Smoothness of solutions of almost hypoelliptic equations

The paper studies a class of almost hypoelliptic equations P(D)U = ? in a strip. It is proved that for \(\mathcal{H}\) great enough and for δ > 0 small enough all solutions of this equation, which are square summable with the weight e ^?δ|x| and for which \(D_2^{\alpha _2 } U\), where α ₂ = 0, …, \(ord_{\alpha _2 } P\), are infinitely differentiable in x ₁ functions, provided D ₁ ^j ? ∈ L ₂(\(\Omega _\mathcal{H} \)) for any j.     

Function classes defined from local approximations by solutions to hypoelliptic equations

We describe anisotropic function classes of the Campanato—Morrey type in terms of local approximations by solutions to the equation P(D)f = 0 in integral metrics, where P(D) is a quasihomogeneous hypoelliptic linear differential operator with constant coefficients.     

Weighted pseudo almost periodic solutions of N-th order neutral differential equations with piecewise constant arguments

In this work, we present some existence theorems of weighted pseudo almost periodic solutions for N-th order neutral differential equations with piecewise constant argument by means of weighted pseudo almost periodic solutions of relevant difference equations.     

Existence of weighted pseudo almost periodic solutions to some classes of hyperbolic evolution equations

This paper is concerned with the study of weighted pseudo almost periodic solutions to some classes of hyperbolic differential equations. Upon making some suitable assumptions, the existence and uniqueness of a weighted pseudo almost periodic solution is obtained. To illustrate our main result, we study the existence of a weighted pseudo almost periodic solution to a system of partial differential equations arising in control systems described by abstract retarded functional-differential equations with feedback control.     

On solutions of almost hypoelliptic equations in weighted Sobolev spaces

It is proved that if P(D) is a regular, almost hypoelliptic operator and

Existence of asymptotically almost periodic solutions for damped wave equations

In this paper, a class of nonlinear damped wave equations of the form αu^?(t)+u^″(t)=βAu(t)+γAu^′(t)+f(t,u(t)), t?0, satisfying αβ<γ with prescribed initial conditions are studied. Some sufficient conditions are established for the existence and uniqueness of an asymptotically almost periodic solution. These results have significance in the study of vibrations of flexible structures possessing internal material damping. Finally, an example is presented to illustrate the feasibility and effectiveness of the results.     

On smooth solutions of a class of almost hypoelliptic equations

For a class of non-degenerate (regular) almost hypoelliptic equations, which are partially hypoelliptic with respect to certain variables, infinitely differentiable in a specific strip solutions are extracted.     

Pseudo almost automorphic solutions for dissipative differential equations in Banach spaces

This work aims to study the existence and uniqueness of pseudo compact almost automorphic solution for some dissipative ordinary and functional differential equations. We prove the existence and uniqueness of pseudo compact almost automorphic solution for dissipative differential equations in Banach spaces and then we apply this result to show the existence of pseudo compact almost automorphic solutions for some functional differential equations.     

Positive almost automorphic solutions for a class of non-linear delay integral equations

This article is concerned with a class of non-linear delay integral equations, which unify some extensively studied delay integral equations. We establish a new existence and uniqueness theorem about positive almost automorphic solutions of the delay integral equations. Our theorem can deal with some cases to which many known results are not applicable. Two examples are given to illustrate our results.     

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.     

Existence of almost periodic solutions of second order neutral delay differential equations with piecewise constant argument

By constructing almost periodic sequence solutions to difference equations, the existence of almost periodic solutions of neutral delay differential equations with piecewise constant argument

On a class of differential operators with constant coefficients

A linear differential operator P(D) = P(D ₁, …, D _n ) with constant coefficients is called almost hypoelliptic if all the derivatives D ^α P of the characteristic polynomial P(ξ ₁, …, ξ _n ) can be estimated by P. The paper proves that if P is an almost hypoelliptic operator and f is an infinitely differentiable function, square-summable with a definite exponential weight, then any square summable with the same weight solution u of the equation P(D)u = f is again an infinitely differentiable function and P(ξ) → ∞ as ξ → ∞.     

Existence of pseudo almost automorphic mild solutions to stochastic fractional differential equations

Fractional stochastic differential equations have gained considerable importance due to their application in various fields of science and engineering. This paper is concerned with the square-mean pseudo almost automorphic solutions for a class of fractional stochastic differential equations in a Hilbert space. The main objective of this paper is to establish the existence and uniqueness of square-mean pseudo almost automorphic mild solutions to a linear and semilinear case of these equations. A new set of sufficient conditions is obtained to achieve the required result by using the stochastic analysis theory and fixed point strategy. Finally, an example is provided to illustrate the obtained theory.     

Pseudo almost automorphic solutions to semilinear differential equations in Banach spaces

In this paper, we prove that the space is complete. This not only gives an affirmative answer to a basic problem in this field, but also enables us to obtain an existence and uniqueness theorem of pseudo almost automorphic mild solutions to semilinear differential equations in Banach spaces. An example is given to illustrate our theorem. The work was supported partly by the National Natural Science Foundation of China, the NCET-04-0572 and Research Fund for the Key Program of the Chinese Academy of Sciences.     

Pseudo almost periodic solutions for a class of first order differential iterative equations

This paper is concerned with a class of first order differential iterative equations. Under proper conditions, we employ a novel argument to establish a criterion on the existence of pseudo almost periodic solutions. The obtained result complements with some existing ones.     

Existence of almost periodic solutions to some stochastic differential equations

The article introduces and studies the concept of p-mean almost periodicity for stochastic processes. Our abstract results are, subsequently, applied to studying the existence of square-mean almost periodic solutions to some semilinear stochastic equations.     

Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations

In this paper, applying the theory of semigroups of operators to evolution family and Banach fixed point theorem, we prove the existence and uniqueness of an (a) almost automorphic (weighted pseudo almost automorphic) mild solution of the semilinear evolution equation x^′(t)=A(t)x(t)+f(t,x(t)) in Banach space under conditions.     

Quasielliptic operators and Sobolev type equations

We consider a class of matrix quasielliptic operators on the n-dimensional space. For these operators, we establish the isomorphism properties in some special scales of weighted Sobolev spaces. Basing on these properties, we prove the unique solvability of the initial value problem for a class of Sobolev type equations.     

A new almost periodic type of solutions of second order neutral delay differential equations with piecewise constant argument

A definition of pseudo almost periodic sequence is given and the existence of pseudo almost periodic sequence to difference equation is studied. Based on these, the existence of pseudo almost periodic solutions to neutral delay differential equations with piecewise constant argument is investigated