Positive periodic solutions for neutral functional differential equations

By applying Green's function of third-order differential equation and a fixed point theorem in cones, we obtain some sufficient conditions for existence, nonexistence, multiplicity, and Lyapunov stability of positive periodic solutions for a third-order neutral differential equation.

In this paper, we study the existence of positive periodic solutions to the equation x″ = f (t, x). It is proved that such a equation has more than one positive periodic solution when the nonlinearity changes sign. The proof relies on a fixed point theorem in cones.

Existence of multiple positive periodic solutions for delay differential equation whose order is a multiple of 4

Existence of positive periodic solutions for neutral functional differential equations

We study the existence of positive periodic solutions of second-order singular differential equations. The proof relies on Schauder’s fixed point theorem. Our results generalized and extended those results contained in the studies by Chu and Torres (2007) and Torres (2007) . In some suitable weak singularities, the existence of periodic solutions may help.

Positive periodic solutions for neutral functional differential equations with distributed delays and feedback control

Multiple positive periodic solutions for a class of nonlinear functional differential equations

Existence and multiplicity of positive periodic solutions to functional differential equations with impulse effects

In this article, we first investigate the linear difference operator (Ax)(t):=x(t)-sum_{i=1}^{n}c_{i}(t)x(t- delta _{i}(t)) in a continuous periodic function space. The existence condition and some properties of the inverse of the operator A are explicitly pointed out. Afterwards, as applications of properties of the operator A, we study the existence of periodic solutions for two kinds of second-order functional differential equations with this operator. One is a kind of second-order functional differential equation, by applications of Krasnoselskii’s fixed point theorem, some sufficient conditions for the existence of positive periodic solutions are established. Another one is a kind of second-order quasi-linear differential equation, we establish the existence of periodic solutions of this equation by an extension of Mawhin’s continuous theorem.

By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation $$\frac{{dN}}{{dt}} = N(t)\left[ {a(t) - \beta (t)N(t) - b(t)N(t - \sigma (t)) - c(t)N'(t - \tau (t))} \right]$$

Abstract. In this article we study the existence of positive periodic solutions for a fourth-order nonlinear neutral differential equation with variable delay. The main tool employed here is the Krasnoselskii's fixed point theorem dealing with a sum of two mappings, one is a contraction and the other one is completely continuous.

The existence and multiplicity of non-collision periodic solutions for second order singular dynamical systems are discussed in this paper. Using the Green’s function of linear differential equation, we consider general singularity and do not need any kind of strong force condition. The proof relies on a nonlinear alternative principle of Leray–Schauder and a fixed point theorem in cones.

This paper studies positive ω-periodic solutions for a class of neutral-type impulsive neural networks with distributed delays and two parameters. Using cone-theoretic fixed point methods, Green's function techniques, a Green-weighted norm, and a Lyapunov-Razumikhin approach, we establish sufficient conditions for the existence of positive periodic solutions. We further derive explicit small-gain conditions under which the positive periodic solution is unique and globally exponentially stable. These conditions reflect the combined effects of leakage, distributed delays, neutral feedback, and impulsive perturbations. In addition, computable estimates are obtained for the contraction constant of the Poincaré map and the exponential convergence rate to the periodic attractor. The main contribution is a unified framework linking cone-based existence theory with weighted small-gain stability analysis for neutral-type impulsive neural networks with distributed delays. Numerical examples illustrate the applicability of the results.

In this paper, we are concerned with the existence of positive periodic solutions for second-order differential equations with a singularity at resonance. A Landesman-Lazer type condition is given to obtain the existence of positive periodic solutions using phase-plane analysis method and topological degree method.

In this paper, we consider the following third-order singular differential equation with variable coefficients: \t\t\tx‴+a(t)x=f(t,x)+e(t).\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ x'''+a(t)x=f(t,x)+e(t). $$\\end{document} By using the Green’s function of the linear differential equation with constant coefficients and some fixed point theorems, i.e., Leray–Schauder alternative principle and Schauder’s fixed point theorem, we prove the existence of positive periodic solutions of the above equation.