Characterization for the Convergence of Krasnoselskij Iteration for...

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2008, Article ID 630589, 5 pagesdoi:10.1155/2008/630589

Research ArticleCharacterization for the Convergence ofKrasnoselskij Iteration for Non-LipschitzianOperators

Stefan M. Soltuz1, 2 and B. E. Rhoades3

1Departamento de Matematicas, Universidad de Los Andes, Carrera 1 no. 18A-10, Bogota, Colombia2 “Tiberiu Popoviciu” Institute of Numerical Analysis, 400110 Cluj-Napoca, Romania3Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

Correspondence should be addressed to Stefan M. Soltuz, [email protected]

Received 13 August 2007; Revised 2 February 2008; Accepted 24 February 2008

Recommended by Enrico Obrecht

We establish the convergence of Krasnoselskij iteration for various classes of non-Lipschitzian op-erators.

Copyright q 2008 S. M. Soltuz and B. E. Rhoades. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

1. Introduction

Let X be a real Banach space; B a nonempty, convex subset of X; and T : B → B an operator.Let x0 ∈ B. The following iteration is known as Krasnoselskij iteration (see [1]):

xn+1 = (1 − λ)xn + λTxn. (1.1)

The map J : X → 2X∗given by Jx := {f ∈ X∗ : 〈x, f〉 = ‖x‖2, ‖f‖ = ‖x‖}, for all x ∈ X, is

called the normalized duality mapping. It is easy to see that we have⟨y, j(x)

⟩ ≤ ‖x‖‖y‖, ∀x, y ∈ X, ∀j(x) ∈ J(x). (1.2)

Denote

Ψ :={ψ | ψ : [0,+∞) −→ [0,+∞) is astrictly increasing map with ψ(0) = 0

}. (1.3)

Definition 1.1. Let X be a real Banach space, and let B be a nonempty subset of X. A map T :B → B is called uniformly pseudocontractive if there exists a map ψ ∈ Ψ and j(x−y) ∈ J(x−y)such that

⟨Tx − Ty, j(x − y)⟩ ≤ ‖x − y‖2 − ψ(‖x − y‖), ∀x, y ∈ B. (1.4)

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2 International Journal of Mathematics and Mathematical Sciences

A map S : X → X is called uniformly accretive if there exists a map ψ ∈ Ψ and j(x −y) ∈J(x − y) such that

⟨Sx − Sy, j(x − y)⟩ ≥ ψ(‖x − y‖), ∀x, y ∈ X. (1.5)

Taking ψ(a) := ψ(a) · a, for all a ∈ [0,+∞), (ψ ∈ Ψ), reduces to the usual definitions ofψ-strongly pseudocontractive and ψ-strongly accretive. Taking ψ(a) := γ · a2, γ ∈ (0, 1), for alla ∈ [0,+∞), (ψ ∈ Ψ), we get the usual definitions of strongly pseudocontractive and stronglyaccretive. Therefore, the class of strongly pseudocontractive maps is included stricly in theclass of ψ-strongly pseudocontractive maps. The example from [2] shows that this inclusion isproper. Remark, further, that the class of ψ-strongly pseudocontractive maps is also includedstrictly in the class of uniformly pseudocontractive maps (see also [3]).

We will give a characterization for the convergence of (1.1) when applied to uniformlypseudocontractive operators. For this purpose, we need the following lemma similar to [4,Lemma 1]. Next, N denotes the set of all natural numbers.

Lemma 1.2. Let {an} be a positive bounded sequence and assume that there exists n0 ∈ N such that

an+1 ≤ (1 − λ)an + λan+1 − λψ(an+1

)

an+1+ λεn, ∀n ≥ n0, (1.6)

where λ ∈ (0, 1), εn ≥ 0, for all n ∈ N and limn→∞εn = 0. Then limn→∞an = 0.

Proof. There exists an M > 0 such that an ≤ M, for all n ∈ N. Denote a := lim inf an. We willprove that a = 0. Suppose on the contrary that a > 0. Then there exists anN1 ∈ N such that

an ≥ a

2, ∀n ≥N1. (1.7)

From limn→∞εn = 0,we know that there exists anN2 ∈ N such that

εn ≤ ψ(a/2)2M

, ∀n ≥N2. (1.8)

SetN0 := max{N1,N2}. Using the fact that −(1/M) ≥ −(1/an+1),we get the following:

an+1 ≤ (1 − λ)an + λan+1 − λψ(an+1

)

an+1+ λεn

≤ (1 − λ)an + λan+1 − λψ(a/2)M

+ λψ(a/2)2M

≤ (1 − λ)an + λan+1 − λψ(a/2)2M

,

(1.9)

which implies that (1 − λ)an+1 ≤ (1 − λ)an − λ((ψ(a/2))/2M), or

an+1 ≤ an − λ

1 − λψ(a/2)2M

≤ an − λψ(a/2)2M

, (1.10)

S. M. Soltuz and B. E. Rhoades 3

since −(λ/(1 − λ)) ≤ −λ. Thus λ(ψ(a/2))/2M ≤ an − an+1, which implies that∑λ < ∞, in

contradiction to∑λ = ∞. Therefore, lim inf an = 0. Hence there exists a subsequence {anj} ⊂

{an} such that limj→∞anj = 0. Fix ε > 0. Then there exists an n3 ∈ N such that

anj <ε

4, ∀j ≥ n3. (1.11)

Also there exists an n4 ∈ N such that

εn <ψ(ε/4)2M

, ∀n ≥ n4. (1.12)

