 Open Journal of Applied Sciences, 2013, 3, 49-52 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 k-set contractive operator and some analysis technique. Keywords: Neutral-type Neural Networks; k-Set Contractive Operator; Periodic Solution 1. Introduction (H1) Functions ()jgu(1,2,,j)n are globally Lipschitz continuous with the Lipschitz constant 0jL, that is, Man-made 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. g12()( )jj 12juguLuu, for all 12,uu R. (H2) Functions ()ij t are nonnega-tive, bounded and continuously differentiable defined on R(,1,2,, )ij n and 1ij, where ()ij t express the derivative of ()ij t. (H3) There exist constant 0j, 0j, such that xR, ()jjjgx x, 1, 2,,. jnThe existence of periodic oscillatory solutions of neu-ral networks model has been studied by many researchers [1-6]. Some authors [3-5] used the well-known Hopf bifurcation theory to discuss the bifurcating periodic so-lutions. However, the usual Hopf bifurcation theory cannot be applied to non-autonomous system. In , Li and Lu applied the theory of coincidence degree to non-autonomous 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 k-set contrac-tive operator. 2. Preliminaries We consider the following model for neutral type neural networks with periodic coefficient: 1()() ()()((()))(),niiiijjjijijxta 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 AE, let 1()inf{0|there is a finite number of subsets,such that , and diam()}EiiiAwhere n is the number of neurons in the network, ()ixt () 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()iatiAAAAA n()ijbtdenotes the (Kuratoskii) measure of noncompactness, where diam( )iA denotes the diameter of set iA. Let X, Y be two Banach spaces and be a bounded open subset of X. A continuous and bounded map :112 2() ((),(), ,())Tnngxg xgxgx: nnRRis the activation function. 12()((),(),,())TnItItIt Itis the -periodic external input to the neuron. thiNY is called k-set contractive if for any bounded set A we have X((NA)) ()Yk A, where k is a constant. In addition, for a Fredholm operator with in-dex zero, according to , we may define :LXYThroughout this paper, we assume that: *Corresponding author. Copyright © 2013 SciRes. OJAppS W. X. ZHANG ET AL. 50 ()sup{0|()(()), for all bounded subset }.XYlLrr ALAAX  Lemma 2.1 Let be a Fredholm operator with index zero, and be a fixed point. Suppose that is a k-set contractive with :LX YaY:NY()klL, where X 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 Qker ,xL ]()xrx where [, is a bilinear form on YX and Q is the project of Y onto . Co ker()LThen there is a x, such that . Lx NxrIn order to use Lemma (2.1) for studying Equation (1.1), we set 1{ ()((),,())(,):()(),1,,},nniiYCxtxtxt CRRxtxt in   with the norm defined by [0, ]10max( )ntiixxt, and 111{ ()((),,())(,):()(),1,,},nniiXCxtxtxtCRRxtxt in   with the norm 00max{ , }xxx. Th en 1,CC are all Banach spaces. Let 1:LC C defined by 1(, , )TdxndtLxx x; 1:NC C, 11 11111() ()()((()))() ()()( (()))njjjjjnnnnnjj jnjjatxtbtgx ttxNxatxtb tg xtt   (2.1) It is easy to see from  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1xC, where . 