Một vài phương pháp tìm điểm bất động chung của họ hữu hạn các ánh xạ không giãn

I express my sincere gratitude to my thesis advisor Prof. Pham Ky Anh, who has introduced me to the field of Numerical Analysis. I am especially grateful for his patience and ability of making abstract mathematics so easily to be perceived. I also want to thank my family since they always motivate, encourage and create favorable conditions for my study and research. Finally, I want to thank my friends in K53 Advanced Maths. They always stay by my side to encourage and to help me.

VIETNAM NATIONAL UNIVERSITYUNIVERSITY OF SCIENCEFACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS

Nguyen Ly Vinh Hanh

SOME METHODS FOR COMMON FIXED POINTS OF

A FAMILY OF NONEXPANSIVE MAPPINGS

Undergraduate Thesis Advanced Undergraduate Program in Mathematics

Hanoi - 2012

VIETNAM NATIONAL UNIVERSITYUNIVERSITY OF SCIENCEFACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS

Nguyen Ly Vinh Hanh

SOME METHODS FOR COMMON FIXED POINTS OF

A FAMILY OF NONEXPANSIVE MAPPINGS

Undergraduate Thesis Advanced Undergraduate Program in Mathematics

Thesis Advisor: Prof Dr Sc Pham Ky Anh

Hanoi - 2012

I express my sincere gratitude to my thesis advisor Prof Pham Ky Anh, who has duced me to the field of Numerical Analysis I am especially grateful for his patience andability of making abstract mathematics so easily to be perceived

intro-I also want to thank my family since they always motivate, encourage and create able conditions for my study and research

favor-Finally, I want to thank my friends in K53 Advanced Maths They always stay by myside to encourage and to help me

Ha Noi, May, 2012.Nguyen Ly Vinh Hanh

2.1 Krasnoselskij iteration for nonexpansive mappings 14

2.2 Mann iterations for nonexpansive mappings 20

2.2.1 Strongly Pseudocontractive Operators 23

2.2.2 Nonexpansive and quasi-nonexpansive operators 26

3 Iterations for relatively nonexpansive mappings 34 3.1 A hybrid method for relatively nonexpansive mappings 34

3.2 Strong Convergence theorems for a Finite Family of Relatively Nonexpan-sive Mappings 39

3.3 Parallel hybrid methods for a finite family of relatively nonexpansive mappings 44 4 Applications 52 4.1 Some basic facts about projections onto hyperplanes 52

4.2 The algebraic reconstruction technique 54

4.2.1 Introduction 54

4.2.2 The convergence of the ART 55

4.3 Reconstruction by successive approximation 57

4.3.1 Introduction 57

4.3.2 The convergence of the reconstruction by successive approximation 584.4 A numerical example 624.4.1 Applying the ART method 624.4.2 Applying Liu’s method 64

In this thesis, we deal with iteration methods for finding fixed points of nonexpansive pings on normed spaces The origin of these methods dates back to 1920, when Stefan Ba-nach(1892 − 1945) formulated his famous contraction mapping principle Banach proved

map-that if(X , d) is a complete metric space and T : X → X is a given contraction, then T has a

unique fixed point p , i.e., T (p) = p and T n (x) → p (as n →∞)

Many scientific problems in game theory, theory of phase transitions, optimization theory,differential equations, differential geometry, image processing, etc lead to a problem offinding fixed points of nonexpansive mappings

This minor thesis will attempt to highlight some achievements in the theory of nonexpansivemappings and concentrate to the following problems

• The existence of fixed points;

• The approximation of fixed points;

• Applications of the fixed point theory

The thesis consists of four chapters

The first chapter is devoted to some minimal functional analysis background

The second chapter is devoted to Krasnoselskij and Mann iterations for nonexpansivemappings and the third chapter deals with hybrid methods for relatively nonexpansivemappings

The last chapter provides a crucial application of nonexpansive mappings in imageprocessing

Chapter 1

Preliminary

In this chapter we collect some facts on nonlinear operators and geometry of Banachspaces

Definition 1.1 Let C be a nonempty set of a metric space and T be a mapping from C into

itseft An element x∗∈ C is called a fixed point of T if T x = x and the set of all fixed points

Definition 1.3 A subset C of a real normed space is called bounded if there exists M> 0

such that||x|| ≤ M, for all x ∈ C.

Definition 1.4 A subset C of a real vector space X is called covex if, for any pair of points

x , y in C, the closed segment with the endpoints x, y, i.e., the set {λ x+ (1 −λ)y :λ ∈ [0, 1)}

is contained in C.

