On maximum periodic solutions of integrodifferential equations of volterra type and their stability

Consideration was given to the question of asymptotic (exponential) stability of the maximum periodic solutions of the integrodifferential equations which have an asymptotically stable linear part and small periodic (exponential maximum periodic) perturbation. Under the unlimitedly increasing time, these solutions tend to the periodic modes. The sufficient conditions for asymptotic stability were indicated. In the resonance case where the linearized equation has a pair of purely imaginary roots with the corresponding oscillation frequency coinciding with the oscillation frequency of the periodic part of small perturbation (time function) and the coefficients of the power series expansion of the nonlinear terms, consideration was given to the problem of existence for the maximum periodic solutions of the integrodifferential equation. Conditions were established for existence of such solutions representable by the power series in the fractional degrees of the small parameter characterizing the value of small perturbation in the equation.     

Constant-sign periodic and almost periodic solutions of a system of difference equations

We consider the following system of difference equations, ,where I is a subset of . Our aim is to establish criteria such that the above system has a constantsign periodic and almost periodic solution (u₁, u₂, …, u_n). The above problem is also extended to that on , .     

Periodic solutions of periodic Riccati equations

For periodically time-varying matrix Riccati equations, controllability and observability (in the usual sense) are shown to be sufficient for the existence of a unique positive definite periodic solution.     

Bessel polynomial solutions of high-order linear Volterra integro-differential equations

In this study, a practical matrix method, which is based on collocation points, is presented to find approximate solutions of high-order linear Volterra integro-differential equations (VIDEs) under the mixed conditions in terms of Bessel polynomials. Numerical examples are included to demonstrate the validity and applicability of the technique and comparisons are made with the existing results. The results show the efficiency and accuracy of the present work. All of the numerical computations have been performed on the computer using a program written in MATLAB v7.6.0 (R2008a).     

Existence of solutions for a class of nonlinear Volterra singular integral equations

In this paper we prove some results concerning the existence of solutions for a large class of nonlinear Volterra singular integral equations in the space C[0,1] consisting of real functions defined and continuous on the interval [0,1]. The main tool used in the proof is the concept of a measure of noncompactness. We also present some examples of nonlinear singular integral equations of Volterra type to show the efficiency of our results. Moreover, we compare our theory with the approach depending on the use of the theory of Volterra-Stieltjes integral equations. We also show that the results of the paper are applicable in the study of the so-called fractional integral equations which are recently intensively investigated and find numerous applications in describing some real world problems.     

Monotonic solutions of a class of quadratic integral equations of Volterra type

Using the technique associated with measures of noncompactness we prove the existence of monotonic solutions of a class of quadratic integral equation of Volterra type in the Banach space of real functions defined and continuous on a bounded and closed interval.     

Time symmetry and periodic solutions of the state equations

The constraint of time symmetry applied to the linear state equations is used to investigate periodic solutions. Specifically, it is shown that the state is an even function of time if and only if the system has odd time symmetry. Furthermore, if the system isT-periodic, then so is the state.     

Existence of periodic solutions in n-dimensional retarded functional differential equations

The existence of periodic orbits of n-dimensional delay systems of the form [xdot](t) = ?f(x(t ? p)) is proved and applied to systems of the form [xdot](t) = ?x(t ? 1)N(x(t)), and to a certain type of hamiltonian system.     

Periodic conjugate direction algorithm for symmetric periodic solutions of general coupled periodic matrix equations

Analysis and design of linear periodic control systems are closely related to the periodic matrix equations. The conjugate direction (CD) method is a famous iterative algorithm to find the solution to nonsymmetric linear systems $Ax=b$. In this work, a new method based on the CD method is proposed for computing the symmetric periodic solutions $(X_1,X_2,\dots ,X_{\lambda })$ and $(Y_1,Y_2,\dots ,Y_{\lambda })$ of general coupled periodic matrix equations

for $i=1,2,\dots ,\lambda$. The key idea of the scheme is to extend the CD method by means of Kronecker product and vectorization operator. In order to assess the convergence properties of the method, some theoretical results are given. Finally two numerical examples are included to illustrate the efficiency and effectiveness of the method.     

Periodic solutions of a class of nonlinear difference equations via critical point method

In this paper, we obtain some new sufficient conditions for the existence of nontrivial m-periodic solutions of the following nonlinear difference equation

by using the critical point method, where f: Z × R → R is continuous in the second variable, m ≥ 2 is a given positive integer, p_n+m = p_n for any n  Z and f(t + m, z) = f(t, z) for any (t, z)  Z × R, (−1)^δ = −1 and δ > 0.     

Numeric-analytic method for finding the periodic solutions of nonlinear differential-difference equations with impulses

A non-linear system of differential-difference equations with impulses is considered. An approximate method for finding the periodic solution of the considered system is constructed and justified.     

Multiplicity of positive solutions for the Kirchhoff-type equations with critical exponent in RN

In this work we study the existence and multiplicity of solutions to the following Kirchhoff-type problem with critical nonlinearity in ${\mathbb{R}}^N$

where $N\ge 2p$, $\mu ,\lambda ,a,b>0$ and the nonlinearity $f(x,u)$ satisfies certain subcritical growth conditions. By using topological and variational methods, infinitely many positive solutions are obtained.     

Infinitely many solutions for a class of nonlinear impulsive differential equations with periodic boundary conditions

In this paper, we consider the existence of solutions for a class of nonlinear impulsive problems with periodic boundary conditions. By using critical point theory, we obtain some existence theorems of infinitely many solutions for the nonlinear impulsive problem when the impulsive functions are superlinear. We extend and improve some recent results.     

Differential periodic Riccati equations: Existence and uniqueness of nonnegative definite solutions

In this paper we consider the differential periodic Riccati equation. All the periodic nonnegative definite solutions are characterized in the more general case, providing a method for constructing them. The method is obtained from the study of the invariant subspaces of the monodromy matrix of the associated Hamiltonian system, and from the relations between these invariant subspaces and the controllability and unobservability subspaces. Finally, the method is applied to obtain necessary and sufficient conditions for the existence of any periodic nonnegative definite solution and to study the existence and uniqueness of minimal, maximal, stabilizing, and strong solutions.This work has been partially supported by Spanish DGICYT Grant No. PB91-O535.