Available online at www.tjnsa.com
J. Nonlinear Sci. Appl. 9 (2016), 757–765
Research Article
Monotone hybrid methods for a common solution
problem in Hilbert spaces
Dongfeng Lia , Juan Zhaob,∗
a
School of Information Engineering, North China University of Water Resources and Electric Power, Zhengzhou, China.
b
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power University,
Zhengzhou 450011, China.
Communicated by X. Qin
Abstract
The purpose of this article is to investigate generalized mixed equilibrium problems and uniformly LLipschitz continuous asymptotically κ-strict pseudocontractions in the intermediate sense based on a monotone hybrid method. Strong convergence theorems of common solutions are established in the framework of
c
Hilbert spaces. 2016
All rights reserved.
Keywords: Asymptotically strict pseudocontraction, asymptotically nonexpansive mapping, generalized
mixed equilibrium problem, solution, fixed point.
2010 MSC: 47H10, 90C33.
1. Introduction-preliminaries
Throughout this paper, we always assume that H is a real Hilbert space with inner product h·, ·i and
norm k · k. Let C be a nonempty closed convex subset of H. Let P rojC be the metric projection onto C.
From now on, we use * and → to stand for the weak convergence and strong convergence, respectively.
Let S be a mapping on C. In this paper, we use F (S) to denote the set of fixed points of S. Recall that
S is said to be closed at y if for any sequence {xn } ⊂ C such that xn → x and Sxn → y, then Sx = y.
Recall that S is said to be demiclosed at y if for any sequence {xn } ⊂ C such that xn * x and Sxn → y,
then Sx = y. Recall that S is said to be nonexpansive iff
kSx − Syk ≤ kx − yk,
∗
∀x, y ∈ C.
Corresponding author
Email addresses: sylidf@yeah.net (Dongfeng Li), zhaojuanyu@126.com (Juan Zhao)
Received 2015-08-17
D. Li, J. Zhao, J. Nonlinear Sci. Appl. 9 (2016), 757–765
758
S is said to be asymptotically nonexpansive iff there exists a sequence {kn } ⊂ [1, ∞) with kn → 1 as
n → ∞ such that
kS n x − S n yk ≤ kn kx − yk, ∀x, y ∈ C, n ≥ 1.
Asymptotically nonexpansive mapping was introduced by Goebel and Kirk [10] as a generalization of nonexpansive mappings. If C is weakly compact, then F (T ) is not empty.
S is said to be asymptotically nonexpansive in the intermediate sense iff it is continuous and the following
inequality holds:
lim sup sup (kS n x − S n yk − kx − yk) ≤ 0.
(1.1)
n→∞ x,y∈C
Setting µn = max{0, sup (kS n x − S n yk − kx − yk)}, we see that µn → 0 as n → ∞. Then (1.1) is reduced
x,y∈C
to the following:
kS n x − S n yk ≤ kx − yk + µn ,
∀x, y ∈ C.
(1.2)
Asymptotically nonexpansive mappings in the intermediate sense were introduced by Kirk [14] as a generalization of asymptotically nonexpansive mappings; see also [5].
S is said to be strictly pseudocontractive iff there exists a constant κ ∈ [0, 1) such that
kSx − Syk2 ≤ kx − yk2 + κk(I − S)x − (I − S)yk2 ,
∀x, y ∈ C.
Strict pseudocontractions is introduced by Brower and Petryshyn [4] as a generalization of the class of
nonexpansive mappings. It is clear that every nonexpansive mapping is 0-strictly pseudocontractive.
S is said to be an asymptotically strict pseudocontraction iff there exist a sequence {kn } ⊂ [1, ∞) with
kn → 1 as n → ∞ and a constant κ ∈ [0, 1) such that
kS n x − S n yk2 ≤ kn kx − yk2 + κk(I − S n )x − (I − S n )yk2 ,
∀x, y ∈ C, n ≥ 1.
Asymptotically strict pseudocontractions were introduced by Qihou [17] as a generalization of strict pseudocontractions. Note that both asymptotically strict pseudocontractions and strict pseudocontractions are
Lipschitz continuous.
S is said to be an asymptotically strict pseudocontraction in the intermediate sense if there exist a
sequence {kn } ⊂ [1, ∞) with kn → 1 as n → ∞ and a constant κ ∈ [0, 1) such that
lim sup sup (kS n x − S n yk2 − kn kx − yk2 − κk(I − S n )x − (I − S n )yk2 ) ≤ 0.
(1.3)
n→∞ x,y∈C
Setting
µn = max{0, sup (kS n x − S n yk2 − kn kx − yk2 − κk(I − S n )x − (I − S n )yk2 )},
x,y∈C
we see that µn → 0 as n → ∞. Then (1.3) is reduced to the following:
kS n x − S n yk2 ≤ kn kx − yk2 + κk(I − S n )x − (I − S n )yk2 + µn ,
∀x, y ∈ C, n ≥ 1.
(1.4)
Asymptotically strict pseudocontractions in the intermediate sense were introduced by Sahu, Xu and Yao
