some sufficient conditions for stability of equilibrium of nonlinear Caputo fractional neural networks with time dependent transmission delays, time varying self-regulating parameters of all units and time varying functions of the connection between two neurons in the network are obtained. The cases of time varying Lipschitz coefficients as well as nonLipschitz activation functions are studied. We illustrate our theory on particular nonlinear Caputo fractional neural networks.

The existence of positive solutions for Kirchhoff-type problems via the sub-supersolution method

Abstract In this paper we discuss the existence of a solution between wellordered subsolution and supersolution of the Kirchhoff equation. Using the sub-supersolution method together with a Rabinowitz-type global bifurcation theory, we establish the existence of positive solutions for Kirchhoff-type problems when the nonlinearity is singular or sign-changing. Moreover, we obtain some necessary and sufficient conditions for the existence of positive solutions for the problem when N = 1.

Dynamics of the stochastic chemostat with Monod-Haldane response function

The stochastic chemostat model with Monod-Haldane response function is perturbed by environmental white noise. This model has a global positive solution. We demonstrate that there is a stationary distribution of the stochastic model and the system is ergodic under appropriate conditions, on the basis of Khasminskii’s theory on ergodicity. Sufficient criteria for extinction of the microbial population in the stochastic system are established. These conditions depend strongly on the Brownian motion. We find that even small scale white noise can promote the survival of microorganism populations, while large scale noise can lead to extinction. Numerical simulations are carried out to illustrate our theoretical results.

Fixed-point theorems for nonlinear operators with singular perturbations and applications

In this paper, using fixed-point index theory and approximation techniques, we consider the existence and multiplicity of fixed points of some nonlinear operators with singular perturbation. As an application we consider the existence and multiplicity of positive solutions of singular systems of multi-point boundary value problems, which improve the results in the literature.