We introduce a new iterative scheme with Meir-Keeler contractions for strict pseudocontractions in q-uniformly smooth Banach spaces.. We also discuss the strong convergence theorems for
Trang 1Volume 2010, Article ID 354202, 19 pages
doi:10.1155/2010/354202
Research Article
Strong Convergence Theorems of a New General Iterative Process with Meir-Keeler Contractions for
Yanlai Song and Changsong Hu
Department of Mathematics, Hubei Normal University, Huangshi 435002, China
Correspondence should be addressed to Yanlai Song,songyanlai2009@163.com
Received 9 August 2010; Revised 2 October 2010; Accepted 14 November 2010
Academic Editor: Mohamed Amine Khamsi
Copyrightq 2010 Y Song and C Hu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited
We introduce a new iterative scheme with Meir-Keeler contractions for strict pseudocontractions
in q-uniformly smooth Banach spaces We also discuss the strong convergence theorems for the new iterative scheme in q-uniformly smooth Banach space Our results improve and extend the
corresponding results announced by many others
1 Introduction
Throughout this paper, we denote by E and E∗a real Banach space and the dual space of E, respectively Let C be a subset of E, and lrt T be a non-self-mapping of C We use FT to denote the set of fixed points of T.
The norm of a Banach space E is said to be Gˆateaux differentiable if the limit
lim
t → 0
x ty − x
exists for all x, y on the unit sphere SE {x ∈ E : x 1} If, for each y ∈ SE, the
limit 1.1 is uniformly attained for x ∈ SE, then the norm of E is said to be uniformly
Gˆateaux differentiable The norm of E is said to be Fr´echet differentiable if, for each x ∈ SE, the limit 1.1 is attained uniformly for y ∈ SE The norm of E is said to be uniformly
Fr´echet differentiable or uniformly smooth if the limit 1.1 is attained uniformly for x, y ∈
SE × SE.
Trang 2Let ρ E:0, 1 → 0, 1 be the modulus of smoothness of E defined by
ρ E t sup
1
2x y x − y − 1 : x ∈ SE, y ≤ t. 1.2
A Banach space E is said to be uniformly smooth if ρ E t/t → 0 as t → 0 Let q > 1.
A Banach space E is said to be q-uniformly smooth, if there exists a fixed constant c > 0 such that ρ E t ≤ ct q It is well known that E is uniformly smooth if and only if the norm of E is
uniformly Fr´echet differentiable If E is q-uniformly smooth, then q ≤ 2 and E is uniformly
smooth, and hence the norm of E is uniformly Fr´echet differentiable, in particular, the norm of
E is Fr´echet differentiable Typical examples of both uniformly convex and uniformly smooth
Banach spaces are L p , where p > 1 More precisely, L pis min{p, 2}-uniformly smooth for every
p > 1.
By a gauge we mean a continuous strictly increasing function ϕ defined R : 0, ∞
such that ϕ0 0 and lim r → ∞ ϕr ∞ We associate with a gauge ϕ a generally
multivalued duality map Jϕ : E → E∗defined by
J ϕ x x∗∈ E∗:x, x∗ xϕx, x∗ ϕx. 1.3
In particular, the duality mapping with gauge function ϕt t q−1 denoted by J q, is referred
to the generalized duality mapping The duality mapping with gauge function ϕt t denoted by J, is referred to the normalized duality mapping Browder 1 initiated the study
J ϕ Set for t ≥ 0
Φt
t
0
Then it is known that J ϕ x is the subdifferential of the convex function Φ · at x It is well known that if E is smooth, then J q is single valued, which is denoted by j q
The duality mapping J q is said to be weakly sequentially continuous if the duality
mapping J q is single valued and for any{x n } ∈ E with x n x, J q x n J∗ q x Every
l p 1 < p < ∞ space has a weakly sequentially continuous duality map with the gauge
ϕt t p−1 Gossez and Lami Dozo2 proved that a space with a weakly continuous duality mapping satisfies Opial’s condition Conversely, if a space satisfies Opial’s condition and has
a uniformly Gˆateaux differentiable norm, then it has a weakly continuous duality mapping
We already know that in q-uniformly smooth Banach space, there exists a constant C q > 0
such that
x yq
≤ x q q y, J q x C qyq
for all x, y ∈ E.