[21] as a generalization of asymptotically strict pseudocontractions.
From now on, R stands for the set of real numbers. Let B be a bifunction of C × C into R. Consider
the problem: find a p such that
B(p, y) ≥ 0, ∀y ∈ C.
(1.5)
The solution set of the problem is denoted by EP (B). The problem was first introduced by Ky Fan [9]. In
the sense of Blum and Oettli [3], the Ky Fan is also called an equilibrium problem.
Equilibrium problems are revealed as an powerful and effective mathematical modelling for studying real
world problems which arise in many subjects, for instance, network, finance, transportation, elasticity and
D. Li, J. Zhao, J. Nonlinear Sci. Appl. 9 (2016), 757–765
759
optimization; see [1], [2], [6], [7], [8], [11], [16], [22], [23], [24] and the references therein. It is known that
the generalized mixed equilibrium problem is to find p ∈ C such that
B(p, y) + hAp, y − pi + ϕ(y) − ϕ(p) ≥ 0,
∀y ∈ C,
(1.6)
where ϕ : C → R is a real valued function and A : C → H is a nonlinear mapping. We use GM EP (B, A, ϕ)
to denote the solution set of the equilibrium problem. That is,
GM EP (B, A, ϕ) := {p ∈ C : B(p, y) + hAp, y − pi + ϕ(y) − ϕ(z) ≥ 0,
∀y ∈ C}.
Next, we give some special cases:
If A = 0 and ϕ = 0, then problem (1.6) is equivalent to (1.5).
If B = 0, then problem (1.6) is equivalent to find p ∈ C such that
hAp, y − pi + ϕ(y) − ϕ(z) ≥ 0,
∀y ∈ C,
(1.7)
which is called the mixed variational inequality of Browder type.
If A = 0, then problem (1.6) is equivalent to find p ∈ C such that
B(p, y) + ϕ(y) − ϕ(z) ≥ 0,
∀y ∈ C,
(1.8)
which is called the mixed equilibrium problem.
If ϕ = 0, then problem (1.6) is equivalent to find p ∈ C such that
B(p, y) + hAp, y − pi ≥ 0,
∀y ∈ C,
(1.9)
which is called the generalized equilibrium problem.
For solving the above equilibrium problems, let us assume that bifunction B : C × C → R satisfies the
following conditions:
(B-1)
(B-2)
(B-3)
(B-4)
B is monotone;
B(x, x) = 0, ∀x ∈ C;
for each x ∈ C, y 7→ B(x, y) is convex and weakly lower semi-continuous;
lim sup B(tz + (1 − t)x, y) ≤ B(x, y), ∀x, y, z ∈ C.
t↓0
Recently, above problems have been extensively investigated based on iterative methods in different
spaces by many authors; see, for instance, [8], [12], [13], [15], [19, 20, 25], and [26]–[29]. In this paper,
motivated by the research mentioned above, we investigate fixed points of an asymptotically strict pseudocontraction in the intermediate sense and generalized mixed equilibrium problem (1.6) based an a hybrid
method.
Next, we give the following tools which play an important role in our paper.
Lemma 1.1 ([18], [21]). Let S : C → C be a uniformly L-Lipschitz continuous and asymptotically strict
pseudocontraction in the intermediate sense. Then the mapping I − S is demiclosed at zero, that is, if {xn }
is a sequence in C such that xn * x̄ and xn − Sxn → 0, then x̄ ∈ F (S).
Lemma 1.2 ([18], [21]). Let C be a nonempty closed convex subset of H. Let S : C → C be an asymptotically
κ-strict pseudocontraction in the intermediate sense. Then F (S) is closed and convex.
Lemma 1.3 ([3]). Let C be a nonempty closed convex subset of H and let B : C × C → R be a bifunction
with (B-1), (B-2), (B-3) and (B-4) . Then, for any r > 0 and x ∈ H, there exists z ∈ C such that
1
B(z, y) + hy − z, z − xi ≥ 0,
r
∀y ∈ C.
Further, define
n
1
Tr x = z ∈ C : B(z, y) + hy − z, z − xi ≥ 0,
r
for all r > 0 and x ∈ H. Then, the following hold:
∀y ∈ C
o
D. Li, J. Zhao, J. Nonlinear Sci. Appl. 9 (2016), 757–765
760
(a) Tr is single-valued firmly nonexpansive;
(b) EP (B) is convex and closed;
(c) F (Tr ) = EP (B).
2. Main results
Theorem 2.1. Let C be a nonempty closed convex subset of H. Let B be a bifunction from C × C to R
enjoys (B-1), (B-2), (B-3) and (B-4). Let ϕ : C → R be a lower semicontinuous and convex function and
let A : C → H be a continuous and monotone mapping. Let S be a uniformly L-Lipschitz continuous and
asymptotically κ-strict pseudocontraction in the intermediate sense on C. Assume that the common solution
set GM EP (B, A, ϕ) ∩ F (S) is nonempty and bounded. Let {αn } and {βn } be sequences in [0, 1]. Let {rn }
be a real sequence in (0, ∞). Let {xn } be a sequence generated as follows:


x1 ∈ H, C1 = C,




B(un , u) + ϕ(u) − ϕ(un ) + hAun , u − un i + r1n hu − un , un − xn i ≥ 0, ∀u ∈ C,



y = (1 − α ) β u + (1 − β )S n u + α x ,
n
n
n n
n
n
n n
2 ≤ (k − 1) sup{kx − wk2 : w ∈ GM EP (B, A, ϕ) ∩ F (S)}

C
=
{w
∈
C
:
ky
−
wk
n+1
n
n
n
n




2 },

+µ
+
kx
−
wk
n
n



x
=
P
roj
x
,
n
≥
1,
n+1
Cn+1 1
where P rojCn+1 is the metric projection from H onto Cn+1 . Assume that the control sequences {rn }, {αn }
and {βn } satisfy the following restrictions: 0 ≤ αn ≤ a < 1, lim inf rn > 0 and 0 < κ ≤ βn ≤ b < 1, where a
n→∞
and b are two real numbers. Then sequence {xn } converges strongly to x̄ = PGM EP (B,A,ϕ)∩F (S) x1 .
Proof. First, we define G(p, y) = B(p, y)+hAp, y −pi+ϕ(y)−ϕ(p), ∀p, y ∈ C. Next, we prove that bifunction
G satisfies (B-1), (B-2), (B-3) and (B-4). Therefore, generalized mixed equilibrium problem is equivalent to
the following equilibrium problem:
find p ∈ C such that G(p, y) ≥ 0, ∀y ∈ C.
Next, we prove G is monotone. Since A is a continuous and monotone operator, we find from the definition
of G that
G(y, z) + G(z, y) = B(y, z) + hAy, z − yi + ϕ(z) − ϕ(y) + B(z, y)
+ hAz, y − zi + ϕ(y) − ϕ(z)
= B(z, y) + hAz, y − zi + B(y, z) + hAy, z − yi
≤ hAz − Ay, y − zi
≤ 0.
It is clear that G satisfies (B-2). Next, we show that for each x ∈ C, y 7−→ G(x, y) is convex and lower
semicontinuous. For each x ∈ C, for all t ∈ (0, 1) and for all y, z ∈ C, since ϕ is convex, we have
G(x, ty + (1 − t)z) = B(x, ty + (1 − t)z) + hAx, ty + (1 − t)z − xi + ϕ(ty + (1 − t)z) − ϕ(x)
≤ t B(x, y) + ϕ(y) − ϕ(x) + hAx, y − xi
+ (1 − t) B(x, z) + ϕ(z) − ϕ(x) + hAx, z − xi
= (1 − t)G(x, z) + tG(x, y).
So, y 7−→ G(x, y) is convex. Similarly, we find that y 7−→ G(x, y) is also lower semicontinuous. Next, we
show G satisfies (B-4), that is,
lim sup G(tz + (1 − t)x, y) ≤ G(x, y), ∀x, y, z ∈ C.
t↓0
Since A is continuous and ϕ is lower semicontinuous, we have
D. Li, J. Zhao, J. Nonlinear Sci. Appl. 9 (2016), 757–765
761
lim sup G(tz + (1 − t)x, y) = lim sup B(tz + (1 − t)x, y)
t↓0
t↓0
+ lim sup ϕ(y) − ϕ(tz + (1 − t)x)big)
t↓0
+ lim suphA tz + (1 − t)x , y − tz + (1 − t)x i
t↓0
≤ B(x, y) + ϕ(y) − ϕ(x) + hAx, y − xi
= G(x, y).
Now, we are in a position to show that Cn is closed and convex. It is easy to see that Cn is closed. We
only show that Cn is convex. Note that C1 = H is convex. Suppose that Ci is convex for some positive
integer i ≥ 1. Next, we show that Ci+1 is convex for the same i. Put
Θn = sup{kxn − wk2 : w ∈ GM EP (B, A, ϕ) ∩ F (S)}.
Since
kyi − wk2 ≤ (ki − 1)Θi + µi + kxi − wk2
is equivalent to
2hxi − yi , wi ≤ kxi k2 − kyi k2 + (ki − 1)Θi + µi ,
