 Journal of Applied Mathematics and Physics, 2013, 1, 1-5 http://dx.doi.org/10.4236/jamp.2013.14001 Published Online October 2013 (http://www.scirp.org/journal/jamp) Copyright © 2013 SciRes. JAMP Existence of Periodic Solutions for Neutral-Type Neural Networks with Delays on Time Scales Zhenkun Huang, Jinxiang Cai School of Science, Jimei University, Xiamen, China Email: hzk974226@jmu.edu.cn Received June 2013 ABSTRACT In this paper, we employ a fixed point theorem due to Krasnosel’skii to attain the existence of periodic solutions for neutral-type neural networks with delays on a periodic time scale. Some new sufficient conditions are established to show that there exists a unique periodic solution by the contraction mapping principle. Keywords: Neutral-Type; Neural Networks; On Time Scales; Periodic Solution 1. Introduction Recently, scholars and researchers have paid more atten-tion to the discussion of neural networks described by neutral-type differential equations with delays (see [1-6]). Meanwhile, difference equations or discrete-time analo-gues of differential equations can preserve the conver-gence dynamics of their continuous time counterparts in some degree . Due to their usage in applications, these discrete-type neural networks with or without delays have been discussed by [8,9] and references therein. It is interesting to study that neural systems on time scales can unify the continuous and discrete situations. The theory of time scales initiated by S. Hilger [10,11] has been incorporated to investigate neural networks [12,13] and so on. However, few works have considered for neutral-type neural networks on time scales [14,15 ]. In this paper, we consider the existence of periodic solutions for the neu-tral networks with delays 11()()()() () ()(()) (), :{1,2,}niii ijjjnijj jjixtatxtc txtkbtgxt kIt in  (1) For a review of dynamic equations o-time scales, we direct the reader to the monographs [9,10] and begin with a few definitions. Definition 1.1. A time scale  is p-periodic if there exists 0p and p such that if t then tp. For , the smallest positive p is called the period of. Definition 1.2. Let  be a p-periodic time scale. f:  is periodic with period  if there exists a natural number n such ,() ()np f tTf t for all t and  is the smallest number such that ()()ftT ft. Without other statements, let  be a p-periodic time scale such that 0. We will show the existence of periodic solutions for (1) where ,.kmpm 2. Preliminaries Theorem 2.1. ([10,11]) Assume : is strictly increasing and :() is a time scale. Let .： If t and t exist for ,kt then one has . Theorem 2.2. ([10,11]) Assume : is strictly increasing and :() is a time scale. If : f is a rd-continuous and  is differentiable with rd-con- tinuous derivative , then one gets  1, ,.bbaaftt tfs sab A function : p is said to be regressive pro-vided 10tpt for all .kt The set of all regressive rd-continuous function : f is devote by R while the set R is given by :10, t.RfRtpt  Let pR. The exponential function is defined by ()(,)exp(( ))tpsets p (2) where hz is called cylinder transformation. Z. K. HUANG, J. X. CAI Copyright © 2013 SciRes. JAMP 2 Lemma 2.1. Let ,pqR. One gets that (i) 0(,)1 and e(,)1pets tt; (ii) ((),)(1()())(,)ppets tptets ; (iii) 1()(,), where (, )1()()ppptets petstpt; (iv) (, )1(,)(,)pppetseste st; (v) (,)(,)(,)pp petsesr etr; (vi) (,)(,)(,)pq pqetsets ets; (vii) (, )(, )(,)ppqqets etsets ; (viii) 1()(, )(, )ppptes es ; Finally, we state Krasnosel’skii fixed point theorem which enables us to prov e the existence of period ic solu-tions. Theorem 2.3. () Let  be a closed convex nonempty subset of a Banach space B, . Suppose that A and B map  into B such that (i) ,xy imply xy; (ii)  is compact and continuous; (iii)  is a contraction mapping; Then there exists z with .zzz  3. Existence of Periodic Solutions Let 0, , TTkT be fixed and if , np for some n. By the notation ,ab, we mean ,:.abtatb The intervals ,, ,ab ab and ,ab are defined similarly, Defined (, ):()()nTPCR tTt , where (, )nCR is the space of all real valued conti-nuous functions. Then TP is a Banach space when it is endowed with norm 0,sup( )tTxxt. For each ,ij, we make basic assumption 1()H: jaR is continuous, ()0iat and ()()iiat Tat for all tT. ()(),()()ijij iictTct ItTIt  and ()ijct is con- tinuous. ()jgu is continuous, (0)0jg and () ()jj jgugv Luv for some 0jL. Lemma 3.1. Assume that 1()H holds. {()}ii TxtP  is a solu-tion of (1) if and only if 111()() () (1(,))[()() ()(()()](, )iiniijjajtijtTnijj jiajxtc txtkettTrsx s kbtgxskIsets s ， (3) where 1():() ()()niijiijjrsc sascs and i. Proof. Let ()iTixtP is a solution of (1). It fol-lows from (1) that 11() ()() () ()(())(),niiiijjjnijj jijxatxt ctxtkbtgxt kIt  (4) where i. Multiply both sides of (4) by (,0)iaet and integrate from tT to t, one obt a i n that 11[(,0)()][()( )()( ())()](,0).iinttai ijjtTtT jnijj jiajes