A correction for a result on convergence of Ishikawa iteration for strongly pseudocontractive maps

Stefan Soltuz
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Abstract

We give a correction to the main result from [13].

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Soltuz M. Stefan
Tiberiu Popoviciu Institute of Numerical analysis

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Ishikawa iteration, strongly pseudocontractive maps

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Ş.M. Şoltuz, A correction for a result on convergence of Ishikawa iteration for strongly pseudocontractive maps, Math. Commun. 7 (2002) no. 1, 61-64.

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Mathematical Communications

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[4] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44(1974), 147-150.
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[7] M. O. Osilike, A note on the stability of iteration procedures for strongly pseudocontractions and strongly accretive type equations, J. Math. Anal. Appl.
250(2000), 726-730.
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[9] S. M. S¸oltuz, Some sequences supplied by inequalities and their applications, Revue d’analyse num´erique et de th´eorie de l’approximation, Tome 29(2000),
207-212.
[10] S. M. Soltuz, Three proofs for the convergence of a sequence, OCTOGON Math. Mag. 9(2001), 503-505.
[11] H. Y. Zhou, Y. Jia, Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption, Proc. Amer. Math. Soc. 125(1997), 1705-1709.
[12] H. Y. Zhou, Y. J. Cho, Ishikawa and Mann iterative process with errors for nonlinear φ− strongly quasi-accretive mappings in normed spaces, J. Korean Math. Soc. 36(1999), 1061-1073.
[13] H. Y. Zhou, Stable iteration procedures for strong pseudocontractions and nonlinear equations involving accretive operators without Lipschitz assumption, J. Math. Anal. Appl. 230(1999), 1-30.

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mc13_sol

A correction for a result on convergence of Ishikawa iteration for strongly pseudocontractive maps

Ştefan M. Şoltuz*

Abstract

We give a correction to the main result from [13]. Key words: Ishikawa iteration, strongly pseudocontractive maps AMS subject classifications: Primary 47H10, Secondary 47H06

