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

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

Existence of positive periodic solutions for neutral functional differential equations

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

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.

Positive periodic solutions for neutral functional differential equations

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.

Positive periodic solution for a neutral delay competitive system

We prove that the totally nonlinear second-order neutral differential equation \[\frac{d^2}{dt^2}x(t)+p(t)\frac{d}{dt}x(t)+q(t)h(x(t))\] \[=\frac{d}{dt}c(t,x(t-\tau(t)))+f(t,\rho(x(t)),g(x(t-\tau(t))))\] has positive periodic solutions by employing the Krasnoselskii-Burton hybrid fixed point theorem.

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.

This article concerns the positive periodic solutions for a system of second-order nonlinear ordinary differential equations, in which the nonlinear term is sublinear in one equation and superlinear in the other equation.By using the fixed point theorem of cone expansion and compression we obtain the existence of positive periodic solutions.

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.

Positive periodic solutions of a class of delay differential system with feedback control

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.

Let T⊂R be a periodic time scale in shifts δ ± with period P∈ [ t 0 , ∞ ) T . We consider the existence of positive periodic solutions in shifts δ ± for the nonlinear functional dynamic equation of the form x Δ (t)=−a(t) x σ (t)+λb(t)f ( t , x ( h ( t ) ) ) ,t∈T using the cone theory techniques. We extend and unify periodic differential, difference, h-difference and q-difference equations and more by a new periodicity concept on time scales.MSC:34N05, 39A12, 35B10.

<abstract><p>This paper aims to study a class of enterprise cluster models with feedback controls and time-varying delays on time scales. Based on periodic time scales theory and the fixed point theorem of strict-set-contraction, some new sufficient conditions for the existence of positive periodic solutions are obtained. Finally, two examples are presented to verify the validity and applicability of the main results in this paper.</p></abstract>