Selçuk J. Appl. Math. Selçuk Journal of Vol. 12. No. 2. pp. 3-19, 2011 Applied Mathematics
A New Two-Step Iterative Process with Errors for Common Fixed Points in Banach Spaces
Esref Turkmen, Murat Ozdemir, Sezgin Akbulut
Department of Mathematics, Faculty of Science, Ataturk University, Erzurum, 25240, Turkiye
e-mail: eturkm en@ atauni.edu.tr,m ozdem ir@ atauni.edu.tr,sezginakbulut@atauni.edu.tr
Received Date: July 16, 2010 Accepted Date: March 29, 2011
Abstract. In this paper, a new two-step iterative scheme with errors is in-troduced for two asymptotically nonexpansive nonself-mappings. Several con-vergence theorems are established in real Banach spaces and real uniformly convex Banach spaces. Our theorems improve and extend the results due to Agarwal-O’Regan-Sahu [R.P. Agarwal, Donal O’Regan and D.R. Sahu, Itera-tive construction of fixed points of nearly asymptotically nonexpansive map-pings, J.Nonliear Convex. Anal.8(1)(2007) 61—79] and many other papers. Key words: Nonself asymptotically nonexpansive mapping; Strong and weak convergence; Common fixed point.
2000 Mathematics Subject Classification: 47H09, 47J25. 1. Introduction
Let K be a nonempty closed convex subset of a real normed linear space E, and T : K → K a self-mapping. Denote by F (T ) the set of fixed points of T , that is, F (T ) = {x ∈ K : T x = x} and by F := F (T1) ∩ F (T2) =
{x ∈ K : T1x = T2x = x}, the set of common fixed points of the mappings T1
and T2. A self-mapping T : K → K is said to be nonexpansive if kT x − T yk ≤
kx − yk for all x, y ∈ K. A self-mapping T : K → K is called asymptotically nonexpansive if there exists a sequence {kn} ∈ [1, ∞) satisfying kn → 1 as
n → ∞ such that
(1.1) kTnx − Tnyk ≤ knkx − yk , ∀x, y ∈ K, n ≥ 1.
A self-mapping T : K → K is said to be uniformly L-Lipschitzian if there exists constant L ≥ 0 such that
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [4] in 1972. They proved that if K is nonempty bounded closed convex subset of real uniformly convex Banach space and T is an asymptotically self-mapping of K, then T has a fixed point.
The notion of nonself asymptotically nonexpansive mappings was defined by Chidume et al. [2] in 2003 as a generalization of asymptotically nonexpansive self-mappings. They defined a nonself asymptotically nonexpansive mapping as follows:
Definition 1. [2] Let K be a nonempty subset of real normed linear space E. Let P : E → K be the nonexpansive retraction of E into K.
(i) A nonself mapping T : K → E is called asymptotically nonexpansive if there exists a sequence {kn} ∈ [1, ∞) with kn → 1 as n → ∞ such that
(1.3) °°T (P T )n−1x − T (P T )n−1y°° ≤ knkx − yk , ∀x, y ∈ K, n ≥ 1.
(ii) A nonself mapping T : K → E is said to be uniformly L-Lipschitzian if there exists a constant L ≥ 0 such that
(1.4) °°T (P T )n−1x − T (P T )n−1y°° ≤ L kx − yk , ∀x, y ∈ K, n ≥ 1.
By using the following iterative algorithm:
x1∈ K, xn+1= P ((1 − αn)xn+ αnT (P T )n−1xn), ∀n ≥ 1.
Chidume et al. [2] established a demiclosed principle, weak and strong con-vergence results for such mappings in a uniformly convex Banach space. Since then, many authors including [6, 9, 15, 16] have studied the weak and strong convergence for such mappings.
As a matter of fact, if T is a self-mapping, then P is a identity mapping. Thus (1.3) and (1.4) reduce to (1.1) and (1.2) respectively in this case. In addition, if T : K → E is asymptotically nonexpansive in the light of (1.3) and P : E → K is a nonexpansive retraction, then P T : K → K is asymptotically nonexpansive in the light of (1.1). Indeed, for all x, y ∈ K and n ≥ 1, by (1.3), it follows that
k(P T )nx − (P T )nyk = °°P T (P T )n−1x − P T (P T )n−1y°° ≤ °°T (P T )n−1x − T (P T )n−1y°° ≤ knkx − yk
The converse, however, may not be true. Therefore, Zhou et al. [17] introduced the following generalized definition recently.
Definition 2. [17] Let K be a nonempty subset of real normed linear space E. Let P : E → K be the nonexpansive retraction of E into K.
