37 results
Search Results
2. Lyapunov-Razumikhin method for existence of almost periodic solutions of impulsive differential-difference equations.
- Author
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Stamov, Gani Tr. and Petrov, Nikolay
- Subjects
DIFFERENTIAL-difference equations ,LYAPUNOV functions ,DIFFERENTIAL equations ,DIFFERENCE equations ,EQUATIONS - Abstract
In the present paper sufficient conditions about the existence of almost periodic solutions of systems of impulsive differential-difference equations are obtained. The investigations are carried out by means piecewise continuous Lyapunov functions and Razumikhin techniques. [ABSTRACT FROM AUTHOR]
- Published
- 2008
3. Almost periodic solutions for neutral delay Hopfield neural networks with time-varying delays in the leakage term on time scales
- Author
-
Li, Ling, Li, Yongkun, and Yang, Li
- Published
- 2014
- Full Text
- View/download PDF
4. Almost periodic solutions of impulsive differential equations with time-varying delay on the PC-Space.
- Author
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Stamov, Gani Tr.
- Subjects
DIFFERENTIAL equations ,EQUATIONS ,CALCULUS ,LYAPUNOV functions ,MATHEMATICAL analysis - Abstract
In the present paper sufficient conditions for tile existence of almost periodic solutions of systems of impulsive differential equations with time varying delay obtained. The investigations are carried out by means piecewise continuous Lyapunov functions and Razumikhin techniques. [ABSTRACT FROM AUTHOR]
- Published
- 2007
5. Almost periodic solutions for an asymmetric oscillation.
- Author
-
Huang, Peng, Li, Xiong, and Liu, Bin
- Subjects
- *
DIFFERENTIAL equations , *MATHEMATICAL constants , *PERIODIC functions , *DIFFERENTIAL invariants , *MATHEMATICAL functions - Abstract
In this paper we study the dynamical behavior of the differential equation x ″ + a x + − b x − = f ( t ) , where x + = max { x , 0 } , x − = max { − x , 0 } , a and b are two different positive constants, f ( t ) is a real analytic almost periodic function. For this purpose, firstly, we have to establish some variants of the invariant curve theorem of planar almost periodic mappings, which was proved recently by the authors (see [11] ). Then we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the above asymmetric oscillation. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
6. Some periodic type solutions for stochastic reaction–diffusion equation with cubic nonlinearities.
- Author
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Gao, Peng
- Subjects
- *
REACTION-diffusion equations , *STOCHASTIC analysis , *NONLINEAR theories , *SEMIGROUPS (Algebra) , *MATHEMATICAL inequalities - Abstract
In this paper, we discuss the bounded solutions, stationary solutions, periodic solutions, almost periodic solutions, almost automorphic solutions for stochastic reaction–diffusion equation with cubic nonlinearities. The main difficulty is the cubic nonlinearities, we overcome this difficulty by the semigroup approach, the energy estimate method and refined inequality technique. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
7. The Existence and Global Exponential Stability of Almost Periodic Solutions for Neutral-Type CNNs on Time Scales.
- Author
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Li, Bing, Li, Yongkun, and Meng, Xiaofang
- Subjects
EXPONENTIAL stability ,ARTIFICIAL neural networks ,TIME-varying networks ,CONTINUOUS time models - Abstract
In this paper, neutral-type competitive neural networks with mixed time-varying delays and leakage delays on time scales are proposed. Based on the contraction fixed-point theorem, some sufficient conditions that are independent of the backwards graininess function of the time scale are obtained for the existence and global exponential stability of almost periodic solutions of neural networks under consideration. The results obtained are brand new, indicating that the continuous time and discrete-time conditions of the network share the same dynamic behavior. Finally, two examples are given to illustrate the validity of the results obtained. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
8. Limit periodic linear difference systems with coefficient matrices from commutative groups.
- Author
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Hasil, Petr and Veselý, Michal
- Subjects
- *
LINEAR differential equations , *SYSTEMS theory , *MATRICES (Mathematics) , *ABELIAN groups , *ALGEBRAIC field theory - Abstract
In this paper, limit periodic and almost periodic homogeneous linear difference systems are studied. The coefficient matrices of the considered systems belong to a given commutative group. We find a condition on the group under which the systems, whose fundamental matrices are not almost periodic, form an everywhere dense subset in the space of all considered systems. The treated problem is discussed for the elements of the coefficient matrices from an arbitrary infinite field with an absolute value. Nevertheless, the presented results are new even for the field of complex numbers. [ABSTRACT FROM AUTHOR]
