Recurrence of the integral of a smooth conditionally periodic function

We prove that the integral of a smooth multifrequency conditionally periodic function with zero mean oscillates. Translated fromMatematicheskie Zametki, Vol. 63, No. 5, pp. 737–748, May, 1998. The author wishes to express his deep gratitude to S. V. Konyagin for fruitful discussions, verification of the results, and the correction of some misprints. This research was supported by the Russian Foundation for Basic Research under grants No. 96-01-00378 and No. 96-15-96072 and by the EFAKOD Foundation.     

Reflexive periodic solutions of general periodic matrix equations

Analysis and design of linear periodic control systems are closely related to the periodic matrix equations. The objective of this paper is to provide four new iterative methods based on the conjugate gradient normal equation error (CGNE), conjugate gradient normal equation residual (CGNR), and least‐squares QR factorization (LSQR) algorithms to find the reflexive periodic solutions (X₁,Y₁,X₂,Y₂,…,X_σ,Y_σ) of the general periodic matrix equations for i = 1,2,…,σ. The iterative methods are guaranteed to converge in a finite number of steps in the absence of round‐off errors. Finally, some numerical results are performed to illustrate the efficiency and feasibility of new methods.     

Gabor systems on discrete periodic sets

Due to its good potential for digital signal processing, discrete Gabor analysis has interested some mathematicians. This paper addresses Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. Complete Gabor systems and Gabor frames on discrete periodic sets are characterized; a sufficient and necessary condition on what periodic sets admit complete Gabor systems is obtained; this condition is also proved to be sufficient and necessary for the existence of sets E such that the Gabor systems generated by χ _E are tight frames on these periodic sets; our proof is constructive, and all tight frames of the above form with a special frame bound can be obtained by our method; periodic sets admitting Gabor Riesz bases are characterized; some examples are also provided to illustrate the general theory. This work was supported by National Natural Science Foundation of China (Grant No. 10671008), Beijing Natural Science Foundation (Grant No. 1092001), PHR (IHLB) and the project sponsored by SRF for ROCS, SEM of China     

The existence of periodic solutions of higher order nonlinear periodic difference equations

Sufficient conditions for the existence of at least one periodic solution of two classes of nonlinear higher order periodic difference equations are established, respectively. The results show us that sufficient conditions for the existence of T ? periodic solutions of difference equation are different from those ones for the existence of T ? periodic solutions of differential equation. Copyright © 2012 John Wiley & Sons, Ltd.     

On the existence and uniqueness of time periodic solutions for a semilinear heat equation in the whole space

This paper is concerned with the existence and uniqueness of time periodic solutions in the whole‐space for a heat equation with nonlinear term. The nonlinear term we considered is of this type, |u |^{q ? 1}u + f (x ,t ), with , N > 2. We show that there exists a unique time periodic solution when the source term f is small. In fact, is a critical exponent; when , there is no time periodic solution. Copyright © 2016 John Wiley & Sons, Ltd.     

The number of periodic solutions of polynomial differential equations

We estimate the number of periodic solutions for special classes ofnth-order ordinary differential equations with variable coefficients. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 720–727, November, 1998. The author thanks Yu. S. Il'yashenko for setting the problems, permanent advice, and overall support. The author is also thankful to D. A. Panov for numerous discussions. This research was supported by the CRDF Foundation under grant MR1-220, by the INTAS Foundation under grant No. 93-05-07, and by the Russian Foundation for Basic Research under grant No. 95-01-01258.     

Lyapunov stability of periodic solutions of the quadratic Newtonian equation

We will find a positive constant Σ₂ such that for any 2π ‐periodic function h (t) with zero mean value, the quadratic Newtonian equation x ″ + x² = σ + h (t) will have exactly two 2π ‐periodic solutions with one being unstable and another being twist (and therefore being Lyapunov stable), provided that the parameter σ is bigger than the first bifurcation value and is smaller than the constant Σ₂. The construction of Σ₂ is obtained by examining carefully the twist coefficients of periodic solutions (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

First‐order systems in on with periodic matrix potentials and vanishing instability intervals

First‐order systems in on with absolutely continuous real symmetric π‐periodic matrix potentials are considered. A thorough analysis of the discriminant is given. Interlacing of the eigenvalues of the periodic, antiperiodic and Dirichlet‐type boundary value problems on [0,π] is shown for a suitable indexing of the eigenvalues. The periodic and antiperiodic eigenvalues are characterized in terms of Dirichlet‐type eigenvalues. It is shown that all instability intervals vanish if and only if the potential is the product of an absolutely continuous real scalar valued function with the identity matrix. Copyright © 2014 John Wiley & Sons, Ltd.     