(i) A nonself mapping T : K → E is called asymptotically nonexpansive with respect to P if there exists sequences {kn} ∈ [1, ∞) with kn→ 1 as n → ∞
such that
k(P T )nx − (P T )nyk ≤ k
(ii) A nonself mapping T : K → E is said to be uniformly L-Lipschitzian with respect to P if there exists a constant L ≥ 0 such that
k(P T )nx − (P T )nyk ≤ L kx − yk , ∀x, y ∈ K, n ≥ 1.
Futhermore, by studying the following iterative process:
(1.5) x1∈ K, xn+1= αnxn+ βn(P T1)nxn+ γn(P T2)nxn, ∀n ≥ 1,
where {αn} , {βn} and {γn} are three sequences in [a, 1 − a] for some a ∈ (0, 1),
satisfying αn+ βn+ γn = 1, Zhou et al. [17] obtained some strong and weak
convergence theorems for common fixed points of nonself asymptotically non-expansive mappings with respect to P in uniformly convex Banach spaces. As a consequence, the main results of Chidume et al. [2] were deduced.
Agarwal-O’Regan-Sahu [1] recently introduced the iteration process:
(1.6) ⎧ ⎨ ⎩ xn+1= (1 − αn)Tnxn+ αnTnyn, yn= (1 − βn) xn+ βnTnxn, n ∈ N
They showed that their process is independent of Mann and Ishikawa and con-verges faster than both of these. See Proposition 3.1 [1].
Obviously the above process deals with one self mapping only. The case of two mappings in iteration processes has also remained under study since Das and Debata [3] gave and studied a two mappings scheme. Also see, for exam-ple, Takahashi and Tamura [13] and Khan and Takahashi [7]. Note that two mappings case, that is, approximating the common fixed points, has its own im-portance as it has a direct link with the minimization problem, see for example Takahashi [12].
The purpose of this paper is to generalize the iterative process (1.6) from not only self to nonself mappings but also from one to two asymptotically nonex-pansive mappings and to prove several weak and strong convergence theorems for such mappings in a uniformly convex Banach space. Our results presented in this paper improve and extend some existing results. Our iteration process reads as follows:
Let K be a nonempty closed convex subset of a real normed linear space E with retraction P. Let T1, T2 : K → E be two nonself asymptotically nonexpansive
mappings with respect to P .
(1.7) ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ x1∈ K, xn+1= αn(P T1)nxn+ βn(P T2)nyn+ γnun, yn= α0nxn+ β0n(P T1)nxn+ γ0nvn, ∀n ≥ 1, where {αn}, {βn}, {γn},{α0n}, © β0nª, {γ0
n} are real sequences in [0, 1] satisfying
It is also to be noted that (1.5) reduces to another extension of Agarwal-O’Regan-Sahu (1.7) process for one asymptotically nonexpansive mapping when T1= T2= T and γn= γ0n = 0, namely (1.8) ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ x1= x ∈ C, xn+1= (1 − αn) (P T )nxn+ αn(P T )nyn, yn= (1 − βn) xn+ βn(P T ) n xn, ∀n ≥ 1,
Not even this has been considered yet.
Zhou et al. process (1.5) and our process (1.7) are independent: neither reduces to the other. Following Agarwal-O’Regan-Sahu [1], we can say that our process is independent of Zhou et al. and converges faster than it.
Note that Agarwal-O’Regan-Sahu process (1.6) does not reduce to any of the Mann type processes xn+1 = (1 − αn) xn+ αnTnxn or xn+1 = (1 − αn) xn+
αn(P T )nxn but our process (1.7) does. It means that the results proved by
using (1.7) not only contain the corresponding results of Agarwal-O’Regan-Sahu using (1.6) extended to nonself case but also covers the left over ones using aforemetioned Mann type processes. Moreover, it is able to compute common fixed points like (1.5) but at a better rate.
2. Preliminaries
For the sake of convenience, we restate the following concepts and results. Let E be a Banach space with its dimension greater than or equal to 2. The modulus of convexity E is the function δE(ε) : (0, 2] → [0, 1] defined by
δE(ε) = inf ½ 1 − ° ° ° ° 1 2(x + y) ° ° ° ° : kxk = 1, kyk = 1, ε = kx − yk ¾ . A Banach space E is uniformly convex if and only if δE(ε) > 0 for all ε ∈ (0, 2].
Let E be a Banach space and S(E) = {x ∈ E : kxk = 1} . The space E said to be smooth if
lim
t→0
kx + tyk − kxk t exists for all x, y ∈ S(E).
A subset K of E is said to be retract if there exists a continuous mapping P : E → K such that P x = x for all x ∈ K. A mapping P : E → E is said to be a retraction if P2= P. Let C and K be subsets of a Banach space E. A
mapping P from C into K is called sunny if P (P x + t(x − P x)) = P x for x ∈ C with P x + t(x − P x) ∈ C and t ≥ 0.