Definition 1.5 A Banach space(X , ||.||) is called strictly convex if, for all x, y ∈ X satisfying

||x|| ≤ 1, ||y|| ≤ 1 and x 6= y, we have ||x + y|| < 2.

Example 1.2.

• All inner product spaces are strictly convex.

• Let X = R2 Then (X , ||.||2) is strictly covex, (X , ||.||1) and (X , ||.||∞) are not strictly

convex

Example 1.3 C [a, b] is not stritly convex.

Proof Choose, x ≡ 1 and y = b t −a −a Clearly, x 6= y.

On the other hand,

Definition 1.6 A Banach space(X , ||.||) is called uniformly convex if, given anyε> 0, there

existsδ > 0 such that for all x, y ∈ X satisfying ||x|| ≤ 1, ||y|| ≤ 1, and ||x − y|| ≥ε, we have

1

2||x + y|| < 1 −δ

Definition 1.7 Let X be a real Banach space The space X∗of all linear continous functionals

on X is called the dual space of X For f ∈ X∗ and x ∈ X the value of f at x is denoted by

h f , xi and is called the duality pairing.

• The dual X∗is a Banach space with respect to the norm

Lemma 1.1 If X∗is strictly convex then

∀x ∈ X ∃!Jx ∈ X∗: hJx, xi = ||x||2X = ||Jx||2X∗

Proof For any fixed element x ∈ X , x 6= 0 consider the subspace X0and the linear

func-tional f defined as follows:

Now we prove that Jx is unique.

Assuming that there exist f i ∈ X∗: h f i , xi = ||x||2= || f i||2 (i = 1, 2) we get

|| f1+ f2||||x|| ≥ h f1+ f2, xi = || f1||2+ || f2||2= (|| f1|| + || f2||)||x||

≥ || f1+ f2||||x||.

Therefore,|| f1+ f2|| = || f1|| + || f2|| Since X∗is stritly convex, f1≡ f2

Definition 1.9 A Banach space X is called smooth if, for every x ∈ X with ||x|| = 1, there

exists a unique f ∈ X∗ such that|| f || = h f , xi = 1 The modulus of smoothness of X is the

fuctionρX :[0,∞) → [0,∞), defined by

ρX(τ) = sup 1

2(||x + y|| + ||x − y||) − 1 : x, y ∈ X , ||x|| = 1, ||y|| =τ

The Banach space X is called uniformly smooth if

Example 1.4.

1 A Hilbert space is uniformly convex and uniformly smooth

2 L p (G) is a uniformly convex and uniformly smooth B-space, where 1 < q <∞; G ⊂ R n

measurable set

3 l pis uniformly convex and uniformly smooth for 1< p <∞

4 W m p (G) is uniformly convex and uniformly smooth for 1 < p <∞

Proposition 1.1 If X is uniformly smooth, then J is uniformly norm-to-norm continous on

each bounded subset of X

Theorem 1.2 A uniformly convex Banach space X has the Klee-Kadec (E f imov−Stechkin)

property, i.e., from x n ⇀ x and ||x n || → ||x|| it follows x n → x.

Proof.

The case, when x = 0 is trivial Let x 6= 0 Without loss of generality assuming that ||x|| = 1

and||x n || 6= 0, then we have y n:= x n

• E (E∗) is uniformly convex if and only if E∗(E) is uniformly smooth.

• If E is uniformly convex, then it is reflexive and strictly convex and it satisfies the

Kadec-Klee property

• If E is uniformly smooth and uniformly convex then J and J−1= J∗are single-valued

and uniformly norm-to-norm continuous on bounded subsets of E and E∗, respectively

Definition 1.10 Let C be a subset of a normed space (X , ||.||) A mapping T : C → C is

(C5) contractive if ||T x − Ty|| < ||x − y||, for all x, y ∈ C, x 6= y;

(C6) isometry if ||T x − Ty|| = ||x − y||, for all x, y ∈ C.

Definition 1.11 Let E be a smooth Banach space and let E∗ be the dual of E The funtion

φ : E × E → R is defined by

φ(y, x) = ||y||2− 2hy, Jxi + ||x||2,

for all x , y ∈ X , where J is the normalized duality mapping from X → X∗

Therefore, we have

(||x|| − ||y||)2≤φ(y, x) ≤ (||y||2+ ||x||2), for allx , y ∈ E. (1.1)

Definition 1.12 The generalized projectionΠC : E → C is a map that assigns to an arbitrary

point x ∈ E the minimum point of the funtional φ(x, y), that is ΠC x = ¯x, where ¯x is the

solution to the minimization problemφ( ¯x, x) = min y ∈Cφ(y, x).

