Characterization for the convergence of Krasnoselskij iteration for non-Lipschitzian operators

Stefan Soltuz
(original), paper

Abstract

Authors

Stefan M. Soltuz
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

B.E. Roades
Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

Keywords

?

Paper coordinates

Stefan M. Soltuz, B.E. Roades, Characterization for the Convergence of Krasnoselskij Iteration for Non-Lipschitzian Operators, International Journal of Mathematics and Mathematical Sciences, Volume 2008, Article ID 630589, 5 pages, doi:10.1155/2008/630589

PDF

https://onlinelibrary.wiley.com/doi/epdf/10.1155/2008/630589

About this paper

Journal

International Journal of Mathematics and Mathematical Sciences

Publisher Name
DOI
Print ISSN

0161-1712

Online ISSN

1687-0425

google scholar link

[1] M. A. Krasnosel’skii, “Two remarks on the method of successive approximations,” Uspekhi Matematich-eskikh Nauk, vol. 10, no. 1, pp. 123–127, 1955.
[2]  M. O. Osilike, “Iterative solution of nonlinear equations of the φ-strongly accretive type,” Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 259–271, 1996.
[3] C. E. Chidume and C. O. Chidume, “Convergence theorems for fixed points of uniformly continuousgeneralized φ-hemi-contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 303,no. 2, pp. 545–554, 2005.
[4] S. M. Soltuz, “New technique for proving the equivalence of Mann and Ishikawa iterations,” Revued’Analyse Numerique et de Theorie de l’Approximation, vol. 34, no. 1, pp. 103–108, 2005.

Paper (preprint) in HTML form

International Journal of Mathematics and Mathematical Sciences - 2008 - Şoltuz - Characterization fo

Characterization for the Convergence of Krasnoselskij Iteration for Non-Lipschitzian Operators

Ştefan M. Şoltuz 1 , 2 1 , 2 ^(1,2){ }^{\mathbf{1 , 2}}1,2 and B. E. Rhoades 3 3 ^(3){ }^{\mathbf{3}}3 1 1 ^(1){ }^{1}1 Departamento de Matematicas, Universidad de Los Andes, Carrera 1 no. 18A-10, Bogota, Colombia2 "Tiberiu Popoviciu" Institute of Numerical Analysis, 400110 Cluj-Napoca, Romania 3 3 ^(3){ }^{3}3 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