Note that, if P is a retraction, then P z = z for every z ∈ R(P ) (the range of P ). It is well-known that every closed convex subset of a uniformly convex Banach space is a retract.
For any x ∈ K, the inward set IK(x) is defined as follows:
A mapping T : K → E is said to satisfy the inward condition if T x ∈ IK(x) for
all x ∈ K. T is said to be weakly inward if T x ∈ clIK(x) for each x ∈ K, where
clIK(x) is the closure of IK(x).
A Banach space E is said to satisfy Opial’s condition if, for any sequence {xn}
in E, xn x implies that
lim sup
n→∞ kxn− xk < lim supn→∞ kxn− yk
for all y ∈ E with y 6= x, where xn x means that {xn} converges weakly to
x.
Recall that the mapping T : K → K with F (T ) 6= ∅ is said to satisfy condition (A) [11] if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (t) > 0 for all t ∈ (0, ∞) such that kx − T xk ≥ f (d (x, F (T ))) for all x ∈ K, where d (x, F (T )) = inf kx − pk : p ∈ F (T ). Khan and Fukharuddin [5] modified the condition (A) for two mappings as follows: Two mappings T1,
T2 : K → K are said to satisfy condition (A0) [5] if there is a nondecreasing
function f : [0, ∞) → [0, ∞) with f (0) = 0, f (t) > 0 for all t ∈ (0, ∞) such that 1
2(kx − T1xk + kx − T2xk) ≥ f (d (x, F ))
for all x ∈ K, where d (x, F ) = inf {kx − pk : p ∈ F := F (T1) ∩ F (T2)}.
Note that condition (A0) reduces to condition (A) when T1= T2. It is also
well-known that Condition (A) is weaker than demicompactness or semicompactness, see [11].
A mapping T with domain D(T ) and range R(T ) in E is said to be demiclosed at p if whenever {xn} is a sequence in D(T ) such that {xn} converges weakly
to x∗ ∈ D(T ) and {T x
n} converges strongly to p, then T x∗= p.
We need the following lemmas for our main results.
Lemma 1. [8] Let {an}, {bn} and {δn} be sequences of nonnegative real
numbers satisfying the inequality
an+1≤ (1 + δn) an+ bn, n ≥ 1.
IfP∞n=1bn< ∞ andP∞n=1δn < ∞, then
(i) limn→∞an exists;
(ii) In particular, if {an} has a subsequence which converges strongly to
zero, then limn→∞an= 0.
Lemma 2. [10] Suppose that E is a uniformly convex Banach space and 0 < p ≤ tn ≤ q < 1 for all n ≥ 1. Also, suppose that {xn} and {yn} are
sequences of E such that lim sup
n→∞ kxnk ≤ r, lim supn→∞ kynk ≤ r and limn→∞ktnxn+ (1 − tn) ynk = r
Lemma 3. [12] Let E be real smooth Banach space, let K be nonempty closed convex subset of E with P as a sunny nonexpansive retraction, and let T : K → E be a mapping satisfying weakly inward condition. Then F (P T ) = F (T ). Lemma 4. [2] Let E be a uniformly convex Banach space and let C be a nonempty closed convex subset of E. Let T be a nonself asymptotically non-expansive mapping. Then I − T is demiclosed with respect to zero, that is, xn x and xn− T xn → 0 imply that T x = x.
3. Main Results
3.1. Convergence Theorems in Real Banach Spaces
In this section, we will prove the strong convergence of the iteration scheme (1.7) to a common fixed point of asymptotically nonexpansive mappings T1 and T2
with respect to P in real Banach spaces. We first prove the following lemmas: Lemma 5. Let E be a real normed linear space and K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1, T2: K → E
be two asymptotically nonexpansive mappings with respect to P with sequences n kn(i) o ⊂ [1, ∞) satisfyingP∞n=1 ³ kn(i)− 1 ´ < ∞ (i = 1, 2), respectively. Sup-pose that {xn} is defined by (1.7) with P∞n=1γn < ∞,
P∞
n=1γ0n < ∞. If
F 6= ∅, then limn→∞kxn− pk exists for all p ∈ F .