A point p in C is said to be an asymptotic fixed point of T if C contains a sequence {x n}

which converges weakly to p such that lim n→∞||T x n − x n|| = 0 The set of asymptotic fixed

point of T will be denoted by ˆ F (T ).

Definition 1.13 Let C be a closed convex subset of E, and let T be a mapping from C into

itself A mapping T is called relatively nonexpansive if

• F (T ) is nonempty,

• ˆF (T ) = F(T ),

• φ(p, T x) ≤φ(p, x) for all x ∈ C and p ∈ F(T ).

A point p in C is said to be a strong asymptotic fixed point of T if C contains a sequence

{x n } which converges strongly to p such that lim n→ ∞||T x n − x n|| = 0 The set of strong

asymptotic fixed points of T will be denoted by ¯ F (T ).

Definition 1.14 A mapping T from C into itself is called weakly relatively nonexpansive if

Definition 1.16 Let H be a real Hilbert space with norm||.|| and an inner product h., i, and

K be a nonempty subset of H An operator T : K → K is said to be a generalized

pseudo-contraction if for all x , y ∈ K, there exists a constant r > 0 such that

||T x − Ty||2≤ r2||x − y||2+ ||T x − Ty − r(x − y)||2

It is equivalent to

hT x − Ty, x − yi ≤ r||x − y||2

or

h(I − T )x − (I − T )y, x − yi ≥ (1 − r)||x − y||2,

where I is the identity mapping.

Definition 1.17 Let X be an arbitrary real Banach space A mapping T with domain D (T )

and range R (T ) in X is called

• pseudocontractive if for each x , y ∈ D(T ) there exists j(x − y) ∈ J(x − y) such that

h(I − T )x − (I − T )y, j(x − y)i ≥ 0,

where J is the normalized duality mapping.

• strong pseudocontractive if the exists k > 0 such that for all x, y ∈ D(T ) there exists

j (x, y) ∈ J(x − y) such that

h(I − T )x − (I − T )y, j(x − y)i ≥ k||x − y||2

Definition 1.18 A mapping U with domain and range in X is called

• Accretive if for each x , y ∈ D(U ), we have

hU x −U y, j(x − y)i ≥ 0.

• Strongly accretive if there exists a positive number k such that for each x , y ∈ D(U ),

there exists a j (x − y) ∈ J(x − y) such that

hU x −U y, j(x − y)i ≥ k||x − y||2

Proposition 1.2.

1 An operator T is (strongly) pseudo-contractive if and only if (I − T ) is (strongly)

ac-cretive.

2 T is strongly pseudo-contractive if there exists t > 1 such that, for all x, y ∈ D(T ) and

k > 0, the following inequality holds

||x − y|| ≤ ||(1 + r)(x + y) − rt(T x − Ty)||. (1.2)

3 T is pseudo-contractive if t = 1 in inequality (1.2).

4 T is strongly accretive if there exists k > 0 such that the inequality

||x − y|| ≤ ||(x − y) + r[(T − kI)x − (T − kI)y]|| (1.3)

holds for all x , y ∈ D(U ) and r > 0.

5 T is accretive if k = 0 in the inequality (1.3)

Definition 1.19 Let H be a Hilbert space and C a subset of H A mapping T : C → H is

called demicompact if it has the property that whenever{u n } is a bounded sequence in H

and{Tu n − u n } is strongly convergent, then there exists a subsequence {u n k } of {u n} which

is strongly convergent

Definition 1.20 A set of points is defined to be convex if it contains the line segments

connecting each pair of its points The convex hull of a given set X is defined as the (unique) minimal convex set containing X , and denoted by coX

Theorem 1.3. ( The Schauder Fixed Point Theorem ) Let E be a closed, bounded, convex

subset of a normed space X If T : E → X is a compact map such that T (E) ⊆ E, then there

is an x ∈ E such that T (x) = x.

Theorem 1.4. ( Mazur Theorem ) If K is a compact subset of a Banach space X , then the

closed convex hull co (K) is compact.

Definition 1.21 Let X be a normed linear space and f ∈ X∗

Chapter 2

Iterations for nonexpansive mappings

2.1 Krasnoselskij iteration for nonexpansive mappings

Based probably on ideas of Cauchy and Liouville, Picard developed the following method

of successive approximations,

x n = T (x n−1) = T n (x0) n = 1, 2, (2.1)

where T is an L-contraction.