- Published
- 2014
9. Almost periodic solutions to dynamic equations on time scales.
- Author
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Zhang, Hongtao and Li, Yongkun
- Subjects
LYAPUNOV functions ,MATHEMATICAL analysis ,EXISTENCE theorems ,PERIODIC functions ,FUNCTIONAL analysis - Abstract
Abstract: In this paper, we first present a notion of almost periodic functions on time scales and study their basic properties. Then by means of Liapunov functionals, we study the existence of almost periodic solutions for an almost periodic dynamic equation on time scales. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
10. Positive almost periodic solutions for a predator-prey Lotka-Volterra system with delays.
- Author
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Yuan Ye
- Subjects
- *
PREDATION , *LOTKA-Volterra equations , *EXISTENCE theorems , *CONTINUATION methods , *PERIODIC functions , *MATHEMATICAL analysis - Abstract
In this paper, by using Mawhin's continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive almost periodic solutions are obtained for the predator-prey Lotka-Volterra competition system with delays {dui(t)/dt = ui(t)[ai(t) - ... ail(t)ul(t - σil(t)) - ... bij(t)vj(t - τij(t))], i = 1,..., n, {dvj(t)/dt = vj(t) [ - rj(t) + ... djl(t)ul(t)ul(t - δjl(t)) - ... ejh(t)vh(t - ϑjh(t))], j = 1,..., m, where ai, rj, ail, bij, djl, ejh ∈ C (ℝ, (0, ∞)), σil, τij, δjl, ϑjh, ∈ c (ℝ, ℝ) (i, l = 1,..., n, j, h = l,..., m) are almost periodic functions. [ABSTRACT FROM AUTHOR]
- Published
- 2012
11. Almost periodic solutions for a class of fourth-order nonlinear differential equations with a deviating argument
- Author
-
Xiong, Wanmin and Yue, Guangxue
- Subjects
- *
NUMERICAL solutions to nonlinear differential equations , *LYAPUNOV stability , *EXPONENTS , *NUMERICAL solutions to differential equations , *MATHEMATICAL analysis , *MATHEMATICS - Abstract
Abstract: This paper considers the fourth-order nonlinear differential equations with a deviating argument. Sufficient conditions for the existence and exponential stability of the almost periodic solutions are established, which are new and complement previously known results. [ABSTRACT FROM AUTHOR]
- Published
- 2010
- Full Text
- View/download PDF
12. Almost periodic solutions for a class of Liénard-type systems with multiple varying time delays
- Author
-
Gao, Hong and Liu, Bingwen
- Subjects
- *
DIFFERENTIABLE dynamical systems , *TIME delay systems , *STABILITY (Mechanics) , *DIFFERENTIAL equations , *PROCESS control systems , *DYNAMICS - Abstract
Abstract: This paper considers the Liénard-type systems with multiple varying time delays. Some sufficient conditions for the existence and exponential stability of the almost periodic solutions are established, which are new and complement previously known results. [Copyright &y& Elsevier]
- Published
- 2010
- Full Text
- View/download PDF
13. Liouville type results for periodic and almost periodic linear operators
- Author
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Rossi, Luca
- Subjects
- *
STURM-Liouville equation , *LINEAR operators , *HARMONIC functions , *NUMERICAL solutions to partial differential equations , *MATHEMATICAL proofs , *DIRICHLET problem , *ALMOST periodic functions - Abstract
Abstract: This paper is concerned with some extensions of the classical Liouville theorem for bounded harmonic functions to solutions of more general equations. We deal with entire solutions of periodic and almost periodic parabolic equations including the elliptic framework as a particular case. We derive a Liouville type result for periodic operators as a consequence of a result for operators periodic in just one variable, which is new even in the elliptic case. More precisely, we show that if and , , c, f are periodic in the same space direction or in time, with the same period, then any bounded solution u of is periodic in that direction or in time. We then derive the following Liouville type result: if , and , , c are periodic in all the space/time variables, with the same periods, then the space of bounded solutions of the above equation has at most dimension one. In the case of the equation , with L periodic elliptic operator independent of t, the hypothesis can be weakened by requiring that the periodic principal eigenvalue of −L is nonnegative. Instead, the periodicity assumption cannot be relaxed, because we explicitly exhibit an almost periodic function b such that the space of bounded solutions of in has dimension 2, and it is generated by the constant solution and a non-almost periodic solution. The above counterexample leads us to consider the following problem: under which conditions are bounded solutions necessarily almost periodic? We show that a sufficient condition in the case of the equation is: f is almost periodic and L is periodic with . Finally, we consider problems in general periodic domains under either Dirichlet or Robin boundary conditions. We prove analogous properties as in the whole space, together with some existence and uniqueness results for entire solutions. [Copyright &y& Elsevier]
- Published
- 2009
- Full Text
- View/download PDF
14. Almost periodic solutions of neutral functional differential equations
- Author
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Abbas, S. and Bahuguna, D.
