﻿ Approximative Method of Fixed Point for Φ-Pseudocontractive Operator and an Application to Equation with Accretive Operator Journal of Applied Mathematics and Physics, 2014, 2, 21-25 Published Online January 2014 (http://www.scirp.org/journal/jamp) http://dx.doi.org/10.4236/jamp.2014.21004 OPEN ACCESS JAMP Approximative Method of Fixed Point for Φ-Pseudocontractive Operator and an Application to Equation with Accretive Operator Yixin Wen, Aifang Feng, Yuguang Xu Department of Mathematics, Kunming University, Kunming, China Email: donwen-620@163.com Received October 2013 ABSTRACT In this paper, Φ-pseudo-contractive operators and Φ-accretive operators, more general than the strongly pseu-do-contractive operators and strongly accretive operators, are introduced. By setting up a new inequality, au-thors proved that if →:TX X is a uniformly continuous Φ-pseudo-contractive operator then T has unique fixed point q and the Mann iterative sequence with random errors approximates to q. As an application, the iterative solution of nonlinear equation with Φ-accretive operator is obtained. The results presented in this paper improve and generalize some corresponding results in recent literature. KEYWORDS Duality Mapping; Φ-Pseudo-Contractive Operator; Φ-Accretive Operator; Mann Iterative Sequence with Random Error Terms 1. Introduction and Preliminaries In 1994, Chidume  solved a problem dealt with the fixed point for the class of Lipschitz strictly (strongly) pseudo-contractive operators in uniformly smooth Banach space X. That is, he proved that the Ishikawa itera-tive sequence converges strongly to the unique fixed point of T in K where KX⊂ and T:K K→ is Lipschitz strictly (strongly) pseudo-contractive. Chang , in 1998, improved the result, i.e., he proved that the conclusion of Chidume holds if T is uniformly continuous and the fixed point set of T is nonempty (i.e., ( )FT ≠∅). Recently, Liu  proved, if the strongly pseudocontractive operators are replaced by the more general φ-strongly pseudo-contractive operators then the conclusion of Chidume still holds. The objective of this paper is to introduce Φ-pseudo-contractive operators—a class of operators which are more general than the φ-strongly pseudo-contractive operators and to study the problems of existence, unique-ness and the iterative approximate method of fixed point by setting up a new inequality in arbitrary Banach space. As an application, the iterative solution of nonlinear equation with Φ-accretive operator is obtained. The results presented in this paper improve and generalize the conclusions of Chidame, Chang and Liu. To set the framework, we recall some basic notations as follows. Throughout this paper, we assume that X is a real Banach space with dual X∗, ( ),⋅⋅ denotes the genera-lized duality pairing. The mapping :2xJX∗→ defined by ( ){ }= :,,JxjXx jxjjxxX∗∈== ∀∈ (1) is called the normalized duality mapping . Now, we introduce Φ-pseudo-contractive operators as follows. Definition 1. Let K be nonempty subset of X. An operator :TK K→ is said to be Φ-pseudo-contractive, if there exists a strictly increasing function [)[): 0,0,Φ∞→ ∞ with ( )00Φ= and ( )( )jxy Jxy−∈ − such that Y. X. WEN ET AL. OPEN ACCESS JAMP 22 ( )( )( )2,,TxTyjxyxyxy xyKΦ−−≤−−−∀∈. (2) An operator :AKK→ is said to be Φ-accretive, if ( )( )( ),,Ax Ayjx yx yxyKΦ−− ≥−∀∈. (3) It is easy to verify that the operator T is Φ-pseudo-accretive if and only if IT− is Φ-accretive where I is an identity mapping on X. Hence, the mapping theory for accretive operators is intimately connected with the fixed