Recall that a mapping T is said to be nonexpansive, if
Tx − Ty ≤ x − y ∀x,y ∈ C. 1.6
Trang 3T is said to be a λ-strict pseudocontraction in the terminology of Browder and
Petryshyn3, if there exists a constant λ > 0 such that
Tx − Ty, j q
x − y
≤x − yq − λI − Tx − I − Ty q
for every x, y, and C for some j q x − y ∈ J q x − y It is clear that 1.7 is equivalent to the following:
I − Tx − I − Ty, j q
x − y ≥ λI − Tx − I − Ty q
The following famous theorem is referred to as the Banach contraction principle
Theorem 1.1 Banach 4 Let X, d be a complete metric space and let f be a contraction on X,
that is, there exists r ∈ 0, 1 such that dfx, fy ≤ rdx, y for all x, y ∈ X Then f has a unique fixed point.
Theorem 1.2 Meir and Keeler 5 Let X, d be a complete metric space and let φ be a Meir-Keeler
contraction (MKC, for short) on X, that is, for every ε > 0, there exists δ > 0 such that dx, y < ε δ implies dφx, φy < ε for all x, y ∈ X Then φ has a unique fixed point.
This theorem is one of generalizations ofTheorem 1.1, because contractions are Meir-Keeler contractions
In a smooth Banach space, we define an operator A is strongly positive if there exists
a constantγ > 0 with the property
Ax, Jx ≥ γx2, aI − bA sup
x≤1 {|aI − bAx, Jx | : a ∈ 0, 1, b ∈ 0, 1}, 1.9
where I is the identity mapping and J is the normalized duality mapping.
Attempts to modify the normal Mann’s iteration method for nonexpansive mappings
and λ-strictly pseudocontractions so that strong convergence is guaranteed have recently
been made; see, for example,6 11 and the references therein
Kim and Xu6 introduced the following iteration process:
x1 x ∈ C,
y n β n x n1− β n
Tx n ,
x n1 α n u 1 − α n y n , n ≥ 0,
1.10
where T is a nonexpansive mapping of C into itself u ∈ C is a given point They proved the
sequence{x n} defined by 1.10 converges strongly to a fixed point of T, provided the control
sequences{α n } and {β n} satisfy appropriate conditions
Trang 4Hu and Cai12 introduced the following iteration process:
x1 x ∈ C,
y n P C
β n x n1− β n
N i1
η n i T i x n
,
x n1 α n γf x n γ n x n1− γ n
I − α n A
y n , n ≥ 1.
1.11
where T i is non-self-λ i -strictly pseudocontraction, f is a contraction and A is a strong positive
linear bounded operator in Banach space They have proved, under certain appropriate assumptions on the sequences {α n }, {γ n }, and {β n }, that {x n} defined by 1.11 converges
strongly to a common fixed point of a finite family of λ i-strictly pseudocontractions, which solves some variational inequality
Question 1 Can Theorem 3.1 of Zhou 8, Theorem 2.2 of Hu and Cai 12 and so on be
extended from finite λ i -strictly pseudocontraction to infinite λ i-strictly pseudocontraction?
Question 2 We know that the Meir-Keeler contractionMKC, for short is more general than the contraction What happens if the contraction is replaced by the Meir-Keeler contraction? The purpose of this paper is to give the affirmative answers to these questions mentioned above In this paper we study a general iterative scheme as follows:
x1 x ∈ C,
y n P C
β n x n1− β n
∞
i1
η n i T i x n
,
x n1 α n γφ x n γ n x n1− γ n
I − α n A
y n , n ≥ 1,
1.12
where T n is non-self λ n -strictly pseudocontraction, φ is a MKC contraction and A is a strong
positive linear bounded operator in Banach space Under certain appropriate assumptions on the sequences{α n }, {β n }, {γ n }, and {μ n
i }, that {x n} defined by 1.12 converges strongly to a
common fixed point of an infinite family of λ i-strictly pseudocontractions, which solves some variational inequality
2 Preliminaries
In order to prove our main results, we need the following lemmas
Lemma 2.1 see 13 Let {x n }, {z n } be bounded sequences in a Banach space E and {β n } be a
sequence in 0, 1 which satisfies the following condition: 0 < lim inf n → ∞ β n ≤ lim supn → ∞ β n < 1 Suppose that x n1 1 − β n x n β n z n for all n ≥ 0 and lim sup n → ∞ z n1 − z n − x n1 − x n ≤ 0.