1Lemma 2.2  The differential operator L is a Fred-holm operator with index zero, and satisfies . :()((),rIt It , ())TnIt() 1lL Lemma 2.3 If , here 1max{,1,, }1nij jjkbLin[0,]maxij tbbC( )ij t, :N is a k-contractive map. Proof. Let A be a bounded subset and let  1()CA. Then, for any 0, there is a finite family of subsets Ai satisfying 1iiAA with diam( )iA. Now let 11(, ,,,)()().niin iijjjVtx yyatxgy Since 1 are uniformly continuous on any compact subset of R(, ,,,)ii nVtx yy1nR, A and iA are precom-pact in 0C, it follows that there is a finite family of sub-sets ijA of iA such that i1jijAA( )),,(()), , (i ni ntu ttu t with 1111(),(( )))(,(()))|,ininVtt uttVt utt |,(, ()iiiixut for any ,ijxuA. Therefore we have 0[0,]11[0,]11 sup|(, (), ((())) (,(),(()))|sup|(,(),((())) (,(),(iin intii iniin intii inNx NuVtxt xttVtut uttVtxt xttVtxt ut1111()), ,()), , (()), ,()), , (ii nii nt txtu tt txtu t11(()),,()), , ()))( ())((ii nj jj ijt ttu tt gutu tt[0, ]1111()))| sup|(,(),(())) (,(),(()))|(((( )))(( ))(iin intii innijjjijijjnij jjijjtVtxt uu ttVtut uttbgxt tbL x ttkk 1) . As  is arbitrary small, it is easy to see that 01C((CNA)) ()kA. 0 1C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 1ij C, () 1ij t (,1,, )ij n. So that (ij t) has a unique inverse, and we set ij to represent the inverse of function()tv().ijtt. Meanwhile, we denote 0:(1/ )()hhsds and 1/ 220:()hhsds, 3. Main Results Set 21max :1(())ijij ijptvtR, 121221diag(,,,), (), ,diag(,,,), (),.ll lmnijnnijijjmmmnijnnnmiijjijijAaaa B bbbpCaaaHhhb I Theorem 3.1 Assume that (H1), (H2) hold, further-more, assume that (H4) ABC is non-singular M-ma-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. 51Corresponding to Equation (3.1), we have That is, 1()() ()()((()))()niiiijjjijjixta tx tbtgxttI t  22 2121 nmmiii ijjijjjnmij jijxaxb pxbI (3.5) (3.2) Suppose that 1(( ),,( ))Tnxtxt(0,1) is a solution of system (3.2) for a parameter i. Multiplying the Equation of system (3.2) by thix and integrating over [0,] gives Then we may rewrite Equation (3.5) as .YCXH (3.6) Substituting (3.6) into (3.4), we obtain 22101101/2 1/222001()()( (()))() ()(())() ()(()) nliiiij jjijijnnmmiijjjij ijjijjnmij jijijjaxxtbtgxtt Itdtxtbxttb Itdbxtdtxttdtb t() .ABC XBHH (3.7) It follows from Lemma 2.2 of paper  that 1(, ,)TnXHh h  . That is 2, 1,,,iixhi n (3.8) where 11(,,) ()()TnHhh ABCBHH   Substituting (3.8) into (3.6), we obtain 1/22011/2 1/22200() ()().nmij jijiixt dtxt dtIt dt 2, 1,,.iixhi n (3.9) It is easy to see that there exist two positive constants (1,2iNi ) such that And according to Lemma 2.4, we have 100, .2xNx N (3.10) 20202 (())()1(()).jijjij ijij jxttdxs dsvspxt Let 1100{: ,xC xNxN 2}, and define a bounded bilinear form [,] on 1CC by 0[,]()()yx ytxtdt. Thus 22211nnlm miiijjijjijj ijjaxb pxbIAlso we define by . Obviously, :Coker(Qy L)dt0()yyt (3.3) 11{|ker} {|, or }xxL xxNxN . Without loss of generality, we may assume that 1xN. Thus Then we may rewrite Equation (3.3) as ,AXBYH (3.4) where 122(,,)TnXx x, 122(,,)TnYx x. Multiplying the Equation of system (3.2) by thiix and integrating over [0, ] gives 22011011/2 1/22200() ()() ,()((()))()()(()) (),() ()()iiniiijj jijijnmmiiijjjijjinmij jijmii iatxtxxt dtbtg xttItaxtbxttxtdtbIta xtdtxtdt 2211111 1111111111[() ,][(),][(0)][([(0)][(nnjj jjjjnnnnjjnnnjjjjQNxQrxQNxQrx NaNb gIaNb gIaNbgIaNbgI0)]0)]n   (3.11) If 111max 0nij jijin ibINa, then 111(0)nniijjiijjjjaNbIbgIi, (3.12) So from (3.12) we get 1/2 1/2220011/22011/2 1/22200 ()(()) () ()().nmij jijijjnmij jijiibxtdtxttdtbxtdtx tdtI tdt [()(),][() (),]0iii iiiQNx 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. 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