However, if T is assumed to be only a nonexpansive map, then the Picard iterations {T n x0}n≥0

need no longer converge (to a fixed point of T ) Then Krasnoselskij used the averaged ping associated to T , namely Tλ = (1 −λ)I +λT Clearly, Tλ has the same fixed point set

map-as T Furthermore, it will be proved that Tλn x0converges to a fixed point as n→∞

Theorem 2.1 Let C be a closed bounded convex subset of the Hilbert space H and T : C → C

be a nonexpansive operator Then T has at least on fixed point.

Proof For a fixed element v0in C and a numberλ with 0<λ < 1, we denote

Tλ(x) = (1 −λ)v0+λT x, x ∈ C. (2.2)

Since C is convex and closed, Tλ maps C into itself Moreover, we obtain

||Tλ(x) − Tλ(y)|| = ||(1 −λ)v0+λT x− (1 −λ)v0−λTy|| = ||λ(T x − Ty)||

≤ |λ|||T x − Ty|| ≤ |λ |||x − y||.

Therefore, Tλ : C → C is aλ-contraction, then it has a unique fixed point, say xλ On the other

hand, since C is closed, convex and bounded in the Hilbert space H, it is weakly compact.

Hence we may find a sequence λj in (0, 1) such that λj → 1 (as j →∞ ) and x j = xλj

converges weakly to an element p of H.

Since C is weakly closed, p lies in C We shall prove that p is a fixed point of T If x is any arbitrary point in H, we have

and due to the boundedness of C, we have also

lim sup(||x j −T p||2−||x j − p||2) = lim sup(||x j −T p||−||x j − p||)(||x j −T p||+||x j − p||) ≤ 0,

Lemma 2.1 Let C be a bounded closed convex subset of a Hilbert space H and T : C → C

be a nonexpansive and demicompact operator Then the set F (T ) of fixed points of T is a

nonempty convex set.

Proof Since T is nonexpansive then T has fixed points in C Therefore, F (T ) 6= /0.

Let x , y ∈ F(T ) andλ ∈ [0, 1] We need to prove that

z= (1 −λ)x +λy ∈ F(T ).

Indeed, since x , y ∈ F(T ) then

||x − T z|| = ||T x − T z|| ≤ ||x − z||, and ||T z − y|| = ||T z − Ty|| ≤ ||z − y||.

Theorem 2.2 Let C be a bounded closed convex subset of a Hilbert space H and T : C → C

be a nonexpansive and demicompact operator Then the set F (T ) of fixed points of T is a

nonempty convex set and for any given x0 in C and any fixed numberλ with 0<λ < 1, the

Krasnoselskij iteration {x n}∞n=0given by

x n+1= (1 −λ)x n+λT x n, n= 0, 1, 2, (2.3)

converges (strongly) to a fixed point of T

Proof The proof is divided into two steps.

1 F (T ) is a nonempty convex set (Lemma 2.1).

2 The Krasnoselskij iteration converges

For the second step, observe that for any fixed x0∈ C, the sequence {x n}∞n=0 given by (2.3)

lies in C and is bounded Let p be a fixed point of T , and consider the averaged map Tλ,

given by

where I is the identity map.

We first prove that the sequence{x n − T x n}n ∈Nconverges strongly to zero Indeed,

x n+1− p = (1 −λ)x n+λT x n − p = (1 −λ)(x n − p) +λ(T x n − p).

On the other hand, for any constant a,

which shows that∑∞n=0||x n − T x n||2<∞and hence||x n − T x n || → 0, as n →∞

Since T is demicompact, there exists a strongly convergent subsequence {x n i} such that

x n i → p ∈ F(T ).

Since T is nonexpansive, T x n j → T p and T p = p.

The convergence of the entire sequence{x n}∞n=0 to p now follows from the inequality

||x n+1− p|| ≤ ||x n − p||,

which can be deduce from the nonexpansiveness of T and is valid for each n.

Algorithm 2.1 Let x0be an initial approximation, n be maximum number of iterations,λ be

a given number in[0, 1], and ε be max change of x value to allow abort For k= 1, 2, we

If err≤ε then print x new , else x0= x new and repeat

Remark 2.1 In general, there is no estimation for the convergence rate of the Krasnoselskij

method In fact, it is typical that the convergence of iteration methods involving sive mappings may be arbitrary slow

nonexpan-Corollary 2.1 [2] Let X be a uniformly convex Banach space, D a closed bounded convex

set in X , and T a nonexpansive mapping of D into D such that T satisfies any one of the following two conditions.