1. Introduction

Let X X XXX be a real Banach space; B B BBB a nonempty, convex subset of X X XXX; and T : B → B T : B → B T:B rarr BT: B \rightarrow BT:B→B an operator. Let x 0 ∈ B x 0 ∈ B x_(0)in Bx_{0} \in Bx0∈B. The following iteration is known as Krasnoselskij iteration (see [1]):
(1.1) x n + 1 = ( 1 − λ ) x n + λ T x n . (1.1) x n + 1 = ( 1 − λ ) x n + λ T x n . {:(1.1)x_(n+1)=(1-lambda)x_(n)+lambda Tx_(n).:}\begin{equation*} x_{n+1}=(1-\lambda) x_{n}+\lambda T x_{n} . \tag{1.1} \end{equation*}(1.1)xn+1=(1−λ)xn+λTxn.
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 ‖ } J x := f ∈ X ∗ : ⟨ x , f ⟩ = ‖ x ‖ 2 , ‖ f ‖ = ‖ x ‖ Jx:={f inX^(**):(:x,f:)=||x||^(2),||f||=||x||}J x:=\left\{f \in X^{*}:\langle x, f\rangle=\|x\|^{2},\|f\|=\|x\|\right\}Jx:={f∈X∗:⟨x,f⟩=‖x‖2,‖f‖=‖x‖}, for all x ∈ X x ∈ X x in Xx \in Xx∈X, is called the normalized duality mapping. It is easy to see that we have
(1.2) ⟨ y , j ( x ) ⟩ ≤ ‖ x ‖ ‖ y ‖ , ∀ x , y ∈ X , ∀ j ( x ) ∈ J ( x ) . (1.2) ⟨ y , j ( x ) ⟩ ≤ ‖ x ‖ ‖ y ‖ , ∀ x , y ∈ X , ∀ j ( x ) ∈ J ( x ) . {:(1.2)(:y","j(x):) <= ||x||||y||","quad AA x","y in X","AA j(x)in J(x).:}\begin{equation*} \langle y, j(x)\rangle \leq\|x\|\|y\|, \quad \forall x, y \in X, \forall j(x) \in J(x) . \tag{1.2} \end{equation*}(1.2)⟨y,j(x)⟩≤‖x‖‖y‖,∀x,y∈X,∀j(x)∈J(x).
Denote
(1.3) Ψ := { ψ ∣ ψ : [ 0 , + ∞ ) ⟶ [ 0 , + ∞ ) is astrictly increasing map with ψ ( 0 ) = 0 } . (1.3) Ψ := { ψ ∣ ψ : [ 0 , + ∞ ) ⟶ [ 0 , + ∞ )  is astrictly increasing map with  ψ ( 0 ) = 0 } . {:(1.3)Psi:={psi∣psi:[0","+oo)longrightarrow[0","+oo)" is astrictly increasing map with "psi(0)=0}.:}\begin{equation*} \Psi:=\{\psi \mid \psi:[0,+\infty) \longrightarrow[0,+\infty) \text { is astrictly increasing map with } \psi(0)=0\} . \tag{1.3} \end{equation*}(1.3)Ψ:={ψ∣ψ:[0,+∞)⟶[0,+∞) is astrictly increasing map with ψ(0)=0}.
Definition 1.1. Let X X XXX be a real Banach space, and let B B BBB be a nonempty subset of X X XXX. A map T T TTT : B → B B → B B rarr BB \rightarrow BB→B is called uniformly pseudocontractive if there exists a map ψ ∈ Ψ ψ ∈ Ψ psi in Psi\psi \in \Psiψ∈Ψ and 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
(1.4) ⟨ T x − T y , j ( x − y ) ⟩ ≤ ‖ x − y ‖ 2 − ψ ( ‖ x − y ‖ ) , ∀ x , y ∈ B . (1.4) ⟨ T x − T y , j ( x − y ) ⟩ ≤ ‖ x − y ‖ 2 − ψ ( ‖ x − y ‖ ) , ∀ x , y ∈ B . {:(1.4)(:Tx-Ty","j(x-y):) <= ||x-y||^(2)-psi(||x-y||)","quad AA x","y in B.:}\begin{equation*} \langle T x-T y, j(x-y)\rangle \leq\|x-y\|^{2}-\psi(\|x-y\|), \quad \forall x, y \in B . \tag{1.4} \end{equation*}(1.4)⟨Tx−Ty,j(x−y)⟩≤‖x−y‖2−ψ(‖x−y‖),∀x,y∈B.
A map S : X → X S : X → X S:X rarr XS: X \rightarrow XS:X→X is called uniformly accretive if there exists a map ψ ∈ Ψ ψ ∈ Ψ psi in Psi\psi \in \Psiψ∈Ψ and 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
(1.5) ⟨ S x − S y , j ( x − y ) ⟩ ≥ ψ ( ‖ x − y ‖ ) , ∀ x , y ∈ X (1.5) ⟨ S x − S y , j ( x − y ) ⟩ ≥ ψ ( ‖ x − y ‖ ) , ∀ x , y ∈ X {:(1.5)(:Sx-Sy","j(x-y):) >= psi(||x-y||)","quad AA x","y in X:}\begin{equation*} \langle S x-S y, j(x-y)\rangle \geq \psi(\|x-y\|), \quad \forall x, y \in X \tag{1.5} \end{equation*}(1.5)⟨Sx−Sy,j(x−y)⟩≥ψ(‖x−y‖),∀x,y∈X
