scholarly journals Problem for hyperbolic system of equations having constant coefficients with integral conditions with respect to the time variable

2014 ◽  
Vol 6 (2) ◽  
pp. 282-299 ◽  
Author(s):  
A.M. Kuz ◽  
B.Yo. Ptashnyk
Keyword(s):  
Hyperbolic System ◽  
Cartesian Product ◽  
Sufficient Conditions ◽  
Time Variable ◽  
System Of Equations ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Integral Conditions ◽  
Small Denominators ◽  
Constant Coefficients

In a domain specified in the form of a Cartesian product of a segment $\left[0,T\right]$ and the space ${\mathbb R}^{p}$, we study a problem with integral conditions with respect to the time variable for  hyperbolic system with constant coefficients in a class of almost periodic functions in the space variables. A criterion for the unique solvability of this problem and sufficient conditions for the existence of its solution are established. To solve the problem of small denominators arising in the construction of solutions of the posed problem, we use the metric approach.

Semidirect Product Compactifications

1983 ◽  
Vol 35 (1) ◽  
pp. 1-32
Author(s):  
F. Dangello ◽  
R. Lindahl
Keyword(s):  
Direct Product ◽  
Semidirect Product ◽  
Topological Semigroup ◽  
Sufficient Conditions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Topological Semigroups ◽  
Weakly Almost Periodic ◽  
Necessary And Sufficient

1. Introduction. K. Deleeuw and I. Glicksberg [4] proved that if S and T are commutative topological semigroups with identity, then the Bochner almost periodic compactification of S × T is the direct product of the Bochner almost periodic compactifications of S and T. In Section 3 we consider the semidirect product of two semi topological semigroups with identity and two unital C*-subalgebras and of W(S) and W(T) respectively, where W(S) is the weakly almost periodic functions on S. We obtain necessary and sufficient conditions and for a semidirect product compactification of to exist such that this compactification is a semi topological semigroup and such that this compactification is a topological semigroup. Moreover, we obtain the largest such compactifications.

Hilbert transforms and almost periodic functions

1960 ◽  
Vol 56 (4) ◽  
pp. 354-366 ◽  
Author(s):  
J. Cossar
Keyword(s):  
Hilbert Transform ◽  
Sufficient Conditions ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Hilbert Transforms ◽  
Cauchy Principal ◽  
Principal Value ◽  
Image Position ◽  
The Hilbert Transform

The Hilbert transform, Hf, of a function f is defined by Hf = g, whereP denoting the Cauchy principal value and the integral being assumed to exist in some sense. When f is suitably restricted, Hf exists andIn the first part of Theorem 1 sufficient conditions are given for the validity of (1·2) rather more general than those of Wood ((6), p. 31). The present proof is based on the well-known condition of Riesz for the validity of (1·2), namely, that f is Lp(−∞, ∞) for some p > 1, and on the ‘Parseval’ relation (Lemma 3, (b)), which was used in a similar way by Hardy ((3), p. 110).

scholarly journals Almost Periodic Functions on Time Scales and Applications

2011 ◽  
Vol 2011 ◽  
pp. 1-20 ◽  
Author(s):  
Yongkun Li ◽  
Chao Wang
Keyword(s):  
Time Scales ◽  
Sufficient Conditions ◽  
Almost Periodic Solution ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Mathematical Methods ◽  
Variable Delays ◽  
Definition Of ◽  
Almost Periodic Time Scales

We first propose the concept of almost periodic time scales and then give the definition of almost periodic functions on almost periodic time scales, then by using the theory of calculus on time scales and some mathematical methods, some basic results about almost periodic differential equations on almost periodic time scales are established. Based on these results, a class of high-order Hopfield neural networks with variable delays are studied on almost periodic time scales, and some sufficient conditions are established for the existence and global asymptotic stability of the almost periodic solution. Finally, two examples and numerical simulations are presented to illustrate the feasibility and effectiveness of the results.

scholarly journals Almost periodic mild solutions of a class of partial functional differential equations

1998 ◽  
Vol 3 (3-4) ◽  
pp. 425-436 ◽  
Author(s):  
Bernd Aulbach ◽  
Nguyen Van Minh
Keyword(s):  
Differential Equations ◽  
Mild Solutions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Partial Functional Differential Equations ◽  
Semilinear Equation ◽  
Functional Differential ◽  
Lipschitz Conditions

We study the existence of almost periodic mild solutions of a class of partial functional differential equations via semilinear almost periodic abstract functional differential equations of the form(*)                                                                       x′=f(t,x,xt).To this end, we first associate with every almost periodic semilinear equation(**)                                                                       x′=F(t,x).a nonlinear semigroup in the space of almost periodic functions. We then give sufficient conditions (in terms of the accretiveness of the generator of this semigroup) for the existence of almost periodic mild solutions of (**) as fixed points of the semigroup. Those results are then carried over to equation (*). The main results are stated under accretiveness conditions of the functionfin terms ofxand Lipschitz conditions with respect toxt.

scholarly journals Stepanov-Like Asymptotical Almost Periodic Functions and an Application

2018 ◽  
Vol 2018 ◽  
pp. 1-9
Author(s):  
Yongkun Li ◽  
Yaolu Wang ◽  
Jianglian Xiang
Keyword(s):  
Delay Differential Equations ◽  
Existence And Uniqueness ◽  
Contraction Mapping ◽  
Sufficient Conditions ◽  
Contraction Mapping Principle ◽  
Almost Periodic Functions ◽  
Almost Periodic Solutions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Delay Differential

In this paper, we first study some basic properties of Stepanov-like asymptotical almost periodic functions including the completeness of the space of Stepanov-like asymptotical almost periodic functions. Then, as an application, based on these and the contraction mapping principle, we obtain sufficient conditions for the existence and uniqueness of Stepanov-like asymptotical almost periodic solutions for a class of semilinear delay differential equations.

scholarly journals Exchange of equilibria in two species Lotka-Volterra competition models

1982 ◽  
Vol 24 (2) ◽  
pp. 160-170 ◽  
Author(s):  
K. Gopalsamy
Keyword(s):  
Periodic Solution ◽  
Sufficient Conditions ◽  
System Of Equations ◽  
Species Competition ◽  
Asymptotically Stable ◽  
Stable Periodic Solution ◽  
Periodic Functions ◽  
Competition System ◽  
Competition Models ◽  
Stable Periodic

AbstractSufficient conditions are obtained for the existence of a unique asymptotically stable periodic solution for the Lotka-Volterra two species competition system of equations when the intrinsic growth rates are periodic functions of time.

scholarly journals Cn-Almost Periodic Functions and an Application to a Lasota-Wazewska Model on Time Scales

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Li Yang ◽  
Yongkun Li ◽  
Wanqin Wu
Keyword(s):  
Time Scales ◽  
Differential Inequality ◽  
Global Exponential Stability ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Almost Periodic Functions ◽  
Almost Periodic ◽  
Periodic Functions ◽  
Inequality Techniques

We first give the definition and some properties ofCn-almost periodic functions on time scales. Then, as an application, we are concerned with a class of Lasota-Wazewska models on time scales. By means of the fixed point theory and differential inequality techniques on time scales, we obtain some sufficient conditions ensuring the existence and global exponential stability ofC1-almost periodic solutions for the considered model. Our results are essentially new whenT=RorT=Z. Finally, we present a numerical example to show the feasibility of obtained results.

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