Proof. Let p ∈ F and kn = max
n k(1)n , k(2)n o . Since P∞n=1³kn(i)− 1 ´ < ∞ (i = 1, 2), soP∞n=1(kn− 1) < ∞. Since {un} and {vn} are bounded sequences
in K, therefore there exists r > 0 such that, r = max ½ sup n≥1kun− pk , supn≥1kvn− pk ¾ . From (1.7), we have. (3.1) kyn− pk = ° °α0 nxn+ β0n(P T1)nxn+ γ0nvn− p ° ° =°°α0 nxn+ β0n(P T1)nxn+ γ0nvn− ¡ α0 n+ β0n+ γ0n ¢ p°° ≤ α0 nkxn− pk + β0nk(P T1)nxn− pk + γ0nkvn− pk ≤ α0 nkxn− pk + β0nknkxn− pk + γ0nkvn− pk ≤ α0 nknkxn− pk + β0nknkxn− pk + γ0nr =¡α0 n+ β0n ¢ knkxn− pk + γ0nr ≤ knkxn− pk + γ0nr,
By (3.1) and (1.7), we obtain (3.2) kxn+1− pk = kαn(P T1)nxn+ βn(P T2)nyn+ γnvn− pk = kαn(P T1)nxn+βn(P T2)nyn+γnvn− (αn+ βn+ γn) pk ≤ αnk(P T1)nxn−pk + βnk(P T2)nyn−pk + γnkun− pk ≤ αnknkxn− pk + βnknkyn− pk + γnkun− pk ≤ αnknkxn− pk + βnkn(knkxn− pk + γ0nr) + γnr ≤ αnk2nkxn− pk + βnkn2kxn− pk + (βnknγ0n+ γn) r = (αn+ βn) kn2kxn− pk + (βnknγ0n+ γn) r ≤ kn2kxn− pk + dn =¡1 + (kn2− 1)¢kxn− pk + dn, where dn= (βnknγ0n+ γn) r. Since P∞ n=1γn< ∞ and P∞ n=1γ0n< ∞, we have P∞
n=1dn< ∞. Also, note thatP∞n=1kn−1 < ∞ is equivalent toP∞n=1k2n−1 <
∞. We obtained from (3.2) and Lemma 1 that limn→∞kxn− pk exists for all
p ∈ F . This completes the proof.
Lemma 6. Let E be a real normed linear space and K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1, T2: K → E
be two asymptotically nonexpansive mappings with respect to P with sequences n kn(i) o ⊂ [1, ∞) satisfyingP∞n=1 ³ kn(i)− 1 ´ < ∞ (i = 1, 2), respectively. Sup-pose that {xn} is defined by (1.7) with P∞n=1γn < ∞,
P∞ n=1γ0n < ∞. If F 6= ∅, then (3.3) kxn+m− pk ≤ Q ⎛ ⎝kxn− pk + ∞ X j=n dj ⎞ ⎠
for all m, n ≥ 1, for all p ∈ F and for some Q > 0 and dj=
¡
βjkjγ0j+ γj
¢ r. Proof. As calculated in Lemma 1,
kxn+1− pk ≤¡1 + (kn2− 1)
¢
kxn− pk + dn,
where dn = (βnknγ0n+ γn) r. It is well known that 1 + x ≤ ex for all x ≥ 0.
Using it for the above inequality, we get
kxn+m− pk ≤ ¡ 1 + (k2n+m−1− 1)¢kxn+m−1− pk + dn+m−1 ≤ ek2n+m−1−1kxn+m −1− pk + dn+m−1 ≤ ek2n+m−1−1£¡1 + (k2 n+m−2− 1) ¢ kxn+m−2− pk + dn+m−2¤ +dn+m−1
≤ ek2n+m−1−1+k2n+m−2−1kx n+m−2− pk +ek2n+m−1−1[d n+m−2+ dn+m−1] .. . ≤ eSn+m−1j=n (k 2 j−1)kx n− pk + e Sn+m−1 j=n+1(k 2 j−1) n+mX−1 j=n dj ≤ Q ⎡ ⎣kxn− pk + n+mX−1 j=n dj ⎤ ⎦ , where Q = eSn+m−1j=n (k 2 j−1). That is kxn+m− pk ≤ Q ⎛ ⎝kxn− pk + n+mX−1 j=n dj ⎞ ⎠
for all m, n ≥ 1, for all p ∈ F .
Theorem 1. Let E be a real Banach space and K be a nonempty closed convex subset of E which is also a nonexpansive retract of E. Let T1, T2: K → E be
two asymptotically nonexpansive mappings with respect to P with sequences n kn(i) o ⊂ [1, ∞) satisfyingP∞n=1 ³ kn(i)− 1 ´ < ∞ (i = 1, 2), respectively. Sup-pose that {xn} is defined by (1.7) with P∞n=1γn < ∞,
P∞
n=1γ0n < ∞ and
F 6= ∅, Then, {xn} converges strongly to a common fixed point of T1 and T2if
and only if lim infn→∞d (xn, F ) = 0, where d (xn, F ) = inf {kx − pk : p ∈ F }.
