ITERATIVE PROCESS FOR CERTAIN NONLINEAR MAPPINGS WITH LIPSCHITZ CONDITION

Applied Mathemnatics and MechanicsPublished by Shanghai University.( English Edition, Vol 22, No 12, Dec 2001)Shanghai, ChinaArticle D: 0253 4827(2001)12-1458-10ITERATIVE PROCESS FOR CERTAIN NONLINEARMAPPINGS WITH LIPSCHITZ CONDITION *GU Feng(谷峰)( Department of Mathematics, Qiqihar University, Qiqihar ，Heilongjiang 161006, P R China)(Communicated by ZHANG Shi-sheng)Abstract: Using the new analysis techniques，the problem of iterative approximation ofsolutions of the equation for Lipschitz φ strongly accretive operators and of fixed points forLipschitz φ strongly pseudo contractive mappings are discussed. The main results of thispaper improve and extend the corresponding results obtained by Chang，Chidume， Deng ,Ding，Tan- Xu and Osilike .Key words: φ-strongly accretive operator; 中 . strongly pseudo-contractive mapping;Ishikawa itcrative sequence ; Mann iterative sequenceCLC number: 0177.91Document code: A1 Introduction and PreliminariesThroughout this paper, we assume that X is a real Banach space and X* is the dual space ofX,<', .> denotes the pairing of X and X* . The mappingJ:X→2X defined byJ(x) = {j∈X*: = |I.lljl| ,|j1 = IIxll}， x∈Xis called the normalized duality mapping .Definition 1.1 Let X be a real normal space and let K be a nonempty subset of X. LetT:K→X be a single-valued mapping.1) T is said to be accretive if for anyx,y ∈K, there existsj(x - y)∈J(x - y) such that≥0.(1)2) T is said to be strongly accretive if for any x,y∈K, there existsj(x一y)∈J(x -y) such that 0. Without loss of generality, we can assume that k∈(0,1) and such anumber k is called the strongly accretive constant of T.3) T is said to be(strongly ) pseudo-contractive if I- T ( where I denote the identitymapping) is a (strongly ) accretive mapping .中国煤化工* Received date: 1999-09-17; Revised date: 2(MYHCNMHG.Biography: GU Feng (1960- ), Associate Professor1458Iterative Process for Certain Nonlinear Mappings14594) T is said to be φ strongly accretive if for any x,y∈K, there exists j(x - y)∈J(x - y) and a strictly increasing function φ:[0,∞)→[0,∞) with φ(0) = 0 such that≥中(Ix- ylI). x-ylI，(3)where φ is called the strongly accretive function of T.5) T is said to be φ_ strongly pseudo-contractive if I- T is a φ- strongly accretive mapping .Obviously, every strongly accretive operator is φ strongly accretive， and every stronglypseudo-contractive mapping is φ strongly pseudo-contractive with φ:[0,∞)→[0,∞) definedby中(s) = ks (wherek∈(0,1)，s≥0).The concept of accretive mapping was introduced independently by Browder' and Kato' 2:in 1967. An early fundamental result in the theory' of accretive mappings which is due to Browderstates that the following initial value problemdu(t)+ Tu(t) = 0,u(0) = uo(4)dtis solvable if T is locally Lipschitzian and accretive on X .The following two iterative processes owe to Ishikawals! and Mannl41，respectively.Definition 1.2 Let X be a real Banach space, let K be a nonempty convex subset of Xand let T: K→K be a mapping. For any given xo∈K, the sequence { xn } defined by{xn+I = (1 - an)xn + anTy..