Taking ψ ( a ) := ψ ( a ) ⋅ a ψ ( a ) := ψ ( a ) ⋅ a psi(a):=psi(a)*a\psi(a):=\psi(a) \cdot aψ(a):=ψ(a)⋅a, for all a ∈ [ 0 , + ∞ ) , ( ψ ∈ Ψ ) a ∈ [ 0 , + ∞ ) , ( ψ ∈ Ψ ) a in[0,+oo),(psi in Psi)a \in[0,+\infty),(\psi \in \Psi)a∈[0,+∞),(ψ∈Ψ), reduces to the usual definitions of ψ ψ psi\psiψ-strongly pseudocontractive and ψ ψ psi\psiψ-strongly accretive. Taking ψ ( a ) := γ ⋅ a 2 , γ ∈ ( 0 , 1 ) ψ ( a ) := γ ⋅ a 2 , γ ∈ ( 0 , 1 ) psi(a):=gamma*a^(2),gamma in(0,1)\psi(a):=\gamma \cdot a^{2}, \gamma \in(0,1)ψ(a):=γ⋅a2,γ∈(0,1), for all a ∈ [ 0 , + ∞ ) , ( ψ ∈ Ψ ) a ∈ [ 0 , + ∞ ) , ( ψ ∈ Ψ ) a in[0,+oo),(psi in Psi)a \in[0,+\infty),(\psi \in \Psi)a∈[0,+∞),(ψ∈Ψ), we get the usual definitions of strongly pseudocontractive and strongly accretive. Therefore, the class of strongly pseudocontractive maps is included stricly in the class of ψ ψ psi\psiψ-strongly pseudocontractive maps. The example from [2] shows that this inclusion is proper. Remark, further, that the class of ψ ψ psi\psiψ-strongly pseudocontractive maps is also included strictly in the class of uniformly pseudocontractive maps (see also [3]).
We will give a characterization for the convergence of (1.1) when applied to uniformly pseudocontractive operators. For this purpose, we need the following lemma similar to [4, Lemma 1]. Next, N N N\mathbb{N}N denotes the set of all natural numbers.
Lemma 1.2. Let { a n } a n {a_(n)}\left\{a_{n}\right\}{an} be a positive bounded sequence and assume that there exists n 0 ∈ N n 0 ∈ N n_(0)inNn_{0} \in \mathbb{N}n0∈N such that
(1.6) a n + 1 ≤ ( 1 − λ ) a n + λ a n + 1 − λ ψ ( a n + 1 ) a n + 1 + λ ε n , ∀ n ≥ n 0 (1.6) a n + 1 ≤ ( 1 − λ ) a n + λ a n + 1 − λ ψ a n + 1 a n + 1 + λ ε n , ∀ n ≥ n 0 {:(1.6)a_(n+1) <= (1-lambda)a_(n)+lambdaa_(n+1)-lambda(psi(a_(n+1)))/(a_(n+1))+lambdaepsi_(n)","quad AA n >= n_(0):}\begin{equation*} a_{n+1} \leq(1-\lambda) a_{n}+\lambda a_{n+1}-\lambda \frac{\psi\left(a_{n+1}\right)}{a_{n+1}}+\lambda \varepsilon_{n}, \quad \forall n \geq n_{0} \tag{1.6} \end{equation*}(1.6)an+1≤(1−λ)an+λan+1−λψ(an+1)an+1+λεn,∀n≥n0
where λ ∈ ( 0 , 1 ) , ε n ≥ 0 λ ∈ ( 0 , 1 ) , ε n ≥ 0 lambda in(0,1),epsi_(n) >= 0\lambda \in(0,1), \varepsilon_{n} \geq 0λ∈(0,1),εn≥0, for all n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N and lim n → ∞ ε n = 0 lim n → ∞   ε n = 0 lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0limn→∞εn=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.
Proof. There exists an M > 0 M > 0 M > 0M>0M>0 such that a n ≤ M a n ≤ M a_(n) <= Ma_{n} \leq Man≤M, for all n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N. Denote a := lim inf a n a := lim inf a n a:=l i m   i n fa_(n)a:=\liminf a_{n}a:=lim infan. We will prove that a = 0 a = 0 a=0a=0a=0. Suppose on the contrary that a > 0 a > 0 a > 0a>0a>0. Then there exists an N 1 ∈ N N 1 ∈ N N_(1)inNN_{1} \in \mathbb{N}N1∈N such that
(1.7) a n ≥ a 2 , ∀ n ≥ N 1 (1.7) a n ≥ a 2 , ∀ n ≥ N 1 {:(1.7)a_(n) >= (a)/(2)","quad AA n >= N_(1):}\begin{equation*} a_{n} \geq \frac{a}{2}, \quad \forall n \geq N_{1} \tag{1.7} \end{equation*}(1.7)an≥a2,∀n≥N1
From lim n → ∞ ε n = 0 lim n → ∞   ε n = 0 lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0limn→∞εn=0, we know that there exists an N 2 ∈ N N 2 ∈ N N_(2)inNN_{2} \in \mathbb{N}N2∈N such that
(1.8) ε n ≤ ψ ( a / 2 ) 2 M , ∀ n ≥ N 2 (1.8) ε n ≤ ψ ( a / 2 ) 2 M , ∀ n ≥ N 2 {:(1.8)epsi_(n) <= (psi(a//2))/(2M)","quad AA n >= N_(2):}\begin{equation*} \varepsilon_{n} \leq \frac{\psi(a / 2)}{2 M}, \quad \forall n \geq N_{2} \tag{1.8} \end{equation*}(1.8)εn≤ψ(a/2)2M,∀n≥N2
Set N 0 := max { N 1 , N 2 } N 0 := max N 1 , N 2 N_(0):=max{N_(1),N_(2)}N_{0}:=\max \left\{N_{1}, N_{2}\right\}N0:=max{N1,N2}. Using the fact that − ( 1 / M ) ≥ − ( 1 / a n + 1 ) − ( 1 / M ) ≥ − 1 / a n + 1 -(1//M) >= -(1//a_(n+1))-(1 / M) \geq-\left(1 / a_{n+1}\right)−(1/M)≥−(1/an+1),we get the following:
a n + 1 ≤ ( 1 − λ ) a n + λ a n + 1 − λ ψ ( a n + 1 ) a n + 1 + λ ε n (1.9) ≤ ( 1 − λ ) a n + λ a n + 1 − λ ψ ( a / 2 ) M + λ ψ ( a / 2 ) 2 M ≤ ( 1 − λ ) a n + λ a n + 1 − λ ψ ( a / 2 ) 2 M , a n + 1 ≤ ( 1 − λ ) a n + λ a n + 1 − λ ψ a n + 1 a n + 1 + λ ε n (1.9) ≤ ( 1 − λ ) a n + λ a n + 1 − λ ψ ( a / 2 ) M + λ ψ ( a / 2 ) 2 M ≤ ( 1 − λ ) a n + λ a n + 1 − λ ψ ( a / 2 ) 2 M , {:[a_(n+1) <= (1-lambda)a_(n)+lambdaa_(n+1)-lambda(psi(a_(n+1)))/(a_(n+1))+lambdaepsi_(n)],[(1.9) <= (1-lambda)a_(n)+lambdaa_(n+1)-lambda(psi(a//2))/(M)+lambda(psi(a//2))/(2M)],[ <= (1-lambda)a_(n)+lambdaa_(n+1)-lambda(psi(a//2))/(2M)","]:}\begin{align*} a_{n+1} & \leq(1-\lambda) a_{n}+\lambda a_{n+1}-\lambda \frac{\psi\left(a_{n+1}\right)}{a_{n+1}}+\lambda \varepsilon_{n} \\ & \leq(1-\lambda) a_{n}+\lambda a_{n+1}-\lambda \frac{\psi(a / 2)}{M}+\lambda \frac{\psi(a / 2)}{2 M} \tag{1.9}\\ & \leq(1-\lambda) a_{n}+\lambda a_{n+1}-\lambda \frac{\psi(a / 2)}{2 M}, \end{align*}an+1≤(1−λ)an+λan+1−λψ(an+1)an+1+λεn(1.9)≤(1−λ)an+λan+1−λψ(a/2)M+λψ(a/2)2M≤(1−λ)an+λan+1−λψ(a/2)2M,
which implies that ( 1 − λ ) a n + 1 ≤ ( 1 − λ ) a n − λ ( ( ψ ( a / 2 ) ) / 2 M ) ( 1 − λ ) a n + 1 ≤ ( 1 − λ ) a n − λ ( ( ψ ( a / 2 ) ) / 2 M ) (1-lambda)a_(n+1) <= (1-lambda)a_(n)-lambda((psi(a//2))//2M)(1-\lambda) a_{n+1} \leq(1-\lambda) a_{n}-\lambda((\psi(a / 2)) / 2 M)(1−λ)an+1≤(1−λ)an−λ((ψ(a/2))/2M), or
(1.10) a n + 1 ≤ a n − λ 1 − λ ψ ( a / 2 ) 2 M ≤ a n − λ ψ ( a / 2 ) 2 M (1.10) a n + 1 ≤ a n − λ 1 − λ ψ ( a / 2 ) 2 M ≤ a n − λ ψ ( a / 2 ) 2 M {:(1.10)a_(n+1) <= a_(n)-(lambda)/(1-lambda)(psi(a//2))/(2M) <= a_(n)-lambda(psi(a//2))/(2M):}\begin{equation*} a_{n+1} \leq a_{n}-\frac{\lambda}{1-\lambda} \frac{\psi(a / 2)}{2 M} \leq a_{n}-\lambda \frac{\psi(a / 2)}{2 M} \tag{1.10} \end{equation*}(1.10)an+1≤an−λ1−λψ(a/2)2M≤an−λψ(a/2)2M
since − ( λ / ( 1 − λ ) ) ≤ − λ − ( λ / ( 1 − λ ) ) ≤ − λ -(lambda//(1-lambda)) <= -lambda-(\lambda /(1-\lambda)) \leq-\lambda−(λ/(1−λ))≤−λ. Thus λ ( ψ ( a / 2 ) ) / 2 M ≤ a n − a n + 1 λ ( ψ ( a / 2 ) ) / 2 M ≤ a n − a n + 1 lambda(psi(a//2))//2M <= a_(n)-a_(n+1)\lambda(\psi(a / 2)) / 2 M \leq a_{n}-a_{n+1}λ(ψ(a/2))/2M≤an−an+1, which implies that ∑ λ < ∞ ∑ λ < ∞ sum lambda < oo\sum \lambda<\infty∑λ<∞, in contradiction to ∑ λ = ∞ ∑ λ = ∞ sum lambda=oo\sum \lambda=\infty∑λ=∞. Therefore, lim inf a n = 0 lim inf a n = 0 l i m   i n fa_(n)=0\liminf a_{n}=0lim infan=0. Hence there exists a subsequence { a n j } ⊂ { a n } a n j ⊂ a n {a_(n_(j))}sub{a_(n)}\left\{a_{n_{j}}\right\} \subset \left\{a_{n}\right\}{anj}⊂{an} such that lim j → ∞ a n j = 0 lim j → ∞   a n j = 0 lim_(j rarr oo)a_(n_(j))=0\lim _{j \rightarrow \infty} a_{n_{j}}=0limj→∞anj=0. Fix ε > 0 ε > 0 epsi > 0\varepsilon>0ε>0. Then there exists an n 3 ∈ N n 3 ∈ N n_(3)inNn_{3} \in \mathbb{N}n3∈N such that
(1.11) a n j < ε 4 , ∀ j ≥ n 3 . (1.11) a n j < ε 4 , ∀ j ≥ n 3 . {:(1.11)a_(n_(j)) < (epsi)/(4)","quad AA j >= n_(3).:}\begin{equation*} a_{n_{j}}<\frac{\varepsilon}{4}, \quad \forall j \geq n_{3} . \tag{1.11} \end{equation*}(1.11)anj<ε4,∀j≥n3.
Also there exists an n 4 ∈ N n 4 ∈ N n_(4)inNn_{4} \in \mathbb{N}n4∈N such that
(1.12) ε n < ψ ( ε / 4 ) 2 M , ∀ n ≥ n 4 (1.12) ε n < ψ ( ε / 4 ) 2 M , ∀ n ≥ n 4 {:(1.12)epsi_(n) < (psi(epsi//4))/(2M)","quad AA n >= n_(4):}\begin{equation*} \varepsilon_{n}<\frac{\psi(\varepsilon / 4)}{2 M}, \quad \forall n \geq n_{4} \tag{1.12} \end{equation*}(1.12)εn<ψ(ε/4)2M,∀n≥n4