Received December 11, 2001 Accepted June 27, 2002

1. Introduction

Let X X XXX be a real Banach space. Let B B BBB be a nonempty, convex subset of X X XXX. Let T : B → B T : B → B T:B rarr BT: B \rightarrow BT:B→B be a map. Let x 1 ∈ B x 1 ∈ B x_(1)in Bx_{1} \in Bx1∈B. We consider the following iteration, see [4]:
(I) x n + 1 = ( 1 − α n ) x n + α n T y n , y n = ( 1 − β n ) x n + β n T x n , n = 1 , 2 , … . (I) x n + 1 = 1 − α n x n + α n T y n , y n = 1 − β n x n + β n T x n , n = 1 , 2 , … . {:[(I)x_(n+1)=(1-alpha_(n))x_(n)+alpha_(n)Ty_(n)","],[y_(n)=(1-beta_(n))x_(n)+beta_(n)Tx_(n)","quad n=1","2","dots.]:}\begin{align*} x_{n+1} & =\left(1-\alpha_{n}\right) x_{n}+\alpha_{n} T y_{n}, \tag{I}\\ y_{n} & =\left(1-\beta_{n}\right) x_{n}+\beta_{n} T x_{n}, \quad n=1,2, \ldots . \end{align*}(I)xn+1=(1−αn)xn+αnTyn,yn=(1−βn)xn+βnTxn,n=1,2,….
We suppose that ( α n ) n , ( β n ) n ⊂ ( 0 , 1 ) α n n , β n n ⊂ ( 0 , 1 ) (alpha_(n))_(n),(beta_(n))_(n)sub(0,1)\left(\alpha_{n}\right)_{n},\left(\beta_{n}\right)_{n} \subset(0,1)(αn)n,(βn)n⊂(0,1), and the sequence ( α n ) n α n n (alpha_(n))_(n)\left(\alpha_{n}\right)_{n}(αn)n satisfies
(1) 0 < w ≤ α n ≤ 1 (1) 0 < w ≤ α n ≤ 1 {:(1)0 < w <= alpha_(n) <= 1:}\begin{equation*} 0<w \leq \alpha_{n} \leq 1 \tag{1} \end{equation*}(1)0<w≤αn≤1
For β n = 0 , ∀ n ∈ N β n = 0 , ∀ n ∈ N beta_(n)=0,AA n in N\beta_{n}=0, \forall n \in Nβn=0,∀n∈N we get Mann iteration, see [5]. Ishikawa iteration with condition (1) is studied in [13]. In [7] it was proven that two assumptions of the main theorem from [13] are contradictory. In this note we will prove that renouncing to one assumption from [13] and supposing true an assumption à la [3], the above theorem from [13] is true.
The map J : X → 2 X ∗ J : X → 2 X ∗ J:X rarr2^(X^(**))J: X \rightarrow 2^{X^{*}}J:X→2X∗ given by
J x := { f ∈ X ∗ : ⟨ x , f ⟩ = ‖ x ‖ 2 , ‖ f ‖ = ‖ x ‖ } , ∀ x ∈ X , J x := f ∈ X ∗ : ⟨ x , f ⟩ = ‖ x ‖ 2 , ‖ f ‖ = ‖ x ‖ , ∀ x ∈ X , Jx:={f inX^(**):(:x,f:)=||x||^(2),||f||=||x||},AA x in X,J x:=\left\{f \in X^{*}:\langle x, f\rangle=\|x\|^{2},\|f\|=\|x\|\right\}, \forall x \in X,Jx:={f∈X∗:⟨x,f⟩=‖x‖2,‖f‖=‖x‖},∀x∈X,
is called the normalized duality mapping. The Hahn-Banach theorem assures that J x ≠ ∅ , ∀ x ∈ X J x ≠ ∅ , ∀ x ∈ X Jx!=O/,AA x in XJ x \neq \emptyset, \forall x \in XJx≠∅,∀x∈X. It is easy to see that we have
