Applications of Schauder’s Fixed Point Theorem to Semipositone Singular Differential Equations

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.

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

A mathematical model on schistosomiasis governed by periodic differential equations with a time delay was studied. By discussing boundedness of the solutions of this model and construction of a monotonic sequence, the existence of positive periodic solution was shown. The conditions under which the model admits a periodic solution and the conditions under which the zero solution is globally stable are given, respectively. Some numerical analyses show the conditional coexistence of locally stable zero solution and periodic solutions and that it is an effective treatment by simply reducing the population of snails and enlarging the death ratio of snails for the control of schistosomiasis.

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.

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.

Existence of positive periodic solutions for neutral functional differential equations

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.

Existence of positive periodic solutions of several types of biological models with periodic coefficients

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.

In this article we study the existence of positive periodic solutions for two types of second-order nonlinear neutral differential equation with infinite distributed 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 is completely continuous. The results obtained here generalize the work of Han and Ren (Nonlinear Anal, 70:1393–1406, 2009).

We investigate the existence of periodic solutions for a predator-prey system with Holling function response and mutual interference. Our model is more general than others since it has both Holling type IV function and impulsive effects. With some new analytical tricks and the continuation theorem in coincidence degree theory proposed by Gaines and Mawhin, we obtain a set of sufficient conditions on the existence of positive periodic solutions for such a system. In addition, in the remark, we point out some minor errors which appeared in the proof of theorems in some published papers with relevant predator-prey models. An example is given to illustrate our results.

In this paper, we establish the existence and stability of periodic solutions for neutral-type differential equations with piecewise impulses on time scales. We first obtain some sufficient conditions for the existence of a unique periodic solution by using the Banach contraction mapping principle. We also prove the existence of at least one periodic solution using the Schauder fixed point theorem. In addition, we establish the stability results based on the existence of periodic solutions. It is worth noting that the results of this paper are based on time scales, so that they are applicable to continuous, discrete, and other types of systems.

Positive periodic solutions for neutral functional differential equations

In this work, sufficient conditions are established for the existence of positive ω-periodic solutions of the second-order neutral differential equations with distributed deviating arguments. The proof of our results is based on the Krasnoselskii fixed point theorem. An example is given to illustrate the application of the results.