xsscsxskbsgxskIse ss Divide both sides of above equation by (,0)iaet , due to ()()iixtxtT and Lemma 2.1, we have 11()1(,)[()() ()(()()(,). iintia ijjtT jnijj jiajxtettT csxskbsgxsk Ise tss  (5) It follows from integration by parts and the period icity of ()ijc and ()jx, we get 111(, )()()() ()[1(,)]()[(,)()].iiintaijjtT jnij jajntsjaijtTjetscsxsksctxtke ttTxsketscss (6) Substitute (6) into (5) and simplify, we get (3). From Lemma 2.1, we get the desired result and the proof is complete. Define the mapping : PPTTH by 111()()() ()1,()( )()(( ))()(,), iiinij jajntijijjj itT jaHtcttke ttTrss kbsgskIsetss (7) where i. Let ()()()()()(), iiiHtBtAt where ,AB are given by 1()()() (),niijjjBt cttk (8) Z. K. HUANG, J. X. CAI Copyright © 2013 SciRes. JAMP 3and 1()()1(,)()( )() ( ())()(,),iitia ijtTnijj jiajAtettT rsskbsgskIse tss  (9) where ()irs is defined in Lemma 3.1 and i. Next, we will prove that A is compact and B is a contrac-tion mapping in Lemma 3.2 and Lemma 3.3, respective-ly. Lemma 3.2. Assume that 1()H holds. : PPTTA defined by (9) is compact. Proof. We first show that A maps PT into PT. It follows from (9) that 11()(+)1( ,)()()()( ())()(,),iiiatijtTnijj jiajAtT etTtru TukbugukIuetTuT u  (10) and 11() ()()() =()()()().niijiijjnij iijijruTcuTauT cuTcuau curu  i.e., ()iru is T-periodic. It follows from (2) and Theo-rem 2.2 that (, )=(,)iiaaetTuTetu and (,)(,).iiaaetTtettT  Thus (10) becomes 11()(+)1(, )()( )()(())()(,) =()().iiiantij ijjjtT jia iAtT ettTruukb ugukIuetu uAt   where i. That is, A: PPTT. Secondly, we will show that is continuous. Let , TP with , CC and define 1[0,][,][,] [0,]:max1(, ):max(,) :max() :max():maxiiiatTiautTtijij iittTtt TjjjettTetubbt rt   ， (11) Given 0 and take ()nM such that . By making use of Lipschitz inequality of 1()H, we get 1()() (),ntiiii ijjtTjij jjjAAbLdu M    where max{ }iMM and 1().niii iijjjMTnbL  Hence, it follows that max ()()iiiAAA A, That proves A is continuous. Thirdly, we need to show A is compact. Consider the sequence {}nTP and assume that the sequence is uniformly bounded. Let 0R be such ()nR for all n. It is easy to estimate that 1()() ()1() ()11()() 1(,)()()()(( ))() (,)(() )[],iiniantnnij ijjjtT jiantnniiijij jjitT jniiiij jijAt ettTrss kbsgs kIsetssrsbLIsnRbLRIT D     where sup{( )}.iisRIIs Thus the sequence (){}nA is uniformly bounded. Now, it can be easily checked that () ()()1()11( )()()()( )()1()() {[(()()())]() ()(())()},nnii iiinnij iijijnnijj jijAtatAtat tctat cttkbtgtk It   which leads to ()1()().nniiijjijAtDanRbLRI  For all n.That is ()()nAF for some positive constant F.Thus the sequence (){}nA is uniformly bounded and equi-continuous. The Arzela-Ascoli theo-rem  implies that (){}knA uniformly co nverg es to a continuous T-periodic function . Thus A is com-pact. Lemma 3.3. Let B is defined by (8) and assume that 2[0,]1(): sup(), max 1.ij ijtTnijijHct  Then : PPTTB is a contraction. Proof. Trivially, : PPTTB. For any , TP, we have Z. K. HUANG, J. X. CAI Copyright © 2013 SciRes. JAMP 4 [0, ]11()()max(() ()) .niiijj jtTjnijjBBctk tk   which leads to .BB Hence, B de-fines a contraction mapping with contraction constant . Theorem 3.1. Assume that 1()H and 2()H hold. If all solutions {x (t)}PiT of (1) satisfy with x(t)iG and 1max [()]niiiij jiiiijTnbLG ITG    wher e 0G and sup{( )},iisRIIs then (1) has a T-periodic solution. Proof. Define {:}.TPG  Lemma 3.1 implies : PPTTA and A is compact and conti-nuous. Lemma 3.2 implies B is a contraction and : PPTTB. Now, we need to show that if , , we have .iiABG Let ,  with ,.G It follows from (8) and (9) that 11(() ()) [()] tii ijii tTnij jjijniiiij jjiiiAB rsbLIssTnbLGITG  All the conditions of Krasnosel’skii theorem are satis-fied on the set . Thus there exists a fixed point z in  such that .zAzBz By Lemma 3.1, this fixed point is a solution of (1). Hence (1) has a T-periodic so-lution. Theorem 3.2. Assume that 1()H and 2()H hold. Let ,,iii be given by ( 11). If 1()1niiiij jjTn bL  holds, then (1) has a unique T-periodic solution. Proof. Let the mapping H be given by (7). For , ,TP we have 11( ) [()],ntiii jjii tTjij jjjniiiij jjHHbL sTn bL    which leads to 1max [()].iiiniiiij jjHHH HTn bL     That is, H defines a contraction mapping and there exists a unique fixed point which is a T-periodic solution of (1). This completes the proof. 4. Conclusion Due to time scales calculus theory and the fixed point theorem, we obtained some more generalized results to ensure the existence of the periodic solutions for neutral- type neural networks with delays. The conditions can be easily checked in practice by simple algebraic methods. The method in this paper can be applied to prove the ex-istence of the periodic solutions of some other similar systems such as neutral-type networks with leakage- terms . REFERENCES  C. J. Cheng, T. L. Liao, J. J. Yan and C. C. 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