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Overf the last few years, by utilizing Mawhin’s continuation theorem of coincidence degree theory and Lyapunov functional, many scholars have been concerned with the global asymptotical stability of positive periodic solutions for the non-linear ecosystems. In the real world, almost periodicity is usually more realistic and more general than periodicity, but there are scarcely any papers concerning the issue of the global asymptotical stability of positive almost periodic solutions of non-linear ecosystems. In this paper we consider a kind of delayed two-species competitive model with stage structure. By means of Mawhin’s continuation theorem of coincidence degree theory, some sufficient conditions are obtained for the existence of at least one positive almost periodic solutions for the above model with nonnegative coefficients. Furthermore, the global asymptotical stability of positive almost periodic solution of the model is also studied. The work of this paper extends and improves some results in recent years. An example and simulations are employed to illustrate the main results of this paper.

In this paper, a dynamic model of Middle East Respiratory Syndrome Coronavirus (MERS-CoV) with periodic coefficients is proposed and studied. By using the continuation theorem of the coincidence degree theory, we obtain some sufficient conditions for the existence of positive periodic solutions of the model. The periodic model degenerates to an autonomous case, and our conditions can be degenerated to the basic reproductive number $ R_0 > 1 $. Finally, we give some numerical simulations to illustrate our main theoretical results.

In this paper, we establish a non-autonomous Hassell-Varley-Holling type predator-prey system with mutual interference. We construct some sufficient conditions for the permanence, extinction and globally asymptotic stability of system by use of the comparison theorem and an appropriate Liapunov function. Then the sufficient and necessary conditions for a periodic solution of the system are obtained via coincidence degree theorem. Finally, the correctness of the previous conclusions are demonstrated by some numerical cases.

The existence of a solution for a second-order p-Laplacian boundary value problem at resonance with two dimensional kernel will be considered in this paper. A semi-projector, the Ge and Ren extension of Mawhin’s coincidence degree theory, and algebraic processes will be used to establish existence results, while an example will be given to validate our result.

In this paper, we are concerned with a class of quaternion-valued cellular neural networks with time-varying transmission delays and leakage delays. By applying a continuation theorem of coincidence degree theory and the Wirtinger inequality as well as constructing a suitable Lyapunov functional, sufficient conditions are derived to ensure the existence and global exponential stability of anti-periodic solutions via direct approaches. Our results are completely new. Finally, numerical examples are also provided to show the effectiveness of our results.

In this paper, we investigate the dynamics of a class of the so-called semi-ratio-dependent predator–prey interaction models with functional responses based on systems of nonautonomous differential equations with time-dependent parameters. The functional responses are classified into five types and typical examples of each type are provided. Then we establish sufficient criteria for the boundedness of solutions, the permanence of system, and the existence, uniqueness and globally asymptotic stability of positive periodic solution and positive almost periodic solution. Some conclusive discussion is presented at the end of this paper.

By using the coincidence degree theorem, we obtain a new result on the existence of solutions for a class of fractional differential equations with periodic boundary value conditions, where a certain nonlinear growth condition of the nonlinearity needs to be satisfied. Furthermore, we study another class of differential equations of fractional order with periodic boundary conditions at resonance. A new result on the existence of positive solutions is presented by use of a Leggett–Williams norm-type theorem for coincidences. Two examples are given to illustrate the main result at the end of this paper.

