Open Journal of Applied Sciences, 2013, 3, 4952 doi:10.4236/ojapps.2013.31B1010 Published Online April 2013 (http://www.scirp.org/journal/ojapps) Periodic Solution for Neutral Type Neural Networks Wenxiang Zhang, Yan Yan, Zhanji Gui, Kaihua Wang* School of Mathematics and Statistics, Hainan Normal University, Haikou, Hainan Email: *kaihuawang@qq.com Received 2013 ABSTRACT The principle aim of this paper is to explore the existence of periodic solution of neural networks model with neutral delay. Sufficient and realistic conditions are obtained by means of an abstract continuous theorem of kset contractive operator and some analysis technique. Keywords: Neutraltype Neural Networks; kSet Contractive Operator; Periodic Solution 1. Introduction (H1) Functions () j u(1,2,,j)n are globally Lipschitz continuous with the Lipschitz constant 0 j L, that is, Manmade neural networks have been widely used in the fields of pattern recognition, image processing, associa tion, optimal computation, and others. However, owing to the unavoidable finite switching speed of amplifiers, time delays in the electronic implementations of analog neural networks are inevitable, which may cause undesirable dynamic network behaviors such as oscillation and insta bility. Thus, it is very important to investigate the dy namics of delay neural networks. 12 ()( ) jj 12j uguLuu , for all 12 ,uu R. (H2) Functions () ij t are nonnega tive, bounded and continuously differentiable defined on R (,1,2,, )ij n and 1 ij , where () ij t express the derivative of () ij t . (H3) There exist constant 0 j , 0 j , such that R , () jj gx x , 1, 2,,. jn The existence of periodic oscillatory solutions of neu ral networks model has been studied by many researchers [16]. Some authors [35] used the wellknown Hopf bifurcation theory to discuss the bifurcating periodic so lutions. However, the usual Hopf bifurcation theory cannot be applied to nonautonomous system. In [6], Li and Lu applied the theory of coincidence degree to nonautonomous neural networks system and obtained some new criteria for the existence of periodic solutions. Remark 1.1 It is easy to verify that if condition (H1) is satisfied then condition (H3) holds. The organization of this paper is as follows. Prelimi naries will be given in the next section. In section 3, we will study the existence of periodic solutions of system (1.1) by the abstract continuous theorem of kset contrac tive operator. 2. Preliminaries We consider the following model for neutral type neural networks with periodic coefficient: 1 ()() ()()((()))(), n iiiijjjiji j ta tx tbtgxttIt (1) In order to study Equation (1.1), we should make some preparations. Let E be a Banach space. For a bounded subset E, let 1 ()inf{0there is a finite number of subsets,such that , and diam()} E ii i A where n is the number of neurons in the network, () i t () corresponds to the state of the ith unit at time t, denotes the neuron firing rate, repre sents the neutral delayed connection weight. 1, 2,,i () i at i AA A A n () ij bt denotes the (Kuratoskii) measure of noncompactness, where diam( ) i denotes the diameter of set i . Let X, Y be two Banach spaces and be a bounded open subset of X. A continuous and bounded map : 112 2 () ((),(), ,()) T nn gxg xgxgx: nn RR is the activation function. 12 ()((),(),,()) T n tItIt Itis the periodic external input to the neuron. thi NY is called kset contractive if for any bounded set we have X ((NA)) () Y k A , where k is a constant. In addition, for a Fredholm operator with in dex zero, according to [7], we may define :LXY Throughout this paper, we assume that: *Corresponding author. Copyright © 2013 SciRes. OJAppS