(2) ⟨ j ( x ) , y ⟩ ≤ ‖ x ‖ ‖ y ‖ , ∀ x , y ∈ X , ∀ j ( x ) ∈ J ( x ) (2) ⟨ j ( x ) , y ⟩ ≤ ‖ x ‖ ‖ y ‖ , ∀ x , y ∈ X , ∀ j ( x ) ∈ J ( x ) {:(2)(:j(x)","y:) <= ||x||||y||","AA x","y in X","AA j(x)in J(x):}\begin{equation*} \langle j(x), y\rangle \leq\|x\|\|y\|, \forall x, y \in X, \forall j(x) \in J(x) \tag{2} \end{equation*}(2)⟨j(x),y⟩≤‖x‖‖y‖,∀x,y∈X,∀j(x)∈J(x)
Definition 1. Let X X XXX be a real Banach space, let B B BBB be a nonempty subset. A A AAA map T : B → B T : B → B T:B rarr BT: B \rightarrow BT:B→B is called strongly pseudocontractive if for all x , y ∈ B x , y ∈ B x,y in Bx, y \in Bx,y∈B, there exists j ( x − y ) ∈ J ( x − y ) j ( x − y ) ∈ J ( x − y ) j(x-y)in J(x-y)j(x-y) \in J(x-y)j(x−y)∈J(x−y) such that
(3) ∃ γ ∈ ( 0 , 1 ) : ⟨ T x − T y , j ( x − y ) ⟩ ≤ γ ‖ x − y ‖ 2 . (3) ∃ γ ∈ ( 0 , 1 ) : ⟨ T x − T y , j ( x − y ) ⟩ ≤ γ ‖ x − y ‖ 2 . {:(3)EE gamma in(0","1):(:Tx-Ty","j(x-y):) <= gamma||x-y||^(2).:}\begin{equation*} \exists \gamma \in(0,1):\langle T x-T y, j(x-y)\rangle \leq \gamma\|x-y\|^{2} . \tag{3} \end{equation*}(3)∃γ∈(0,1):⟨Tx−Ty,j(x−y)⟩≤γ‖x−y‖2.
The following Lemma could be found in [6], [12], with different proofs. A particular form of this lemma is in [11].
Lemma 1. [6], [11], [12] If X X XXX is a real Banach space, then the following relation is true
(4) ‖ x + y ‖ 2 ≤ ‖ x ‖ 2 + 2 ⟨ y , j ( x + y ) ⟩ , ∀ x , y ∈ X , ∀ j ( x + y ) ∈ J ( x + y ) . (4) ‖ x + y ‖ 2 ≤ ‖ x ‖ 2 + 2 ⟨ y , j ( x + y ) ⟩ , ∀ x , y ∈ X , ∀ j ( x + y ) ∈ J ( x + y ) . {:(4)||x+y||^(2) <= ||x||^(2)+2(:y","j(x+y):)","quad AA x","y in X","AA j(x+y)in J(x+y).:}\begin{equation*} \|x+y\|^{2} \leq\|x\|^{2}+2\langle y, j(x+y)\rangle, \quad \forall x, y \in X, \forall j(x+y) \in J(x+y) . \tag{4} \end{equation*}(4)‖x+y‖2≤‖x‖2+2⟨y,j(x+y)⟩,∀x,y∈X,∀j(x+y)∈J(x+y).
The following result is from [9]. Three other proofs could be found in [10].
Proposition 1. [9], [10]. Let ( a n ) n a n n (a_(n))_(n)\left(a_{n}\right)_{n}(an)n be a nonnegative sequence which satisfies
(5) a n + 1 ≤ ( 1 − w ) a n + σ n S , (5) a n + 1 ≤ ( 1 − w ) a n + σ n S ,  {:(5)a_(n+1) <= (1-w)a_(n)+sigma_(n)S", ":}\begin{equation*} a_{n+1} \leq(1-w) a_{n}+\sigma_{n} S \text {, } \tag{5} \end{equation*}(5)an+1≤(1−w)an+σnS, 
where w ∈ ( 0 , 1 ) , S > 0 w ∈ ( 0 , 1 ) , S > 0 w in(0,1),S > 0w \in(0,1), S>0w∈(0,1),S>0 are fixed numbers, σ n ≥ 0 , ∀ n ∈ N , lim n → ∞ σ n = 0 σ n ≥ 0 , ∀ n ∈ N , lim n → ∞   σ n = 0 sigma_(n) >= 0,AA n in N,lim_(n rarr oo)sigma_(n)=0\sigma_{n} \geq 0, \forall n \in N, \lim _{n \rightarrow \infty} \sigma_{n}=0σn≥0,∀n∈N,limn→∞σn=0. Then lim n → ∞ a n = 0 lim n → ∞   a n = 0 lim_(n rarr oo)a_(n)=0\lim _{n \rightarrow \infty} a_{n}=0limn→∞an=0.