In this paper, by using Mawhin-s continuation theorem of coincidence degree and a method based on delay differential inequality, some sufficient conditions are obtained for the existence and global exponential stability of periodic solutions of cellular neural networks with distributed delays and impulses on time scales. The results of this paper generalized previously known results., {"references":["J. Cao, New results concerning exponential stability and periodic solutions\nof delayed cellular neural networks with delays, Phys Lett A 307 (2003)\n136-147.","Y. Li, S. Gao, Global Exponential Stability for Impulsive BAM Neural\nNetworks with Distributed Delays on Time Scales, Neural Processing\nLetters 31 (1) (2010) 65-91.","Z. Liu and A. Chen, Existence and global exponential stability of periodic\nsolutions of cellular neural networks with time-vary delays, J. Math Anal.\nAppl. 290 (2004) 247-262.","Y. Li, Existence and stability of periodic solutions for Cohen¨CGrossberg\nneural networks with multiple delays. Chaos, Solitons & Fractals 20\n(2004) 459-466.","Y. Li, P. Liu, Existence and stability of positive periodic solution for BAM\nneural networks with delays, Math Comput Model 40 (2004) 757-770.","Y. Li, C. Liu, L. Zhu, Global exponential stability of periodic solution for\nshunting inhibitory CNNs with delays. Phys Lett A 337 (2005) 46-54.","Y. Li, T. Zhang, Global exponential stability of fuzzy internal delayed\nneural networks with impulses on time scales, Int. J. Neural Syst.19 (6)\n(2009) 449-456.","Y. Chen, Global stability of neural networks with distributed delays,\nNeural Networks 15 (2002) 867-871.","C. Feng, R. Plamondon, On the stability analysis of delayed neural\nnetworks systems, Neural Networks 14 (2011) 1181-1188.\n[10] J. Zhang, Globally exponential stability of neural networks with variables\ndelays. IEEE Trans Circuit Syst-I 50 (2003) 288-291.\n[11] Y. Li, Global exponential stability of BAM neural networks with delays\nand impulses, Chaos, Solitons & Fractals 24 (2005) 279-285.\n[12] Y. Ren, Y. Li, Periodic solutions of recurrent neural networks with\ndistributed delays and impulses on time scales, Int. J. Comp. Math. Sci.\n5 (2011) 209-218.\n[13] Z. Guan, L. James, G. Chen, On impulsive auto-associative neural\nnetworks, Neural Networks 13 (2000) 63-69.\n[14] Y. Li, W. Xing, L. Lu, Existence and global exponential stability of\nperiodic solution of a class of neural networks with impulses, Chaos,\nSolitons & Fractals 27 (2006) 437-445.\n[15] Y. Li, W. Xing, Existence and global exponential stability of periodic\nsolution of CNNs with impulses, Chaos, Solitons & Fractals 33 (2007)\n1686-1693.\n[16] R. Gains, J. Mawhin, Coincidence degree and nonlinear differential\nequations, Springer-Verlag, Berlin, 1977.\n[17] H. Tokumarn, N. Adachi, T. Amemiya, Macroscopic stability of interconnected\nsystems, 6th IFAC Congress, paper ID44.4, 1975, 1-7."]}

In this article, we present a new method for converting the boundary value problems for impulsive fractional differential systems involved with the Riemann-Liouville type derivatives to integral systems, some existence results for solutions of a class of boundary value problems for nonlinear impulsive fractional differential systems at resonance case and non-resonance case are established respectively. Our analysis relies on the well known Schauder’s fixed point theorem and coincidence degree theory. Examples are given to illustrate main results. This paper is motivated by [Solvability of multi-point boundary value problem of nonlinear impulsive fractional differential equation at resonance, Electron. J. Qual. Theory Differ. Equ. 89(2011), 1-19], [Existence result for boundary value problem of nonlinear impulsive fractional differential equation at resonance, J, Appl, Math, Comput. 39(2012) 421-443] and [Solvability for a coupled system of fractional differential equations with impulses at resonance, Bound. Value Probl. 2013, 2013: 80].

We prove that the superlinear indefinite equation \begin{equation*} u" + a(t)u^{p} = 0, \end{equation*} where $p > 1$ and $a(t)$ is a $T$-periodic sign-changing function satisfying the (sharp) mean value condition $\int_{0}^{T} a(t)~\!dt < 0$, has positive subharmonic solutions of order $k$ for any large integer $k$, thus providing a further contribution to a problem raised by G. J. Butler in its pioneering paper (JDE, 1976). The proof, which applies to a larger class of indefinite equations, combines coincidence degree theory (yielding a positive harmonic solution) with the Poincar��-Birkhoff fixed point theorem (giving subharmonic solutions oscillating around it)., 26 pages

In this paper we focus on the periodic boundary value problem associated with the Liénard differential equation x ′′ + f ⁢ ( x ) ⁢ x ′ + g ⁢ ( t , x ) = s {x^{\prime\prime}+f(x)x^{\prime}+g(t,x)=s} , where s is a real parameter, f and g are continuous functions and g is T-periodic in the variable t. The classical framework of Fabry, Mawhin and Nkashama, related to the Ambrosetti–Prodi periodic problem, is modified to include conditions without uniformity, in order to achieve the same multiplicity result under local coercivity conditions on g. Analogous results are also obtained for Neumann boundary conditions.