(5)lyn = (1- β)xn + βnTx.n≥0is called the Ishikawa iterative sequence of T, where {an} and {β} are two real sequences in[0, 1] satisfying some additive conditions .In particular, ifβn = 0for alln ≥0, then the {xn} defined byxn+1 = (1- an)xn + anTxn，(6)is called the Mann iterative sequence of T.The convergence problems of Ishikawa and Mann iterative sequences were studiedextensively by many authors (see, for example, [3- 13]).In this article, by using the new analysis techniques, we study the convergence problems ofthe Ishikawa and Mann iterative sequences for 中-strongly accretive operators and φ -stronglypseudo contractive mappings with Lipschitz condition， respectively. The main results in thisarticle improve and extend the corresponding results in Changf6] ，Chidume'7.8， Deng andDing9] , Deng'0.11]，Tan and Xu[12] and Osilike[13]2 LemmasThe following Lemmas will be needed in the proofs of our main results .Lemma 2.151Let X be a real Banach space and J be a normality duality mapping. Thenfor any given x,y∈X, the following inequality holds:|x + yl|°≤|xI|2 + 2(y,j(x + y)),中国煤化工y). (7)Lemma 2.2[14]X is a uniformly smooth BanachMYHCNMHGsauniformlyconvex Banach space) if and only if J is single-valued and uniformly continuous on any bounded1460GU Fengsubset of X.Lemma 2.36) Let {an} be a nonnegative real sequence and let {in } be a real sequencein[0,1] such hat 2i in =∞。If there exists a positive integer no such thatan+1≤(1- tn)an + tnPn,Vn≥no,(8)whereρn≥0,Vn≥0andρn→0(n→∞), then we havean→0(n→∞).3 Convergence of Ishikawa Iterative Sequences for φ Strongly Pseudo-Contractive MappingsLemma 3.1 Let X be a real Banach space, let K be a nonempty subset of X and letT: K→X be a φ-strongly pseudo-contractive mapping. Then, for any given x,y∈K, thereexistsj(x - y)∈J(x - y) such that(Tx- Ty,(x-y)≤|x-yl2_φ(\x-yl)x-yW，where中is the strongly accretive function of I- T.Proof Since T is φ strongly pseudo-contractive, there is a strictly increasing functionφ:[0,∞)→[0,∞) with中(0) = 0 such that for each x,y∈K, there exists j(x - y)∈J(x - y) with<(1- T)x- (I- T)y,j(x-y)>≥$(|x- ylI)Ix- yI，that is-≤Ix-yI2_φ(Hx- ylI)IIx- ylI.The conclusion is proved.Theorem3.2 Let X be a uniformly smooth real Banach space, let K be a nonemptyclosed convex subset of X (it need not be bounded) and let T: K→K be a Lipschitian φstrongly pseudo-contractive mapping. Let L≥1 be the Lipschitz constant of T and let {an}.{β} be two real sequences in [0,1 ] satisfying the following conditions:(i) an→0，β。→0(n→∞);( |i)=∞IfF(T)≠0(thesetofallfixedpointsofTinK),then,foranygivenxo∈K,theIshikawa iterative sequence { xn } defined by[xn+1 = (1- an)xn + anTyn，<9)lyn = (1- βn)x。+ β,Txn,n≥0converges strongly to the unique fixed point of T in K中国煤化工Proof Takeq ∈F(T) and henceq = Tq. IfMHCN MH Ger no such that%n。= q,then we haveTterative Process for Certain Nonlinear Mappings1461yn。 = (1- Bn,)xn。+ βn。Txn。= (1-。)q+βn。9= 9,xn。+1 = (1-an.)xn。+ anTy。= (1-an,)q+an。9= q.By induction, we can prove that xn+. = q for all i≥1. This implies that xn→q(n→∞). Consequently，without loss of generality, we can assume thatxn≠qfor alln≥0, i.e..| x。- qH|> Ofor alln≥0. Because X is uniformly smooth, by Lemma 2.2, J is single-valued and uniformly continuous on any bounded subset of X. It follows from (9) and Lemma2.1 that .l|xn+1- ql|'2= I(1- an)(x。-q) + an(Tyn- q)|°≤(1- an)2Ilxn - q||2 +2an.b。=〈π°x-qT,xn-q(I ) First we consider the second term on the night side of (10). From Lemma 3.1 wehave≤lIxn- qll2- $(H|xn- q|)l|x。- qll.