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Algorithm of Iterative Process for Some Mappings and Iterative Solution of Some Diffusion Equation Liu Wenjun Department of Mathematics Jiujiang University Jiujiang , China liuwj4573@163.com Meng Jinghua Department of Mathematics Jiujiang University Jiujiang , China mengjh1956@sina.com Abstract—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. Introduction Let E be Banach spaceˈand k be a nonempty closed convex subset of E . Suppose that T is a mapping from K to K ,and )(TF is a set of fixed point of T with () FT I z . Assume that E EJ 2: o is regular dual mapping on E , and ^` .,,,)( *ExxffxfxEfxJ ! As HE is Hilbert spaceˈthe internal product of H is donate by the symbol !xx , , and the norm of H is designated by symbol x . Definition 1 Mapping KKT o: is said to be pseudo contraction if for arbitrary ,xy K , there exits )()( yxJyxj such that 2 )(, yxyxjTyTx !d . T is said to be strong pseudo contraction if there is (0,1)k such that 2 )(, yxkyxjTyTx !d for arbitrary ,xy K . Definition 2 Mapping KKT o: is said to be nonexpanxive if for abitrary ,xy K , there is Tx Tyxyd . As we all know, that T is pseudo contraction is equivalent to that for every 0s! and every Kyx , ,there is ])()[( yTIxTIsyxyx d (1) When HE is Hilbert space, E EJ 2: o is single value , and for abitrary ,xy K ,there Is !! yxyxyxjyx ,)(, . Obviously, nonexpansive mapping is pseudo contraction. 2. Lemmas and Methods Lemma 1 [1,2] Let E be a real Banach space , and K be nonempty closed convex subset of E . Assume that KKT o: is continuous strong pseudo constraction mapping . Then T is unique fixed point in K . Lemma2 [3] Let E be a real reflexive Banach space satisfying opial condition, and K be a nonempty closed convex subset of E . Supposethat KKTo: is continuous strong pseudo constraction mapping . Then for abitrary ^` Ex n , n x weakly converge to y , and 0 nn xTxo . So there is 0)( xTI . Lemma 3[4] Let 1, 0pr!! be two certain real number, then Banach space is ()0ITx if and only if there is a strictly increasing continuous function :[0, )[0, )gf of , (0) 0g , such that (1 )(1 )()() pp p p xyx yWgxy OOO OO d , for all ,r xy B , where [0,1] O , and r B is a closed spheroid which center is origin and radius is r , and ()(1 )(1 ) pp p W OOOOO . Lemma 4 [5] Let nonnegative real sequence ^` n a satisfy the inequaltiy : 1 (1 ) nnnn aa JG d , 0nt ,where [0,1), n J 1 , n n J f f ¦ lim 0 n nn G J of or 1 , n n G f f ¦ then lim 0 n na of . Open Journal of Applied Sciences Supplement：2012 world Congress on Engineering and Technology 62 Cop y ri g ht © 2012 SciRes. In Hilbert space, [6] Moudaf has get strong convergence theorem of implicit iteration process of nonexpansive mapping, and [7] Xu has improved and extended some relative results in Reference [7]. In this paper, by applying a new implicit iteration sequence nnnnnnn Txxxfx JED )( ,and explicit iterative sequence nnnnnnn Tyyyfy JED )( 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, ^` n x and ^` n y all strongly converge to the fixed point of T . When the conditions for ^` n D and f in Reference [6],[7] are widened, and as 0 n E , we can obtain the iterative sequence in Reference [6],[7] , 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 :fKKo be continuous strong pseudo constraction mapping with constant D (0 1) D . Suppose that 1 nnn DEJ for ,, (0,1) nnn DEJ , and we stucture mapping : n SK Ko , () nn nn Sxf xxTx DEJ .Then n S is continuous strong pseudo constraction mapping . By virtue of Lemma 1 , n S hasunique fixed point n x , then we have () nnn nnnnnn xSxfxx Tx DEJ (2) 3. Main Results Theorem 1 Let E be a Hilbert space , and K be a nonempty closed convex subset of E . Assume that :fKKo is continuous strong pseudo constraction mapping with constant D (0 1) D , and f is bounded on bounded set , and :TK Ko is continuous pseudo constraction mapping .Then (a) If 0 1 n n n D VE o or limsup 1 n n V of ˈand there is ()pFT such that 22 () 0 nn fxpx po , 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 ,as 10 nn nn VV DV o and n D f ¦ , the explicit iterative sequence 1() nnnnnnn yfyyTy DEJ strongly converges to the point of ()FT . Proof. ˄a˅Because ()pFT , 2 n xp (( ))()(), nnnnnn n fxpxpTxpxp DEJ ! 