W. X. ZHANG ET AL. 50 ()sup{0()(()), for all bounded subset }. XY lLrr ALA AX Lemma 2.1 Let be a Fredholm operator with index zero, and be a fixed point. Suppose that is a kset contractive with :LX Y aY :NY()kl L , where is bounded, open, and symmetric about . Furthermore, we also assume that 0 1) LxNx r , for x, (0,1) ; 2) [( ,]0, for ) ,QNxQr x][QN Q ker ,xL ] ()xrx where [, is a bilinear form on YX and Q is the project of Y onto . Co ker()L Then there is a x, such that . Lx Nxr In order to use Lemma (2.1) for studying Equation (1.1), we set 1 { ()((),,())(,): ()(),1,,}, n n ii YCxtxtxt CRR xtxt in with the norm defined by [0, ] 1 0max( ) n ti i xt , and 11 1 { ()((),,())(,): ()(),1,,}, n n ii XCxtxtxtCRR xtxt in with the norm 00 max{ , } xx. Th en 1 ,CC are all Banach spaces. Let 1 :LC C defined by 1 (, , ) T dx n dt Lxx x ; 1 :NC C , 11 11 1 1 1 () ()()((())) () ()()( (())) n jjjj j n nnnnjj jnj j atxtbtgx tt x N xatxtb tg xtt (2.1) It is easy to see from [8] that L is a Fredholm operator with index zero. Clearly, Equation (1.1) has a peri odic solution if and only if for some LxNx r 1 C , where . 1 Lemma 2.2 [9] The differential operator L is a Fred holm operator with index zero, and satisfies . :()((),rIt It , ()) T n It () 1lL Lemma 2.3 If , here 1 max{,1,, }1 n ij j j kbLi n [0,] max ij t bb C( ) ij t, :N is a kcontractive map. Proof. Let be a bounded subset and let 1() C . Then, for any 0 , there is a finite family of subsets Ai satisfying 1ii A with diam( ) i A . Now let 1 1 (, ,,,)()(). n iin iijj j Vtx yyatxgy Since 1 are uniformly continuous on any compact subset of R (, ,,,) ii n Vtx yy 1n R , A and i are precom pact in 0 C , it follows that there is a finite family of sub sets ij of i such that i1jij A ( )),,( ()), , ( i n i n tu t tu t with 11 11 (),(( ))) (,(())), in in Vtt utt Vt utt ,( , () ii ii x ut for any ,ij uA. Therefore we have 0 [0,] 11 [0,] 11 sup(, (), ((())) (,(),(())) sup(,(),((())) (,(),( iin in t ii in iin in t ii in Nx Nu Vtxt xtt Vtut utt Vtxt xtt Vtxt ut 11 11 ()), , ()), , ( ()), , ()), , ( i i n i i n t tx tu t t tx tu t 11 (()),, ()), , ( )))( ( ))(( i i n j j j ij t t tu t t gut u tt [0, ] 11 1 1 ())) sup(,(),(())) (,(),(())) (((( ))) (( )) ( iin in t ii in n ijjjijij j n ij jjij j t Vtxt uu tt Vtut utt bgxt t bL x tt kk 1) . As is arbitrary small, it is easy to see that 01 C (( C NA)) ()kA . 0 1 C Lemma 2.4 Let C, , and ()1t , then 0 (())vt C ()tt , where is the inverse function of ()vt . Throughout this paper, we assume that 1 ij C , () 1 ij t (,1,, )ij n . So that ( ij t) has a unique inverse, and we set ij to represent the inverse of function ()tv (). ij tt . Meanwhile, we denote 0 :(1/ )()hhsds and 1/ 2 20 :()hhsds , 3. Main Results Set 21 max : 1(()) ij ij ij pt vt R , 12 12 2 1 diag(,,,), (), , diag(,,,), (), . ll lm nijnnijijj mmm nijnn n m iijji j ij aaa B bbbp CaaaHh hb I Theorem 3.1 Assume that (H1), (H2) hold, further more, assume that (H4) BC is nonsingular Mma trix, then Equation (1.1) has at least one positive  periodic solution. Proof. We consider the operator Equation , (0,1)LxNx r (3.1) Copyright © 2013 SciRes. OJAppS
W. X. ZHANG ET AL. 51 Corresponding to Equation (3.1), we have That is, 1 ()() ()()((()))() n iiiijjjij j i ta tx tbtgxttI t 22 2 1 2 1 n mm iii ijjijj j n m ij ji j axb px bI (3.5) (3.2) Suppose that 1 (( ),,( ))T n txt (0,1) is a solution of system (3.2) for a parameter i . Multiplying the Equation of system (3.2) by thi and integrating over [0,] gives Then we may rewrite Equation (3.5) as .YCXH (3.6) Substituting (3.6) into (3.4), we obtain 2 2 1 0 11 0 1/2 1/2 2 2 00 1 ()()( (()))() ()(())() ()(()) n l iiiij jjiji j nn mm iijjjij ijji jj n m ij jijij j axxtbtgxtt Itdt tbxttb Itd bxtdtxttdt b t () . BC XBHH (3.7) It follows from Lemma 2.2 of paper [10] that 1 (, ,) T n Hh h . That is 2, 1,,, ii hi n (3.8) where 1 1 (,,) ()() T n hh ABCBHH Substituting (3.8) into (3.6), we obtain 1/2 2 0 1 1/2 1/2 22 00 () ()(). n m ij ji j ii xt dt xt dtIt dt 2, 1,,. ii hi n (3.9) It is easy to see that there exist two positive constants (1,2 i Ni ) such that And according to Lemma 2.4, we have 1 00 , . 