In this paper, by using a continuation theorem of coincidence degree theory and a differential inequality, we establish some sufficient conditions ensuring the existence and global exponential stability of anti-periodic solutions for a class of fuzzy Cohen–Grossberg neural networks with time-varying and distributed delays. In addition, we present an illustrative example to show the feasibility of obtained results.

Made available in DSpace on 2018-12-11T17:13:45Z (GMT). No. of bitstreams: 0 Previous issue date: 2017-08-01 Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) In this paper, some new results on the existence and uniqueness of anti-periodic solution for a first-order impulsive functional differential equation with delay are obtained. The main tool to be employed here is the coincidence degree theory. Departamento de Matemática Instituto de Geociências e Ciências Exatas UNESP - Univ Estadual Paulista, Avenida 24-A, 1515, Caixa Postal 178 Departamento de Análise Matemática Instituto de Matemática e Estatística Universidade do Estado do Rio de Janeiro Departamento de Matemática Instituto de Geociências e Ciências Exatas UNESP - Univ Estadual Paulista, Avenida 24-A, 1515, Caixa Postal 178 FAPESP: 2014/14607-7

This paper gives an application of Mawhin’s coincidence degree and matrix spectral theory to a predator-prey model with M-predators and N-preys. The method is different from that used in the previous work. Some new sufficient conditions are obtained for the existence and global asymptotic stability of the periodic solution. The existence and stability conditions are given in terms of spectral radius of explicit matrices which are much different from the conditions given by the algebraic inequalities. Finally, an example is given to show the feasibility of our results.

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. Xu, \"Positive periodic solutions of n-species neutral delayed\nLotka-Volterra competition system with impulsive perturbations\", Applied\nMathematics and Computation, Vol.243, pp. 379-391, 2014.\n[36] Z.J. Du, Y.S. Lv, \"Permanence and almost periodic solution of a\nLotka-Volterra model with mutual interference and time\", Applied\nMathematical Modelling, Vol.37 no.3, pp. 1054-1068, 2013.\n[37] Z.J. Liu, J.H. Wu, Y.P. Chen, M. Haque, \"Impulsive perturbations in\na periodic delay differential equation model of plankton allelopathy\",\nNonlinear Analysis: Real World Applications, Vol.11, pp. 432-445, 2010.\n[38] Z.J. Liu, L.S. Chen, \"Positive periodic solution of a general discrete\nnon-autonomous difference system of plankton allelopathy with delays\",\nJournal of Computational and Applied Mathematics, Vol.197, pp.\n446-456,2006.\n[39] Z.L. He, L.F. Nie, Z.D. Teng, \"Dynamics analysis of a two-species\ncompetitive model with state-dependent impulsive effects\", Journal of\nthe Franklin Institute, Vol.352 no.5, pp. 2090-2112, 2015.\n[40] Z. Li, M.A. Han, F.D. Chen, \"Almost periodic solutions of a discrete\nalmost periodic logistic equation with delay\", Applied Mathematics and\nComputation, Vol.232, pp. 743-751, 2014.\n[41] Z.Q. Zhang, Z. Hou, \"Existence of four positive periodic solutions\nfor a ratio-dependent predator-prey system with multiple exploited (or\nharvesting) terms\", Nonlinear Anal. RWA., Vol.11, pp. 1560-1571, 2010.\n[42] Z. Zhang, T. Tian, \"Multiple positive periodic solutions for a generalized\npredator-prey system with exploited terms\", Nonlinear Anal. RWA., Vol.9,\npp. 26-39, 2008."]}

In this paper we study the periodic boundary value problem associated with a first order ODE of the form x' + g(t, x) = s where s is a real parameter and g is a continuous function, T-periodic in the variable t. We prove an Ambrosetti-Prodi type result in which the classical uniformity condition on g(t, x) at infinity is considerably relaxed. The Carathéodory case is also discussed.

