 Algorithm of Iterative Process for Some Mappings and Iterative Solution of Some Diffusion EquationLiu Wenjun Department of Mathematics Jiujiang University Jiujiang , China liuwj4573@163.com Meng Jinghua Department of Mathematics Jiujiang University Jiujiang , China mengjh1956@sina.comAbstract—In Hilbert spaces , through improving some corresponding conditions in some literature and extending some recent relevent results, a strong convergence theorem of some implicit iteration process for pesudocon-traction mappings and explicit iteration process for nonexpansive mappings were established. And by using the result, some iterative solution for some equation of response diffusion were obtained. Keywords—pesudocon-traction mappings; nonexpansive mappings; implit iteration process; explicit iteration process; diffusion equation. 1. IntroductionLetEbeBanachspaceˈand kbe a nonempty closed convex subset of E. Suppose that Tis a mapping fromKto K,and)(TFis a set of fixed point of Twith ()FTIz. Assume that EEJ 2: o is regular dual mapping onE, and ^`.,,,)( *ExxffxfxEfxJ  !  As HE isHilbertspaceˈthe internal product of H is donate by the symbol!xx ,, and the norm ofHis designated by symbol x. Definition 1 MappingKKT o: is said to be pseudo contraction if for arbitrary ,xy K, there exits )()( yxJyxj  such that 2)(, yxyxjTyTx !d. Tis said to be strong pseudo contraction if there is(0,1)k such that2)(, yxkyxjTyTx !d for arbitrary,xy K. Definition 2 MappingKKT o: is said to be nonexpanxive if for abitrary ,xy K, there isTx Tyxyd . As we all know, that T is pseudo contraction is equivalent to that for every 0s!and everyKyx ,,there is ])()[( yTIxTIsyxyx d (1) WhenHE isHilbertspace,EEJ 2: ois single value , and for abitrary ,xy K,there Is !! yxyxyxjyx ,)(,. Obviously, nonexpansive mapping is pseudo contraction. 2. Lemmas and MethodsLemma 1[1,2]LetEbe a realBanachspace , andKbe nonempty closed convex subset of E. Assume that KKT o: is continuous strong pseudo constraction mapping . Then Tis unique fixed point in K. Lemma2 LetEbe a real reflexiveBanach space satisfying opialcondition, andKbe a nonempty closed convex subset of E. Supposethat KKTo: is continuous strong pseudo constraction mapping . Then for abitrary^`Exn, nx weakly converge toy, and0nnxTxo. So there is 0)( xTI. Lemma 3 Let1, 0pr!!be two certain real number, then Banachspace is()0ITx if and only if there is a strictly increasing continuous function :[0, )[0, )gf of,(0) 0g , such that (1 )(1 )()()pp ppxyx yWgxyOOO OO d , for all,rxy B, where[0,1]O, andrBis a closed spheroid which center is origin and radius is r, and ()(1 )(1 )pppWOOOOO . Lemma 4 Let nonnegative real sequence^`na satisfy the inequaltiy :1(1 )nnnnaaJGd , 0nt,where [0,1),nJ 1,nnJf f¦lim 0nnnGJof or 1,nnGf f¦ then lim 0nnaof . Open Journal of Applied Sciences Supplement：2012 world Congress on Engineering and Technology62 Copyright © 2012 SciRes. InHilbertspace,Moudafhas get strong convergence theorem of implicit iteration process of nonexpansive mapping, and Xu has improved and extended some relative