On classical almost-periodic solutions of nonlinear hyperbolic equations

Translated from Matematicheskie Zametki, Vol. 54, No. 6, pp. 146–148, December, 1993.     

Tight estimates for general averaging applied to almost-periodic differential equations

Estimates are presented for the averaging method of ordinary differential equations. Previous results are improved by relaxing the conditions under which they hold, and by providing tight bounds for the estimations for almost-periodic differential equations and quasi-periodic differential equations.     

Complete guiding sets for a class of almost-periodic differential equations

A functional differential equation admitting a complete guiding set always has an a priori bounded solution. The introduction of this concept and the idea of reducibility allow the authors to apply a well-known method for the autonomous case to a non-autonomous problem. A previous result of Ortega and Tineo is extended to a non-hyperbolic semilinear functional differential equation with almost-periodic coefficients in its lineal part. The non-hyperbolicity condition is established in terms of the Sacker-Sell spectrum of the associated linear system.     

Infinite-order differential equations with analytic solutions

An infinite-order linear differential equation with constant coefficients and characteristic equation of the class [1, 0] is investigated, and a class of solutions is introduced. It is shown that, if the zeros _k = _k+ i _k of the characteristic function satisfy the condition , then all solutions of the class under consideration are analytic functions.Translated from Matematicheskie Zametki, Vol. 10, No. 3, pp. 269–278, September, 1971.     

Linear integro-differential equations with almost-periodic coefficients

A class of integro-differential equations is described, for which the regularity of the corresponding operator is equivalent to the exponential dichotomy of solutions of the homogeneous equation.Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 463–473, October, 1970.     

Nonlinear differential equations with asymptotically stable solutions

We establish conditions of asymptotic stability of all solutions of the equation , t≥0in a Banach space E in the case where σ(A(x)⊂{λ:Reλ<0}∀x∈E. We give an example of an equation with unstable solutions. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 2, pp. 264–273, February, 1998.     

Abel-like differential equations with no periodic solutions

We present various criteria for the non-existence of positive periodic solutions of generalized Abel differential equations with periodic coefficients that can change sign. As an application, we obtain some families of planar vector fields without limit cycles.     

Approximation of solutions of stochastic differential equations with fractional Brownian motion by solutions of random ordinary differential equations

We prove a general theorem on the convergence of solutions of stochastic differential equations. As a corollary, we obtain a result concerning the convergence of solutions of stochastic differential equations with absolutely continuous processes to a solution of an equation with Brownian motion.     

Explicit solutions for differential equations

Advantages exist in use of the decomposition method [1, 2] for solutions of differential equations. Even for the trivial case of solution of first-order separable differential equations the decomposition solutions are more useful because of the resulting convenient computable explicit solutions. The same techniques and benefits apply to the algebraic equations obtained by transform methods in solving differential equations. A comparison is made also between solutions by integrating factor and decomposition, and it is shown that decomposition is an obvious recourse when an integrating factor is not available. To show advantages of the procedure, a differential equation solvable by several methods and involving a logarithmic nonlinearity is solved by Adomian's decomposition for comparisons. The decomposition method will also solve higher-order differential equations and partial differential equations with logarithmic or even composite nonlinearities [2] when the other methods fail.     

On solutions of stochastic differential equations with drift

Summary We study stochastic differential equations of the formdX _t=(X _t)dM_t+b(X_t)d_t whereM is a continuous local martingale and <M> stands for its quadratic variation process. The conditions introduced by Engelbert and Schmidt, which ensure the existence and uniqueness in law of solutions of SDE's driven by the Wiener process without drift (or with generalized drift) are shown to be no longer valid.