In this paper, we will consider a discrete non-autonomous predator-prey system with nonconstant death rate. We give sufficient conditions in order to get a dissipative and permanent system. By using the continuation theorem based on Gaines and Mawhin's coincidence degree, we study the existence of positive periodic solutions when the coecients of system are periodic. A numerical example is given to illustrate the effectiveness of the main result. En este artículo consideramos un sistema depredador-presa discreto no autónomo con tasa de mortalidad no constante. Damos condiciones sucientes para que el sistema sea disipativo y permanente. Usando el Teorema de Continuación basado en el grado de coincidencia de Gaines y Mawhin, estudiamos la existencia de soluciones periódicas positivas cuando los coecientes del sistema son periódicos. Un ejemplo numérico es dado para ilustrar la efectividad del resultado principal.

This paper is concerned with the periodic solutions for a class of new Nicholson-type delay system with nonlinear density-dependent mortality terms. By using coincidence degree theory, some criteria are obtained to guarantee the existence of positive periodic solutions of the model. Moreover, an example is employed to illustrate the main results.

By means of a fixed point theorem of coincidence degree theory, sufficient conditions are established for the existence of a positive almost periodic solution to a kind of delayed predator–prey model with Hassell-Varley type functional response. The method used in this paper offers a possible means to study the existence of positive almost periodic solutions to the models in biological populations. Finally, an example as well as numerical simulations are given to illustrate the feasibility and effectiveness of our results.

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 \begin{eqnarray*} \left\{\begin{array}{lll} \frac{{\mathrm{d}} u_i(t)}{{\mathrm{d}}t}&=&u_i(t)\bigg [a_i(t)-\sum\limits_{ l=1}^{n}a_{il}(t)u_l(t-\sigma_{il}(t))-\sum\limits_{j=1}^{m}b_{ij}(t)v_j(t-\tau_{ij}(t))\bigg],i=1,\ldots,n,\\ \frac{{\mathrm{d}} v_j(t)}{{\mathrm{d}} t}&=&v_j(t)\bigg[-r_j(t)+\sum\limits_{l=1}^{n}d_{jl}(t)u_l(t-\delta_{jl}(t))-\sum\limits_{h=1}^{m}e_{jh}(t)v_h(t-\theta_{jh}(t))\bigg ], j=1,\ldots,m, \end{array}\right. \end{eqnarray*} where $a_i,r_j, a_{il}, b_{ij},d_{jl},e_{jh}\in C(\mathbb{R},(0,\infty)\sigma_{il},\tau_{ij},\delta_{jl},\theta_{jh}\in C(\mathbb{R},\mathbb{R})(i,l=1,\ldots,n, j,h=1,\ldots,m)$ are almost periodic functions.

By using Mawhin's coincidence degree theory, this paper establishes some sufficient conditions on the existence of four positive periodic solutions for a food-limited two-species ratio-dependent predator-prey patch system with delay and harvesting. Some novel techniques are employed to obtain the appropriate a priori estimates. An example is given to illustrate our results.

This paper is concerned with a class of Nicholson's blowflies models with a nonlinear density-dependent mortality term. We use coincidence degree theory and give several sufficient conditions which guarantee the existence of positive periodic solutions of the model. Moreover, we give an example to illustrate our main results.

In this paper, we study periodic oscillation of seasonally forced epidemiological models with impulse vaccination where periodicity occurs in contact rate. Using the famous Mawhin's coincidence degree method, we get the existence of positive periodic solutions of seasonally forced SIR models with impulse vaccination at fixed time. Some numerical simulations are presented to illustrate the effectiveness of such pulse vaccination strategy.

In this paper, the following fractional ordinary differential equation boundary value problem: D0+αu(t)=f(t,u(t),D0+α−1u(t))+e(t),0

In this paper, a modified delayed predator–prey system with stage structure for predator is proposed and studied, which generalizes and incorporates as special cases some known models. With the help of continuation theorem based on the Gaines and Mawhin coincidence degree theory, we investigate the existence of periodic solutions for the proposed delayed predator–prey system with stage-structured predator and two kinds of functional responses (called Holling III functional response and Beddington–DeAngelis functional response) on time scales. In particular, when the time scale is chosen as the set of the real numbers or the integers, the existence of the periodic solutions of the corresponding continuous-time and discrete-time models follows.