1 (I − T ) maps closed sets in D into closed sets in X ;

2 T is demicompact at 0.

Then for any given x0in C and any fixed numberλwith 0<λ< 1, the Krasnoselskij iteration {x n}∞n=0given by (2.3) coverges (strongly) to a fixed point of T

Theorem 2.3 Suppose T is a nonexpansive operator that maps a bounded closed convex set

C of H into C and that F (T ) = {p} Then the Krasnoselskij iteration converges weakly to p,

Tλn ⇀ p, for any x0∈ C. (2.5)

Proof It suffices to show that if {x n j}∞j=0, x n j = T n j

λ x converges weakly to a certain p0, then

p0is a fixed point of T or of Tλ and therefore p0= p.

Suppose that{x n j}∞j=0does not converge weakly to p Then

||x n j − Tλp0|| ≤ ||Tλx n j − Tλp0|| + ||x n j − Tλx n j ≤ ||x n j − p0|| + ||x n j − Tλx n j||

and, using the arguments in the proof of Theorem 2.2, it results

||x n j − Tλx n j|| → 0, as n→∞

The last inequality implies that

lim sup(||x n j − Tλp0|| − ||x n j − p0||) ≤ 0 (2.6)But, like in the proof of Theorem 2.2, we have

Theorem 2.4 Let C be a bounded closed convex subset of a Hilbert space and T : C → C be

a nonexpansive operator Then, for any x0in C, the Krasnoselskij iteration converges weakly

to a fixed point of T

Proof Let F (T ) be the set of all fixed points of T in C (which is nonempty, by Theorem

2.1) As T is nonexpansive, for each p ∈ F(T ) and each n we have

||x n+1− p|| ≤ ||x n − p||,

which shows that the function g (p) = lim n→ ∞||x n − p|| is well defined and is a lower

semi-continuous covex function on F (T ) Let

Hence, ε >0Fε 6= /0 Moreover, F0 contains exactly one point Indeed, since F0 is convex

and closed, for p0, p1∈ F0, and pλ = (1 −λ)p0+λp1,

which leads to a contraction

Now, in order to show that x n = T n

λx0⇀ p0, is suffices to assume that x n j ⇀ p for an infinite

subsequence and then prove that p = p0 By the arguments in Theorem 2.3, p ∈ F(T ).

Taking into account the definition of g and the fact that x n j → p, we have

2.2 Mann iterations for nonexpansive mappings

The Mann iteration was chronologically introduced two years earlier than the Krasnoselskijiteration, even so it is generalization of the latter and in its normal form is obtained byreplacing the parameterλ in the Krasnoselskij iteration formula by a sequence{a n}

Definition 2.1 Let E be a linear space, C a convex subset of E and let T : C → C be a

mapping and x1∈ C, arbitrary Let A = [a n j] be an infinite real matrix satisfying

(A1) a n j ≥ 0 for all n, j and a n j = 0 for j > n.

(A2)∑n

j=1a n j = 1 for all n ≥ 1.

(A3) limn→ ∞a n j = 0 for all j ≥ 1.

The sequence{x n}∞n=1 defined by x n+1= T (v n), where

1 The Mann iterative process {x n}∞n=1 can be constructed by choosing the initial guess

x1, the matrix A and the operator T , then computing x nand{v n} So, we will denote

the Mann iteration by M (x1, A, T ).

2 There exists a rich literature on the convergence of Mann iteration for different classes

of operators considered on various spaces We will state without proof some results onMann iterations

Theorem 2.5 [2] Suppose E is a locally convex Hausdorff linear topological space, C is a

closed convex subset of E, T : C → C is continuous, x1∈ C and A = [a n j ] satisfies conditions (A1), (A2) and (A3) If either of the sequences {x n } or {v n } in the Mann iterative process

M (x1, A, T ) converges to a point p, then the other sequence also converges to p, and p is a

fixed point of T

Definition 2.2 A Mann process M (x1, A, T ) is said to be normal provided that A = [a n j]

satisfies the following conditions

(A1) a n j ≥ 0 for all n, j and a n j = 0 for j > n.

• The matrices A = [a n j ] (other than the infinite identity matrix) in all normal Mann

process M (x1, A, T ) are constructed as follows.

Choose {c n } such that 0 ≤ c n < 1 for all n and the series∑∞n=1c n diverges, and define

Example 2.1 Let c n = 1 ∀n ≥ 1 then Mann iteration corresponds to Picard iteration.

Example 2.2 Let c n= n+11 then the obtained matrix A is the Cesaro matrix.

Example 2.3 Letλ ∈ (0, 1), and Aλ = [a n j] be defined by

• a n1=λn−1, a n j =λn − j(1 −λ), for j = 2, 3, , n.