point theory for pseudo-contractive operators. We like to point out: every strongly pseudo-contractive operator is φ-strongly pseudo-contractive with [)[): 0,0,φ∞→ ∞ defined by ( )s ksφ= where ( )0, 1k∈, and every φ-strongly pseudo-contractive operator must be the Φ-pseudo-contractive operator with [)[): 0,0,Φ∞→ ∞ defined by ( )( )s ssΦφ=. Obviously, if a Φ-pseudo-contractive operator has a fixed point then it is unique. Definition 2. Let :TK K→ be an operator. For any given 0xK∈ the sequence { }nx defined by ( )( )110nnnnnnnxxTxu nααγ+=−++≥ (4) is called Mann iteration sequence with random errors. Here { }nu is a bounded sequence in X and is said random error terms of iterative process, and the parameters { }nα and { }nγ both are sequences in [ ]0, 1. By the way, Xu introduced another definition of Mann iterative sequence with random errors in . In particular, the parameters 0nγ= for all 0n≥ in Equation (4) then { }nx is called Mann  iteration sequence. 2. Main Results First, we have an existence theorem of fixed point as follow. Theorem 1. If :TX X→ is a continuous Φ-pseudo-contractive operator with bounded range then T has an unique fixed point in X. Proof. Define :nTX X→ by 1nnTxxc TxxXand n= −∀∈≥ where ( )0, 1nc∈ and 1nnlim c→∞=. Note that T is Φ-pseudo-contractive, thus ( )( )( )( )( )( )( )22,,1nn nnnnTxTyjxyxy cTxTyjxyc xycxyc xyΦΦ−−=−− −−≥−− +−≥− (5) for all , ,1xy Xn≥≥ and some () ()jx yJxy−∈ +. Clearly, nT is a continuous strongly accretive operator. It follows from the Theorem 13.1 of Deimling  that there exists an nxX∈ such that 0nnTx = for any1n≥. Next, the sequence { }1nnx− is bounded. In fact, if Tx M≤ then nxM≤ for all xX∈ and ( )limlim 10nn nnnnxTxc Tx→∞ →∞−= −= (6) Since T is Φ-pseudo-contractive, that is, ( )IT− is Φ-accretive, so ( )( )( )( )( )( )( ),2mkmk mkmk mkmkmm kkxxITxI Txjxxx Txx TxxxM xTxxTxΦ−≤ −−−−≤− +−−≤− +− (7) for any { }mn nxx x−∈. Equations of (6) and (7) ensure that { }1nnx∞= is a Cauchy sequence. Consequently, { }1nnx∞= converges to some qX∈. By the continuity of T we have 1nnnnnTqlim Txlimxqc→∞ →∞= ==. Suppose that there exists a qX∗∈ such that Tq q∗∗∈ then Y. X. WEN ET AL. OPEN ACCESS JAMP 23 ( )( )( )2, =0qq qq TqTqjqqΦ∗ ∗∗∗−≤−−−− which means that qq∗=. i.e., q is unique fixed point of T. The proof is completed. The following two Lemmas will play crucial roles in the proof of Theorem 2. Lemma 1.  If X be a real Banach space then there exists ( )( )jxyJxy+∈ + such that ( )222, ,xyxyjxy xyX+ ≤++∀∈. (8) Second, to set up a new inequality as follows. Lemma 2. Let [)[): 0,0,Φ∞→ ∞ be a strictly increasing function with ( )00Φ= and let { }nb and { }nc be two nonnegative real sequences satisfying ,and linnn nncb b→∞<∞ =∞∑∑. (9) Suppose { }na is a nonnegative real sequences. If there exists a integer 00N> satisfying ( )()221 10nnnnnnaaobcban NΦ++≤++ −∀≥ (10) then lim 0nna→∞=. Proof. Let { }inf 2naσ=. If 0σ>, then ( )( )1naΦ Φσ+> for all 0n≥. From the conditions of Equation (9) there exists an integer 0N> 0 such that ( )( )2nnobbnNΦσ≤ ∀≥. (11) So, we have ( )2211Φ2n nnnaacb nNσ+≤+ −∀≥. By induction, we obtain ( )2==1Φ2jN jjN jNba cσ+∞ +∞≤+≤ +∞∑∑. (12) Equation (12) is in contradiction with =0 nNb+∞= +∞∑. It implies that 0σ=. Therefore, there exists a subse-quence { }{ }jnnaa⊂ such that lin 0jnja→∞=. So, for any given 0ε> there exists an integer 00j≥ such that ( )( )000jnnnn jajj andobcbn nεε<∀≥+≤Φ∀ ≥>. (13) If 0J is fixed, we will prove that 0jnkaε+< for all integers 1k≥. The proof is by induction. For 1k=, suppose 0jnkaε+<. It follows from Equations (10) and (13) that ( )( )000 0002222 21Φjjj