Then, lim n → ∞ z n − x n 0.
Trang 5Lemma 2.2 see Xu 14 Assume that {α n } is a sequence of nonnegative real numbers such that
α n1 ≤ 1 − γ n α n δ n , where γ n is a sequence in (0, 1) and {δ n } is a sequence in R such that
i∞n1 γ n ∞,
ii lim supn → ∞ δ n /γ n ≤ 0 or∞n1 |δ n | < ∞.
Then lim n → ∞ α n 0.
Lemma 2.3 see 15 demiclosedness principle Let C be a nonempty closed convex subset of a
reflexive Banach space E which satisfies Opial’s condition, and suppose T : C → E is nonexpansive Then the mapping I − T is demiclosed at zero, that is, x n x, x n − Tx n → 0 implies x Tx.
Lemma 2.4 see 16, Lemmas 3.1, 3.3 Let E be real smooth and strictly convex Banach space,
and C be a nonempty closed convex subset of E which is also a sunny nonexpansive retraction of E Assume that T : C → E is a nonexpansive mapping and P is a sunny nonexpansive retraction of E onto C, then FT FP T.
Lemma 2.5 see 17, Lemma 2.2 Let C be a nonempty convex subset of a real q-uniformly smooth
Banach space E and T : C → C be a λ-strict pseudocontraction For α ∈ 0, 1, we define T α x
1 − αx αTx Then, as α ∈ 0, μ, μ min{1, {qλ/C q}1/q−1 }, T α : C → C is nonexpansive such
that FT α FT.
Lemma 2.6 see 12, Remark 2.6 When T is non-self-mapping, theLemma 2.5 also holds.
Lemma 2.7 see 12, Lemma 2.8 Assume that A is a strongly positive linear bounded operator
on a smooth Banach space E with coefficient γ > 0 and 0 < ρ ≤ A−1 Then,
I − ρA ≤ 1 − ργ. 2.1
Lemma 2.8 see 18, Lemma 2.3 Let φ be an MKC on a convex subset C of a Banach space E
Then for each ε > 0, there exists r ∈ 0, 1 such that
x − y ≥ ε implies φx − φy ≤ rx − y ∀x,y ∈ C. 2.2
Lemma 2.9 Let C be a closed convex subset of a reflexive Banach space E which admits a weakly
sequentially continuous duality mapping J q from E to E∗ Let T : C → C be a nonexpansive mapping with FT / ∅ and φ : C → C be a MKC, A is strongly positive linear bounded operator with
coefficient γ > 0 Assume that 0 < γ < γ Then the sequence {x t } define by x t tγφx t 1 − tATx t
converges strongly as t → 0 to a fixed point x of T which solves the variational inequality:
A − γφ
Proof The definition of {x t} is well definition Indeed, from the definition of MKC, we can
see MKC is also a nonexpansive mapping Consider a mapping S t on C defined by
S t x tγφ x I − tATx, x ∈ C. 2.4
Trang 6It is easy to see that S tis a contraction Indeed, byLemma 2.8, we have
S t x − S t y ≤ tγφx − φy I − tATx − Ty
≤ tγφ x − φ
y 1 − tγx − y
≤ tγx − y 1 − tγx − y
≤1− tγ − γx − y.
2.5
Hence, S t has a unique fixed point, denoted by x t, which uniquely solves the fixed point equation
We next show the uniqueness of a solution of the variational inequality2.3 Suppose both x ∈ FT and x ∈ FT are solutions to 2.3, not lost generality, we may assume there
is a number ε such that x − x ≥ ε Then by Lemma 2.8, there is a number r such that
φx − φx ≤ rx − x From 2.3, we know
A − γφ
x, J q x − x ≤ 0,
A − γφ
Adding up2.7 gets
A − γφ
x −A − γφ
Noticing that
A − γφ
x −A − γφ
x, J q x − x Ax − x, J q x − x − γ φ x − φ x, J q x − x
≥ γx − x q − γφ x − φ x x − x q−1
≥ γx − x q − γrx − x q
≥γ − γr
x − x q
≥γ − γr
ε q
> 0.
2.9
Thereforex x and the uniqueness is proved Below, we use x to denote the unique solution
of2.3
We observe that{x t} is bounded Indeed, we may assume, with no loss of generality,
t < A−1, for all p ∈ FT, fixed ε1, for each t ∈ 0, 1.