This paper is concerned with a class of Nicholson's blowflies model with a linear harvesting term. By applying the method of coincidence degree, some criteria are established for the existence and uniqueness of positive periodic solutions of the model. Moreover, an example is employed to illustrate the main results.

In this paper, some new Banach spaces be introduced. Based on those new Banach spaces and by using the coincidence degree theory, we present two existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions at resonance case.

In this paper, the stability and existence of periodic solutions for an unidirectionally coupled nonlinear system with delays are investigated by combining the linear stability theory and the embedding technique of asymptotically autonomous semiflows, Hopf bifurcation theory and a contin- uation theorem of coincidence degree theory, respectively. Some numerical simulations are carried out for illustrating the analytical results.

In this paper, a delay model of plankton allelopathy is investigated. By using the coincidence degree theory, sufficient conditions for existence of periodic solutions are obtained. The presented criteria improve and extend previous results in the literature.

In this paper, the existence and global exponential stability of periodic solutions for a class of numerical discretization neural networks are considered. Using coincidence degree theory and the Lyapunov method, sufficient conditions for the existence and global exponential stability of periodic solutions are obtained. A numerical simulation is given to illustrate the results.

In this paper, we rigorously establish an existence theorem of periodic solutions for the competition of Lotka–Volterra dynamic systems with a time delay and diffusion on time scales. It is shown that the existence of periodic solutions depend on the parameters of the model. It is also shown that a known result in the literature can can carry over quite easily to its discrete counterpart, and a much more accurate result can be obtained when studying the dynamic system on time scales. Moreover, one example is given to illustrate the result obtained.

In this paper, the Lienard type p-Laplacian equation with two deviating arguments (@f"p(x^'(t)))^'+f(x(t))x^'(t)+g"1(t,x(t-@t"1(t)))+g"2(t,x(t-@t"2(t)))=e(t) is studied. By applying the coincidence degree theory, we obtain some new results on the existence of periodic solutions to this equation. Our results improve and extend some existing ones in the literature.

Using inequality techniques and coincidence degree theory, new results are provided concerning the existence and uniqueness of T-periodic solutions for a Liénard equations with delay. An illustrative example is provided to demonstrate that the results in this paper hold under weaker conditions than existing results, and are more effective.

In this paper, we consider a kind of Rayleigh equation with two deviating arguments of the form x^''+f(t,x^'(t))+g"1(t,x([email protected]"1(t)))+g"2(t,x([email protected]"2(t)))=p(t). By using the coincidence degree theory, we establish new results on the existence and uniqueness of periodic solutions for the above equation.

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Liénard type p-Laplacian equation with a deviating argument of the form:(ϕp(x′(t)))′+f(x(t))x′(t)+g(t,x(t-τ(t)))=e(t).

In this paper, we derive a sufficient condition as well as a necessary condition for existence of positive periodic solutions of the hematopoiesis model with periodic coefficients x^'(t)[email protected](t)x(t)+p(t)@?x([email protected](t))1+[x([email protected](t))]^n,t>=0, where @d,[email protected]?C(R^+,(0,~)) and @[email protected]?C(R^+,R^+) are T-periodic functions. When p(t)[email protected]@d(t) with @c>0, a sufficient and necessary condition for the existence of a positive T-periodic solution is established.

In this paper we study the existence of ω-periodic solutions for some functional–differential equations of mixed type. Among the main results are the averaging principle and existence theorems for some equations with homogeneous nonlinearities. We use here the coincidence degree theory of Mawhin.

In this paper, by using the Mawhin's continuation theorem, we prove the existence of periodic solutions for a stage-structure ecological model on time scales. This unifies the results for differential and difference equations.

In this paper, we consider quasilinear fractional differential equation with resonance boundary condition. After translating the quasilinear equation into the linear fractional differential system, by using coincidence degree theory, the existence result is established.