2. Main result

We are able now to give the following result.
Theorem 1. Let X X XXX be a real Banach space, and let T : X → X T : X → X T:X rarr XT: X \rightarrow XT:X→X be a continuous, strongly pseudocontractive with bounded range map. If lim n → ∞ ‖ T y n − T x n + 1 ‖ = 0 lim n → ∞   T y n − T x n + 1 = 0 lim_(n rarr oo)||Ty_(n)-Tx_(n+1)||=0\lim _{n \rightarrow \infty}\left\|T y_{n}-T x_{n+1}\right\|=0limn→∞‖Tyn−Txn+1‖=0, and γ ∈ ( 0 , 1 / 2 ) γ ∈ ( 0 , 1 / 2 ) gamma in(0,1//2)\gamma \in(0,1 / 2)γ∈(0,1/2), then the iteration ( I ) ( I ) (I)(I)(I) strongly converges to the unique fixed point of T T TTT.
Proof. . The existence follows from [2] and the uniqueness from strongly pseudocontractivity. Let x ∗ = T x ∗ x ∗ = T x ∗ x^(**)=Tx^(**)x^{*}=T x^{*}x∗=Tx∗. Using (4) for the first inequality, (2) and (3) for the third one, we can see:
‖ x n + 1 − x ∗ ‖ 2 = ‖ ( 1 − α n ) ( x n − x ∗ ) + α n ( T y n − x ∗ ) ‖ 2 ≤ ( 1 − α n ) 2 ‖ x n − x ∗ ‖ 2 + 2 α n ⟨ T y n − x ∗ , j ( x n + 1 − x ∗ ) ⟩ ≤ ( 1 − α n ) 2 ‖ x n − x ∗ ‖ 2 + 2 α n ⟨ T y n − T x n + 1 , j ( x n + 1 − x ∗ ) ⟩ + 2 α n ⟨ T x n + 1 − x ∗ , J ( x n + 1 − x ∗ ) ⟩ ≤ ( 1 − α n ) 2 ‖ x n − x ∗ ‖ 2 + 2 α n ‖ T y n − T x n + 1 ‖ ‖ x n + 1 − x ∗ ‖ + 2 α n γ ‖ x n + 1 − x ∗ ‖ 2 , ∀ j ( x n + 1 − x ⋆ ) ∈ J ( x n + 1 − x ⋆ ) . x n + 1 − x ∗ 2 = 1 − α n x n − x ∗ + α n T y n − x ∗ 2 ≤ 1 − α n 2 x n − x ∗ 2 + 2 α n T y n − x ∗ , j x n + 1 − x ∗ ≤ 1 − α n 2 x n − x ∗ 2 + 2 α n T y n − T x n + 1 , j x n + 1 − x ∗ + 2 α n T x n + 1 − x ∗ , J x n + 1 − x ∗ ≤ 1 − α n 2 x n − x ∗ 2 + 2 α n T y n − T x n + 1 x n + 1 − x ∗ + 2 α n γ x n + 1 − x ∗ 2 , ∀ j x n + 1 − x ⋆ ∈ J x n + 1 − x ⋆ . {:[||x_(n+1)-x^(**)||^(2)=||(1-alpha_(n))(x_(n)-x^(**))+alpha_(n)(Ty_(n)-x^(**))||^(2)],[ <= (1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)(:Ty_(n)-x^(**),j(x_(n+1)-x^(**)):)],[ <= (1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)(:Ty_(n)-Tx_(n+1),j(x_(n+1)-x^(**)):)],[+2alpha_(n)(:Tx_(n+1)-x^(**),J(x_(n+1)-x^(**)):)],[ <= (1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)||Ty_(n)-Tx_(n+1)||||x_(n+1)-x^(**)||],[+2alpha_(n)gamma||x_(n+1)-x^(**)||^(2)","quad AA j(x_(n+1)-x^(***))in J(x_(n+1)-x^(***)).]:}\begin{aligned} & \left\|x_{n+1}-x^{*}\right\|^{2}=\left\|\left(1-\alpha_{n}\right)\left(x_{n}-x^{*}\right)+\alpha_{n}\left(T