W.M. Kozlowski
Strong convergence of implicit iteration processes for nonexpansive semigroups in Banach spaces
Abstract. Let C be a convex compact subset of a uniformly convex Banach space.
Let {Tt}t0 be a strongly-continuous nonexpansive semigroup on C. Consider the iterative process defined by the sequence of equations
xk+1= ckTtk+1(xk+1) + (1 − ck)xk.
We prove that, under certain conditions on {ck} and {tk}, the sequence {xk}∞n=1 converges strongly to a common fixed point of the semigroup {Tt}t0. There are known results on convergence of such iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property, and also weak con- vergence results in Banach spaces that are simultaneously uniformly convex and uniformly smooth. In this paper, we do not assume the Opial property or uniform smoothness of the norm.
1991 Mathematics Subject Classification: Primary 47H09; Secondary 47H10.
Key words and phrases: Fixed point, nonexpansive mapping, nonexpansive semi- group, fixed point iteration process, implicit iterative process, strong convergence, uniformly convex Banach space.
1. Introduction. Let C be a nonempty, closed, bounded and convex subset of a uniformly convex Banach space X. Let a mapping T : C → C be nonexpansive.
By the 1965 Browder Theorem [2], T has a fixed point in C. An immediate question is how to construct such a fixed point. One of such fixed point construction methods is based on an observation that for each 0 < c < 1 and a x0∈ C the equation
(1) x = cT (x) + (1− c)x0
has a unique solution xc∈ C guaranteed by the Banach Contraction Principle and obtained as a strong limit of the Picard iterates. Browder [1] (Reich [9], respectively)
proved that as c → 1, xc converges strongly to a fixed point of T in a Hilbert space (uniformly smooth Banach space, respectively).
It is interesting and important to know whether such fixed point construction methods, frequently called implicit iterative processes, can be extended to the semi- groups of nonexpansive mappings.
Let us recall that a one-parameter family F = {Tt : t ∈ [0, ∞)} of mappings from C into itself is said to be a strongly-continuous nonexpansive semigroup on C if F satisfies the following conditions:
(i) T0(x) = x for x ∈ C;
(ii) Tt+s(x) = Tt(Ts(x)) for x ∈ C and t, s ∈ [0, ∞);
(iii) for each t ∈ [0, ∞), Ttis a nonexpansive mapping;
(iv) for each x ∈ C, the mapping t → Tt(x) is strong continuous.
For each t ∈ [0, ∞) let F (Tt) denote the set of its fixed points. Define then the set of all common set points for mappings from F as the following intersection
F (F) = \
t∈[0,∞)
F (Tt).
We know that F (F) is nonempty provided X is uniformly convex [2] (see also [5]).
There are known results on weak or strong convergence of implicit iterative processes for nonexpansive semigroups in Hilbert spaces and Banach spaces with the Opial property, see e.g. [12, 14, 3, 10, 13], and on weak convergence in Banach spaces that are simultaneously uniformly convex and uniformly smooth [8]. However, many important spaces like Lp for 1 ¬ p 6= 2 do not possess the Opial property.
In the current paper we prove the strong convergence of the implicit iterative processes for nonexpansive semigroups in Banach spaces, Theorem 2.6. We do not assume the Opial property, we assume only that X is uniformly convex. Hence, our results cover important cases like Lp for p > 1.
The strong convergence of the implicit iterative processes in uniformly convex Banach spaces without additional assumptions, like Opial property, has not been previously established. Hence the current paper opens a new research direction as well as introduces to this field new techniques based solely on the uniform convexity of Banach spaces.
2. Results. The following elementary lemma will be used in this paper.
Lemma 2.1 ([12]) Let {tn}n∈Nbe a sequence of real numbers and let τ ∈ R be such that
lim inf
n→∞ tn¬ τ ¬ lim sup
n→∞ tn
and
nlim→∞(tn+1− tn) = 0.
Then τ is a cluster point of the sequence {tn}n∈N.
Let us recall the following property of uniformly convex Banach spaces which will be used throughout this paper.
Lemma 2.2 ([11, 15]) Let X be a uniformly convex Banach space. Let {cn} ⊂ (0, 1) be bounded away from 0 and 1, and {un}, {vn} ⊂ X be such that
lim sup
n→∞ kunk ¬ a, lim sup
n→∞ kvnk ¬ a, limn→∞kcnun+ (1 − cn)vnk = a.
Then lim
n→∞kun− vnk = 0.
Let us start with the following definition of the implicit iteration process.
Definition 2.3 Given a nonexpansive semigroup F = {Tt: t ∈ [0, ∞)} on C, the implicit iteration process P (C, F, x0,{ck}, {tk}) is defined by the following formula:
(2)
(x0∈ C
xk+1= ckTtk+1(xk+1) + (1 − ck)xk, f or k 0,
where {ck} ⊂ (0, 1) is bounded away from 0 and 1 and {tk} ⊂ (0, ∞). We will also say that the sequence {xk}k∈N0 is generated by the process P (C, F, x0,{ck}, {tk}) and write
(3) {xk}k∈N0 = P (C, F, x0,{ck}, {tk}).
For k ∈ N0, u ∈ C, w ∈ C let us introduce the following notation (4) Pk,w(u) = ckTtk+1(u) + (1 − ck)w.
Since each Pk,w(u) : C → C is a contraction, it follows by the Banach Contraction Principle that each xk+1 in (2) is uniquely defined.
