Multiple Positive Periodic Solutions Research Articles

Positive periodic solutions of first-order delay differential equations with impulses

Using global bifurcation techniques, we show the existence and multiplicity of positive periodic solutions for a class of first-order delay differential equation with impulses.

Existence of Multiple Positive Periodic Solutions to Two Species Parasitical Model with Impulsive Effects and Harvesting Terms

By applying Mawhin’s continuation theorem of coincidence degree theory and some skills of inequalities, we establish the existence of four positive solutions for two species parasitical system with impulsive effects and harvesting terms. Finally, an example is given to illustrate the effectiveness of our results.

Multiple positive periodic solutions for an n-species competition predator-prey system on time scales

In this paper, we consider an n-species competition predator-prey system on time scales with Holling-type II functional response and multiple exploited (or harvesting) terms, which contains n−1 competing preys and one predator. By using the continuation theorem based on Gaines and Mawhin’s coincidence degree theory, easily verifiable criteria are established for global existence of multiple positive periodic solutions to the above system. In addition, our results generalize the corresponding results of Zhang and Hou (Nonlinear Anal. (RWA) 11:1560–1571, 2010), Fan and Wang (J. Math. Anal. Appl. 262:179–190, 2001), Ding and Lu (Appl. Math. Model. 33:2748–2756, 2009).

Multiple positive periodic solutions for a food-limited two-species Gilpin-Ayala competition patch system with periodic harvesting terms

By using Mawhin’s coincidence degree theory and some inequality techniques, this paper establishes a new sufficient condition on the existence of at least eight positive periodic solutions for a food-limited two-species Gilpin-Ayala competition patch system with periodic harvesting terms. An example is given to illustrate the effectiveness of the result.

Existence and multiplicity of periodic solutions of semilinear resonant Duffing equations with singularities

In this paper, we deal with the existence of positive periodic solutions of singular resonant Duffing equations where g has a singularity at x = 0 and n is a positive integer. We give an explicit condition to ensure the existence of positive 2π-periodic solutions when the limit limx→+∞g(x) = g(+∞) exists and is finite. On the basis of this conclusion, we give an answer to the problem raised by Del Pino and Manásevich. We also study the multiplicity of positive periodic solutions of singular Duffing equations When g satisfies the semilinear condition at infinity and the time map satisfies an oscillation condition, we prove that the given equation possesses infinitely many positive 2π-periodic solutions by using the Poincaré–Birkhoff theorem.

Multiple Periodic Solutions of a Ratio‐Dependent Predator‐Prey Discrete Model

A delayed ratio‐dependent predator‐prey discrete‐time model with nonmonotone functional response is investigated in this paper. By using the continuation theorem of Mawhins coincidence degree theory, some new sufficient conditions are obtained for the existence of multiple positive periodic solutions of the discrete model. An example is given to illustrate the feasibility of the obtained result.

Multiple Periodic Solutions of Delayed Predator‐Prey Systems with Type IV Functional Responses on Time Scales

With the help of a continuation theorem based on Gaines and Mawhin′s coincidence degree, easily verifiable criteria are established for the existence of multiple positive periodic solutions of delayed predator‐prey systems with type IV functional responses on time scales. Our results not only unify the existing ones but also widen the range of applications.

Multiple Positive Periodic Solutions for a Gilpin‐Ayala Competition Predator‐Prey System with Harvesting Terms

By applying Mawhin’s continuation theorem of coincidence degree theory, we study the existence of multiple positive periodic solutions for a Gilpin‐Ayala competition predator‐prey system with harvesting terms and obtain some sufficient conditions for the existence of multiple positive periodic solutions for the system under consideration. The result of this paper is completely new. An example is employed to illustrate our result.

Four positive periodic solutions of a delayed plankton allelopathy system on time scales with multiple exploited (or harvesting) terms

In this paper, we consider a delayed periodic plankton allelopathy system on time scales with multiple exploited (or harvesting) terms. Using the continuation theorem based on Gaines and Mawhin's coincidence degree theory, easily verifiable criteria are established for global existence of multiple positive periodic solutions to the above system on time scales.

Pairs of positive periodic solutions of second order nonlinear equations with indefinite weight

We study the problem of the existence and multiplicity of positive periodic solutions to the scalar ODE u ″ + λ a ( t ) g ( u ) = 0 , λ > 0 , where g ( x ) is a positive function on R + , superlinear at zero and sublinear at infinity, and a ( t ) is a T-periodic and sign indefinite weight with negative mean value. We first show the nonexistence of solutions for some classes of nonlinearities g ( x ) when λ is small. Then, using critical point theory, we prove the existence of at least two positive T-periodic solutions for λ large. Some examples are also provided.

Positive periodic solutions of singular systems of first order ordinary differential equations

The existence and multiplicity of positive periodic solutions for first non-autonomous singular systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. The proof of our results is based on the Krasnoselskii fixed point theorem in a cone.

Existence of Periodic Positive Solutions for Abstract Difference Equations

We will consider the existence of multiple positive periodic solutions for a class of abstract difference equations by using the well‐known fixed point theorem (due to Krasnoselskii).

Positive periodic solutions of singular systems with a parameter

The existence and multiplicity of positive periodic solutions for second-order non-autonomous singular dynamical systems are established with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter. Our results provide a unified treatment for the problem and significantly improve several results in the literature. The proof of our results is based on the Krasnoselskii fixed point theorem in a cone.

Multiple positive periodic solutions for a nonlinear first order functional difference equation

We obtain sufficient conditions for the existence of at least three non-negative periodic solutions for the first order functional difference equation Our main tool is the Leggett–Williams fixed point theorem, and our main application is a hematopoiesis model in population dynamics.

Existence of multiple positive periodic solutions to a periodic predator–prey system with harvesting terms and Holling III type functional response

This paper is concerned with a periodic predator–prey system with Holling III functional response and harvesting terms. By means of the coincidence degree theory, we establish the existence of at least eight positive periodic solutions for the system. An example is given to illustrate the effectiveness of our results.

Multiple positive periodic solutions of [formula omitted] species delay competition systems with harvesting terms

This paper considers a delay nonautonomous Lotka–Volterra type multispecies competitive system with harvesting terms. The model has the intraspecific competition terms defined by functions which depend on population densities. An existence theorem of 2 n positive periodic solutions is established by using the coincidence degree theory. An example is given to illustrate the effectiveness of our result.

Periodic solutions for a system of the first order nonautonomous differential equations

In this paper, we rigorously show the existence of multiple positive periodic solutions for a system of the first order nonautonomous differential equations in the form of { x ̇ = x F 1 ( t , x , y ) , y ̇ = y F 2 ( t , x , y ) . The proof relies on some analytical techniques and coincidence degree theory. As an application, we study two classes of biological population models, both of which have at least four positive periodic solutions.

Positive periodic solutions for higher order functional difference equations

In this work, we apply a fixed point theorem in a cone to obtain sufficient conditions for the existence of multiple positive periodic solutions to a class of higher order functional difference equations.

Multiple Positive Periodic Solutions of a Discrete Time Delayed Predator–Prey System with Harvesting Terms

In this paper, by using Mawhin’s continuation theorem of coincidence degree theory, we establish the existence of at least two positive periodic solutions for a discrete time delayed predator–prey system with harvesting terms. An example is given to illustrate the effectiveness of our results.

Multiple periodic solutions of a delayed predator–prey system with stage structure for the predator

By using the continuation theorem of coincidence degree theory, the existence of multiple positive periodic solutions for a delayed predator–prey system with stage structure for the predator is established.