- Subjects
- *
NUMERICAL solutions to differential equations , *NUMERICAL analysis , *BANACH spaces , *SEMIGROUPS of operators , *EVOLUTION equations - Abstract
Abstract: In this paper we study a non-autonomous neutral functional differential equation in a Banach space. Applying the theory of semigroups of operators to evolution equations and Krasnoselskii’s fixed point theorem we establish the existence and uniqueness of a mild almost periodic solution of the problem under consideration. [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
15. Invariant manifolds, global attractors, almost automorphic and almost periodic solutions of non-autonomous differential equations
- Author
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Cheban, David and Schmalfuss, Bjoern
- Subjects
- *
DIFFERENTIAL equations , *COCYCLES , *HOMOLOGICAL algebra , *BESSEL functions - Abstract
Abstract: The paper is devoted to the study of non-autonomous evolution equations: invariant manifolds, compact global attractors, almost periodic and almost automorphic solutions. We study this problem in the framework of general non-autonomous (cocycle) dynamical systems. First, we prove that under some conditions such systems admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non-autonomous differential equations (both ODEs (in finite and infinite spaces) and PDEs). [Copyright &y& Elsevier]
- Published
- 2008
- Full Text
- View/download PDF
16. Almost periodic solutions for a class of Liénard-type systems with continuously distributed delays
- Author
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Liu, Bingwen
- Subjects
- *
MATHEMATICAL analysis , *COMPUTER simulation , *ELECTROMECHANICAL analogies , *MATHEMATICAL models - Abstract
Abstract: This paper considers the Liénard-type systems with continuously distributed delays. Some sufficient conditions for the existence and exponential stability of the almost periodic solutions are established, which are new and complement previously known results. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
17. On almost periodic solutions of logistic delay differential equations with almost periodic time dependence
- Author
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Yuan, Rong
- Subjects
- *
DELAY differential equations , *FUNCTIONAL differential equations , *EQUATIONS , *CALCULUS - Abstract
Abstract: In this paper, we study almost periodic logistic delay differential equations. The existence and module of almost periodic solutions are investigated. In particular, we extend some results of Seifert in [G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Differential Equations 164 (2000) 451–458]. [Copyright &y& Elsevier]
- Published
- 2007
- Full Text
- View/download PDF
18. Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales
- Author
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Xue Yalong, Xie Xiangdong, and Lin Qifa
- Subjects
almost periodic solutions ,commensal symbiosis model ,lyapunov functional ,michaelis-menten type harvesting ,time scales ,34c25 ,92d25 ,34d20 ,34d40 ,Mathematics ,QA1-939 - Abstract
In this paper, we consider an almost periodic commensal symbiosis model with nonlinear harvesting on time scales. We establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. Our results show that the continuous system and discrete system can be unify well. Examples and their numerical simulations are carried out to illustrate the feasibility of our main results.
- Published
- 2019
- Full Text
- View/download PDF
19. Common asymptotic behavior of solutions and almost periodicity for discontinuous, delayed, and impulsive neural networks
- Author
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Walter Allegretto, Duccio Papini, and Mauro Forti
- Subjects
Periodicity ,HOPFIELD NEURAL NETWORKS ,Time Factors ,Neural Networks ,Computer Networks and Communications ,Models, Neurological ,common asymptotic behavior ,Computer ,Exponential stability ,Artificial Intelligence ,Control theory ,Robustness (computer science) ,Models ,Cellular neural network ,Attractor ,Applied mathematics ,Humans ,discontinuous neural networks ,Mathematics ,Almost periodic function ,Neurons ,Sequence ,Almost periodic functions ,Artificial neural network ,delays ,DISCONTINUOUS ACTIVATIONS ,ALMOST PERIODIC SOLUTIONS ,General Medicine ,global exponential stability ,IMPULSES ,GLOBAL STABILITY ,DELAYS ,Computer Science Applications ,Periodic function ,impulses ,Neural Networks, Computer ,Nonlinear Dynamics ,Neurological ,Software - Abstract
The paper considers a general neural network model with impulses at a given sequence of instants, discontinuous neuron activations, delays, and time-varying data and inputs. It is shown that when the neuron interconnections satisfy an M-matrix condition, or a dominance condition, then the state solutions and the output solutions display a common asymptotic behavior as time t--+infinity. It is also shown, via a new technique based on prolonging the solutions of the delayed neural network to -infinity, that it is possible to select a unique special solution that is globally exponentially stable and can be considered as the unique global attractor for the network. Finally, this paper shows that for almost periodic data and inputs the selected solution is almost periodic; moreover, it is robust with respect to a large class of perturbations of the data. Analogous results also hold for periodic data and inputs. A by-product of the analysis is that a sequence of almost periodic impulses is able to induce in the generic case (nonstationary) almost periodic solutions in an otherwise globally convergent nonimpulsive neural network. To the authors' knowledge the results in this paper are the only available results on global exponential stability of the unique periodic or almost periodic solution for a general neural network model combining three main features, i.e., impulses, discontinuous neuron activations and delays. The results in this paper are compared with several results in the literature dealing with periodicity or almost periodicity of some subclasses of the neural network model here considered and some hints for future work are given.