Define n0 := max{n3, n4,N0}.We claim that anj+k < ε/4 for each j > n0 and each k > 0. Supposenot. Then there exists an n0 and a k > 0 such that

anj+k ≥ε

4. (1.13)

For this nj, let k denote the smallest positive integer for which (1.13) is true. Then anj+k−1 ≤ ε/4.From (1.6),

anj+k ≤ (1 − λ)anj+k−1 + λanj+k − λψ(anj+k

)

anj+k+ λεnj+k−1

≤ (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

,

(1.14)

which implies that anj+k ≤ (ε/4) − (λ/(1 − λ))(ψ(ε/4)/2M). This leads to the contradiction:

ε

4≤ anj+k ≤

ε

4− λ

1 − λψ(ε/4)2M

<ε

4. (1.15)

Therefore, anj+k < ε/4, for all k ∈ N, and each j > n0, hence limn→∞an = 0.

2. Main result

Theorem 2.1. LetX be a real Banach space, B a nonempty, closed, convex, bounded subset ofX. Let T :B → B be a uniformly pseudocontractive and uniformly continuous operator with F(T)/=∅. Then forx0 ∈ B, the Krasnoselskij iteration (1.1) converges to the fixed point of T if and only if limn→∞‖xn+1 −xn‖ = 0.

Proof. Since T is a self-map of B, which is bounded and convex, then, from (1.1), each xn ∈ B,so {xn} is bounded for each n ∈ N. Uniqueness of the fixed point follows from (1.4). If {xn}converges to the fixed point of T, that is, limn→∞xn = x∗, then, obviously, limn→∞‖xn+1−xn‖ = 0.Conversely, we will prove that if limn→∞‖xn+1 − xn‖ = 0, then limn→∞xn = x∗. Suppose that

4 International Journal of Mathematics and Mathematical Sciences

xn = x∗ for some n ∈ N. Then from (1.1), it follows that xm = x∗ for each m > n, and thetheorem is proved. Now suppose that xn /=x∗ for each n ∈ N. Using (1.1) and (1.2),

∥∥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∗∥∥

≤ ∥∥xn+1 − x∗∥∥(

(1 − λ)∥∥xn − x∗∥∥ + λ‖xn+1 − x∗‖ − λψ(‖xn+1 − x∗‖)

‖xn+1 − x∗‖ + λ‖Txn − Txn+1‖)

.

(2.1)

Hence

∥∥xn+1 − x∗∥∥ ≤ (1 − λ)∥∥xn − x∗∥∥ + λ∥∥xn+1 − x∗∥∥ − λψ

(∥∥xn+1 − x∗∥∥)

∥∥xn+1 − x∗∥∥ + λ∥∥Txn − Txn+1

∥∥. (2.2)

Since limn→∞‖xn+1 − xn‖ = 0 and T is uniformly continuous, it follows that

limn→∞

∥∥Txn − Txn+1∥∥ = 0. (2.3)

Set an = ‖xn − x∗‖, εn = ‖Txn − Txn+1‖ and use Lemma 1.2 to obtain the conlcusion.

Remark 2.2. (1) If B is not bounded, then Theorem 2.1 holds under the assumption that {xn} isbounded.

(2) If T(B) is bounded, then {xn} is bounded.(3) If T is strongly pseudocontractive, then automatically F(T)/=∅.

3. Further results

Let I denote the identity map. A map T : B → B is called pseudocontractive if there existsj(x − y) ∈ J(x − y) such that 〈Tx − Ty, j(x − y)〉 ≤ ‖x − y‖2.

Remark 3.1. The operator T is a (uniformly, strongly) pseudocontractive map if and only if(I − T) is a (uniformly, strongly) accretive map.

Remark 3.2. (1) Let T, S : X → X, and let f ∈ X be given. A fixed point for the map Tx =f + (I − S)x, for all x ∈ X, is a solution for Sx = f.

(2) Let f ∈ X be a given point. If S is an accretive map, then T = f − S is a stronglypseudocontractive map.

S. M. Soltuz and B. E. Rhoades 5

Consider Krasnoselskij iteration with Tx = f + (I − S)x,

xn+1 = (1 − λ)xn + λ(f + (I − S)xn

). (3.1)

Remarks 3.1 and 3.2 and Theorem 2.1 lead to the following result.

Corollary 3.3. Let X be a real Banach space and let S : X → X be a uniformly accretive and uniformlycontinuous operator, with (I − S)(X) bounded. Suppose that Sx = f has a solution. Then for any x0 ∈X, the Krasnoselskij iteration (3.1) converges to the solution of Sx = f if and only if limn→∞‖xn+1 −xn‖ = 0.

Let S be an accretive operator. The operator Tx = f−Sx is strongly pseudocontractive fora given f ∈ X. A solution for Tx = x becomes a solution for x + Sx = f. Consider Krasnoselskijiteration with Tx := f − Sx,

xn+1 = (1 − λ)xn + λ(f − Sxn

). (3.2)

Again, using Remarks 3.1 and 3.2 and Theorem 2.1, we obtain the following result.

Corollary 3.4. Let X be a real Banach space and let S : X → X be an accretive and uniformly contin-uous operator, with (I − S)(X) bounded. Suppose that x + Sx = f has a solution. Then for x0 ∈ X, theKrasnoselskij iteration (3.2) converges to the solution of x+Sx = f if and only if limn→∞‖xn+1−xn‖ = 0.

Remark 3.5. If (1.4) holds for all x ∈ B and y := x∗ ∈ F(T), then such a map is called uniformlyhemicontractive. 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 sug-gestions.

References

[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 ofMathematical 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.

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