Define n 0 := max { n 3 , n 4 , N 0 } n 0 := max n 3 , n 4 , N 0 n_(0):=max{n_(3),n_(4),N_(0)}n_{0}:=\max \left\{n_{3}, n_{4}, N_{0}\right\}n0:=max{n3,n4,N0}. We claim that a n j + k < ε / 4 a n j + k < ε / 4 a_(n_(j)+k) < epsi//4a_{n_{j}+k}<\varepsilon / 4anj+k<ε/4 for each j > n 0 j > n 0 j > n_(0)j>n_{0}j>n0 and each k > 0 k > 0 k > 0k>0k>0. Suppose not. Then there exists an n 0 n 0 n_(0)n_{0}n0 and a k > 0 k > 0 k > 0k>0k>0 such that
(1.13) a n j + k ≥ ε 4 (1.13) a n j + k ≥ ε 4 {:(1.13)a_(n_(j)+k) >= (epsi)/(4):}\begin{equation*} a_{n_{j}+k} \geq \frac{\varepsilon}{4} \tag{1.13} \end{equation*}(1.13)anj+k≥ε4
For this n j n j n_(j)n_{j}nj, let k k kkk denote the smallest positive integer for which (1.13) is true. Then a n j + k − 1 ≤ ε / 4 a n j + k − 1 ≤ ε / 4 a_(n_(j)+k-1) <= epsi//4a_{n_{j}+k-1} \leq \varepsilon / 4anj+k−1≤ε/4.
From (1.6),
a n j + k ≤ ( 1 − λ ) a n j + k − 1 + λ a n j + k − λ ψ ( a n j + k ) a n j + k + λ ε n j + k − 1 (1.14) ≤ ( 1 − λ ) a n j + k − 1 + λ a n j + k − λ ψ ( ε / 4 ) a n j + k + λ ψ ( ε / 4 ) 2 M ≤ ( 1 − λ ) a n j + k − 1 + λ a n j + k − λ ψ ( ε / 4 ) 2 M , a n j + k ≤ ( 1 − λ ) a n j + k − 1 + λ a n j + k − λ ψ a n j + k a n j + k + λ ε n j + k − 1 (1.14) ≤ ( 1 − λ ) a n j + k − 1 + λ a n j + k − λ ψ ( ε / 4 ) a n j + k + λ ψ ( ε / 4 ) 2 M ≤ ( 1 − λ ) a n j + k − 1 + λ a n j + k − λ ψ ( ε / 4 ) 2 M , {:[a_(n_(j)+k) <= (1-lambda)a_(n_(j)+k-1)+lambdaa_(n_(j)+k)-lambda(psi(a_(n_(j)+k)))/(a_(n_(j)+k))+lambdaepsi_(n_(j)+k-1)],[(1.14) <= (1-lambda)a_(n_(j)+k-1)+lambdaa_(n_(j)+k)-(lambda psi(epsi//4))/(a_(n_(j)+k))+lambda(psi(epsi//4))/(2M)],[ <= (1-lambda)a_(n_(j)+k-1)+lambdaa_(n_(j)+k)-lambda(psi(epsi//4))/(2M)","]:}\begin{align*} a_{n_{j}+k} & \leq(1-\lambda) a_{n_{j}+k-1}+\lambda a_{n_{j}+k}-\lambda \frac{\psi\left(a_{n_{j}+k}\right)}{a_{n_{j}+k}}+\lambda \varepsilon_{n_{j}+k-1} \\ & \leq(1-\lambda) a_{n_{j}+k-1}+\lambda a_{n_{j}+k}-\frac{\lambda \psi(\varepsilon / 4)}{a_{n_{j}+k}}+\lambda \frac{\psi(\varepsilon / 4)}{2 M} \tag{1.14}\\ & \leq(1-\lambda) a_{n_{j}+k-1}+\lambda a_{n_{j}+k}-\lambda \frac{\psi(\varepsilon / 4)}{2 M}, \end{align*}anj+k≤(1−λ)anj+k−1+λanj+k−λψ(anj+k)anj+k+λεnj+k−1(1.14)≤(1−λ)anj+k−1+λanj+k−λψ(ε/4)anj+k+λψ(ε/4)2M≤(1−λ)anj+k−1+λanj+k−λψ(ε/4)2M,
which implies that a n j + k ≤ ( ε / 4 ) − ( λ / ( 1 − λ ) ) ( ψ ( ε / 4 ) / 2 M ) a n j + k ≤ ( ε / 4 ) − ( λ / ( 1 − λ ) ) ( ψ ( ε / 4 ) / 2 M ) a_(n_(j)+k) <= (epsi//4)-(lambda//(1-lambda))(psi(epsi//4)//2M)a_{n_{j}+k} \leq(\varepsilon / 4)-(\lambda /(1-\lambda))(\psi(\varepsilon / 4) / 2 M)anj+k≤(ε/4)−(λ/(1−λ))(ψ(ε/4)/2M). This leads to the contradiction:
(1.15) ε 4 ≤ a n j + k ≤ ε 4 − λ 1 − λ ψ ( ε / 4 ) 2 M < ε 4 (1.15) ε 4 ≤ a n j + k ≤ ε 4 − λ 1 − λ ψ ( ε / 4 ) 2 M < ε 4 {:(1.15)(epsi)/(4) <= a_(n_(j)+k) <= (epsi)/(4)-(lambda)/(1-lambda)(psi(epsi//4))/(2M) < (epsi)/(4):}\begin{equation*} \frac{\varepsilon}{4} \leq a_{n_{j}+k} \leq \frac{\varepsilon}{4}-\frac{\lambda}{1-\lambda} \frac{\psi(\varepsilon / 4)}{2 M}<\frac{\varepsilon}{4} \tag{1.15} \end{equation*}(1.15)ε4≤anj+k≤ε4−λ1−λψ(ε/4)2M<ε4
Therefore, a n j + k < ε / 4 a n j + k < ε / 4 a_(n_(j)+k) < epsi//4a_{n_{j}+k}<\varepsilon / 4anj+k<ε/4, for all k ∈ N k ∈ N k inNk \in \mathbb{N}k∈N, and each j > n 0 j > n 0 j > n_(0)j>n_{0}j>n0, hence 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