Case 1 x t − p < ε1 In this case, we can see easily that {x t} is bounded
Trang 7Case 2 x t − p ≥ ε1 In this case, by Lemmas2.7and2.8, there is a number r1such that
φ x t − φ
p< r1x t − p,
x t − p tγφx t I − tATx t − p
t
γφ x t − Ap I − tATx t − p
≤ tγφ x t − Ap 1 − tγx t − p
≤ tγφ x t − γφ
p γφp − Ap 1 − tγx t − p
≤ tγr1x t − p tγφp − Ap 1 − tγx t − p,
2.10
therefore,x t − p ≤ γφp − Ap/γ − γr1 This implies the {x t} is bounded
To prove that x t → x x ∈ FT as t → 0.
Since{x t } is bounded and E is reflexive, there exists a subsequence {x t n } of {x t} such
that x t n x∗ By x t − Tx t tγφx t − ATx t We have x t n − Tx t n → 0, as t n → 0 Since E
satisfies Opial’s condition, it follows fromLemma 2.3that x∗∈ FT We claim
By contradiction, there is a number ε0and a subsequence{x t m } of {x t n} such that x t m − x∗ ≥
ε0 FromLemma 2.8, there is a number r ε0 > 0 such that φx t m − φx∗ ≤ r ε0x t m − x∗, we write
x t m − x∗ t m
γφ x t m − Ax∗
I − t m A Tx t m − x∗, 2.12
to derive that
x t m − x∗q t m γφx t m − Ax∗, J q x t m − x∗ I − t m A Tx t m − x∗, J q x t m − x∗
≤ t m γφ x t m − Ax∗, J q x t m − x∗ 1− t m γ
It follows that
x t m − x∗q≤ 1
γ γφ x t m − Ax∗, J q x t m − x∗
1
γ γφ x t m − γφx∗, J q x t m − x∗ γφ x∗ − Ax∗, J q x t m − x∗
≤ 1
γ
γr ε0x t m − x∗q γφ x∗ − Ax∗, J q x t m − x∗ .
2.14
Therefore,
x t m − x∗q≤ γφ x∗ − Ax∗, J q x t m − x∗
Trang 8Using that the duality map J q is single valued and weakly sequentially continuous from E to
E∗, by2.15, we get that x t m → x∗ It is a contradiction Hence, we have x t n → x∗
We next prove that x∗solves the variational inequality2.3 Since
we derive that
A − γφ
x t −1
Notice
I − Tx t − I − Tz, J q x t − z ≥ x t − z q − Tx t − Tzx t − z q−1
≥ x t − z q − x t − z q
0.
2.18
It follows that, for z ∈ FT,
A − γφ
x t , J q x t − z −1
t I − tAI − Tx t , J q x t − z
−1
t I − Tx t − I − Tz, J q x t − z A I − Tx t , J q x t − z
≤ A I − Tx t , J q x t − z .
2.19
Now replacing t in 2.19 with t n and letting n → ∞, noticing I−Tx t n → I−Tx∗ 0
for x∗∈ FT, we obtain A − γφx∗, J q x∗− z ≤ 0 That is, x∗∈ FT is a solution of 2.3; Hencex x∗by uniqueness In a summary, we have shown that each cluster point of{x t} at
t → 0 equals x, therefore, x t → x as t → 0.
Lemma 2.10 see, e.g., Mitrinovi´c 19, page 63 Let q > 1 Then the following inequality holds:
ab ≤ 1
q a
qq − 1
q b
q/q−1 , 2.20
for arbitrary positive real numbers a, b.
Lemma 2.11 Let E be a q-uniformly smooth Banach space which admits a weakly sequentially
continuous duality mapping J q from E to E∗ and C be a nonempty convex subset of E Assume that T i : C → E is a countable family of λ i -strict pseudocontraction for some 0 < λ i < 1 and
inf{λi : i ∈ N} > 0 such that F ∞
i1 FT i / ∅ Assume that {η i}∞i1 is a positive sequence such that
∞
i1 η i 1 Then∞
i1 η i T i : C → E is a λ-strict pseudocontraction with λ inf{λ i : i ∈ N} and
F∞
i1 η i T i F.