- Published
- 2010
20. Limit periodic linear difference systems with coefficient matrices from commutative groups
- Author
-
Petr Hasil and Michal Veselý
- Subjects
limit periodicity ,almost periodicity ,almost periodic sequences ,almost periodic solutions ,linear difference equations ,Mathematics ,QA1-939 - Abstract
In this paper, limit periodic and almost periodic homogeneous linear difference systems are studied. The coefficient matrices of the considered systems belong to a given commutative group. We find a condition on the group under which the systems, whose fundamental matrices are not almost periodic, form an everywhere dense subset in the space of all considered systems. The treated problem is discussed for the elements of the coefficient matrices from an arbitrary infinite field with an absolute value. Nevertheless, the presented results are new even for the field of complex numbers.
- Published
- 2014
- Full Text
- View/download PDF
21. Decay estimate of viscosity solutions of nonlinear parabolic PDEs and applications
- Author
-
Silvana Marchi
- Subjects
Nonlinear parabolic equations ,Viscosity solutions ,Almost periodic solutions ,Mathematics ,QA1-939 - Abstract
The aim of this paper is to establish a decay estimate for viscosity solutions of nonlinear PDEs. As an application we prove existence and uniqueness for time almost periodic viscosity solutions.
- Published
- 2014
22. The Existence and Global Exponential Stability of Almost Periodic Solutions for Neutral-Type CNNs on Time Scales
- Author
-
Bing Li, Yongkun Li, and Xiaofang Meng
- Subjects
competitive neural networks ,leakage delays ,almost periodic solutions ,time scales ,Mathematics ,QA1-939 - Abstract
In this paper, neutral-type competitive neural networks with mixed time-varying delays and leakage delays on time scales are proposed. Based on the contraction fixed-point theorem, some sufficient conditions that are independent of the backwards graininess function of the time scale are obtained for the existence and global exponential stability of almost periodic solutions of neural networks under consideration. The results obtained are brand new, indicating that the continuous time and discrete-time conditions of the network share the same dynamic behavior. Finally, two examples are given to illustrate the validity of the results obtained.
- Published
- 2019
- Full Text
- View/download PDF
23. Almost periodic solutions to dynamic equations on time scales
- Author
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Hongtao Zhang and Yongkun Li
- Subjects
Almost periodic functions ,Almost periodic solutions ,Time scales ,Dynamic equations ,Liapunov functionals ,Mathematics ,QA1-939 - Abstract
In this paper, we first present a notion of almost periodic functions on time scales and study their basic properties. Then by means of Liapunov functionals, we study the existence of almost periodic solutions for an almost periodic dynamic equation on time scales.
- Published
- 2013
- Full Text
- View/download PDF
24. Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales
- Author
-
Xiangdong Xie, Yalong Xue, and Qifa Lin
- Subjects
34d40 ,time scales ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,34c25 ,commensal symbiosis model ,Type (model theory) ,92d25 ,Commensalism ,34d20 ,01 natural sciences ,Michaelis–Menten kinetics ,010101 applied mathematics ,QA1-939 ,michaelis-menten type harvesting ,almost periodic solutions ,0101 mathematics ,Mathematics ,lyapunov functional - Abstract
In this paper, we consider an almost periodic commensal symbiosis model with nonlinear harvesting on time scales. We establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. Our results show that the continuous system and discrete system can be unify well. Examples and their numerical simulations are carried out to illustrate the feasibility of our main results.
