Periodic Solutions of Linear Integro-Differential Equations

Using a degree-theoretic result of Granas, a homotopy is constructed enabling us to show that if there is an a priori bound on all possible T-periodic solutions of a Volterra equation, then there is a T-periodic solution. The a priori bound is established by means of a Liapunov functional. The latter result is unusual in that no bounds on the Liapunov functional are required. Thus, in addition to the periodic solution, the equation may have both bounded and unbounded Solutions.     

On Systems of Boundary Value Problems for Differential Inclusions

Herein we consider the existence of solutions to second-order, two-point boundary value problems （BVPs） for systems of ordinary differential inclusions. Some new Bernstein-Nagumo conditions are presented that ensure a priori bounds on the derivative of solutions to the differential inclusion. These a priori bound results are then applied, in conjunction with appropriate topological methods, to prove some new existence theorems for solutions to systems of BVPs for differential inclusions. The new conditions allow of the treatment of systems of BVPs without growth restrictions.     

Nonlinear integrodifferential equations anda priori bounds on periodic solutions

Summary This paper studies the existence of aperiodic solution of a nonlinear integrodifferential system of the form , for each continuous periodic function p and under suitable assumptions on f, k and g. A topological transversality method is employed to obtain the existence of periodic solutions. This method relies ona priori bounds on periodic solutions. Several examples are provided where a variant of Liapunov's direct method is employed to obtaina priori bounds on periodic solutions.     

6-Periodic travelling waves in an artificial neural network with bang–bang control

Doubly periodic travelling waves can be used to describe dynamic patterns of signals that govern movements of animals. In this paper, we study the existence of such waves in cellular networks involving the discontinuous Heaviside step function. This is done by finding ω-periodic solutions of an accompanying recurrence relation with a priori unknown parameters and the Heaviside function. Since analytic tools cannot be used to handle discontinuous models such as ours, existence of periodic solutions is investigated by means of symmetry, combinatorial techniques and accompanying linear systems. By such means, we are able to obtain all periodic solutions with least periods 1 through 6. Our techniques are new and good for other periodic solutions with relatively small periods.     

Exponential attractor of the 3D derivative Ginzburg-Landau equation

In this paper, we consider a derivative Ginzburg-Landau-type equation with periodic initial-value condition in three-dimensional spaces. Sufficient conditions for existence and uniqueness of a global solution are obtained by uniform a priori estimates of the solution. Furthermore, the existence of a global attractor and an exponential attractor with finite dimensions are proved.     

On the existence of periodic solutions in the thermistor problem with degenerate thermal conductivity

Abstract This paper deals with the existence of weak periodic solutions for a parabolic-elliptic system proposed as a model for a time dependent thermistor with degenerate thermal conductivity. Applying the maximal monotone mappings theory, we prove an existence result for weak periodic solutions. Keywords: Nonlinear parabolic-elliptic system of degenerate type, Periodic solutions, Thermistor problem Mathematics Subject Classification (2000): 35B10, 35J60, 35K65     

Estimates of solutions of initial-irregular oblique derivative problems for linear parabolic equations of second order with measurable coefficients

The initial-irregular oblique derivative boundary value problems for linear and nondivergence parabolic complex equations of second order in multiply connected domains are dealt with, where the coefficients of equations are measurable. Firstly the uniqueness of solutions for the above problems is introduced, and then somea priori estimates of solutions for the problems are given. By using the above estimates and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems can be proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

Existence of periodic solutions to a system with functional response on time scales

This paper investigates the existence of periodic solutions of a three-species food-chain diffusive system with Beddington-DeAngelis functional responses and time delays in a two-patch environment on time scales. By using a continuation theorem based on coincidence degree theory, we obtain sufficient criteria for the existence of periodic solutions for the system. Moreover, when the time scale T is chosen as R or Z, the existence of the periodic solutions of the corresponding continuous and discrete models follows. Therefore, the method is unified to provide the existence of the desired solutions for continuous differential equations and discrete difference equations.     

A study on periodic solutions of higher order nonlinear functional difference equations with p-Laplacian†

The study on the existence of periodic solutions of higher order nonlinear functional difference equations with p-Laplacian is made. Sufficient conditions for the existence of at least one periodic solution of these equations are established, respectively. We give no restriction on the deviating function, which is the significance of the paper. Our result is based on Mawhin's continuation theorem. The methods we used to estimate a priori bound on periodic solutions are also new.     