Theorem 2.1. Let X X XXX be a real Banach space, B B BBB a nonempty, closed, convex, bounded subset of X X XXX. Let T T TTT : B → B B → B B rarr BB \rightarrow BB→B be a uniformly pseudocontractive and uniformly continuous operator with F ( T ) ≠ ∅ F ( T ) ≠ ∅ F(T)!=O/F(T) \neq \varnothingF(T)≠∅. Then for x 0 ∈ B x 0 ∈ B x_(0)in Bx_{0} \in Bx0∈B, the Krasnoselskij iteration (1.1) converges to the fixed point of T T TTT if and only if lim n → ∞ ‖ x n + 1 − x n ‖ = 0 lim n → ∞   ‖ x n + 1 − x n ‖ = 0 lim_(n rarr oo)||x_(n+1)-x_(n)||=0\lim _{n \rightarrow \infty} \| x_{n+1}- x_{n} \|=0limn→∞‖xn+1−xn‖=0.
Proof. Since T T TTT is a self-map of B B BBB, which is bounded and convex, then, from (1.1), each x n ∈ B x n ∈ B x_(n)in Bx_{n} \in Bxn∈B, so { x n } x n {x_(n)}\left\{x_{n}\right\}{xn} is bounded for each n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N. Uniqueness of the fixed point follows from (1.4). If { x n } x n {x_(n)}\left\{x_{n}\right\}{xn} converges to the fixed point of T T TTT, that is, 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∗, then, obviously, lim n → ∞ ‖ x n + 1 − x n ‖ = 0 lim n → ∞   x n + 1 − x n = 0 lim_(n rarr oo)||x_(n+1)-x_(n)||=0\lim _{n \rightarrow \infty}\left\|x_{n+1}-x_{n}\right\|=0limn→∞‖xn+1−xn‖=0. Conversely, we will prove that if lim n → ∞ ‖ x n + 1 − x n ‖ = 0 lim n → ∞   x n + 1 − x n = 0 lim_(n rarr oo)||x_(n+1)-x_(n)||=0\lim _{n \rightarrow \infty}\left\|x_{n+1}-x_{n}\right\|=0limn→∞‖xn+1−xn‖=0, then 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∗. Suppose that
x n = x ∗ x n = x ∗ x_(n)=x^(**)x_{n}=x^{*}xn=x∗ for some n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N. Then from (1.1), it follows that x m = x ∗ x m = x ∗ x_(m)=x^(**)x_{m}=x^{*}xm=x∗ for each m > n m > n m > nm>nm>n, and the theorem is proved. Now suppose that x n ≠ x ∗ x n ≠ x ∗ x_(n)!=x^(**)x_{n} \neq x^{*}xn≠x∗ for each n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N. Using (1.1) and (1.2),
‖ x n + 1 − x ∗ ‖ 2 = ⟨ x n + 1 − x ∗ , j ( x n + 1 − x ∗ ) ⟩ = ⟨ ( 1 − λ ) ( x n − x ∗ ) + λ ( T x n − T x ∗ ) , j ( x n + 1 − x ∗ ) ⟩ = ( 1 − λ ) ⟨ ( x n − x ∗ ) , j ( x n + 1 − x ∗ ) ⟩ + λ ⟨ T x n − T x ∗ , j ( x n + 1 − x ∗ ) ⟩ ≤ ( 1 − λ ) ‖ x n − x ∗ ‖ ‖ x n + 1 − x ∗ ‖ + λ ⟨ T x n + 1 − T x ∗ , j ( x n + 1 − x ∗ ) ⟩ + λ ⟨ T x n − T x n + 1 , j ( x n + 1 − x ∗ ) ⟩ ≤ ( 1 − λ ) ‖ x n − x ∗ ‖ ‖ x n + 1 − x ∗ ‖ + λ ‖ x n + 1 − x ∗ ‖ 2 − λ ψ ( ‖ x n + 1 − x ∗ ‖ ) + λ ‖ T x n − T x n + 1 ‖ ‖ x n + 1 − x ∗ ‖ (2.1) ≤ ‖ x n + 1 − x ∗ ‖ ( ( 1 − λ ) ‖ x n − x ∗ ‖ + λ ‖ x n + 1 − x ∗ ‖ − λ ψ ( ‖ x n + 1 − x ∗ ‖ ) ‖ x n + 1 − x ∗ ‖ + λ ‖ T x n − T x n + 1 ‖ ) x n + 1 − x ∗ 2 = x n + 1 − x ∗ , j x n + 1 − x ∗ = ( 1 − λ ) x n − x ∗ + λ T x n − T x ∗ , j x n + 1 − x ∗ = ( 1 − λ ) x n − x ∗ , j x n + 1 − x ∗ + λ T x n − T x ∗ , j x n + 1 − x ∗ ≤ ( 1 − λ ) x n − x ∗ x n + 1 − x ∗ + λ T x n + 1 − T x ∗ , j x n + 1 − x ∗ + λ T x n − T x n + 1 , j x n + 1 − x ∗ ≤ ( 1 − λ ) x n − x ∗ x n + 1 − x ∗ + λ x n + 1 − x ∗ 2 − λ ψ x n + 1 − x ∗ + λ T x n − T x n + 1 x n + 1 − x ∗ (2.1) ≤ x n + 1 − x ∗ ( 1 − λ ) x n − x ∗ + λ x n + 1 − x ∗ − λ ψ x n + 1 − x ∗ x n + 1 − x ∗ + λ T x n − T x n + 1 {:[||x_(n+1)-x^(**)||^(2)],[quad=(:x_(n+1)-x^(**),j(x_(n+1)-x^(**)):)],[quad=(:(1-lambda)(x_(n)-x^(**))+lambda(Tx_(n)-Tx^(**)),j(x_(n+1)-x^(**)):)],[quad=(1-lambda)(:(x_(n)-x^(**)),j(x_(n+1)-x^(**)):)+lambda(:Tx_(n)-Tx^(**),j(x_(n+1)-x^(**)):)],[ <= (1-lambda)||x_(n)-x^(**)||||x_(n+1)-x^(**)||+lambda(:Tx_(n+1)-Tx^(**),j(x_(n+1)-x^(**)):)+lambda(:Tx_(n)-Tx_(n+1),j(x_(n+1)-x^(**)):)],[ <= (1-lambda)||x_(n)-x^(**)||||x_(n+1)-x^(**)||+lambda||x_(n+1)-x^(**)||^(2)-lambda