 Open Journal of Applied Sciences, 2013, 3, 62-64 doi:10.4236/ojapps.2013.31B1012 Published Online April 2013 (http://www.scirp.org/journal/ojapps) Periodic Solution of Impulsive Lotka-Volterra Recurrent Neural Networks with Delays Yan Yan, Kaihua Wang, Zhanji Gui* School of Mathematics and Statistics, Hainan Normal University, Haikou, Hainan, China Email: *zhanjigui@sohu.com Received 2013 ABSTRACT In this paper, periodic solution of impulsive Lotka-Volterra recurrent neural networks with delays is studied. Using the continuation theorem of coincidence degree theory and analysis techniques, we establish criteria for the existence of periodic solution of impulsive Lotka-Volterra recurrent neural networks with delays. Keywords: Lotka-Volterra; Delays; Periodic Solution; Impulsive 1. Introduction In recent years, applications of theory differential equa-tions in mathematical ecology have been developed rap-idly. Various mathematical models have been proposed in the study of population dynamics. The Lotka-Volterra competition system is the most famous models for dy-namics of population. Owing to its theoretical and prac-tical significance, the Lotka-Volterra systems have been studied extensively [1,2]. The Lotka-Volterra type neural networks, derived from conventional membrane dynam-ics of competing neurons, provide a mathematical basis for understanding neural selection mechanisms. Recently, periodic solutions of impulsive Lotka-Volterra recurrent neural networks have be en reported. It is well known that delays are important phenomenon in neural networks . Thus, studying the dynamic prop-erties of neural networks with delays has interesting im-plications in both theory and applications [4-7]. In this paper, we will study the following impulsive Lotka-Volterra recurrent neural networks system with delays: 1()()[ ()()()()( )(( ))],,()(),1,2,,1,2, ,,niiii ijjjii ikik ikikxtxthtxta txtctxttttxtbxti nkn  (1) where each ()ixt denotes the activity of neuron, ()ijn nAa is real matrices, each of their elements denotes a synaptic weight and represents the strength of the synaptic connection from neuron j to neuron i, denotes ex-ternal inputs. The variable delays nn()iht()it for are nonnegative continuous functions satisfying 1, 2,,in0()it for , where 0t0 is a constant. i, , i are all positive periodic continuous functions with period . ()ht ()ijat(ct)T02. Existence of Positive Periodic Solutions Lemma 1  Let X and Y be two Banach spaces. Con-sider an operator equation Lx Nx where :DomLLX[0,1] Y is a Fredholm operator of index zero and  is a parameter. Let P and Q denote two pro-jectors such that and QY . Assume that :PX KerL:/ImY L:NY is L-compact on , where  is open bounded in X. Furthermore, assume that (a) For each (0,1), Dom ,xL ,Lx Nx (b) For each Ker xL, , 0QNx(c) deg{,,0} 0KerJQNKL, where :ImerJQL is an isomorphism and repre-sents the Brouwer degree. deg{*}Then the equation LxNx has at least one solution in Dom L. For the sake of convenience, we introduce the follow-ing notation: 0[0, ][0, ]111(), m), (:with resand (in (),max(1,2,,),() is continpect to,...,)(,) ) exsitat,...,and()(),1,2,...,uous; (;,TiitTuiitTkkuu gtTggi nRxtttPC JRtxtxt kpltdtgtxJxtpptxtt *Corresponding author. Copyright © 2013 SciRes. OJAppS Y. YAN ET AL. 63()ut , ()gt()ztwhere are T-perio dic fu n ctions. is an T-periodic solution of (1) if and only if an T -periodic solution of Lemma 2i( )}zt is ln{ i1()()exp{ ()}()exp{()}( )exp{(( ))},,() ln(1),1,2,,1,2, ,.niiiij jjiii kik ikzt htztatztctzttttxtbink   n (2) nresent paTheorem Assume that where 12((ln{()},ln{( )},,ln{izt ztzln{)}( )})ztt e reaNow we ardy to state and prove the main results of the ppe r. 1ij iacution. , then system (1) has at least one T-periodic solProof. To complete the proof, we only need to search for an appropriate open bounded subset verifyingrequirements in Lemma 1. Let all the 12((),(),,()),{(,)| ()() Tnnzztzt zt2},, pZzPCRRztT ztYZR that both Ze norms en it is standard to show th and Y are Banach space when they are endowed with th[0,]||||sup|()|ctTzzt and 1222 211|| (,,,) ||(||||||||).pp pzc czcc Set Y as jprove tL is a Fredholm mapping of inero. Consider the operator equation :DomLL1()()((),(),,()), pLztztztzt where Dom{|()(,)}. nLZzZztPCRR At the same time, we denote :NZ Y as 11()()((()exp{ ()}()exp{()} ()exp{(())}),(, )) niiijjiii pNzth tztatztctzttI I It is easily to hat dex z(0,1).Lz Nz (3) al , we obtain Integra t i ng (3) ov er the interv[0,]T0101ij jjT (4) 0()exp{(1,2,,iict zinln(1)exp{( )}()exp{ ()}(( ))},). pTiikiknTihTbz tdtat ztdttt dtThen, we can deriv e 01|()|2 ln(11,2,,). pTiiizt hTbn ),(kkiSince there exist ( )([0,],),niztPCT R,i [0, ]iT 2,12[,tt(, ,],pt such that )inf(), ()[0, ][0, ]sup(), (1,,).iii iiitT tTzinzztz t For (4) we can see 01ln(1iikkhT b000101)exp{( )}()exp{ ()}()exp{ ( exp{()}xp{ ()}()exp{ ()} ln(1)(1)exp{pTiTii iTiipi ikTi iiiipiiik ikz tdtat ztdtct ztdzdtz dtctz dtbac z ()},iiT which implies 0()eiiTat())}ln(1 )iTikt tb 11ln(1)() ln:1piikkii iiihbTzAac　 Thus, [0, ]tT, we have 01()( )ln(1)|()|2:.pTiiiik iikztzbztdt AhTM  Similarly, according to (4), we have 01010010 ()exp{}nTijat Mdt101ln(1)exp{( )}()exp{ }())} ln(1)exp{()} ()exp{()} ln(1) pTiik iknTijjTiiipTiki ikjTiiipikkhTbz tdtat Mdtcttdtbzdtctz dtb()exp{ (tz1()exp{ } (1)exp{()},nijjiiiat MTczT Copyright © 2013 SciRes. OJAppS Y. YAN ET AL. Copyright © 2013 SciRes. OJAppS 64 which implies, Software Technology (HrZD201101). 111ln(1)exp{ }() ln1:.pniijikkjii ihbaTzcB MThus, , we have REFERENCES  Z. Jin and M. Zhen, “Periodic Solutions for Delay Dif-ferential Equations Model of Plankton Allelopathy,” Computers & Mathematics with Applications, Vol. 44, No. 3-4, 2002, pp. 491-500. doi: 10.1016/S0898-1221(02)00163-3 [0,]tT  S. Battaaz and F. Zanolin, “Coexistence States for Peri-odic Competitive Kolmogorov Systems,” Journal of Mathematical Analysis and Applications, Vol. 219, 1998, pp. 178-199. 011()( )ln(1)|()| 22ln(1):.  pTiiiikikpiikkzt zbztdtBhTb N  S. Arik, “Stability Analysis of Delayed Neural Network,” Fundam. Theory Applications, Vol. 52, 2000, pp. 1089- 1092. Now, we can derive Obviously,  Y. Zhang and K. T. 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Hence  X. Z. Meng and L. S. A. Chen, “Permanence and Global Stability in an Impulsive Lotka-Volterra Species Com-petitive System with both Discrete Delays and Continu-ous Delays,” International Journal of Biomathematics, Vol. 1, No. 2, 2008, pp. 179-196. doi: 10.1142/S1793524508000151 12()((), (),,())Tnztz tztztDom L. The 3. Acknowledgements Th by the National Natural Science Foo: 60963025); the Foundation of the Office of Education of Hainan Province (No:Hd f proof is completes. is work is supportedundation of China (N R. E. Gaines and J. L. Mawhin, “Coincidence Degree and Nonlinear Deferential Equations,” Berlin: Springer Ver-lag, 1977. j2009-36) anthe Foundation oHainan College of