• a n j = 0 for j > n, n = 1, 2, 3,

then M (x1, Aλ, T ) is the normal Mann process Indeed, it can be easy seen that the matrix Aλ

satisfies all of five above conditions

Remark 2.3 If we consider

T n = (1 − c n )I + c n T,

then we have F (T ) = F(T n ), for all c n∈ (0, 1]

If the sequence c n=λ (constant), then the Mann iteration obviously reduces to the

Kras-noselskij iteration So we can denote KrasKras-noselskij iteration by K (v0,λ, T ).

Moreover, most of the literature deals with the specialized Mann iteration method defined by

x1∈ E and (2.9), where {c n } satisfies c1= 1, 0 < c n < 1, n ≥ 2 and∑∞n=1c n=∞

Theorem 2.7 Let T be a selfmap of a closed convex subset K of a real Banach space

(E, ||.||) Let {x n}∞n=1 be a general Mann iteration of T with a matrix A given in part b of Theorem 2.6 Suppose that {x n}∞n=1 converges to a point p ∈ K If there exist the constants

α,β,γ,δ ≥ 0,δ < 1 such that

||T x n − T p|| ≥α||x n − p|| +β||x n − T x n|| +γ||p − T x n||+

+δmax{||p − T p||, ||x n − T p||}, (2.11)

then p is a fixed point of T

Proof Since A is equivalent to convergence then A is limit-preseving over c, the space of

convergent sequences Therefore, let

Remark 2.4 The general Mann iteration method can be written in the form x = Aw, where

x = {x n }, w = {T x n } an A = [a nk ] is the weighted mean matrix generated by a nk = p k /P n,where

2.2.1 Strongly Pseudocontractive Operators

Lemma 2.2 [2] Let {x n}∞n=0be a sequence of nonnegative real numbers and let {a n}∞n=0be

a real sequence in [0, 1] such that

∞

∑

n=0

a n=∞

If there exists a positive integer N0such that

x n+1≤ (1 −αn )x n + a n b n, for all n ≥ N0,

where b n ≥ 0 for all n = 0, 1, 2, and b n → 0 as n →∞, then we get

lim

n→ ∞x n= 0.

Theorem 2.8 Let E be a Banach space and K a nonempty closed convex and bounded subset

of E If T : K → K is a Lipschitzian strongly pseudo-contractive operator such that the fixed

point set of T , F (T ), is nonempty, then the Mann iteration {x n } ⊂ K generated by

||x − y|| ≤ ||(x − y) + r[(I − T − kI)x − (I − T − kI)y]|| (2.13)

holds for any x , y ∈ K and r > 0 Let L > 0 be the Lipschitz constant Therefore, from the

Since T is Lipschitzian, it follows that

for some constant M> 0

Sinceαn → 0, there exists N0≥ 0 such that

Mαn ≤ k(1 − k), ∀n ≥ N0

Therefore,

||x n+1− p|| ≤ (1 − k2αn )||x n − p||, ∀n ≥ N0

Using Lemma 2.2, we conclude that the sequence {||x n − p||} converges to 0, i.e., {x n}

converges strongly to the fixed point of T

Thus, we can obtain a convergence theorem for the Mann iteration method in the class ofLipschitzian strictly pseudo-contractive operators in a Banach space setting In addition, theerror estimate of this method will be given in the next Corollary

Corollary 2.2 Let K and T be as in Theorem 2.8 Ifαn= k

2(3+3L+L2 ), where k= t−1

t and

F (T ) = {p}, then the sequence {x n } generated by (2.12) converges strongly to the unique

fixed point of T and we have the estimate

Proof We have 0<αn < 1 Since p = T p, then

Lemma 2.3 [2] If E is a strictly convex Banach space and u , v ∈ E such that ||v|| ≤ ||u||

and for 0 < t < 1, ||(1 − t)u + tv|| = ||u||, then u = v.

Lemma 2.4 [2] Let C be a closed convex subset of a Banach space E and T : C → C be a

quasi nonexpansive operator, p a fixed point of T , and x1∈ C If M(x1, A, T ) is any normal

Mann process (with the sequences {x n }, {v n } ), then the following are true.

1 ||v n+1− p|| ≤ ||v n − p||, for each n = 1, 2, 3,

2 If {v n } clusters at p, then {v n } converges to p.

3 If {v n } clusters at y and z, then ||y − p|| = ||z − p||.

Theorem 2.9 Let E be a strictly convex Banach space, C be a closed convex subset of E,

and T : C → C be a continuous and quasi nonexpansive operator, such that T (C) ⊂ K ⊂ C,

where K is compact Let x1∈ C and M(x1, A, T ) be a normal Mann process such that the

sequence {c n } given by (2.10) clusters at some c ∈ (0, 1).