psi(||x_(n+1)-x^(**)||)+lambda||Tx_(n)-Tx_(n+1)||||x_(n+1)-x^(**)||],[(2.1) <= ||x_(n+1)-x^(**)||((1-lambda)||x_(n)-x^(**)||+lambda||x_(n+1)-x^(**)||-lambda(psi(||x_(n+1)-x^(**)||))/(||x_(n+1)-x^(**)||)+lambda||Tx_(n)-Tx_(n+1)||)]:}\begin{align*} & \left\|x_{n+1}-x^{*}\right\|^{2} \\ & \quad=\left\langle x_{n+1}-x^{*}, j\left(x_{n+1}-x^{*}\right)\right\rangle \\ & \quad=\left\langle(1-\lambda)\left(x_{n}-x^{*}\right)+\lambda\left(T x_{n}-T x^{*}\right), j\left(x_{n+1}-x^{*}\right)\right\rangle \\ & \quad=(1-\lambda)\left\langle\left(x_{n}-x^{*}\right), j\left(x_{n+1}-x^{*}\right)\right\rangle+\lambda\left\langle T x_{n}-T x^{*}, j\left(x_{n+1}-x^{*}\right)\right\rangle \\ & \leq(1-\lambda)\left\|x_{n}-x^{*}\right\|\left\|x_{n+1}-x^{*}\right\|+\lambda\left\langle T x_{n+1}-T x^{*}, j\left(x_{n+1}-x^{*}\right)\right\rangle+\lambda\left\langle T x_{n}-T x_{n+1}, j\left(x_{n+1}-x^{*}\right)\right\rangle \\ & \leq(1-\lambda)\left\|x_{n}-x^{*}\right\|\left\|x_{n+1}-x^{*}\right\|+\lambda\left\|x_{n+1}-x^{*}\right\|^{2}-\lambda \psi\left(\left\|x_{n+1}-x^{*}\right\|\right)+\lambda\left\|T x_{n}-T x_{n+1}\right\|\left\|x_{n+1}-x^{*}\right\| \\ & \leq\left\|x_{n+1}-x^{*}\right\|\left((1-\lambda)\left\|x_{n}-x^{*}\right\|+\lambda\left\|x_{n+1}-x^{*}\right\|-\lambda \frac{\psi\left(\left\|x_{n+1}-x^{*}\right\|\right)}{\left\|x_{n+1}-x^{*}\right\|}+\lambda\left\|T x_{n}-T x_{n+1}\right\|\right) \tag{2.1} \end{align*}‖xn+1−x∗‖2=⟨xn+1−x∗,j(xn+1−x∗)⟩=⟨(1−λ)(xn−x∗)+λ(Txn−Tx∗),j(xn+1−x∗)⟩=(1−λ)⟨(xn−x∗),j(xn+1−x∗)⟩+λ⟨Txn−Tx∗,j(xn+1−x∗)⟩≤(1−λ)‖xn−x∗‖‖xn+1−x∗‖+λ⟨Txn+1−Tx∗,j(xn+1−x∗)⟩+λ⟨Txn−Txn+1,j(xn+1−x∗)⟩≤(1−λ)‖xn−x∗‖‖xn+1−x∗‖+λ‖xn+1−x∗‖2−λψ(‖xn+1−x∗‖)+λ‖Txn−Txn+1‖‖xn+1−x∗‖(2.1)≤‖xn+1−x∗‖((1−λ)‖xn−x∗‖+λ‖xn+1−x∗‖−λψ(‖xn+1−x∗‖)‖xn+1−x∗‖+λ‖Txn−Txn+1‖)
Hence
(2.2) ‖ x n + 1 − x ∗ ‖ ≤ ( 1 − λ ) ‖ x n − x ∗ ‖ + λ ‖ x n + 1 − x ∗ ‖ − λ ψ ( ‖ x n + 1 − x ∗ ‖ ) ‖ x n + 1 − x ∗ ‖ + λ ‖ T x n − T x n + 1 ‖ (2.2) x n + 1 − x ∗ ≤ ( 1 − λ ) x n − x ∗ + λ x n + 1 − x ∗ − λ ψ x n + 1 − x ∗ x n + 1 − x ∗ + λ T x n − T x n + 1 {:(2.2)||x_(n+1)-x^(**)|| <= (1-lambda)||x_(n)-x^(**)||+lambda||x_(n+1)-x^(**)||-lambda(psi(||x_(n+1)-x^(**)||))/(||x_(n+1)-x^(**)||)+lambda||Tx_(n)-Tx_(n+1)||:}\begin{equation*} \left\|x_{n+1}-x^{*}\right\| \leq(1-\lambda)\left\|x_{n}-x^{*}\right\|+\lambda\left\|x_{n+1}-x^{*}\right\|-\lambda \frac{\psi\left(\left\|x_{n+1}-x^{*}\right\|\right)}{\left\|x_{n+1}-x^{*}\right\|}+\lambda\left\|T x_{n}-T x_{n+1}\right\| \tag{2.2} \end{equation*}(2.2)‖xn+1−x∗‖≤(1−λ)‖xn−x∗‖+λ‖xn+1−x∗‖−λψ(‖xn+1−x∗‖)‖xn+1−x∗‖+λ‖Txn−Txn+1‖
Since lim n → ∞ ‖ x n + 1 − x n ‖ = 0 lim n → ∞   x n + 1 − x n = 0 lim_(n rarr oo)||x_(n+1)-x_(n)||=0\lim _{n \rightarrow \infty}\left\|x_{n+1}-x_{n}\right\|=0limn→∞‖xn+1−xn‖=0 and T T TTT is uniformly continuous, it follows that
(2.3) lim n → ∞ ‖ T x n − T x n + 1 ‖ = 0 (2.3) lim n → ∞   T x n − T x n + 1 = 0 {:(2.3)lim_(n rarr oo)||Tx_(n)-Tx_(n+1)||=0:}\begin{equation*} \lim _{n \rightarrow \infty}\left\|T x_{n}-T x_{n+1}\right\|=0 \tag{2.3} \end{equation*}(2.3)limn→∞‖Txn−Txn+1‖=0
Set a n = ‖ x n − x ∗ ‖ , ε n = ‖ T x n − T x n + 1 ‖ a n = x n − x ∗ , ε n = T x n − T x n + 1 a_(n)=||x_(n)-x^(**)||,epsi_(n)=||Tx_(n)-Tx_(n+1)||a_{n}=\left\|x_{n}-x^{*}\right\|, \varepsilon_{n}=\left\|T x_{n}-T x_{n+1}\right\|an=‖xn−x∗‖,εn=‖Txn−Txn+1‖ and use Lemma 1.2 to obtain the conlcusion.
Remark 2.2. (1) If B B BBB is not bounded, then Theorem 2.1 holds under the assumption that { x n } x n {x_(n)}\left\{x_{n}\right\}{xn} is bounded.
(2) If T ( B ) T ( B ) T(B)T(B)T(B) is bounded, then { x n } x n {x_(n)}\left\{x_{n}\right\}{xn} is bounded.
(3) If T T TTT is strongly pseudocontractive, then automatically F ( T ) ≠ ∅ F ( T ) ≠ ∅ F(T)!=O/F(T) \neq \varnothingF(T)≠∅.

