Almost periodic fuzzy-number-valued functions

In this paper we develop a theory of almost periodic fuzzy functions, i.e. of the almost periodic functions of real variable and with values fuzzy real numbers, Although the class of fuzzy real numbers does not form a linear normed space, the majority of main properties of almost periodic functions with values in Banach spaces are extended to this case. Applications to fuzzy differential equations and to (fuzzy) dynamical systems are given.     

Mappings of higher order and nonlinear equations in some spaces of almost periodic functions

In the first part of the paper we examine mappings of higher order from a general point of view, that is, in normed spaces of bounded real-valued functions defined on R$\mathbb{R}$. Particular attention is paid to the relation of such mappings with the so-called autonomous superposition operators. Next we investigate mappings of higher order in Banach spaces of almost periodic functions and their perturbations. We also give necessary and sufficient conditions guaranteeing that a nonautonomous superposition operator acts in the space of almost periodic functions in the sense of Levitan and is uniformly continuous. In the Banach space of bounded almost periodic functions in the sense of Levitan we discuss mappings of higher order and a convolution operator. Some applications to nonlinear differential and integral equations are given.     

Spaces of Quasi-Periodic Functions and Oscillations in Differential Equations

We build spaces of q.p. (quasi-periodic) functions and we establish some of their properties. They are motivated by the Percival approach to q.p. solutions of Hamiltonian systems. The periodic solutions of an adequatez partial differential equation are related to the q.p. solutions of an ordinary differential equation. We use this approach to obtain some regularization theorems of weak q.p. solutions of differential equations. For a large class of differential equations, the first theorem gives a result of density: a particular form of perturbated equations have strong solutions. The second theorem gives a condition which ensures that any essentially bounded weak solution is a strong one.     

Representation and investigation of solutions of a nonlocal boundary-value problem for a system of partial differential equations

We study the boundary-value problem for a system of partial differential equations with constant coefficients with conditions nonlocal in time. By using a metric approach, we prove the well-posedness of the problem in the scale of Sobolev spaces of functions periodic in space variables. By using matrix calculus, we construct an explicit representation of a solution.     

On periodic matrix-valued Weyl-Titchmarsh functions

We consider a certain class of Herglotz-Nevanlinna matrix-valued functions which can be realized as the Weyl-Titchmarsh matrix-valued function of some symmetric operator and its self-adjoint extension. New properties of Weyl-Titchmarsh matrix-valued functions as well as a new version of the functional model for such realizations are presented. In the case of periodic Herglotz-Nevanlinna matrix-valued functions, we provide a complete characterization of their realizations in terms of the corresponding functional model. We also obtain properties of a symmetric operator and its self-adjoint extension which generate a periodic Weyl-Titchmarsh matrix-valued function. We study pairs of operators (a symmetric operator and its self-adjoint extension) with constant Weyl-Titchmarsh matrix-valued functions and establish connections between such pairs of operators and representations of the canonical commutation relations for unitary groups of operators in Weyl's form. As a consequence of such an approach, we obtain the Stone-von Neumann theorem for two unitary groups of operators satisfying the commutation relations as well as some extension and refinement of the classical functional model for generators of those groups. Our examples include multiplication operators in weighted spaces, first and second order differential operators, as well as the Schrödinger operator with linear potential and its perturbation by bounded periodic potential.     

Elliptic operators and maximal regularity on periodic little-H?lder spaces

We consider one-dimensional inhomogeneous parabolic equations with higher-order elliptic differential operators subject to periodic boundary conditions. In our main result we show that the property of continuous maximal regularity is satisfied in the setting of periodic little-H?lder spaces, provided the coefficients of the differential operator satisfy minimal regularity assumptions. We address parameter-dependent elliptic equations, deriving invertibility and resolvent bounds which lead to results on generation of analytic semigroups. We also demonstrate that the techniques and results of the paper hold for elliptic differential operators with operator-valued coefficients, in the setting of vector-valued functions.     

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.     

μ-Pseudo Almost Periodic and Automorphic Solutions in the α-Norm for Some Partial Functional Differential Equations

In this work, we prove the existence and uniqueness of μ-pseudo almost periodic and μ-pseudo almost automorphic solutions in the α ?norm for some classes of partial differential equations in Banach spaces. For illustration, we study some models arising in physical systems.     