Trang 9Proof Let
G n x η1T1x η2T2x · · · η n T n x 2.21
andn
i1 η i 1 Then, G n : C → E is a λ i -strict pseudocontraction with λ min{λ i : 1≤ i ≤
n} Indeed, we can firstly see the case of n 2.
I − G2x − I − G2y, J q
x − y
η1I − T1x η2I − T2x − η1I − T1y − η2I − T2y, J q
x − y
η1 I − T1x − I − T1y, J q
x − y
η2 I − T2x − I − T2y, J q
x − y
≥ η1λ1I − T1x − I − T1yq η2λ2I − T2x − I − T2yq
≥ λη1I − T1x − I − T1yq η2I − T2x − I − T2yq
≥ λI − G2x − I − G2yq
,
2.22
which shows that G2 : C → E is a λ-strict pseudocontraction with λ min{λ i : i 1, 2} By
the same way, our proof method easily carries over to the general finite case
Next, we prove the infinite case From the definition of λ-strict pseudocontraction, we
know
I − T n x − I − T n y, J q
x − y ≥ λI − T n x − I − T n yq
Hence, we can get
I − T n x − I − T n y ≤1
λ
1/q−1
x − y. 2.24
Taking p ∈ FT n, from 2.24, we have
I − T n x I − T n x − I − T n p ≤1
λ
1/q−1
x − p. 2.25
Consquently, for all x ∈ E, if F ∞
i1 FT i / ∅, η i > 0 i ∈ N and∞
i1 η i 1, then∞i1 η i T i
strongly converges Let
Tx
∞
i1
we have
Tx
∞
i1
η i T i x lim
n → ∞ n
i1
η i T i x lim
n → ∞
1
n
i1 η i
n
i1
Trang 10I − Tx − I − Ty, J q
x − y
lim
n → ∞
I − n1
i1 η i n
i1
η i T i
x
I − n1
i1 η i n
i1
η i T i
y, J q
x − y
lim
n → ∞
1
n
i1 η i
n
i1
η i I − T i x − I − T i y, J q
x − y
≥ lim
n → ∞
1
n
i1 η i
n
i1
η i λ I − T i x − I − T i yq
≥ λ lim
n → ∞
I −n1
i1 η i
n
i1
η i T i
x −
I − n1
i1 η i
n
i1
η i T i
y
q
λI − Tx − I − Ty q
.
2.28
So, we get T is λ-strict pseudocontraction.
Finally, we show F∞
i1 η i T i F Suppose that x ∞
i1 η i T i x, it is sufficient to show
that x ∈ F Indeed, for p ∈ F, we have
x − pq x − p, J q
x − p
i1
η i T i x − p, J q
x − p
i1
η i T i x − p, J q
x − p
≤x − pq − λ∞
i1
η i x − T i xq ,
2.29
where λ inf{λ i : i ∈ N} Hence, x T i x for each i ∈ N, this means that x ∈ F.
3 Main Results
Lemma 3.1 Let E be a real q-uniformly smooth, strictly convex Banach space and C be a closed
convex subset of E such that C ± C ⊂ C Let C be also a sunny nonexpansive retraction of E Let
φ : C → C be a MKC Let A : C → C be a strongly positive linear bounded operator with the coefficient γ > 0 such that 0 < γ < γ and T i : C → E be λ i -strictly pseudo-contractive non-self-mapping such that F ∞
i1 FT i / ∅ Let λ inf {λ i : i ∈ N} > 0 Let {x n } be a sequence of C
generated by1.12 with the sequences {α n },{β n } and {γ n } in 0, 1, assume for each n, {η i n } be an
infinity sequence of positive number such that∞
i1 η n i 1 for all n and η n i > 0 The following control conditions are satisfied
i∞
i1 α n ∞, lim n → ∞ α n 0,
ii 1 − α ≤ 1 − β n ≤ μ, μ min {1, {qλ/C q}1/q−1 } for some α ∈ 0, 1 and for all n ≥ 0,
...linear bounded operator in Banach space They have proved, under certain appropriate assumptions on the sequences {α n }, {γ n }, and {β n }, that...
positive linear bounded operator in Banach space Under certain appropriate assumptions on the sequences{α n }, {β n }, {γ n }, and {μ n...
i }, that {x n} defined by 1.12 converges strongly to a
common fixed point of an infinite family of λ i-strictly pseudocontractions,