jjjnnn nnnaa obcbaε εε+≤≤++−≤<. It is a contradiction. Hence, 01jnaε+≤ holds for 1k=. Assume now that 0jnpaε+≤ for some integer1p>. We prove that 0+1jnpaε+≤. Again, assuming the contrary, Using Equations (10) and (13), as above, it leads to a contradiction as follows ( )( )+1000 00022222npjjj jjnpjnnpnpnpaa obcbaεεε++++ +≤≤ ++−Φ≤< Where ooj jon pnj+≥ >. Therefore, 0jnpaε+< holds for all integers 1k≥, i.e., 0lin lin0jn nknnaa+→∞ →∞= = The Proof is completed. Y. X. WEN ET AL. OPEN ACCESS JAMP 24 Theorem 2. Let :TX X→ be an → a uniformly continuous Φ-pseudo-contractive operator with bounded range. Suppose that the iterative sequence { }nx is defined by Equation (4) satisfying ,lin0 andnn nn.γα α→∞<∞ ==∞∑∑ (14) then { }nx converges strongly to unique fixed point of T. Proof. From Theorem 1, we know that there exists a unique fixed point q of T. Putting { }{ }00: :0nMsup TxxXsupunx=∈ +≥+ for any given 0xX∈, then ( )100 0001-xxTx Mαα≤ +≤. Using induction, we have 00xM≤ for all 0n≥. Let 0MM q= +. Since ( )120n nnnnnxxaxTxMasnα+−=− ≤→→∞, therefore, ( )1:0nnneTxTxasn+− →→∞ by the uniformly continuity of T. From Equation (14) there exists an integer 00N> such that 0106nnNα≤≤ ∀≥. (15) By Equations (4), (8) and (15) we have ()() ()( )( )( )( )( )( )( )( )( )( )( )( )2212211221121222212122211+2 ,2,12,2, 21 2+22 2223 2nn nnnnnn nnnnnn nnnnn nnnn nnn nnnnnnnn nnn nnx qxqaTxquxqu jxqTxq j xqxqTxTxjxqTxq j xqMxqxqMe Mxqx qMMeαγαγααααγαα αα γααα++++++++−= −−+−+≤− −−+− −≤− −+− −+ −−+≤− +−−++−Φ −≤−+ ++( )( )( )2121ΦΦnnnnnnnnM xqxq ocxqγααα++−−= −++−− (16) for all nN≥ where ( )2223nn nnoMMeα αα= + and 22nncMγ=. It follows from the Lemma 2 that limnxq→∞−. The Proof is completed. Last, as an application, the existence, the uniqueness and the approximate method of solution of nonlinear equation with Φ-accretive operator is obtained. That is Theorem 3. Suppose that :AX X→ is an → a uniformly continuous Φ-accretive operator and the range of either A or IA− be bounded. For any given fX∈, the equation Ax f= has unique solution in X, and if { }nx is defined by ( )()()110n nnxxfIAxu nαα γ+=− ++−+≥ satisfying the conditions of equation (14) then it converges strongly to the solution. Proof. We define :SX X→ by Sxfx Ax= +− for allxX∈. Clearly, S is Φ-pseudo-contractive and continuous if A is Φ-accretive and continuous, and the range of S is bounded if ( )IA− is. It is easy to see that x∗ is a solution of the equation Ax f= if and only if that x∗ is a fixed point of S. It follows from the Theorem 2 that Ax f= has an unique solution xX∗∗∈ and the iterative sequence { }nx is defined by Equa-tion (4) converges strongly to x∗. i.e., the sequence { }nx is an iterative solution of the equation Ax f=. The proof is completed. Y. X. WEN ET AL. OPEN ACCESS JAMP 25 Remark. Theorem 2 improves a number of results (for example, Theorem 4.2 of , Theorem 2 of  and Corollary 3.3 and 3.4 of ) in the following senses. 1) The existence and the convergence of the fixed point for Φ-pseudo-contractive operator are studied simul-taneously. 2) The operators may not be strongly pseudo-contractive or φ-strongly pseudo-contractive. 3) The continuity of operator may not be Lipschitzian. 4) The errors come from the iterative process have been considered appropriately. REFERENCES  C. E. Chidume, “Approximation of Fixed Points of Strongly Pseudo-Contractive Mappings,” Proceedings of the American Mathematical Society, Vol. 120, 1994, pp. 545-551. http://dx.doi.org/10.1090/S0002-9939-1994-1165050-6  S. S. Chang, Y. J. Cho and B. S. 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