- Published
- 2019
25. Almost Periodic Solutions of Nonlinear Volterra Difference Equations with Unbounded Delay.
- Author
-
Yoshihiro Hamaya, Tomomi Itokazu, and Kaori Saito
- Subjects
ALMOST periodic functions ,DIFFERENCE equations ,VOLTERRA equations ,ASYMPTOTIC efficiencies ,DIFFERENCE sets - Abstract
In order to obtain the conditions for the existence of periodic and almost periodic solutions of Volterra difference equations, x(n+1) = f(n; x(n))+ Σ
n s=-∞ F(n,s, x(n + s), x(n)), we consider certain stability properties, which are referred to as (K, ρ)-weakly uniformly-asymptotic stability and (K, ρ)-uniformly asymptotic stability. Moreover, we discuss the relationship between the ρ-separation condition and the uniformly-asymptotic stability property in the ρ sense. [ABSTRACT FROM AUTHOR]- Published
- 2015
- Full Text
- View/download PDF
26. Almost Periodicity of All $L^2$-bounded Solutions of a Functional Heat Equation
- Author
-
Qi-Ru Wang and Zhi-Qiang Zhu
- Subjects
Hölder's inequality ,Class (set theory) ,Pure mathematics ,Property (philosophy) ,Differential equation ,Hölder inequality ,General Mathematics ,Poincaré inequality ,35B15 ,symbols.namesake ,35K05 ,Bounded function ,symbols ,functional heat equations ,Heat equation ,almost periodic solutions ,Mathematics - Abstract
In this paper, we continue the investigations done in the literature about the so called Bohr-Neugebauer property for almost periodic differential equations. More specifically, for a class of functional heat equations, we prove that each $L^2$-bounded solution is almost periodic. This extends a result in [5] to the delay case.
- Published
- 2020
27. Uniformly asymptotic stability of almost periodic solutions for a delay difference system of plankton allelopathy
- Author
-
Wang, Qinglong and Liu, Zhijun
- Published
- 2013
- Full Text
- View/download PDF
28. Weak averaging of semilinear stochastic differential equations with almost periodic coefficients
- Author
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Omar Mellah, Mikhail Kamenski, Paul Raynaud de Fitte, Departement of Mathematics (VSU) State University of Voronezh, Departement of Mathematics (VSU), Voronezh State University-Voronezh State University, Laboratoire de Mathématiques Raphaël Salem (LMRS), Université de Rouen Normandie (UNIROUEN), and Normandie Université (NU)-Normandie Université (NU)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Applied Mathematics ,Probability (math.PR) ,Mathematical analysis ,Minor (linear algebra) ,stochastic evolution equations ,Stochastic evolution ,[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] ,Functional Analysis (math.FA) ,Method of averaging ,Mathematics - Functional Analysis ,[MATH.MATH-PR]Mathematics [math]/Probability [math.PR] ,Stochastic partial differential equation ,Stochastic differential equation ,Distribution (mathematics) ,Convergence (routing) ,FOS: Mathematics ,averaging methods ,Wasserstein distance ,almost periodic solutions ,Mathematics - Probability ,Analysis ,Mathematics - Abstract
An averaging result is proved for stochastic evolution equations with highly oscillating coefficients. This result applies in particular to equations with almost periodic coefficients. The convergence to the solution of the averaged equation is obtained in distribution, as in previous works by Khasminskii and Vrko{\v c}.This version corrects two minor errors from our paper published in J. Math. Anal. Appl. 427(1):336--364, 2015.
- Published
- 2015
29. On the existence of almost-periodic solutions for the 2D dissipative Euler equations
- Author
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Luigi C. Berselli and Luca Bisconti
- Subjects
Dini continuous functions ,2D Euler equations ,General Mathematics ,Semi-implicit Euler method ,Mathematical analysis ,Almost-periodic solutions ,Continuous and Dini-continuous functions ,Euler equations ,Transport equation ,Mathematics (all) ,almost periodic solutions ,transport equation ,Vorticity ,Backward Euler method ,Domain (mathematical analysis) ,Modulus of continuity ,symbols.namesake ,Bounded function ,Dissipative system ,symbols ,Mathematics - Abstract
In this paper we study the two-dimensional dissipative Euler equations in a smooth and bounded domain. In the presence of a sufficiently large dissipative term (or equivalently a sufficiently small external force) precise uniform estimates on the modulus of continuity of the vorticity are proved. These allow us to show existence of Stepanov almost-periodic solutions.