(11)Again since T is a Lipschitzian mapping, we have|Tyn - TxnlI ≤Lllyn-xn = Lβ。llTxn-xnII≤Lβ(HTxn-qll + |Ix。-ql|) ≤L(L+ 1)βnlx。- qIl.(12)Thus, in view of (11) and (12), we have + (Txn- q,J(xn - q)>≤L(L+ 1)βnl|xn - ql|12+ lIx。-qH|2-$(Ixn- qi)lxn~qll =[L(L+ 1)β +1}lxn-ql2- (lx-ql)llxo- q|l .(13)( I) Next we consider the third term on the right side of (10). We prove that bn→0(n→∞). In fact, we haveHyn-qli = I|(1-p.)(xn- q)+β(Txn-q)| ≤(1-βn)|xn~q__11中国煤化工IlIxn-qll,1YHCNMHG(14)and so, from (14), it follows that1462GU Feng .Tyn - ql≤y。- q!≤l2.(15)Txn- qTTxn - qllBy the assumptionan-➢0 (n→∞), from (15) we have .xn+1 - qxn-q|l xn+1- xnl|anITyn- xn |I| Ixn-qIf~Txn-qIT|xn-qIT|xn-qTx.-qIan-(|Ty。- q| + lIxn- q|I)≤an(L2 + 1)→0(n→∞)，(16)which implies that1(TXa+1-qr)-(知一g)→0(n→∞).xn-qTlBesides, from (15) we know that {(Tyn - q)/!xn - ql! }nzo is a bounded sequence in X.Therefore we havebn→0 (n→∞).(17)Substituting (13) into (10), we havel|xn+1- q|2≤{(1-an)2 +2an[L(L+ 1)β. + 1]+ 2a,b.}x|xn- q||"-2anH(lxn- qll)l|xn-qll =(1 + Anan)l|xn- qlI2 -2a,$(|xn- qlI)Ixn-qlI， (18)where λ。= an + 2β,L(L + 1) + 2b。. By condition( I ) and(17) we haveλ。→0(n→∞).Letσ = inf{中(H xn- q| )/||xn- qil:n≥0}, thenσ≥0. Next we prove that xn→q (n→∞). For the purpose, we consider the following two cases:Case1 σ > 0, without loss of generality, we can assume thatσ < 1, then中(||xn -q||)/|xn- ql!≥σfor alln≥0. Hence from (18) we have|ixn+1- ql|°≤(1 +入nan)I xn- q2- 2anσllxn- q||'=(1 + λnan - 2anσ)|lx,- q|2.Because λn→0 (n→∞), there exists a positive integer no such that for any n≥no，λn < σ.Therefore, we have||xn+1- q|l'≤(1 - oan)lxn- qlI'2,Vn≥no.Letting It xn - q||2 = an, In = oan, andpn = 0, it follows from Lemma2.3 that an→0(n→∞), i.e., xn→q(n→∞).Case2 σ = 0, i.e., inf{$(ixn- ql)/|xn- qll :n≥0}= 0, so that thereexists a subsequence {x, } of the sequence {xn } such that中国煤化工Ixn, -qlI→0TYHCNMHGSinceλn→0, an→0(n→∞), for any givenε > 0, there exists a positive integern,≥noIterative Process for Certain Nonlinear Mappings1463such thatlxn -q||< E,(19)andforalln≥n;wehave(E/2)1λn< φ2ε’an< 2([2+ 1)(20)Next we prove that for all integer k≥0(21)In fact, from (9) and (14) we havelIxn+t-q||≤Ixn-q|| + anl|Tyo-xnI≤IIxn-qlI +a,{lTyn-qIl + |Ix。- qH}≤|Ix-qll +an{Lllyn -q|! + |x。-q!}≤i|xn-qll + an(L2+ 1)xn-qll，that is|xn- q||≥IIx+1-qll - an(L2 + 1)x。-qH.(22)New we prove that|Ixn+1-qH≤ε，(23)Suppose on the contrary, |xn+1 - ql| > e, then from (22)，(19) and (20) we haveilxn -qI≥|xn+1-qlI -an(L2 +1)lxn -ql >(24)2°= 2If follows from the strictly increasing property ofψ thatp(Ixn - qHl) >中(ε/2) > 0. By(18), (19), (20) and (24) we havelxn11-qI°≤(1+Aan)x -qI2-2an (|x。-q)!xn -qH≤(1+nan,)e2-2an.中(号)号=e2 -an (4(e/2)e/2-λne2)-an中(号)号≤e2.This contradicts the assumption, hence |I xn +1 - qll≤E holds.Suppose|xn ,h。-q||≤ε for some ko≥0, by tbe same method as in the proof of (23),we can prove that II xn +(k.1)-qI|l ≤E holds. Hence (21) is true also for allk≥0. By thearbitrariness of ε > 0, we know that xn→q(n→∞).(匹) Finally, we prove that q is the unique fixed中国煤化工ifq1 is also afixed point of T in K, by Lemma 3.1, we haveMYHCNMHG|q-q||2= ≤|x- yI2- φ(ilx-yl)|x-ylI.Proof Since T is φ- strongly accretive, for any givenx,y∈X there existsaj(x - y)∈(Tx - Ty,j(x- y)>≥φ(||x- yil)l|x- yll.Hence we have =≤||x- yl|'- φ(|x- yll)lx- yII .This completes the proof.Theorem4.2 Let X be a real uniformly smooth, RonachX→XbeaLipschitzian φ_ strongly accretive mapping. Let L > 01中国煤化工of T. For anygivenf∈X, define a mappingS: X→XbySx = fYHCNMHG{a。