Then the sequences {x n }, {v n } in the Mann process M(x1, A, T ) converge strongly to a fixed

point of T

Proof We have coK ⊂ C, then

Therefore, there exists a subsequence{v k } which converges to some y ∈ C.

Thus, v k=1 = (1 − c k )v k + c k T v k → (1 − c)y + cTy Since {v n } clusters at both y and (1 −

c )y + cTy, and p is a fixed point of T , Lemma 2.4 gives that

||y − p|| = ||[(1 − c)y + cTy] − p||.

Therefore,

||(1 − c)(y − p) + c(Ty − p)|| = ||y − p||.

Since||Ty − p|| ≤ ||y − p|| and 0 < c < 1, Lemma 2.3 implies that y − p = Ty − p, that is,

Ty = y So y is a fixed point of T

Since{v n } clusters at y, then by Lemma 2.4 we have v n → y Hence, x n → y.

If we drop the continuity assumption on T , we need to work in a more particular class

of Banach spaces, i.e, in uniformly Banach spaces The following lemma and theorem willshow that

Lemma 2.5 [2] Let E be a uniformly convex Banach space and {c n } a sequence in [a, b],

where 0 < a < b < 1 Suppose {w n }, {y n } are sequences in E such that ||w n || ≤ 1, ||y n≤ 1

for all n We define a sequence {z n } by

z n = (1 − c n )w n + c n y n

If lim ||z n || = 1, then lim ||w n − y n || = 0.

Theorem 2.10 Let C be a closed convex subset of a uniformly convex Banach space E, T :

C → C a quasi nonexpansive operator on C which has at least one fixed point p ∈ C If x1∈ C

and M (x1, c n , T ) is a normal Mann process such that the sequence {c n } is bounded away

from 0 and 1, then each of the sequences {v n+1− v n } and {T v n − v n } converges (strongly) to

0∈ E.

Proof From Theorem 2.6, we have

v n+1= (1 − c n )v n + c n T v n

||v n+1− v n || = c n ||T v n − v n||,

where 0< a ≤ c n ≤ b < 1 So if either one of the sequences {v n+1− v n } or {T v n − v n}

converges to 0 then the other does also

In the first case, lim||v n − p|| = 0, then lim ||v n+1− v n|| = 0

In the second case, lim||v n − p|| 6= 0 Using Lemma 2.4, the sequence (||v n − p||) is

nonin-creasing, and lim||v n − p|| = d > 0 We define the sequences {w n }, {y n } and z nby

Corollary 2.3 Let E be a uniformly convex Banach space and T : E → E a nonexpansive

operator which has at least one fixed point Then for any λ ∈ (0, 1), the Krasnoselskij

iteration K (x1,λ, T ) is asymptotically regular for each x1∈ E.

Proof Since the Krasnoselskij iteration is a particular case of the normal Mann iteration with matrix Aλ, the by Theorem 2.10 we have

v n+1− v n→ 0

Theorem 2.11 Let C be a closed convex subset of a uniformly convex Banach space E and

T : C → C be a quasi nonexpansive operator on C that has at least one fixed point p ∈ C.

If I − T is closed and M(x1, c n , T ) is a normal Mann process with x1 ∈ C, such that {c n}

is bounded away from 0 and 1, then for any sequence {v n } that clusters (strongly) at some

y ∈ C, we have Ty = y and the sequences {x n }, {v n } converge (strongly) to y.

Proof Since {v n } clusters at some y ∈ C, then there exists a subsequence {v n k } of {v n} such

that v n k → y Moreover, C is a closed convex subset of a uniformly convex Banach space E

and T : C → C is a quasi nonexpansive operator on C that has at least one fixed point p ∈ C,

so(I − T )v n → 0 by Theorem 2.9 Thus, (I − T )v n k→ 0

Since I − T is closed, it implies that (I − T )y = 0, i.e Ty = y We also have {v n} clusters at

y Therefore, by Lemma 2.4, we have v n → y.

On the other hand, x n+1= T v n Hence, x n → y.

Remark 2.5 Since C is a closed convex subset of a uniformly convex Banach space E then for any nonexpansive operator T , I − T is closed.

Now we will improve Theorem 2.10 by considering the demiclosedness property instead

of the closedness of the operator I − T , as in the following theorem.

Definition 2.3 A mapping S : C → E is said to be demiclosed if {u n } is a sequence in C

which converges weakly to u ∈ C, while {Su n } converges strongly to v ∈ E, then Su = v.

Remark 2.6 Let C be a closed and convex set Every weakly continuous mapping T : C → C

is weakly closed and every weakly closed mapping of T : C → C is demiclosed.