- Published
- 2015
30. Four Positive Almost Periodic Solutions To An Impulsive Delayed Plankton Allelopathy System With Multiple Exploit (Or Harvesting) Terms
- Author
-
Fengshuo Zhang and Zhouhong Li
- Subjects
Almost periodic solutions ,impulse ,plankton allelopathy system ,coincidence degree - Abstract
In this paper, we obtain sufficient conditions for the existence of at least four positive almost periodic solutions to an impulsive delayed periodic plankton allelopathy system with multiple exploited (or harvesting) terms. This result is obtained through the use of Mawhins continuation theorem of coincidence degree theory along with some properties relating to inequalities., {"references":["A. Mukhopadhyay, J. Chattopadhyay, P.K. Tapaswi, \"A delay differential\nequations model of plankton allelopathy\", Mathematical Biosciences,,\nVol.149, pp. 167-189,1998.","A.M. Samoilenko, N.A. Perestyuk, \"Impulsive Differential\nEquations\",World Scientific, Singapore, 1995.","B.X. Yang, J.L. Li, An almost periodic solution for an impulsive\ntwo-species logarithmic population model with time -varying delay,\nMathematical and Computer Modelling, Vol.55 n0o.7-8, pp. 1963-1968,\n2012.","C.Y. He, \"Almost Periodic Differential Equations\", Higher Education\nPublishing House, Beijing (in Chinese), 1992.","D. Hu, Z. Zhang, \"Four positive periodic solutions to a Lotka-Volterra\ncooperative system with harvesting terms\", Nonlinear Anal. RWA., Vol.11,\npp. 1560-1571, 2010.","D.S. Wang, \"Four positive periodic solutions of a delayed plankton\nallelopathy system on time scales with multipoe exploited (or harvesting)\nterms\", IMA Journal of Applied mathematics, Vol.78, pp. 449-473, 2013.","E. L. Rice, Alleopathy, second ed., Academic Press, New York, 1984.","G.T. Stamov, I.M. Stamova, J.O. Alzaut, \"Existence of almost periodic\nsolutions for strongly stable nonlinear impulsive differential-difference\nequations\", Nonlinear Analysis: Hybrid Systems, Vol.6 no.2, pp. 818-823,\n2012.","J.B. Geng, Y.H. Xia, \"Almost periodic solutions of a nonlinear ecological\nmodel\", Commun Nonlinear Sci Numer Simulat, Vol.16, pp.2575-2597,\n2011.\n[10] J. Chattopadhyay, \"Effect of toxic substances on a two-species\ncompetitive system\", Ecol. Modelling, Vol.84, pp. 287-289, 1996.\n[11] J. Dhar, K. S. Jatav, \"Mathematical analysis of a delayed stage-structured\npredator-prey model with impulsive diffusion between two predators\nterritories\", Ecological Complexity, Vol.16, pp. 59-67, 2013.\n[12] J.G. Jia, M.S. Wang, M.L. Li, \"Periodic solutions for impulsive delay\ndifferential equations in the control model of plankton allelopathy\",\nChaos, Solitons and Fractals, Vol.32, pp. 962-968, 2007.\n[13] J. Hou, Z.D. Teng, S.J. Gao, \"Permanence and global stability\nfor nonautonomous Nspecies Lotka-Volterra competitive system with\nimpulses\", Nonlinear Anal. RWA., Vol.11 no.3, pp. 1882-1896, 2010.\n[14] J.M.Smith, Modles in Ecology, Cambridge University, Cambridge, 1974. [15] J. ZHEN, Z.E. MA, \"Periodic Solutions for Delay Differential Equations\nModel of Plankton Allelopathy\", Computers and Mathematics with\nApplications , Vol.44, pp. 491-500, 2002.\n[16] K.H. Zhao, Y.K. Li, \"Four positive periodic solutions to two species\nparasitical system with harvesting terms\", Comput. Math. with Appl.,\nVol.59 no.8, pp. 2703-2710, 2010.\n[17] K.H. Zhao, Y. Ye, \"Four positive periodic solutions to a periodic\nLotka-Volterra predatoryprey system with harvesting terms\", Nonlinear\nAnal. RWA., Vol.11, pp.2448-2455, 2010.\n[18] L. Yang, S.M. Zhong, \"Dynamics of a delayed stage-structured model\nwith impulsive harvesting and diffusion\", Ecological Complexity, Vol.19,\npp. 111-123, 2014.\n[19] M.X. He, F.D. Chen, Z. Li, \"Almost periodic solution of an impulsive\ndifferential equation model of plankton allelopathy\", Nonlinear Analysis:\nReal World Applications,, Vol.11, pp. 2296-2301, 2010.\n[20] M. Zhao, X.T. Wang, H.G.Yu, J. Zhu, \"Dynamics of an ecological model\nwith impulsive control strategy and distributed time delay\", Mathematics\nand Computers in Simulation, Vol.82 no.8, pp. 1432-1444, 2012.\n[21] Q. Wang, Y.Y. Fang, D.C. Lu, \"Existence of four periodic solutions\nfor a generalized delayed ratio-dependent predator-prey system\", Applied\nMathematics and Computation, Vol.247, pp. 623-630 ,2014.