Proof. The necessity of the conditions is obvious. Thus, we need only prove the sufficiency. Suppose that lim infn→∞d (xn, F ) = 0 From (3.2), we have
d (xn+1, F ) ≤¡1 + (kn2− 1) ¢ kxn− pk + dn. As P∞n=1dn < ∞ and P∞n=1 ¡ k2 n− 1 ¢
< ∞, therefore limn→∞d (xn, F ) exists
by Lemma 1. But by hypothesis lim infn→∞d (xn, F ) = 0, therefore we must
have limn→∞d (xn, F ) = 0.
Next we show that {xn} is a Cauchy sequence. Let ε > 0. Since limn→∞d (xn, F ) =
0 and P∞n=1dn < ∞, therefore there exists a constant n0 such that for all
n ≥ n0, we have d (xn, F ) < ε 4Q and ∞ X j=n0 dj< ε 6Q. In particular, d (xn0, F ) < ε
4Q. That is, inf {kxn0− pk : p ∈ F } <
ε
4Q. There
must exist p0∈ F such that
kxn0− p0k <
ε 3Q.
Now for n ≥ n0, we have from inequality (3.3) of Lemma 6 that kxn+m− xnk ≤ kxn+m− p0k + kxn− p0k ≤ 2Q ⎛ ⎝kxn0− p0k + n0+mX−1 j=n0 dj ⎞ ⎠ ≤ 2Q µ ε 3Q + ε 6Q ¶ = ε.
Hence {xn} is a Cauchy sequence in a closed subset K of a Banach space
E, therefore it must converge to a point in K. Let limn→∞xn = q. Now,
limn→∞d (xn, F ) = 0 gives that d (q, F ) = 0. Since the set of fixed points of
asymptotically nonexpansive mappings is closed, hence F is closed. This implies that q ∈ F , and so q is a common fixed point of T1and T2. This completes the
proof of Theorem.
3.2. Convergence Theorems in Real Uniformly Convex Banach Spaces In this section, we will prove the strong convergence of the iteration scheme (1.7) to a common fixed point of asymptotically nonexpansive mappings T1and
T2 with respect to P in real uniformly convex Banach spaces. We first prove
the following lemma:
Lemma 7. Let E be a real uniformly convex Banach space and K be a non-empty closed convex subset of E which is also a nonexpansive retract of E. Let T1, T2 : K → E be two asymptotically nonexpansive mappings with
re-spect to P with sequences nk(i)n
o ⊂ [1, ∞) satisfying P∞n=1 ³ k(i)n − 1 ´ < ∞ (i = 1, 2), respectively. Suppose that {xn} is defined by (1.7) withP∞n=1γn <
∞, P∞n=1γ0n < ∞, where {αn} and {α0n} are sequences in [a, 1 − a] for some
a ∈ (0, 1). If F 6= ∅, then lim
n→∞kxn− (P T1) xnk = limn→∞kxn− (P T2) xnk = 0.
Proof. For any p ∈ F , it follows from Lemma 5 that limn→∞kxn− pk exists.
Let limn→∞kxn− pk = c for some c ≥ 0. It follows that {xn} and from (3.1)
{yn} are bounded. Hence {(P T1)nxn} and {(P T2)nyn} are bounded. Since
{un} and {vn} are bounded sequences in K, therefore there exists M > 0 such
that, M = max ( supn≥1kun− (P T1)nxnk , supn≥1kvn− (P T1)nxnk , supn≥1kun− (P T2)nynk , supn≥1kvn− xnk ) . From (3.1), we have kyn− pk ≤ knkxn− pk + γ0nr.
Taking lim sup on both sides in the inequality above, we obtain (3.4) lim sup
n→∞ kyn− pk ≤ lim supn→∞ kxn− pk = limn→∞kxn− pk = c.
From k(P T1)nxn− pk ≤ knkxn− pk and k(P T2)nyn− pk ≤ knkyn− pk for all
n ≥ 1, we get that lim supn→∞k(P T1)nxn−pk ≤ c and lim supn→∞k(P T2)nyn−pk ≤
c, respectively. Next, consider
k(P T1)nxn−p + γn(un− (P T2)nyn)k ≤ k(P T1)nxn−pk +γnkun− (P T2)nynk
≤ knkxn− pk + γnM.