y_{n}-x^{*}\right)\right\|^{2} \\ & \leq\left(1-\alpha_{n}\right)^{2}\left\|x_{n}-x^{*}\right\|^{2}+2 \alpha_{n}\left\langle T y_{n}-x^{*}, j\left(x_{n+1}-x^{*}\right)\right\rangle \\ & \leq\left(1-\alpha_{n}\right)^{2}\left\|x_{n}-x^{*}\right\|^{2}+2 \alpha_{n}\left\langle T y_{n}-T x_{n+1}, j\left(x_{n+1}-x^{*}\right)\right\rangle \\ & +2 \alpha_{n}\left\langle T x_{n+1}-x^{*}, J\left(x_{n+1}-x^{*}\right)\right\rangle \\ & \leq\left(1-\alpha_{n}\right)^{2}\left\|x_{n}-x^{*}\right\|^{2}+2 \alpha_{n}\left\|T y_{n}-T x_{n+1}\right\|\left\|x_{n+1}-x^{*}\right\| \\ & +2 \alpha_{n} \gamma\left\|x_{n+1}-x^{*}\right\|^{2}, \quad \forall j\left(x_{n+1}-x^{\star}\right) \in J\left(x_{n+1}-x^{\star}\right) . \end{aligned}‖xn+1−x∗‖2=‖(1−αn)(xn−x∗)+αn(Tyn−x∗)‖2≤(1−αn)2‖xn−x∗‖2+2αn⟨Tyn−x∗,j(xn+1−x∗)⟩≤(1−αn)2‖xn−x∗‖2+2αn⟨Tyn−Txn+1,j(xn+1−x∗)⟩+2αn⟨Txn+1−x∗,J(xn+1−x∗)⟩≤(1−αn)2‖xn−x∗‖2+2αn‖Tyn−Txn+1‖‖xn+1−x∗‖+2αnγ‖xn+1−x∗‖2,∀j(xn+1−x⋆)∈J(xn+1−x⋆).
There results
( 1 − 2 α n γ ) ‖ x n + 1 − x ∗ ‖ 2 ≤ ( 1 − α n ) 2 ‖ x n − x ∗ ‖ 2 + 2 α n ‖ T y n − T x n + 1 ‖ ‖ x n + 1 − x ∗ ‖ , ‖ x n + 1 − x ∗ ‖ 2 ≤ ( 1 − α n ) 2 ( 1 − 2 α n γ ) ‖ x n − x ∗ ‖ 2 + 2 α n ( 1 − 2 α n γ ) ‖ T y n − T x n + 1 ‖ ‖ x n + 1 − x ∗ ‖ . 1 − 2 α n γ x n + 1 − x ∗ 2 ≤ 1 − α n 2 x n − x ∗ 2 + 2 α n T y n − T x n + 1 x n + 1 − x ∗ , x n + 1 − x ∗ 2 ≤ 1 − α n 2 1 − 2 α n γ x n − x ∗ 2 + 2 α n 1 − 2 α n γ T y n − T x n + 1 x n + 1 − x ∗ . {:[(1-2alpha_(n)gamma)||x_(n+1)-x^(**)||^(2) <= (1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)||Ty_(n)-Tx_(n+1)||||x_(n+1)-x^(**)||","],[||x_(n+1)-x^(**)||^(2) <= ((1-alpha_(n))^(2))/((1-2alpha_(n)gamma))||x_(n)-x^(**)||^(2)+(2alpha_(n))/((1-2alpha_(n)gamma))||Ty_(n)-Tx_(n+1)||||x_(n+1)-x^(**)||.]:}\begin{aligned} & \left(1-2 \alpha_{n} \gamma\right)\left\|x_{n+1}-x^{*}\right\|^{2} \leq\left(1-\alpha_{n}\right)^{2}\left\|x_{n}-x^{*}\right\|^{2}+2 \alpha_{n}\left\|T y_{n}-T x_{n+1}\right\|\left\|x_{n+1}-x^{*}\right\|, \\ & \left\|x_{n+1}-x^{*}\right\|^{2} \leq \frac{\left(1-\alpha_{n}\right)^{2}}{\left(1-2 \alpha_{n} \gamma\right)}\left\|x_{n}-x^{*}\right\|^{2}+\frac{2 \alpha_{n}}{\left(1-2 \alpha_{n} \gamma\right)}\left\|T y_{n}-T x_{n+1}\right\|\left\|x_{n+1}-x^{*}\right\| . \end{aligned}(1−2αnγ)‖xn+1−x∗‖2≤(1−αn)2‖xn−x∗‖2+2αn‖Tyn−Txn+1‖‖xn+1−x∗‖,‖xn+1−x∗‖2≤(1−αn)2(1−2αnγ)‖xn−x∗‖2+2αn(1−2αnγ)‖Tyn−Txn+1‖‖xn+1−x∗‖.