\n[22] R. Gaines, J. Mawhin, Coincidence Degree and Nonlinear Differetial\nEquitions, Springer Verlag, Berlin, 1977.\n[23] S.Y. Tang, L.S. Chen, \"The periodic predator-prey Lotka-Volterra model\nwith impulsive effect\", J. Mech. Med. Biol., Vol.2, pp. 1-30, 2002.\n[24] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive\nDifferential Equations, World Scientific, Singapore, 1989.\n[25] X.H. Wang, J.W. Jia, \"Dynamic of a delayed predator-prey model with\nbirth pulse and impulsive harvesting in a polluted environment\", Physica\nA: Statistical Mechanics and its Applications, Vol.422, pp. 1-15, 2015.\n[26] X.Y. Song and L.S. Chen, \"Periodic solution of a delay differential\nequation of plankton allelopathy\", Acta Math. Sci. Ser. A, Vol.23, pp.\n8-13, 2003.\n[27] Y.K. Li, K.H. Zhao, \"2n positive periodic solutions to n species\nnon-autonomous Lotka-Volterra unidirectional food chains with\nharvesting terms\", Math. Model. Anal., Vol.15, pp. 313-326, 2010.\n[28] Y.K. Li, K.H. Zhao, \"Eight positive periodic solutions to three species\nnon-autonomous Lotka-Volterra cooperative systems with harvesting\nterms\", Topol. Methods Nonlinear Anal., Vol.37, pp. 225-234, 2011.\n[29] Y.K. Li, K.H. Zhao, \"Multiple positive periodic solutions to m-layer\nperiodic Lotka-Volterra network-like multidirectional food-chain with\nharvesting terms\", Anal. Appl., Vol.9, pp. 71-96, 2011.\n[30] Y.K. Li, K.H. Zhao, Y. Ye, \"Multiple positive periodic solutions of\nn species delay competition systems with harvesting terms\", Nonlinear\nAnal. RWA., Vol.12, pp. 1013-1022, 2011.\n[31] Y.K. Li, \"Positive periodic solutions of a periodic neutral delay\nlogistic equation with impulses\", Comput. Math. Appl., Vol.56 no.9, pp.\n2189-2196, 2008.\n[32] Y.K. Li, Y. Ye, \"Multiple positive almost periodic solutions to an\nimpulsive non-autonomous Lotka-Volterra predator-prey system with\nharvesting terms\", Commun. Nonlinear Sci. Numer. Simul., Vol.18 no.11,\npp. 3190-3201, 2013.\n[33] Y. Xie, X.G. Li, \"Almost periodic solutions of single population model\nwith hereditary\", Appl. Math. Comput., Vol.203, pp. 690-697, 2008.\n[34] Z.H. Li, K.H. Zhao, Y.K. Li, \"Multiple positive periodic solutions for a\nnon-autonomous stage-structured predatory-prey system with harvesting\nterms\", Commun. Nonlinear Sci. Numer. Simul., Vol.15, pp. 2140-2148,\n2010.\n[35] Z.J. Du, M. 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- 2017
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31. Almost periodic solutions to dynamic equations on time scales
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Yongkun Li and Hongtao Zhang
- Subjects
Almost periodic function ,Almost periodic functions ,Almost periodic solutions ,Dynamic equations ,lcsh:Mathematics ,Liapunov functionals ,Mathematical analysis ,Periodic sequence ,Time scales ,lcsh:QA1-939 ,Dynamic equation ,Mathematics - Abstract
In this paper, we first present a notion of almost periodic functions on time scales and study their basic properties. Then by means of Liapunov functionals, we study the existence of almost periodic solutions for an almost periodic dynamic equation on time scales.
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- 2013
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32. Convergence in Almost Periodic Cooperative Systems with a First Integral
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Shen, Wenxian and Zhao, Xiao-Qiang
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- 2005
33. Almost periodic solutions of neutral functional differential equations
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Dhirendra Bahuguna and Syed Abbas
- Subjects
Equilibrium point ,Almost periodic solutions ,Picard–Lindelöf theorem ,Differential equation ,Semigroup ,Neutral differential equation ,Semigroup of linear operators ,Mathematical analysis ,Banach space ,Fixed-point theorem ,Computational Mathematics ,Computational Theory and Mathematics ,Modelling and Simulation ,Modeling and Simulation ,Initial value problem ,C0-semigroup ,C-semigroup ,Exponentially stable semigroup ,Mathematics - Abstract
In this paper we study a non-autonomous neutral functional differential equation in a Banach space. Applying the theory of semigroups of operators to evolution equations and Krasnoselskii’s fixed point theorem we establish the existence and uniqueness of a mild almost periodic solution of the problem under consideration.