Theorem 2.12 [2] Let C be a closed convex subset of a uniformly convex Banach space E,

T : C → C a nonexpansive operator on C that has at least one fixed point p ∈ C.

Let x1∈ C and M(x1, c n , T ) be the normal Mann process such that {c n } is bounded away

from 0 to 1 Then the following are true.

1 There exists a subsequence of {v n } which converges weakly to some y ∈ C, and if I − T

is demiclosed then each weak subsequential limit point of {v n } is a fixed point of T

2 If I − T is demiclosed and T has only one fixed point p ∈ C, then the sequences {x n }, {v n } converge weakly to p.

3 If I − T is weakly closed, then each weak cluster point of {v n } is a fixed point of T

continu-Now we will extend to Mann iteration by simultaneously weakening the ness property.

demicompact-Definition 2.4 Let E be a Banach space, C a convex subset of E and T : C → C an operator

with the set of fixed points F (T ) T is said to satisfy condition(D) on C if there exists a

nondecreasing functionφ :[0,∞) → [0,∞) withφ(0) = 0 andφ(r) > 0 for r > 0 such that

||x − T x|| ≥φ(inf{||x − z|| : z ∈ F(T )})

for all x ∈ C.

Lemma 2.6 [2] Let C be a closed bounded subset of a Banach space E and T : C → C an

operator with F (T ) 6= /0 If I − T maps closed bounded subsets of C onto closed subsets of

E, then T satisfies condition (D) on C.

Theorem 2.13 [2] Let C be a closed, bounded, convex, nonempty subset of a uniformly

convex Banach space E and T : C → C be a nonexpansive operator with the fixed point set of

T in C denoted by F (T ) If T satisfies condition (D), then for any x1∈ C the Mann iteration

M (x1, c n , T ) given by

x n+1= (1 − c n )x n + c n T x n,

where c n ∈ (0, 1), converges to a point of F(T ).

Definition 2.5 Let(X , ||.||) be a normal space.

A mapping T : X → X is called an a-contraction if

(p1) ||T x − Ty|| ≤ a||x − y|| for all x , y ∈ X , where a∈ [0, 1)

The map T is called a Kannan mapping if there exists b∈ [0, 1/2) such that

(p2) ||T x − Ty|| ≤ b[||x − T x|| + ||y − Ty||] for all x, y ∈ X

A similar definition is due to Chatterjea, there exists c∈ [0, 1/2) such that

(p3) ||T x − Ty|| ≤ c[||x − Ty|| + ||y − T x||] for all x , y ∈ X (p1), (p2), and (p3) are independent contractive conditions An operator T which satisfies at

least one of the contractive conditions(p1), (p2), and (p3) is called a Zamfirescu operator or

Lemma 2.7. ( Groetsch ) Let X be a uniformly convex Banach space and x, y ∈ X such that

||x|| ≤ 1, ||y|| ≤ 1 and ||x − y|| ≥ε> 0 Then for 0 ≤λ ≤ 1, ||λ x+ (1 −λ)y|| ≤ 1 − 2λ(1 −

λ)δX(ε), whereδX (.) is the modulus of convexity of X

Theorem 2.14 Let E be a uniformly convex Banach space, K a closed convex subset of E,

and T : K → K be a Zamfirescu mapping Then the Mann iteration {x n },

x n+1= (1 −αn )x n+αn T x n, n= 1, 2, (2.14)

with{αn } satisfying the conditions

(i)α1= 1; (ii) 0 ≤αn < 1, for n > 1 and (iii)∑αn(1 −αn) =∞, converges to the unique fixed point of T

Proof Let p be a unique fixed point of T For any x1∈ K, we have

Let us assume that there exist a number a > 0 such that ||x n − p|| ≥ a, for all n.

Suppose{||x n − T x n||}n≥1does not converge to zero Then there are two cases

In the first case, there exists anε > 0 such that ||x n − T x n|| ≥ε for all n By using Lemma 2.7 with b= 2δX(ε/||x0− p||), we get

which contradicts (iii).

In the second case,

Indeed, if x n k , x n l satisfy(z1), i.e.,

Therefore in all situations{T x n k} is a Cauchy sequence and hence convergent

Let u be its limit From (2.15) it results that

lim

k→ ∞x n k= lim

k→ ∞T x n k = u.

Moreover,

||u − Tu|| ≤ ||u − x n k || + ||x n k − T x n k || + ||T x n k − Tu||.

We will show that u = Tu, that is, u is a fixed point of T Indeed, if x n k , u satisfies (z1), then

||T x n k − Tu|| ≤α||x n k − u||.