Because γ ∈ ( 0 , 1 / 2 ) , α n ∈ ( 0 , 1 ) ⇒ 2 ( 1 − γ ) − α n 1 − 2 α n γ ≥ 1 γ ∈ ( 0 , 1 / 2 ) , α n ∈ ( 0 , 1 ) ⇒ 2 ( 1 − γ ) − α n 1 − 2 α n γ ≥ 1 gamma in(0,1//2),alpha_(n)in(0,1)=>(2(1-gamma)-alpha_(n))/(1-2alpha_(n)gamma) >= 1\gamma \in(0,1 / 2), \alpha_{n} \in(0,1) \Rightarrow \frac{2(1-\gamma)-\alpha_{n}}{1-2 \alpha_{n} \gamma} \geq 1γ∈(0,1/2),αn∈(0,1)⇒2(1−γ)−αn1−2αnγ≥1 i.e. − ( 2 ( 1 − γ ) − α n 1 − 2 α n γ ) ≤ − 1 − 2 ( 1 − γ ) − α n 1 − 2 α n γ ≤ − 1 -((2(1-gamma)-alpha_(n))/(1-2alpha_(n)gamma)) <= -1-\left(\frac{2(1-\gamma)-\alpha_{n}}{1-2 \alpha_{n} \gamma}\right) \leq-1−(2(1−γ)−αn1−2αnγ)≤−1, we have
( 1 − α n ) 2 ( 1 − 2 α n γ ) = 1 − 2 α n + α n 2 ( 1 − 2 α n γ ) = ( ( 1 − 2 α n γ ) + 2 α n γ − 2 α n + α n 2 ) ( 1 − 2 α n γ ) = (6) = 1 − ( 2 ( 1 − γ ) − α n 1 − 2 α n γ ) α n ≤ 1 − α n 1 − α n 2 1 − 2 α n γ = 1 − 2 α n + α n 2 1 − 2 α n γ = 1 − 2 α n γ + 2 α n γ − 2 α n + α n 2 1 − 2 α n γ = (6) = 1 − 2 ( 1 − γ ) − α n 1 − 2 α n γ α n ≤ 1 − α n {:[((1-alpha_(n))^(2))/((1-2alpha_(n)gamma))=(1-2alpha_(n)+alpha_(n)^(2))/((1-2alpha_(n)gamma))=(((1-2alpha_(n)gamma)+2alpha_(n)gamma-2alpha_(n)+alpha_(n)^(2)))/((1-2alpha_(n)gamma))=],[(6)=1-((2(1-gamma)-alpha_(n))/(1-2alpha_(n)gamma))alpha_(n) <= 1-alpha_(n)]:}\begin{align*} \frac{\left(1-\alpha_{n}\right)^{2}}{\left(1-2 \alpha_{n} \gamma\right)} & =\frac{1-2 \alpha_{n}+\alpha_{n}^{2}}{\left(1-2 \alpha_{n} \gamma\right)}=\frac{\left(\left(1-2 \alpha_{n} \gamma\right)+2 \alpha_{n} \gamma-2 \alpha_{n}+\alpha_{n}^{2}\right)}{\left(1-2 \alpha_{n} \gamma\right)}= \\ & =1-\left(\frac{2(1-\gamma)-\alpha_{n}}{1-2 \alpha_{n} \gamma}\right) \alpha_{n} \leq 1-\alpha_{n} \tag{6} \end{align*}(1−αn)2(1−2αnγ)=1−2αn+αn2(1−2αnγ)=((1−2αnγ)+2αnγ−2αn+αn2)(1−2αnγ)=(6)=1−(2(1−γ)−αn1−2αnγ)αn≤1−αn
Also, the sequence ( x n ) x n (x_(n))\left(x_{n}\right)(xn) is bounded. We will prove that by induction. Let us denote by d := sup { ‖ T x ‖ : x ∈ B } + ‖ x ∗ ‖ d := sup { ‖ T x ‖ : x ∈ B } + x ∗ d:=s u p{||Tx||:x in B}+||x^(**)||d:=\sup \{\|T x\|: x \in B\}+\left\|x^{*}\right\|d:=sup{‖Tx‖:x∈B}+‖x∗‖. Because the range of T T TTT is bounded we have d < ∞ d < ∞ d < ood<\inftyd<∞. We denote by M := d + ‖ x 0 − x ∗ ‖ + 1 M := d + x 0 − x ∗ + 1 M:=d+||x_(0)-x^(**)||+1M:=d+\left\|x_{0}-x^{*}\right\|+1M:=d+‖x0−x∗‖+1. Observe that