results in Reference . In this paper, by applying a new implicit iteration sequence nnnnnnnTxxxfxJED )(,and explicit iterative sequence nnnnnnnTyyyfyJED )(1, we shall consider the problem involving the fixed point of strong pseudo constraction and nonexpansive mapping on closed convex set K . When exact conditions are satisfied, ^`nx and ^`ny all strongly converge to the fixed point of T. When the conditions for ^`nD andfin Reference, are widened, and as 0 nE, we can obtain the iterative sequence in Reference ,, and then we improve and extend some ralative results and obtain some equation of diffusion by applying the above results. Let :TK Ko be continuous pseudo constraction mapping , and :fKKobe continuous strong pseudo constraction mapping with constant D(0 1)D. Suppose that1nnnDEJ for,, (0,1)nnnDEJ, and we stucture mapping :nSK Ko, ()nn nnSxf xxTxDEJ .Then nS is continuous strong pseudo constraction mapping . By virtue of Lemma 1 , nS hasunique fixed point nx , then we have ()nnn nnnnnnxSxfxx TxDEJ  (2) 3. Main Results Theorem 1 LetE be a Hilbert space , and Kbe a nonempty closed convex subset of E. Assume that :fKKois continuous strong pseudo constraction mapping with constant D(0 1)D, and fis bounded on bounded set , and :TK Kois continuous pseudo constraction mapping .Then (a) If 01nnnDVE o or limsup 1nnVofˈand there is()pFTsuch that 22() 0nnfxpx po, then implict iterative sequence˄2˅strongly converges to the point of ()FT. (b) If T is nonexpansive mapping and f is constraction mapping with constant D,as10nnnnVVDVo and nD f¦, the explicit iterative sequence 1()nnnnnnnyfyyTyDEJ  strongly converges to the point of ()FT. Proof.˄a˅Because()pFT, 2nxp (( ))()(),nnnnnn nfxpxpTxpxpDEJ ! 222()nnnn nnnnxpfppxp xpxpDD DE Jd, we have () ()11nnnnnnxpfpp fppDDDDE JDd . Hence ^`^`^ `,(),nnnxfxTx are bounded. If 01nnnDVE o,then using formula (2),we can write ()(1 )nnn nnxfx TxVV ,and then we obtain ()0nn n nnxTxfxTxV o. (3) If limsup 1nnVofand there is ()pFTsuch that 2() 0nnfxpx po, then by virtue of formula (1) and Lemma 3, we obtain 2nxp 21()2nnnnnxp xTxVVd  21(( ))2nnnnxpfx TxV  211(( ))()22nnfxpx p 22111()( ())224nn nnfxpxpg xfxd, and then 221(())()02nn nng xfxfxpxpdo. So we have () 0nnxfxo. Whereas ()() 01nnn nnnnnnxTxfxTxx fxVVV  o (4) Because^`nxis bounded , andEisHilbertspace , we have thatnx weakly converge to qK.By virtue of formula Copyright © 2012 SciRes.63 (3)or (4) and Lemma 2, we have ()qFT. Because 2( ())(1)(),nnn nnnxqfx qTxqxqVV ! 22() ,(1)nnnnn nxq fqqxqxqDV VVd!, we obtain 21(),1nnxq fqqxqDd !. Since nx weakly converges to q,nxstrongly converges to ()qFT. (b) Because 1nnxx 11 1()(1 )()(1 )n nnn nnnnfxTx fxTxVVV V   111 111()(1 )nnnnnnnnnnnnxxfx xxTxVDV VVV V d , we obtain 11(1 )nnnnnxx MVVVDd , (5) where 1() 2nMfxdˈ12nMTx d. 1nnyy () ()nn nnnnnn nnnnfyy Tyfxx TxDEJDEJ  [1(1 )]nn nnn nnn nnn nyxyxyxyxDDEJD Dd 11[1(1 )][1(1 )]nnnnnnyx xxDD DDd  Since nD f¦,10(1 )nnnnVVDV Do, formula (5) and Lemma 4, we obtain 10nnyxo. Hence we have 110nnnnyqy xxqd  o, which means that ^`nystrongly converges to ()qFT. Note. Theorem 