Extremum problems for functions with small support

In this paper we consider an extremum problem for even periodic functions having small intervals as their supports and subject to restrictions on the sum of the absolute values of their Fourier coefficients. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 323–332, September, 1996. We wish to express our gratitude to S. B. Stechkin for his interest in our work. This research was supported by the Russian Foundation for Basic Research under grant No. 96-01-00378, and the work of the first and second authors also by the International Science Foundation under grant No. MC5300.     

Spatially almost periodic complex‐valued solutions of the Boussinesq equations

We extend several ideas developed in E. Dinaburg, Ya. G. Sinai and D. Li's papers (see Lemma 2.2 , Remark   2.1 , Lemma 2.4 , Remark   2.2 , Remark 2.3 ) and construct a unique mild solution (u,θ) of the system 1.1 satisfying 1.3 with spatially almost periodic initial data . Moreover, (u,θ) is also a spatially almost periodic complex‐valued mild solution. Assume further that (see Definition 1.6 ). Then, we overcome the difficulty in the work of Y. Giga et al. (see Introduction below) and show that the above mild solution also satisfies 1.4 for all .     

The Atkinson formula for the periodic zeta-function

We give an Atkinson-type formula for the periodic zeta-function ζ_λ with rational parameter λ. Published in Lietuvos Matematikos Rinkinys, Vol. 47, No. 4, pp. 504–516, October–December, 2007.     

Homoclinic solutions in periodic difference equations with saturable nonlinearity

In this paper, a periodic difference equation with saturable nonlinearity is considered. Using the linking theorem in combination with periodic approximations, we establish sufficient conditions on the nonexistence and on the existence of homoclinic solutions. Our results not only solve an open problem proposed by Pankov, but also greatly improve some existing ones even for some special cases.     

Global periodic conservative solutions of a periodic modified two-component Camassa-Holm equation

In the paper, we first show the existence of global periodic conservative solutions to the Cauchy problem for a periodic modified two-component Camassa-Holm equation. Then we prove that these solutions, which depend continuously on the initial data, construct a semigroup.     

Periodic and almost periodic solutions of conservation laws: Global existence and decay

In this paper we survey recent results on the decay of periodic and almost periodic solutions of conservation laws. We also recall some recent results on the global existence of periodic solutions of conservation laws systems which lie inBV _loc and are constructed through Glimm scheme. The latter motivates a discussion on a possible strategy for solving the open problem of the global existence of periodic solutions of the Euler equations for nonisentropic gas dynamics. We base our decay analysis on a general result about space-time functions which are almost periodic in the space variable, established here for the first time. This result is an abstract version of Theorem 2.1 in [31], which in turn is an extention of the combined result given by Theorems 3.1–3.2 in [9].     

On periodic solutions of ordinary differential equations with discontinuous right-hand side

A new version of the method of translation along trajectories, which does not require the uniqueness of the solution of the Cauchy problem, is applied to the proof of the existence theorem for vector-valued periodic solutions of ordinary differential equations of first and second order. This result is applicable to equations and differential inclusions with discontinuous right-hand side. Several applications of the theorems proved in this paper are considered in cases which are not covered by the classical theory of ordinary differential equations with continuous right-hand side and equations with right-hand side satisfying the Carathéodory conditions.     

Existence and global attractivity of a positive periodic solution of a class of delay differential equation

The existence and the global attractivity of a positive periodic solution of the delay differential equationy(t)=y(t) F[t, y](t-τ ₁ (t)),⋯,y(t−τ _n (t))] are studied by using some techniques of the Mawhin coincidence degree theory and the constructing suitable Liapunov functionals. When these results are applied to some special delay bio-mathematics models, some new results are obtained, and many known results are improved. Project partially supported by the National Natural Science Foundation of China (Grant No. 10572057) and the Applied Basic Research Foundation of Yunnan Province.     

Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations

We establish new theorems for the composition of pseudo almost periodic and pseudo almost automorphic functions in Banach spaces. Our results extend the recent ones [H. Li, F. Huang and J. Li, Composition of pseudo almost-periodic functions and semilinear differential equations, J. Math. Anal. Appl. 255 (2001), pp. 436–446; J. Liang, J. Zhang, T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2001), pp. 1493–1499]. We also study some sufficient conditions for the continuity of the superposition operator. As an application to the abstract results, we give some existence theorems of pseudo almost periodic/automorphic solutions for some semilinear evolution equations and examples with the heat equation.     

Reducibility of a linear system of differential equations with odd almost periodic coefficients

A theorem on the Kolmogorov reducibility of a system of ordinary differential equations with odd almost periodic coefficients is proved. Translated fromMatematicheskie Zametki, Vol. 60, No. 1, pp. 109–119, July, 1996.     

Existence of periodic solutions

An apparatus for proving existence theorems for periodic solutions of equations with discontinuous right-hand side and differential inclusions is developed. Translated fromMatematischeskie Zametki, Vol. 61, No. 5, pp. 769–784, May, 1997. Translated by V. N. Dubrovsky