‖ x 1 − x ∗ ‖ ≤ ( 1 − α 0 ) ‖ x 0 − x ∗ ‖ + α 0 ‖ T y 0 − x ∗ ‖ ≤ ( 1 − α 0 ) M + α 0 ( ‖ T y 0 ‖ + ‖ x ∗ ‖ ) ≤ ( 1 − α 0 ) M + α 0 M = M x 1 − x ∗ ≤ 1 − α 0 x 0 − x ∗ + α 0 T y 0 − x ∗ ≤ 1 − α 0 M + α 0 T y 0 + x ∗ ≤ 1 − α 0 M + α 0 M = M {:[||x_(1)-x^(**)|| <= (1-alpha_(0))||x_(0)-x^(**)||+alpha_(0)||Ty_(0)-x^(**)||],[ <= (1-alpha_(0))M+alpha_(0)(||Ty_(0)||+||x^(**)||) <= (1-alpha_(0))M+alpha_(0)M=M]:}\begin{aligned} \left\|x_{1}-x^{*}\right\| & \leq\left(1-\alpha_{0}\right)\left\|x_{0}-x^{*}\right\|+\alpha_{0}\left\|T y_{0}-x^{*}\right\| \\ & \leq\left(1-\alpha_{0}\right) M+\alpha_{0}\left(\left\|T y_{0}\right\|+\left\|x^{*}\right\|\right) \leq\left(1-\alpha_{0}\right) M+\alpha_{0} M=M \end{aligned}‖x1−x∗‖≤(1−α0)‖x0−x∗‖+α0‖Ty0−x∗‖≤(1−α0)M+α0(‖Ty0‖+‖x∗‖)≤(1−α0)M+α0M=M
Supposing ‖ x n − x ∗ ‖ ≤ M x n − x ∗ ≤ M ||x_(n)-x^(**)|| <= M\left\|x_{n}-x^{*}\right\| \leq M‖xn−x∗‖≤M, we will prove that ‖ x n + 1 − x ∗ ‖ ≤ M x n + 1 − x ∗ ≤ M ||x_(n+1)-x^(**)|| <= M\left\|x_{n+1}-x^{*}\right\| \leq M‖xn+1−x∗‖≤M. Indeed we have
‖ x n + 1 − x ∗ ‖ ≤ ( 1 − α n ) ‖ x n − x ∗ ‖ + α n ‖ T y n − x ∗ ‖ ≤ ( 1 − α n ) M + α n ( ‖ T y n ‖ + ‖ x ∗ ‖ ) ≤ ( 1 − α n ) M + α n M = M x n + 1 − x ∗ ≤ 1 − α n x n − x ∗ + α n T y n − x ∗ ≤ 1 − α n M + α n T y n + x ∗ ≤ 1 − α n M + α n M = M {:[||x_(n+1)-x^(**)|| <= (1-alpha_(n))||x_(n)-x^(**)||+alpha_(n)||Ty_(n)-x^(**)||],[ <= (1-alpha_(n))M+alpha_(n)(||Ty_(n)||+||x^(**)||) <= (1-alpha_(n))M+alpha_(n)M=M]:}\begin{aligned} \left\|x_{n+1}-x^{*}\right\| & \leq\left(1-\alpha_{n}\right)\left\|x_{n}-x^{*}\right\|+\alpha_{n}\left\|T y_{n}-x^{*}\right\| \\ & \leq\left(1-\alpha_{n}\right) M+\alpha_{n}\left(\left\|T y_{n}\right\|+\left\|x^{*}\right\|\right) \leq\left(1-\alpha_{n}\right) M+\alpha_{n} M=M \end{aligned}‖xn+1−x∗‖≤(1−αn)‖xn−x∗‖+αn‖Tyn−x∗‖≤(1−αn)M+αn(‖Tyn‖+‖x∗‖)≤(1−αn)M+αnM=M
Thus we have
(7) ∃ M > 0 : ‖ x n + 1 − x ∗ ‖ ≤ M , ∀ n ≥ 0 (7) ∃ M > 0 : x n + 1 − x ∗ ≤ M , ∀ n ≥ 0 {:(7)EE M > 0:||x_(n+1)-x^(**)|| <= M","AA n >= 0:}\begin{equation*} \exists M>0:\left\|x_{n+1}-x^{*}\right\| \leq M, \forall n \geq 0 \tag{7} \end{equation*}(7)∃M>0:‖xn+1−x∗‖≤M,∀n≥0