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- 2008
34. On almost periodic solutions of logistic delay differential equations with almost periodic time dependence
- Author
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Rong Yuan
- Subjects
Delay ,Almost periodic solutions ,Differential equation ,Logistic equations ,Modulo ,Applied Mathematics ,Mathematical analysis ,Module containment ,Periodic sequence ,Delay differential equation ,Piecewise ,Logistic function ,Constant (mathematics) ,Analysis ,Mathematics - Abstract
In this paper, we study almost periodic logistic delay differential equations. The existence and module of almost periodic solutions are investigated. In particular, we extend some results of Seifert in [G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Differential Equations 164 (2000) 451–458].
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- 2007
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35. Extremal Viscosity Solutions of Almost Periodic Hamilton-Jacobi Equations
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Belloni, Marino and Marchi, Silvana
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35D40 ,Pseudo almost periodic solutions ,viscosity solutions ,Asymptotically almost periodic solutions ,almost periodic solutions ,Hamilton-Jacobi equations ,35B15 ,35F21 - Abstract
This paper deals with viscosity solutions of Hamilton-Jacobi equations in which the Hamiltonian $H$ is weakly monotone with respect to the zero order term: this leads to non-uniqueness of solutions, even in the class of periodic or almost periodic (briefly a.p.) functions. The lack of uniqueness of a.p. solutions leads to introduce the notion of minimal (maximal) a.p. solution and to study its properties. The classes of asymptotically almost periodic (briefly a.a.p.) and pseudo almost periodic (briefly p.a.p.) functions are also considered.
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- 2015
36. Existence of almost periodic solutions of discrete time equations
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Denis Pennequin, CEntre de Recherche en Mathématiques, Statistique et Économie Mathématique (CERMSEM), and Université Paris 1 Panthéon-Sorbonne (UP1)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
Discrete Time Equations ,Dynamical systems theory ,Differential equation ,[MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS] ,Harmonic (mathematics) ,[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA] ,01 natural sciences ,symbols.namesake ,Nonlinear functional analysis ,Discrete Mathematics and Combinatorics ,almost periodic solutions ,0101 mathematics ,Newton's method ,Mathematics ,bounded solutions ,Applied Mathematics ,39A10, 43A60, 93C55 ,010102 general mathematics ,Linear system ,Mathematical analysis ,Lipschitz continuity ,010101 applied mathematics ,Discrete time and continuous time ,symbols ,Analysis - Abstract
International audience; In this paper, we study almost periodic (a.p.) solutions of discrete dynamical systems. We first adapt some results on a.p. differential equations to a.p. difference equations, on the link between boundedness of solutions and existence of a.p. solutions. After, we obtain an existence result in the space of the Harmonic Synthesis for an equation $A_t (x_t,...,x_{t+p})=0$ when the dependance of $A$ on $t$ is a.p. and when $A_t$ and $D A_t$ are uniformly Lipschitz and satisfy another condition which is exactly the extension of a simple one for the basic linear system. The main tools for that are Nonlinear Functional Analysis and the Newton method.
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- 2001
37. Existence of $AP_{r}$-Almost Periodic Solutions For Some Classes of Functional Differential Equations
- Author
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Corduneanu, C. and Mahdavi, M.
- Subjects
34K05 ,34K13 ,34K40 ,Functional equations ,45B05 ,almost periodic solutions ,existence and uniqueness - Abstract
This paper presents a couple of existence results, related to the classes of functional equations of the form $x+k\ast x=f$, or $\frac{d}{dt}[\dot{x}+k\ast x]=f$, with $f, x\in AP_r(R, \,{\mathcal{C}})=$ the space of almost periodic functions defined by \[ AP_r(R,\, {\mathcal{C}})=\left\{f : f\simeq \sum_{j=1}^{\infty} f_j\,e^{i\lambda_j t},\,f_j\in {\mathcal{C}},\lambda_j\in R,\sum_{j=1}^{\infty}|f_j|^r \lt \infty\right\}, \] the norm being given by $|f|_r= \left(\sum_{j=1}^{\infty}|f_j|^r\right)^{\frac{1}{r}}$, for each $r\in [1, 2]$. The convolution product $k\ast x$, $k\in L^1(R,\, {\mathcal{C}})$, $x\in AP_r(R,\, {\mathcal{C}})$ is defined by \[ (k\ast x)(t)= \sum_{j=1}^{\infty} x_j\left( \int_R k(s)\,e^{\lambda_j\,s}\,ds\right)\,e^{i\lambda_j\,t}, \] where $x(t)\simeq \sum_{j=1}^{\infty} x_j\,e^{i\lambda_j\,t}$.
- Published
- 2013
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