1 improves and extends some relative results in Reference  and . As follows, we will discuss iterative solution of some response diffusion equation . Let2()ELI {(, )(,)xts tsI,(, )xtsand 2(, )xts are Lebsgue intergrable on I},where [,][, ]Iabcd u, and ,xy E ,we define ,(,)(,)Ix yxtsyt sdtds! ³³.Then EisHilbert space, and 22,(,)Ixx xxt s dtds ! ³³, ,(),, ,yjxyxxy E! !. Consider the problem involving solution of some first order diffusion equation: 001,(,0)(),(0,)( )xxuxGxhxtsxsxsxtx tww °ww®° ¯ˈ (6) where G is continuous mapping on E, and 00ut is constant , and (, )0hhts t. This problem is equivalent to the integral equation as follows: 000 0000(, )(, )(, )(, )xt tstsxtsds uxtsdthtsxtsdtdsxGxdtds ³³ ³³³³ 00100()() 0stxs dsuxtdt ³³ ˄7˅ Let ^(, )KxExts  is continuous function onI`, then Kis nonempty closed convex subset ofE. Let:HK Ko. 00 00000(, )(,)()()t s tstsHxuxtsdtx tsdsh txx ts dtdsxGxdtds ³ ³³³³³ 00100()() .stxs dsuxtdt³³ IfGsatisfies (A)˖,, ,xy KxGxyGy dthen let :,TKKTxHx xo . IfGsatisfies (B)˖there is 10L!such that 1xGx yGyLx yd for abitrary ,xy K. Then H is Lipschitz mapping on Kˈand then we have 0L! such that ,,xyK HxHyLxyd. Let 12HHL ˈ111:, .TK KTxHxxo  Theorem 2 Let integral equation˄7˅has solutionˈthen (i) IfGsatisfies (A), when 01nnnDVE o or limsup 1nnVof,and there is()pFT such that 22() 0nnfxpxpo, Implicit iterative sequence ()nnn nnnnxfxxTxDEJ  strongly converges to the fixed point of Twhich is solution of equation (7). (ii) IfGsatisfies˄B˅,when nD f¦ and10nnnnVVDVo, explicit iterative sequence1()nnnnnnnyfyyTyDEJ  strongly converges to the fixed point of T which is solution of equation (7). Proof.(i) Now, ,,( )()xyKHxHyxy is nonnegative on 64 Copyright © 2012 SciRes. ^`1(, )0EtsIxy d and ^`2(, )0EtsIxy t. Then we have ,0,HxHy xy !tthat is said that Tis pseudo constraction mapping on K. Using Theorem 1, we obtain the result . (ii) Now ,11,, 2xyK HxHyxyd  ˄8˅ 21) 1()Tx Ty 211[()]Hx xHyy  2211 11()2()()()HxHyyx HyHxxy  2211 11()2()Hy Hxy xHy Hxx y  21111()( 2)()Hy HxHy Hxyxxy  If 112,Hy Hxy xdthen we obtain 2211()()Tx Tyxyd. If 1120,Hy Hxy xtt then we obtain 2 211 1111()( 2)( 2)()TxTyHy Hxy xHy Hxyxxyd  22211()4()( )HyHxxyxy . Hence, by virtue of formula (8), we have 2211Tx Tyxyd. That is said that 1T is nonexpansive mapping onK. Using Theorem 1, we obtain the result. Thereforeˈthrough improving some corresponding conditions in literature ,ˈand extending some recent relevent results, Theorem 1 was established. Theorem 1 is a strong convergence theorem of some implicit iteration process for pesudocon-traction mappings and explicit iteration process for nonexpansive mappings . By applying Theorem 1, the iterative solution for some equation of response diffusion was obtained, Theorem 2 was established. REFERENCES  Deimling K. zeros of accretive operators [J].Manuscripta Math, 1974,13(4):365-374.  Chang S S. Cho Y J. Zhou H Y. Iterutive methods for nonlinear operator Equation in Banach space [M].New York: Science publishers, 2002.  Zhou H Y. 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