Conditions (6), (7) and (8) lead us to
‖ x n + 1 − x ∗ ‖ 2 ≤ ( 1 − α n ) ‖ x n − x ∗ ‖ 2 + ‖ T y n − T x n + 1 ‖ 2 α n ( 1 − 2 α n γ ) M x n + 1 − x ∗ 2 ≤ 1 − α n x n − x ∗ 2 + T y n − T x n + 1 2 α n 1 − 2 α n γ M ||x_(n+1)-x^(**)||^(2) <= (1-alpha_(n))||x_(n)-x^(**)||^(2)+||Ty_(n)-Tx_(n+1)||(2alpha_(n))/((1-2alpha_(n)gamma))M\left\|x_{n+1}-x^{*}\right\|^{2} \leq\left(1-\alpha_{n}\right)\left\|x_{n}-x^{*}\right\|^{2}+\left\|T y_{n}-T x_{n+1}\right\| \frac{2 \alpha_{n}}{\left(1-2 \alpha_{n} \gamma\right)} M‖xn+1−x∗‖2≤(1−αn)‖xn−x∗‖2+‖Tyn−Txn+1‖2αn(1−2αnγ)M
But ( 1 − α n ) ≤ ( 1 − w ) 1 − α n ≤ ( 1 − w ) (1-alpha_(n)) <= (1-w)\left(1-\alpha_{n}\right) \leq(1-w)(1−αn)≤(1−w), and 2 α n ( 1 − 2 α n γ ) ≤ 2 1 − 2 γ 2 α n 1 − 2 α n γ ≤ 2 1 − 2 γ (2alpha_(n))/((1-2alpha_(n)gamma)) <= (2)/(1-2gamma)\frac{2 \alpha_{n}}{\left(1-2 \alpha_{n} \gamma\right)} \leq \frac{2}{1-2 \gamma}2αn(1−2αnγ)≤21−2γ. So, we have
‖ x n + 1 − x ∗ ‖ 2 ≤ ( 1 − w ) ‖ x n − x ∗ ‖ 2 + ‖ T y n − T x n + 1 ‖ 2 1 − 2 γ M x n + 1 − x ∗ 2 ≤ ( 1 − w ) x n − x ∗ 2 + T y n − T x n + 1 2 1 − 2 γ M ||x_(n+1)-x^(**)||^(2) <= (1-w)||x_(n)-x^(**)||^(2)+||Ty_(n)-Tx_(n+1)||(2)/(1-2gamma)M\left\|x_{n+1}-x^{*}\right\|^{2} \leq(1-w)\left\|x_{n}-x^{*}\right\|^{2}+\left\|T y_{n}-T x_{n+1}\right\| \frac{2}{1-2 \gamma} M‖xn+1−x∗‖2≤(1−w)‖xn−x∗‖2+‖Tyn−Txn+1‖21−2γM
Let us denote be a n := ‖ x n − x ∗ ‖ 2 , σ n := ‖ T y n − T x n + 1 ‖ a n := x n − x ∗ 2 , σ n := T y n − T x n + 1 a_(n):=||x_(n)-x^(**)||^(2),sigma_(n):=||Ty_(n)-Tx_(n+1)||a_{n}:=\left\|x_{n}-x^{*}\right\|^{2}, \sigma_{n}:=\left\|T y_{n}-T x_{n+1}\right\|an:=‖xn−x∗‖2,σn:=‖Tyn−Txn+1‖, and S := 2 1 − 2 γ M S := 2 1 − 2 γ M S:=(2)/(1-2gamma)MS:=\frac{2}{1-2 \gamma} MS:=21−2γM. Then we have lim n → ∞ a n = 0 lim n → ∞   a n = 0 lim_(n rarr oo)a_(n)=0\lim _{n \rightarrow \infty} a_{n}=0limn→∞an=0. Thus lim n → ∞ x n = x ∗ lim n → ∞   x n = x ∗ lim_(n rarr oo)x_(n)=x^(**)\lim _{n \rightarrow \infty} x_{n}=x^{*}limn→∞xn=x∗.

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  1. *Ştefan M. Şoltuz, Kurt Schumacher Str. 48, Ap. 38, 67663 Kaiserslautern, Germany, e-mail: ssoltuz@yahoo.com, soltuz@itwm.fhg.de
2002

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