}. {p.}be two real sequences in [0,1] satisfying the following conditions: :Iterative Process for Certain Nonlinear Mappings1465(i ) an→0, β。→0(n→∞);(ii )IfS(T)≠0 (the set of solutions of the equationf = Tx in X), then, for any given xo∈X,the Ishikawa iterative sequence { xn } defined by{*n+1 = (1- an)xn + anSyn，(25)lyn = (1- βn)xn + βnSxn,n≥0converges strongly to the unique solution of the equationf = Tx in X.Proof Takingq∈ S(T), we havef = Tq andsoq = Sq. If there exists a positive integerno such that x。= q，then, by the same way as stated in Theorem 3.2, we can prove thatxn。+i = q foralli≥1 and soxn→q (n→∞). Therefore, without loss of generality, we canassume thatxn≠q for alln≥Oand so | xn- q II > 0for alln≥0. Because X is uniformlysmooth, it follows from Lemma 2.2 that J is single-valued and uniformly continuous on anybounded subset of X. From (25) and Lemma 2.1, it follows thatiIxx1←qI°≤(1-an)}Ixn-qH2 +2a, =(1 -an)2# x。- qH2 + 2an = (Syn -Sxn,J(xn- q)> + (Sxn - q,J(xn - q)). (27)By Lemma4.1, we have≤|Ixn-qH2- (l|xn-ql)|x- qll，(28)and|lSyn -Sx,H ≤(1+ LD)lyx-xnl.(29)Hence we have≤{(1 + l)(2+ L)B。+ 1}Hx_一a!2-中(Hxn-q中国煤化工(31)(I) Next we prove thatdn →0(n→∞). In.MYHCNMH G the proof of(14)- (16), we can prove that1466GU Feng .|yn-qlI≤(1+ l)Hxn-qlI,(32)llSyn- qll(1 + l)ly- qlTxn-qT≤-xn-qIf≤(1+L)，(33)xn+1-q≤an[(1 + L)2+ 1]→0(n→∞).(34)」l|xn-qBy the uniform continuity of J, from (33) and (34)， it follows that dn→0 (n→∞).Substituting (31) into (26), we obtainlxm- ql2≤{(1-a,)2 +2a。[(1 + L)(2+ L)β. + 1] + 2andn}xI|x。- q|I2-2an$(Ixn- ql)lxn- q|I =(1 + 8。an)|xn-q12-2ap(llx.-qI)xn-qI.where δn = an +2βn(1 + L)(2+ L) + 2dn. By condition( i ) andd,-→0(n→∞) we haveδ↔→0(n→∞).Letσ = inf{中( H xn - ql|)/l|xn- qll :n≥0}, then, by the same way as is stated inTheorem 3.2, we can prove that xn→q (n→∞).In addition, it is easy to prove that q is the unique solution of the equation f = Tx in X.This completes the proof .Corollary 4.3 Let X be a real uniformly smooth Banach space and let T: X→Xbe aLipschitzian φ_ strongly accretive mapping. Let L > 0 be the Lipschitz constant of T. For anygivenf∈X, define a mappingS: X→XbySx = f-Tx + x，Vx∈X. Let{an} be a real .sequence in [0,1] satisfying the following conditions :(i ) an→0(n→∞);( li)IfS(T)≠0, then, for any given xo∈X, the Mann iterative sequence {xn } defined byxn+1 = (1- an)xn + anSxn,n≥0converges strongly to the unique solution of the equationf = Tx in X..Proof Taking βn = 0for alln≥0 in Theorem 4.2, then the conclusions of Corollary 4.3can be obtained from Theorem 4 . 2 immediately .Remark2 Since a strongly accretive mapping is a special of φ-strongly accretive mapping with中(s) = ks(k∈(0,1),s≥0). Hence Theorem 4.2 improve and extends the results ofChangb,horen5.2]，Chidumel8:,Dbxcmn2]Deng and Ding9,heorn2]， Deng[10Toro "Tan anXul!2.TDeran4.1] and Osilkel13.Theorem 1].AcknowledgmentThe author would like to express his sincere gratitude to Professor ZHANGShi-sheng for his guidance .References :中国煤化工[ 1 ] Browder F E. Nonlincar mappings of nonexpansive :.MYHCN M H Gpaces[J]. 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J Math Anal Appl ,1991,157:189 - 210.中国煤化工MYHCNMHG