In this paper, by applying the continuation theorem of coincidence degree theory, we establish some new criteria for the existence of multiple positive periodic solutions for the delayed predator-prey model.x^'(t)=x(t)(r(t)-a(t)x(t))-b(t)f(x(t))y(t),y^'(t)=y(t)(c(t)f(x([email protected]))-d(t)), when functional response function f is monotonic or nonmonotonic.

In this paper, we use the coincidence degree theory to establish new results on the existence of $T$-periodic solutions for the Rayleigh equation with two deviating arguments of the form $$ x''+f(x(t), x'(t))+g_{1}(t,x(t-au_{1}(t))) +g_{2}(t,x(t-au_{2}(t)))=p(t). $$

In this paper, we use the coincidence degree theory to establish the existence and uniqueness of T-periodic solutions for the first-order neutral functional differential equation, with two deviating arguments, $$ (x(t)+Bx(t-delta))'= g_{1}(t,x(t-au_{1}(t))) +g_{2}(t,x(t-au_{2}(t))) +p(t). $$

In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Rayleigh equation with a deviating argument of the form x ″ + f ( x ′ ( t ) ) + g ( t , x ( t − τ ( t ) ) ) = p ( t ) .

In this paper, we employ the Mawhin's continuation theorem to study the existence of positive periodic solutions of the nonautonomous periodic model of population with continuous and discrete time. It is interesting that the conditions to guarantee the existence of positive periodic solutions of discrete time model are similar to those for the corresponding continuous time model.

In this paper, the existence of positive periodic solutions of a class of periodic Lotka-Volterra type impulsive systems with distributed delays is studied. By using the continuation theorem of coincidence degree theory, a set of easily verifiable sufficient conditions are obtained, which improve and generalize some existing results.

In this paper, we consider the following delayed periodic differential equation,x′(t)+V(t)x(t)xn(t−τ(t))θ(t)+xn(t−τ(t))=τ(t),which describes a ‘dynamic disease’ involving respiratory disorders. Using the method of coincidence degree, we first show the existence of a positive periodic solution. This result greatly extends the existing ones. Then, several sufficient conditions for the global attractivity of this periodic solution are established. These results are independent of each other and, in general, are much better than the existing ones.

In this paper, we give existence and uniqueness results for solutions of nonlocal boundaryvector value problems of the formx^(^n^)(t)=f(t,x(t),x^'(t),...,x^(^n^-^1^)(t)),t@?[0,1],x(0)=x^'(0)=...=x^(^n^-^2^)(0)=0,x^(^n^-^1^)(1)=@!"0^1[dg(s)]x^(^n^-^1^)(s), where n2, @? : [0, 1] x (R^N^"^1)^n -> R^N^"^1 is a Caratheodory function, g : [0, l] -> R^N^"^1 x R^N^"^1 is a Lebesgue measurable N"1 x N"1-matrix function and it satisfies g(0) = 0, the integral is in sense of Riemann-Stieltjes. The existence of a solutions is proven by the coincidence degree theory. As an application, we also give one example to demonstrate our results.

In this paper we study the existence of positive periodic solutions for a class of neutral type competitive system with delay. Applying the continuation theorem of coincidence degree theory, a general criteria on the existence of positive periodic solutions are established. Some the related results are generalized and improved. It is shown that the arguments delays have no effect on the existence positive periodic solution of system.

In this paper, by using Mawhin's continuation theorem of coincidence degree theory, we obtain some sufficient conditions for the existence of homoclinic solutions for a class of nonlinear second-order differential equations with time-varying delays. Moreover, we give an example to illustrate the feasibility of obtained results. Our results are completely new.

In this paper, we investigate a generalized n -species Gilpin-Ayala competition system with several deviating arguments in periodic environment, which is more general and more realistic than the classical Lotka-Volterra competition systems. By using the method of coincidence degree, a set of easily verifiable sufficient conditions are derived for the existence of at least one strictly positive (componentwise) periodic solution. Some new results are obtained. As applications, we also apply our main results to some special cases of the system we consider here, including the classical n -species Lotka-Volterra competition systems and n -species Gilpin-Ayala competition model, which have been studied extensively in the literature. Some known results are improved and generalized. The examples show that our criteria are new, general, and easily verifiable.