3. Further results

Let I I III denote the identity map. A map T : B → B T : B → B T:B rarr BT: B \rightarrow BT:B→B is called pseudocontractive if 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 ⟨ T x − T y , j ( x − y ) ⟩ ≤ ‖ x − y ‖ 2 ⟨ T x − T y , j ( x − y ) ⟩ ≤ ‖ x − y ‖ 2 (:Tx-Ty,j(x-y):) <= ||x-y||^(2)\langle T x-T y, j(x-y)\rangle \leq\|x-y\|^{2}⟨Tx−Ty,j(x−y)⟩≤‖x−y‖2.
Remark 3.1. The operator T T TTT is a (uniformly, strongly) pseudocontractive map if and only if ( I − T ) ( I − T ) (I-T)(I-T)(I−T) is a (uniformly, strongly) accretive map.
Remark 3.2. (1) Let T , S : X → X T , S : X → X T,S:X rarr XT, S: X \rightarrow XT,S:X→X, and let f ∈ X f ∈ X f in Xf \in Xf∈X be given. A fixed point for the map T x = f + ( I − S ) x T x = f + ( I − S ) x Tx=f+(I-S)xT x= f+(I-S) xTx=f+(I−S)x, for all x ∈ X x ∈ X x in Xx \in Xx∈X, is a solution for S x = f S x = f Sx=fS x=fSx=f.
(2) Let f ∈ X f ∈ X f in Xf \in Xf∈X be a given point. If S S SSS is an accretive map, then T = f − S T = f − S T=f-ST=f-ST=f−S is a strongly pseudocontractive map.
Consider Krasnoselskij iteration with T x = f + ( I − S ) x T x = f + ( I − S ) x Tx=f+(I-S)xT x=f+(I-S) xTx=f+(I−S)x,
(3.1) x n + 1 = ( 1 − λ ) x n + λ ( f + ( I − S ) x n ) . (3.1) x n + 1 = ( 1 − λ ) x n + λ f + ( I − S ) x n . {:(3.1)x_(n+1)=(1-lambda)x_(n)+lambda(f+(I-S)x_(n)).:}\begin{equation*} x_{n+1}=(1-\lambda) x_{n}+\lambda\left(f+(I-S) x_{n}\right) . \tag{3.1} \end{equation*}(3.1)xn+1=(1−λ)xn+λ(f+(I−S)xn).
Remarks 3.1 and 3.2 and Theorem 2.1 lead to the following result.
Corollary 3.3. Let X X XXX be a real Banach space and let S : X → X S : X → X S:X rarr XS: X \rightarrow XS:X→X be a uniformly accretive and uniformly continuous operator, with ( I − S ) ( X ) ( I − S ) ( X ) (I-S)(X)(I-S)(X)(I−S)(X) bounded. Suppose that S x = f S x = f Sx=fS x=fSx=f has a solution. Then for any x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X, the Krasnoselskij iteration (3.1) converges to the solution of S x = f S x = f Sx=fS x=fSx=f if and only if lim n → ∞ ‖ x n + 1 − x n ‖ = 0 lim n → ∞   ‖ x n + 1 − x n ‖ = 0 lim_(n rarr oo)||x_(n+1)-x_(n)||=0\lim _{n \rightarrow \infty} \| x_{n+1}- x_{n} \|=0limn→∞‖xn+1−xn‖=0.
Let S S SSS be an accretive operator. The operator T x = f − S x T x = f − S x Tx=f-SxT x=f-S xTx=f−Sx is strongly pseudocontractive for a given f ∈ X f ∈ X f in Xf \in Xf∈X. A solution for T x = x T x = x Tx=xT x=xTx=x becomes a solution for x + S x = f x + S x = f x+Sx=fx+S x=fx+Sx=f. Consider Krasnoselskij iteration with T x := f − S x T x := f − S x Tx:=f-SxT x:=f-S xTx:=f−Sx,
(3.2) x n + 1 = ( 1 − λ ) x n + λ ( f − S x n ) . (3.2) x n + 1 = ( 1 − λ ) x n + λ f − S x n . {:(3.2)x_(n+1)=(1-lambda)x_(n)+lambda(f-Sx_(n)).:}\begin{equation*} x_{n+1}=(1-\lambda) x_{n}+\lambda\left(f-S x_{n}\right) . \tag{3.2} \end{equation*}(3.2)xn+1=(1−λ)xn+λ(f−Sxn).
Again, using Remarks 3.1 and 3.2 and Theorem 2.1, we obtain the following result.
Corollary 3.4. Let X X XXX be a real Banach space and let S : X → X S : X → X S:X rarr XS: X \rightarrow XS:X→X be an accretive and uniformly continuous operator, with ( I − S ) ( X ) ( I − S ) ( X ) (I-S)(X)(I-S)(X)(I−S)(X) bounded. Suppose that x + S x = f x + S x = f x+Sx=fx+S x=fx+Sx=f has a solution. Then for x 0 ∈ X x 0 ∈ X x_(0)in Xx_{0} \in Xx0∈X, the Krasnoselskij iteration (3.2) converges to the solution of x + S x = f x + S x = f x+Sx=fx+S x=fx+Sx=f if and only if lim n → ∞ ‖ x n + 1 − x n ‖ = 0 lim n → ∞   x n + 1 − x n = 0 lim_(n rarr oo)||x_(n+1)-x_(n)||=0\lim _{n \rightarrow \infty}\left\|x_{n+1}-x_{n}\right\|=0limn→∞‖xn+1−xn‖=0.
Remark 3.5. If (1.4) holds for all x ∈ B x ∈ B x in Bx \in Bx∈B and y := x ∗ ∈ F ( T ) y := x ∗ ∈ F ( T ) y:=x^(**)in F(T)y:=x^{*} \in F(T)y:=x∗∈F(T), then such a map is called uniformly hemicontractive. It is trivial to see that our results hold for the uniformly hemicontractive maps.

Acknowledgment

The authors are indebted to referee for carefully reading the paper and for making useful suggestions.

References

[1] M. A. Krasnosel'skii, "Two remarks on the method of successive approximations," Uspekhi Matematicheskikh Nauk, vol. 10, no. 1, pp. 123-127, 1955.
[2] M. O. Osilike, "Iterative solution of nonlinear equations of the ϕ ϕ phi\phiϕ-strongly accretive type," Journal of Mathematical Analysis and Applications, vol. 200, no. 2, pp. 259-271, 1996.
[3] C. E. Chidume and C. O. Chidume, "Convergence theorems for fixed points of uniformly continuous generalized ϕ ϕ phi\phiϕ-hemi-contractive mappings," Journal of Mathematical Analysis and Applications, vol. 303, no. 2, pp. 545-554, 2005.
[4] Ş. M. Şoltuz, "New technique for proving the equivalence of Mann and Ishikawa iterations," Revue d'Analyse Numérique et de Théorie de l'Approximation, vol. 34, no. 1, pp. 103-108, 2005.
2008

Related Posts