Since limn→∞γn = 0, we have
(3.5) lim sup n→∞ k(P T1 )nxn− p + γn(un− (P T2)nyn)k ≤ c. And k(P T2)nyn−p + γn(un− (P T2)nyn)k ≤ k(P T2)nyn−pk +γnkun− (P T2)nynk ≤ knkyn− pk + γnM. implies that (3.6) lim sup n→∞ k(P T2 )nyn− p + γn(un− (P T2)nyn)k ≤ c. Further c = lim n→∞kxn+1− pk = limn→∞kαn(P T1) n xn+ βn(P T2)nyn+ γnun− pk = lim n→∞kαn(P T1) n xn+ (1 − αn− γn) (P T2)nyn+ γnun− (1 − αn)p − αnpk = lim n→∞ ° ° ° ° ° αn(P T1)nxn+ (1 − αn) (P T2)nyn− γn(P T2)nyn+ γnun −(1 − αn)p − αnp ° ° ° ° ° = lim n→∞ ° ° ° ° ° ° ° ° αn(P T1)nxn− αnp + αnγnun− αnγn(P T2)nyn + (1 − αn) (P T2)nyn− (1 − αn)p − γn(P T2)nyn+ γnun −αnγnun+ αnγn(P T2)nyn ° ° ° ° ° ° ° ° = lim n→∞ ° ° ° ° ° αn[(P T1)nxn− p + γn(un− (P T2)nyn)] + (1 − αn) [(P T2)nyn− p + γn(un− (P T2)nyn)] ° ° ° ° ° ≤ lim n→∞ " αnklim supn→∞((P T1)nxn− p + γn(un− (P T2)nyn))k + (1 − αn) klim supn→∞((P T2)nyn− p + γn(un− (P T2)nyn))k # ≤ lim n→∞[αnc + (1 − αn) c ] = c gives that (3.7) lim n→∞ ° ° ° ° ° αn[(P T1)nxn− p + γn(un− (P T2)nyn)] + (1 − αn) [(P T2)nyn− p + γn(un− (P T2)nyn)] ° ° ° ° °= c.
Hence, using (3.5), (3.6), (3.7) and Lemma 2, we obtain (3.8) lim n→∞k(P T1) n xn− (P T2)nynk = 0. Noting that kxn+1− pk = kαn(P T1)nxn+ (1 − αn− γn) (P T2)nyn+ γnun− pk ≤ k(P T2)nyn− pk + αnk(P T1)nxn− (P T2)nynk +γnkun− (P T2)nynk ≤ knkyn− pk + αnk(P T1)nxn− (P T2)nynk + γnM
which yields that
(3.9) c ≤ lim inf n→∞ kyn− pk . By (3.4) and (3.9), we obtain lim n→∞kyn− pk = c. Moreover, kxn− p + γ0n(vn− (P T1)nxn)k ≤ kxn− pk + γ0nkvn− (P T1)nxnk ≤ kxn− pk + γ0nM. implies that (3.10) lim sup n→∞ kxn− p + γ 0 n(vn− (P T1)nxn)k ≤ c. And k(P T1)nxn−p + γn0 (vn− (P T1)nxn)k ≤ k(P T1)nxn−pk +γ0nkvn− (P T1)nxnk ≤ knkxn− pk + γ0nM. implies that (3.11) lim sup n→∞ k(P T1 )nxn− p + γ0n(vn− (P T1)nxn)k ≤ c. Further c = lim n→∞kyn− pk = limn→∞ ° °α0 nxn+ β0n(P T1)nxn+ γ0nvn− p ° ° = lim n→∞kα 0 nxn+ (1 − α0n− γ0n) (P T1)nxn+ γ0nvn− (1 − α0n)p − α0npk = lim n→∞ ° ° ° ° ° α0 nxn+ (1 − α0n) (P T1)nxn− γ0n(P T1)nxn+ γ0nvn −(1 − α0 n)p − α0np ° ° ° ° ° = lim n→∞ ° ° ° ° ° ° ° ° α0 nxn− α0np + α0nγ0nvn− α0nγ0n(P T1)nxn + (1 − α0 n) (P T1)nxn− (1 − α0n)p − γn0 (P T1)nxn+ γ0nvn −α0nγ0nvn+ αn0γ0n(P T1)nxn ° ° ° ° ° ° ° °
= lim n→∞ ° ° ° ° ° α0 n[xn− p + γ0n(vn− (P T1)nxn)] + (1 − α0 n) [(P T1)nxn− p + γn0 (vn− (P T1)nxn)] ° ° ° ° ° ≤ lim n→∞ " α0 nklim supn→∞(xn− p + γ0n(vn− (P T1)nxn))k + (1 − α0 n) klim supn→∞((P T1)nxn− p + γ0n(vn− (P T1)nxn))k # ≤ lim n→∞[α 0 nc + (1 − α0n) c] = c gives that (3.12) lim n→∞ ° ° ° ° ° α0 n[xn− p + γ0n(vn− (P T1)nxn)] + (1 − α0 n) [(P T1)nxn− p + γn0 (vn− (P T1)nxn)] ° ° ° ° °= c. So again by Lemma 2, we obtain
(3.13) lim
n→∞kxn− (P T1) n
xnk = 0.