Periodic solutions of parabolic inclusions and the averaging method

We obtain an analog of the second Bogolyubov theorem for differential inclusions with multimappings acting in Sobolev spaces and satisfying some monotonicity and compactness conditions. As a consequence, we obtain criteria for the existence of periodic solutions of secondorder parabolic inclusions.     

Bochner’s theorem and Stepanov almost periodic functions

Motivated by a renewed interest in generalizations of classical almost periodicity (originally due to Harald Bohr), we develop a theorem of Bochner within the framework of almost periodic functions in the sense of Stepanov. As a result we establish some conditions that guarantee the existence of Stepanov almost periodic solutions to differential equations with Stepanov almost periodic coefficients. Finally, we extend a now classic theorem of Favard originally stated for classical almost periodic functions to the Stepanov almost periodic case.      

Common fixed point theorems for mappings satisfying property (E.A) on cone metric spaces

The aim of this paper is to give some new common fixed point theorems for mappings satisfying property (E.A) on cone metric spaces. And we prove the existence and uniqueness of solution for a ordinary differential equation with periodic boundary condition.     

Approximation of semigroups and cosine functions in spaces of periodic functions

In this article we furnish a representation of the solutions of some classes of first-order and second-order evolution problems as limit of iterates of classical sequences of approximating operators. The method is based on Trotter's theorem on the approximation of semigroups which is applied here also for the approximation of groups and cosine functions. We apply this method in spaces of continuous periodic functions and using some classical sequences of trigonometric polynomials.     

Almost Periodic and Pseudo Almost Periodic Mild Solutions to a Partial Differential Equation VIA Fractional Operators

This article is mainly concerned with the existence and uniqueness of almost periodic and pseudo almost periodic mild solutions to a class of partial differential equations in Banach spaces. Under some reasonable assumptions, the main results are proved in α-norm. Our results cover the cases that the functions F ₁; F ₂ take values in different spaces such as X, X _α, and X _ß, where α ≤ β ∈ (0; 1).     

Convergence of a class of discrete-time semiflows with application to neutral delay differential equations

In this paper, we introduce a class of pseudo-monotone maps on ordered topological spaces. By exploiting monotonicity methods and the invariance of the omega limit set, we establish a convergence principle for discrete-time semiflows generated by the maps introduced. The convergence principle is then applied to a class of periodic neutral delay differential equations, which leads to some novel and sharper results.     

Asymptotically almost periodic,almost periodic and pseudo-almost periodic mild solutions for neutral differential equations

In this paper we establish the existence and uniqueness of almost periodic, asymptotically almost periodic and pseudo-almost periodic mild solutions for neutral differential equations in Banach spaces.     

Some remarks on functions with values in probabilistic normed spaces

In this paper we consider an enlargement of the notion of the probabilistic normed space. For this new class of probabilistic normed spaces we give some topological properties. By using properties of the probabilistic norm we prove some differential and integral properties of functions with values into probabilistic normed spaces. As special cases, results for deterministic and random functions can be obtained.      

Cauchy problem for parabolic systems with convolution operators in periodic spaces

For a class of periodic systems of parabolic type with pseudodifferential operators containing $\{ \vec p,\vec h\} $ -parabolic systems of partial differential equations, we study the properties of the fundamental matrices of the solutions and establish the well-posed solvability of the Cauchy problem for these systems in the spaces of generalized periodic functions of the type of Gevrey ultradistributions. For a particular subclass of systems, we describe the maximal classes of well-posed solvability of the Cauchy problem.     

抽象泛函微分方程的权伪概周期解

本文利用算子半群理论和Banach不动点定理研究了一类具非局部条件的抽象泛函微分方程权伪概周期解的存在唯一性,所得结论拓展了已有结果.     

Banach空间中一类周期非线性受控系统最优控制的存在性

研究一类强非线性发展方程的周期解及相应的最优控制问题的存在性，首先，证明了Banach空间中一类包含非线性单调算子和非线性非单调扰动的强非线性发展方程周期解的存在性；其次，给出了保证相应的Lagrange最优控制的充分条件；最后，举例说明理论结果在拟线笥抛物方程周期问题及相应的最优控制问题中的应用。     

Banach空间中二阶非线性常微分方程周期边值问题

在Banach空间中利用上下解方法与增算子不动点定理，研究了含间断项的二阶非线性常微分方程周期边值问题的最大解、最小解的存在性，推广和改进了现有的结果．