Boundary value problem for the N‐dimensional time periodic Vlasov–Poisson system

In this work, we study the existence of time periodic weak solution for the N‐dimensional Vlasov–Poisson system with boundary conditions. We start by constructing time periodic solutions with compact support in momentum and bounded electric field for a regularized system. Then, the a priori estimates follow by computations involving the conservation laws of mass, momentum and energy. One of the key point is to impose a geometric hypothesis on the domain: we suppose that its boundary is strictly star‐shaped with respect to some point of the domain. These results apply for both classical or relativistic case and for systems with several species of particles. Copyright © 2006 John Wiley & Sons, Ltd.     

The Existence of Positive Almost Periodic Type Solutions for Some Nonlinear Delay Integral Equations

This paper presents the conditions of existence of positive almost periodic type solutions for some nonlinear delay integral equations, by using a fixed point theorem in the mixed monotone operators （Ma, Y.： On a class of mixed monotone operators and a kind of two-point bounded value problem. Indian J. Math., 41（2）, 211-220 （1999]]. Some known results are operators.     

Bifurcations of travelling wave solutions for the mK(n, n) equation

Using the method of planar dynamical systems to the mK(n, n) equation, the existence of uncountably infinite many smooth and non-smooth periodic wave solutions, solitary wave solutions and kink and anti-kink wave solutions is proved. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All possible exact explicit parametric representations of smooth and non-smooth travelling wave solutions are obtain.     

Multibump homoclinic solutions for a class of second order, almost periodic Hamiltonian systems

We prove the existence of infinitely many homoclinic solutions for a class of second order Hamiltonian systems in R ^N of the form where is almost periodic and W is superquadratic. Received October 17, 1995     

Weak almost periodicity and the strong ergodic limit theorem for contraction semigroups

We show that if (S(t)) _t≧0 is a contraction semigroup on a closed convex subset of a uniformly convex Banach space, then every bounded and “asymptotically isometric” almost-orbit of (S(t)) _t≧0 is weakly almost periodic in the sense of Eberlein. As one consequence, results on the existence of almost periodic solutions to the abstract Cauchy problem are obtained without the need fora priori compactness assumptions. As a further consequence, the known strong ergodic limit theorems for (almost-) orbits of nonlinear contraction semigroups turn out to be special cases of Eberlein’s classical ergodic theorem for weakly almost periodic functions.     

Oscillatory Periodic Solutions of Nonlinear Second Order Ordinary Differential Equations

In this paper the existence results of oscillatory periodic solutions are obtained for a second order ordinary differential equation -u″（t） = f（t, u（t））, where f ： R^2 → R is a continuous odd function and is 2π-periodic in t. The discussion is based on the fixed point index theory in cones.     

On the existence of solutions for nonlinear impulsive periodic viable problems

In this paper we prove the existence of periodic solutions for nonlinear impulsive viable problems monitored by differential inclusions of the type x′(t)∈F(t,x(t))+G(t,x(t)). Our existence theorems extend, in a broad sense, some propositions proved in [10] and improve a result due to Hristova-Bainov in [13].     

Existence of Single and Multiple Solutions for First Order Periodic Boundary Value Problems

This paper is devoted to the study ofthe existence of single and multiple positive solutions forthe first order boundary value problem x′= f(t,x),x(0)=x(T),where f ∈ C([0,T]×R).In addition,weapply our existence theorems to a class of nonlinear periodic boundary value problems with a singularity at theorigin.Our proofs are based on a fixed point theorem in cones.Our results improve some recent results in theliteratures.     

The Global Attractors for the Periodic Initial Value Problem For a Coupled Non-linear Wave Equation

The existence of global attractors of the periodic initial value problem for a coupled non-linear wave equation is proved. We also get the estimates of the upper bounds of Hausdorff and fractal dimensions for the global attractors by means of uniform a priori estimates for time.     

非自治p(t)-拉普拉斯系统周期解的存在性

本文研究一类非自治p(t)-Laplace系统.利用鞍点定理和极小作用原理,获得了周期解存在的充分条件,推广和改进了文献[8]中的结果.     

On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in Lp spaces

In this paper the global existence of weak solutions for the Vlasov-Poisson-Fokker-Planck equations in three dimensions is proved with an L¹ ∩ L^p initial data. Also, the global existence of weak solutions in four dimensions with small initial data is studied. A convergence of the solutions is obtained to those built by E. Horst and R. Hunze when the Fokker-Planck term vanishes. In order to obtain the a priori necessary estimates a sequence of approximate problems is introduced. This sequence is obtained starting from a non-linear regulation of the problem together with a linearization via a time retarded mollification of the non-linear term. The a priori bounds are reached by means of the control of the kinetic energy in the approximate sequence of problems. Then, the proof is completed obtaining the equicontinuity properties which allow to pass to the limit.