In addition, from yn= α0nxn+ β0n(P T1)nxn+ γ0nvn, we have
kyn− xnk = ° °α0 nxn+ β0n(P T1)nxn+ γ0nvn− xn ° ° = °°¡1 − β0n− γ0n ¢ xn+ β0n(P T1)nxn+ γ0nvn− xn ° ° = β0nk(P T1)nxn− xnk + γ0nkvn− xnk ≤ β0nk(P T1)nxn− xnk + γ0nM. Hence by (3.13), (3.14) lim n→∞kyn− xnk = 0. Also, then k(P T2)nyn− xnk ≤ k(P T2)nyn− (P T1)nxnk + k(P T1)nxn− xnk , (3.8) and (3.13), we have (3.15) lim n→∞k(P T2) n yn− xnk = 0.
Using (3.14) and (3.15), we obtain
k(P T2)nxn− xnk = k(P T2)nxn− (P T2)nynk + k(P T2)nyn− xnk ≤ knkxn− ynk + k(P T2)nyn− xnk → 0 as n → ∞, that is, (3.16) lim n→∞k(P T2) n xn− xnk = 0.
Also note that (3.17) kxn+1− xnk = kαn(P T1)nxn+ βn(P T2)nyn+ γnun− xnk = k(1 − βn− γn) (P T1)nxn+ βn(P T2)nyn+ γnun− xnk = k(P T1)nxn− xnk + βnk(P T2)nyn− (P T1)nxnk + γnkun− (P T1)nxnk ≤ k(P T1)nxn− xnk + βnk(P T2)nyn− (P T1)nxnk + γnM → 0 as n → ∞, so that (3.18) kxn+1− ynk = kxn+1− xnk + kxn− ynk → 0 as n → ∞. Furthermore, from kxn+1− (P T1)nynk ≤ kxn+1−xnk + kxn− (P T1)nxnk + k(P T1)nxn− (P T1)nynk ≤ kxn+1− xnk + kxn− (P T1)nxnk + knkxn− ynk we find that (3.19) lim n→∞kxn+1− (P T1) n ynk = 0.
Now we shall make use of the fact that every asymptotically nonexpansive map-ping with respect to P must be uniformly L-Lipschitzian with respect to P combined with (3.13), (3.18) and (3.19), where L = supn≥1{kn} ≥ 1, to reach
at kxn− (P T1) xnk ≤ kxn− (P T1)nxnk + k(P T1)nxn− (P T1)nyn−1k + k(P T1)nyn−1− (P T1) xnk ≤ kxn− (P T1)nxnk + knkxn− yn−1k +L°°°(P T1)n−1yn−1− xn ° ° ° so that lim n→∞kxn− (P T1) xnk = 0.
From (3.16) and (3.17), we have
kxn+1− (P T2)nxnk ≤ kxn+1− xnk + kxn− (P T2)nxnk → 0 as n → ∞, and so (3.20) lim n→∞kxn+1− (P T2) n xnk = 0.
Again making use of the fact that every asymptotically nonexpansive mapping with respect to P must be uniformly L-Lipschitzian with respect to P and (3.16), (3.17) and (3.20), we have kxn+1− (P T2) xn+1k ≤ ° ° °xn+1− (P T2)n+1xn+1 ° ° ° + ° ° °(P T2)n+1xn+1− (P T2)n+1xn ° ° ° + ° ° °(P T2)n+1xn− (P T2) xn+1 ° ° ° ≤°°°xn+1− (P T2)n+1xn+1 ° ° ° + kn+1kxn+1− xnk + L k(P T2)nxn− xn+1k → 0 as n → ∞. That is, lim n→∞kxn− (P T2) xnk = 0.
This completes the proof of the lemma.
Theorem 2. Let K be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retrac-tion. Let T1, T2 : K → E be two weakly inward and asymptotically
nonex-pansive mappings with respect to P with sequencesnkn(i)
o ⊂ [1, ∞) satisfying P∞ n=1 ³ k(i)n − 1 ´
< ∞ (i = 1, 2), respectively. Suppose that {xn} is defined by
(1.7), where {αn} and {α0n} are sequences in [a, 1 − a] for some a ∈ (0, 1). If one
of T1and T2is completely continuous and F 6= ∅, then {xn} converges strongly
to a common fixed point of T1 and T2.
Proof. By Lemma 5, limn→∞kxn− pk exists for any p ∈ F . It is
suf-ficient to show that {xn} has a subsequence which converges strongly to a
common fixed point of T1 and T2. By Lemma 7, limn→∞kxn− (P T1) xnk =
limn→∞kxn− (P T2) xnk = 0. Suppose that T1 is completely continuous.