The next two results deal with the general behavior of the implicit iterative processes in uniformly convex Banach spaces. Observe that the properties of the implicit iterative processes proven in Lemmas 2.4 and 2.4 below, do not depend on the choice of the sequences {tk} and {ck}.
Lemma 2.4 Let X be a uniformly convex Banach space and C be a nonempty, closed, bounded and convex subset of X. Let F be a strongly-continuous nonex- pansive semigroup on C, w ∈ F (F) and {xk}k∈N0 = P (C, F, x0,{ck}, {tk}) be an implicit iteration process. Then there exists r ∈ R such that lim
k→∞kxk− wk = r.
Proof Calculate
kxk+1− wk = kckTtk+1(xk+1) + (1 − ck)xk− wk
¬ ckkTtk+1(xk+1) − Ttk+1(w)k + (1 − ck)kxk− wk
¬ ckkxk+1− wk + (1 − ck)kxk− wk.
(5)
From (5) it follows that kxk+1− wk ¬ kxk − wk which implies that there exists r∈ R with lim
k→∞kxk− wk = r.
Lemma 2.5 below shows that the sequence generated by the implicit iterative process in a uniformly convex Banach space is, in a more general sense, an approx- imate fixed point sequence.
Lemma 2.5 Let X be a uniformly convex Banach space and C be a nonempty, closed, bounded and convex subset of X. Let F be a strongly-continuous nonexpan- sive semigroup on C and {xk}k∈N0= P (C, F, x0,{ck}, {tk}) be an implicit iteration process. Then
(6) lim
k→∞kTtk(xk) − xkk = 0.
Proof By the Browder Theorem there exists w ∈ F (F). It follows from Lemma 2.4 that there exists r ∈ R such that limk
→∞kxk− wk = r. Hence (7) lim sup
k→∞ kTtk(xk) − wk = lim sup
k→∞ kTtk(xk) − Ttk(w)k ¬ lim sup
k→∞ kxk− wk = r.
Denote vk = xk−1− w, uk= Ttk(xk) − w and observe that lim
k→∞kvkk = r. It follows from (7) that lim sup
k→∞ kukk ¬ r. Since
klim→∞kckuk+(1−ck)vkk = lim
k→∞kck(Ttk(xk)−w)+(1−ck)(xk−1−w)k = lim
k→∞kxk−wk = r, it follows from Lemma 2.2 that
(8) lim
k→∞kTtk(xk) − xk−1k = lim
k→∞kuk− vkk = 0.
Observe that
(9) kxk+1− xkk = kckTtk+1(xk+1) + (1 − ck)xk− xkk = ckkTtk+1(xk+1) − xkk.
Hence by (8) we conclude that
(10) lim
k→∞kxk+1− xkk = 0.
Using (8) and (10) we have
klim→∞kTtk(xk) − xkk ¬ limk
→∞kTtk(xk) − xk−1k + limk
→∞kxk−1− xkk = 0.
The proof of the lemma is now complete.
We are now ready to prove our main result, the strong convergence theorem for implicit iterative processes.
Theorem 2.6 Let C be a nonempty, convex and compact subset of a uniformly convex and uniformly smooth Banach space X. Let F be a strongly-continuous non- expansive semigroup on C and {xk}k∈N0 = P (C, F, x0,{ck}, {tk}) be an implicit iterative process. Assume that
(i) tn> 0for every n ∈ N (ii) lim inf
n→∞ tn= 0 (iii) lim sup
n→∞ tn> 0 (iv) lim
n→∞(tn+1− tn) = 0.
Then there exists a common fixed point x ∈ F (F) such that xk → x.
Proof Fix any 0 < t < lim sup
n→∞ tn. By Lemma 2.1, there exists {tkn}) a subsequence of {tn}) such that
(11) tkn→ t.
From Lemma 2.5 it follows that
(12) lim
n→∞kTtkn(xkn) − xknk = 0.
Since C is compact, there exists {xkni} a subsequence of {xkn} and an element x∈ C such that
(13) lim
i→∞kTtkni(xkni) − xk = 0.
Denote si = tkni and wi = xkni and observe that by (12) and (13) we have the following
(14) kwi− xk ¬ kwi− Tsi(wi)k + kTsi(wi) − xk → 0, as i → ∞. Hence
kTsi(x) − xk ¬ kTsi(x) − Tsi(wi)k + kTsi(wi) − wik + kwi− xk
¬ kx − wik + kTsi(wi) − wik + kwi− xk → 0, (15)
as i → ∞. From the strong-continuity of F and from (15), we obtain that (16) kTt(x) − xk ¬ kTt(x) − Tsi(x)k + kTsi(x) − xk → 0,
as i → ∞, which gives us Tt(x) = x. Take now any s > 0. Then there exist 0 < t < lim sup
n→∞ tn, 0 < u < lim sup
n→∞ tn and k ∈ N0 such that
(17) s = t + ku.
Hence
(18) Ts(x) = Tku(Tt(x)) = Tku(x) = Tu+ ··· +u(x) = x,
which means that x ∈ F}. The proof is complete.
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W.M. Kozlowski
School of Mathematics and Statistics, University of New South Wales Sydney, NSW 2052, Australia
E-mail: w.m.kozlowski@unsw.edu.au
(Received: 15.10.2014)