2 Nx N (3.10) 2 0 2 0 2 (()) () 1(()) . jij j ij ij ij j ttd xs ds vs px t Let 1 1 00 {: ,xC xNxN 2 }, and define a bounded bilinear form [,] on 1 CC by 0 [,]()()yx ytxtdt . Thus 22 2 11 nn lm m iiijjijjijj i jj axb pxbI Also we define by . Obviously, :Coker(Qy L)dt 0()yyt (3.3) 11 {ker} {, or } xL xxNxN . Without loss of generality, we may assume that 1 N . Thus Then we may rewrite Equation (3.3) as , XBYH (3.4) where 122 (,,) T n Xx x, 122 (,,) T n Yx x . Multiplying the Equation of system (3.2) by thii and integrating over [0, ] gives 2 2 0 1 1 0 1 1/2 1/2 22 00 () () () , ()((()))() ()(()) (), () ()() ii n ii ijj jiji j n mm iiijjjij j in m ij ji j m ii i atxt xt dt btg xttIt axtbxtt tdt bIt a xtdtxtdt 22 1 1111 1111 11 11 11 [() ,][(),] [(0)][( [(0)][( nn jj jj jj nn nnjjnnnjj jj QNxQrxQNxQrx N aNb gIaNb gI aNbgIaNbgI 0)] 0)] n (3.11) If 1 11 max 0 n ij ji j in i bI Na , then 1 11 (0) nn iijjiijj jj aNbIbgI i , (3.12) So from (3.12) we get 1/2 1/2 2 2 00 1 1/2 2 0 1 1/2 1/2 22 00 ()(()) () ()(). n m ij jijij j n m ij ji j ii bxtdtxttdt bxtdt x tdtI tdt [()(),][() (),]0 iii iii QNx QrxQNx Qrx 1, ,in Therefore, by using Lemma 2.1, we obtain that Equa tion (1.1) has at least one positive periodic solution. The proof is complete. Copyright © 2013 SciRes. OJAppS
W. X. ZHANG ET AL. Copyright © 2013 SciRes. OJAppS 52 4. Acknowledgements This work is supported by the Natural Sciences Founda tion of China under Grant No. 60963025. REFERENCES [1] X. Y. Liu and J. D. Cao, “On Periodic Solution of Neural Networks via Differential Inclusions,” Neural Networks, Vol. 22, No. 4, 2009，pp. 329334. doi：10.1016/j.neunet.2008.11.003 [2] J. Zhang and Z. J. Gui, “Periodic Solutions of Nonautonomous Cellular Neural Networks with Impulses and Delays,” Nonlinear Analysis: Real World Applica tions, Vol. 10, No. 3, 2009，pp. 18911903. doi：10.1016/j.nonrwa.2008.02.029 [3] K. Gopalsamy and I. Leung, “Delay Induced Periodicity in a Neural Netlet of Excitation and Inhibition,” Physica D: Nonlinear Phenomena, Vol. 89, No. 34, 1996，pp. 395426. doi:10.1016/01672789(95)002030 [4] L. P. Shayer and S. A. Campbell, “Stability, Bifurcation and Multistability in a System of Two Coupled Neurons with Multiple Delays,” SIAM Journal on Applied Math ematics, Vol. 61, No. 2, 2000，pp. 673700. doi:0.1137/S0036139998344015 [5] J. Wu, T. Faria and Y. S. Huang, “Synchronization and Stable Phaselocking in a Network of Neurons with Memory,” Mathematical and Computer Modeling, Vol. 30, No. 12, 1999，pp. 117138. doi:0.1016/S08957177(99)00120X [6] Y. Li and L. Lu, “Global Exponential Stability and Exis tence of Periodic Solution of Hopfield type Neural Net works with Impulses,” Physics Letters A, Vol. 333, No. 12, 2004，pp. 6271. doi:0.1016/j.physleta.2004.09.083 [7] Z. J. Gui, S. J. Lu and W. G. Ge, “Existence of Periodic Solution of Two Competition Dynamics System with Neutral Delay and Several Deviating Arguments,” Chi nese Journal of Engrg. Mathematics, Vol. 22, 2005，pp. 703711. [8] R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations, SpringerVerlag, Berlin, 1977. [9] Z. Liu and Y. Mao, “Existence Theorem for Periodic Solutions of Higher Order Nonlinear Differential Equa tions,” Journal of Mathematical Analysis and Appications, Vol. 216, No. 2, 1997，pp. 481–490. doi:.1006/jmaa.1997.5669 [10] Z. Liu and L. Liao, “Existence and Global Expo nential Stability of Periodic Solution of Cellular Neural Networks with Timevarying Delays,” Journal of Mathematical Analysis and Applications, Vol. 290, No. 1, 2004，pp. 247–262. doi:10.1016/j.jmaa.2003.09.052