Not-ing that P is nonexpansive, we conclude that there exists subsequence {P T1xnj}
of {P T1xn} such that P T1xnj → p. Thus
° °xnj − p ° ° ≤ °°xnj− P T1xnj ° ° + ° °P T1xnj − p °
° implies xnj → p as j → ∞. Again limj→∞
°
°xnj− (P T1) xnj
° ° = 0 yields by continuity of P and T1 that p = P T1p. Similarly p = P T2p, and so
p ∈ F by Lemma 3. Thus {xn} converges strongly to a common fixed point p
of T1 and T2. This completes the proof.
Theorem 3. Let K be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E with P as a sunny nonexpansive retrac-tion. Let T1, T2 : K → E be two weakly inward asymptotically
nonexpan-sive mappings with respect to P with sequences nk(i)n
o
P∞ n=1
³ k(i)n − 1
´
< ∞ (i = 1, 2), respectively. Suppose that {xn} is defined by
(1.7),where {αn} and {α0n} are sequences in [a, 1 − a] for some a ∈ (0, 1). If
T1 and T2 satisfy condition (A0) and F 6= ∅, then {xn} converges strongly to a
common fixed point of T1 and T2.
Proof. By Lemma 5, we readily see that limn→∞kxn− pk and so, limn→∞d (xn, F )
exists for all p ∈ F . Also, by Lemma 7,
limn→∞kxn− (P T1) xnk = limn→∞kxn− (P T2) xnk = 0. It follows from
con-dition (A0) and Lemma 3 that
lim n→∞f (d (xn, F )) ≤ limn→∞ µ 1 2(kxn− (P T1) xnk + kxn− (P T2) xnk) ¶ = 0. That is, lim n→∞f (d (xn, F )) = 0.
Since f : [0, ∞) → [0, ∞) is a nondecreasing function satisfying f (0) = 0, f (t) > 0 for all t ∈ (0, ∞), therefore we have
lim
n→∞d (xn, F ) = 0.
Now, we can take a subsequence ©xnj
ª
of {xn} and sequence {yj} ⊂ F such
that °°xnj − yj
°
° < 2−j for all integers j ≥ 1. Following Tan and Xu [14], we
have ° °xnj+1− yj ° ° ≤°°xnj − yj ° ° < 2−j and hence kyj+1− yjk ≤ ° °yj+1− xnj+1 ° ° +°°xnj+1− yj ° ° ≤ 2−(j+1)+ 2−j < 2−j+1.
We conclude that {yj} is a Cauchy sequence in F and so it converges. Let yj→
y. Since F is closed, therefore y ∈ F and then xnj → y. As limn→∞kxn− pk
exists, xn→ y ∈ F, thereby completing the proof.
Theorem 4. Let K be a nonempty closed convex subset of a real uniformly convex and smooth Banach space E satisfying Opial’s condition with P as a sunny nonexpansive retraction. Let T1, T2 : K → E be two weakly inward
and asymptotically nonexpansive mappings with respect to P with sequences n kn(i) o ⊂ [1, ∞) satisfyingP∞n=1 ³ kn(i)− 1 ´ < ∞ (i = 1, 2), respectively. Sup-pose that {xn} is defined by (1.7), where {αn} and {α0n} are sequences in
[a, 1 − a] for some a ∈ (0, 1). If F 6= ∅, then {xn} converges weakly to a
common fixed point of T1 and T2.
Proof. Let p ∈ F . By Lemma 5, limn→∞kxn− pk exists and {xn} is bounded.
Note that P T1and P T2are self-mappings from K into itself. We now prove that
{xn} has a unique weak subsequential limit in F . Suppose that subsequences
{xnk} and
© xnj
ª
7, we have limn→∞kxnk− (P Ti) xnkk = 0, (i = 1, 2). Lemma 4 guarantees that
(I − P T1) p1 = 0, i.e., (P T1) p1 = p1. Similary, (P T2) p1 = p1. Again in the
same way, we can prove that p2 ∈ F . Lemma 3 now assures that p1, p2 ∈ F .
For uniqueness, assume p16= p2, then by Opial’s condition, we have
lim n→∞kxn− p1k = klim→∞kxnk− p1k < lim k→∞kxnk− p2k = lim j→∞ ° °xnj − p2 ° ° < lim j→∞ ° °xnj − p1 ° ° = lim n→∞kxn− p1k ,
which is a contradiction and hence p1= p2. As a result, {xn} converges weakly
to a common fixed point of T1 and T2.
4. Acknowledgement
This work was supported by Ataturk University Rectorship under "The Scien-tific and Research Project of Ataturk University", Project No.: 2010/276. References
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