222 () nnnn nnnn xpfppxp xpxp DD DE J d , we have () () 11 nn n nnn xpfpp fpp DD DDE JD d . Hence ^`^`^ ` ,(), nnn xfxTx are bounded. If 0 1 n n n D VE o ,then using formula (2),we can write ()(1 ) nnn nn xfx Tx VV ,and then we obtain ()0 nn n nn xTxfxTx V o . (3) If limsup 1 n n V of and there is ()pFT such that 2 () 0 nn fxpx po , then by virtue of formula (1) and Lemma 3, we obtain 2 n xp 2 1() 2 n nnn n xp xTx V V d 2 1(( )) 2 n nnn xpfx Tx V 2 11 (( ))() 22 nn fxpx p 22 111 ()( ()) 224 nn nn fxpxpg xfxd , and then 22 1(())()0 2nn nn g xfxfxpxpdo . So we have () 0 nn xfxo . Whereas ()() 0 1 n nn nnnnn n xTxfxTxx fx V VV o (4) Because ^` n x is bounded , and E is Hilbert space , we have that n x weakly converge to qK .By virtue of formula Cop y ri g ht © 2012 SciRes.63 (3)or (4) and Lemma 2, we have ()qFT . Because 2 ( ())(1)(), nnn nnn xqfx qTxqxq VV ! 22 () ,(1) nnnnn n xq fqqxqxq DV VV d! , we obtain 2 1(), 1 nn xq fqqxq D d ! . Since n x weakly converges to q , n x strongly converges to ()qFT . (b) Because 1nn xx 11 1 ()(1 )()(1 ) n nnn nnnn fxTx fxTx VVV V 111 111 ()(1 ) nnnnnnnnnnnn xxfx xxTx VDV VVV V d , we obtain 1 1 (1 ) nn nn n xx M VV VD d , (5) where 1 () 2 n M fx d ˈ 12 n M Tx d . 1nn yy () () nn nnnnnn nnnn fyy Tyfxx Tx DEJDEJ [1(1 )] nn nnn nnn nnn n yxyxyxyx DDEJD D d 11 [1(1 )][1(1 )] nnnnnn yx xx DD DD d Since n D f ¦ , 1 0 (1 ) nn nn VV DV D o , formula (5) and Lemma 4, we obtain 1 0 nn yx o . Hence we have 11 0 nnnn yqy xxq d o , which means that ^` n y strongly converges to ()qFT . Note. Theorem 1 improves and extends some relative results in Reference [6] and [7]. As follows, we will discuss iterative solution of some response diffusion equation . Let 2 ()ELI { (, )(,)xts tsI , (, )xts and 2(, )xts are Lebsgue intergrable on I},where [,][, ]Iabcd u , and ,xy E ,we define ,(,)(,) I x yxtsyt sdtds! ³³ .Then E is Hilbert space, and 22 ,(,) I xx xxt s dtds ! ³³ , ,(),, ,yjxyxxy E! ! . Consider the problem involving solution of some first order diffusion equation: 0 01 , (,0)(),(0,)( ) xx uxGxhx ts xsxsxtx t ww °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: 0 00 0000 (, )(, )(, )(, ) xt tsts xtsds uxtsdthtsxtsdtdsxGxdtds ³³ ³³³³ 001 00 ()() 0 st xs dsuxtdt ³³ ˄7˅ Let ^ (, )KxExts is continuous function on I ` , then K is nonempty closed convex subset of E . Let :HK Ko . 00 00000 (, )(,)()() t s tsts Hxuxtsdtx tsdsh txx ts dtdsxGxdtds ³ ³³³³³ 001 00 ()() . st xs dsuxtdt ³³ If G satisfies (A)˖ ,, ,xy KxGxyGy d then let :,TKKTxHx xo . If G satisfies (B)˖there is 1 0L! such that 1 xGx yGyLx yd for abitrary ,xy K . Then H is Lipschitz mapping on K ˈand then we have 0L! such that ,,xyK HxHyLxyd . Let 1 2 HH L ˈ 111 :, .TK KTxHxxo Theorem 2 Let integral equation˄7˅has solutionˈthen (i) If G satisfies (A), when 0 1 n n n D VE o or limsup 1 n n V of ,and there is ()pFT such that 22 () 0 nn fxpxpo , Implicit iterative sequence () nnn nnnn xfxxTx DEJ strongly converges to the fixed point of T which is solution of equation (7). (ii) If G satisfies˄B˅,when n D f ¦ and 10 nn nn VV DV o , explicit iterative sequence 1() nnnnnnn yfyyTy DEJ strongly converges to the fixed point of T which is solution of equation (7). Proof. (i) Now, ,,( )()xyKHxHyxy is nonnegative on 64 Cop y ri g ht © 2012 SciRes. ^` 1(, )0EtsIxy d and ^` 2 (, )0EtsIxy t . Then we have ,0,HxHy xy !t that is said that T is pseudo constraction mapping on K . Using Theorem 1, we obtain the result . (ii) Now , 11 ,, 2xyK HxHyxyd ˄8˅ 2 1) 1 ()Tx Ty 2 11 [()]Hx xHyy 22 11 11 ()2()()()HxHyyx HyHxxy 22 11 11 ()2()Hy Hxy xHy Hxx y 2 1111 ()( 2)()Hy HxHy Hxyxxy If 11 2,Hy Hxy xd then we obtain 22 11 ()()Tx Tyxyd . If 11 20,Hy Hxy xtt then we obtain 2 2 11 1111 ()( 2)( 2)()TxTyHy Hxy xHy Hxyxxyd 222 11 ()4()( )HyHxxyxy . Hence, by virtue of formula (8), we have 22 11 Tx Tyxyd . That is said that 1 T is nonexpansive mapping on K . Using Theorem 1, we obtain the result. Thereforeˈthrough improving some corresponding conditions in literature [6],[7] 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 [1] Deimling K. zeros of accretive operators [J].Manuscripta Math, 1974,13(4):365-374. [2] Chang S S. Cho Y J. Zhou H Y. Iterutive methods for nonlinear operator Equation in Banach space [M].New York: Science publishers, 2002. [3] Zhou H Y. 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