we take w1 and w2 in Ci+1 and put w̄ = tw1 + (1 − t)w2 to find that w1 ∈ Ci , w2 ∈ Ci ,
2hxi − yi , w1 i ≤ kxi k2 − kyi k2 + (ki − 1)Θi + µi
and
2hxi − yi , w2 i ≤ kxi k2 − kyi k2 + (ki − 1)Θi + µi .
Using the above two inequalities, we obtain that kyi −wk2 ≤ kxi −wk2 +(ki −1)Θi +µi . By use the convexity
of Ci , we find that w̄ ∈ Ci . This shows that w̄ ∈ Ci+1 . This yields that Cn is convex and closed. Next, we
find that GM EP (B, A, ϕ) ∩ F (S) ⊂ Cn . It is clear that GM EP (B, A, ϕ) ∩ F (S) ⊂ C1 = H. Assume that
GM EP (B, A, ϕ) ∩ F (S) ⊂ Ch for some integer h ≥ 1. We intend to claim that GM EP (B, A, ϕ) ∩ F (S) ⊂
Ch+1 for the same h. For any p ∈ GM EP (B, A, ϕ) ∩ F (S) ⊂ Ch , we have
kyh − pk2 ≤ αh kxh − pk2 + (1 − αh )kβh (uh − p) + (1 − βh )(S h uh − p)k2
≤ αh kxh − pk2 + (1 − αh ) βh kuh − pk2
2
h
2
h
2
+ (1 − βh ) kh kuh − pk + κkuh − S uh k + µh − βh (1 − βh )kuh − S uh k
≤ (1 − αh )(kh − 1)kxh − pk2 + (1 − αh )(1 − βh )(κ − βh )kuh − S h uh k2
+ (1 − αh )(1 − βh )µh + kxh − pk2
≤ (kh − 1)Θh + µh + kxh − pk2 .
This implies that GM EP (B, A, ϕ) ∩ F (S) ⊂ Cn . Since xn = PCn x1 , we have kx1 − xn k ≤ kx1 − pk. In
particular, one has
kx1 − xn k ≤ kx1 − P rojGM EP (B,A,ϕ)∩F (S) x1 k.
This obtains the boundedness of {xn }. It follows that
0 ≤ hx1 − xn , xn − xn+1 i ≤ kx1 − xn kkx1 − xn+1 k − kx1 − xn k2 .
This implies kxn − x1 k ≤ kxn+1 − x1 k. Hence, we obtain that lim kxn − x1 k exists. Since
n→∞
kxn − xn+1 k2 = kxn − x1 k2 + 2hxn − x1 , x1 − xn+1 i + kx1 − xn+1 k2
= kxn − x1 k2 + 2hxn − x1 , x1 − xn + xn − xn+1 i + kx1 − xn+1 k2
= kxn − x1 k2 − 2kxn − x1 k2 + 2hxn − x1 , xn − xn+1 i + kx1 − xn+1 k2
≤ kx1 − xn+1 k2 − kxn − x1 k2 ,
D. Li, J. Zhao, J. Nonlinear Sci. Appl. 9 (2016), 757–765
762
we find that
lim kxn − xn+1 k = 0.
n→∞
(2.1)
By use of xn+1 = P rojCn+1 x1 ∈ Cn+1 , we have
kyn − xn+1 k2 ≤ (kn − 1)Θn + µn + kxn − xn+1 k2 .
Hence
lim kxn+1 − yn k = 0.
n→∞
(2.2)
This together with (2.1) yield that
lim kxn − yn k = 0.
(2.3)
Since kxn − yn k = (1 − αn )kxn − βn un + (1 − βn )S n un k, we find from (2.3) and the restriction imposed
on sequence {αn } that
lim kxn − βn un + (1 − βn )S n un k = 0.
(2.4)
n→∞
n→∞
For p ∈ GM EP (B, A, ϕ) ∩ F (S), one has
1
kun − pk2 ≤ hTrn xn − Trn p, xn − pi = (kun − pk2 + kxn − pk2 − kun − xn k2 ).
2
Hence, we have
kun − pk2 ≤ kxn − pk2 − kun − xn k2 .
(2.5)
It follows that
kyn − pk2 ≤ αh kxn − pk2 + (1 − αn ) βn kun − pk2 + (1 − βn )kS n un − S n pk2
− βn (1 − βn )kun − p − (S n un − S n p)k2
≤ αn kxn − pk2 + (1 − αn )βn kun − pk2 + (1 − αn )(1 − βn ) kn kun − pk2
+ κkun − p − (S n un − S n p)k2 + µn
− (1 − αn )βn (1 − βn )kun − p − (S n un − S n p)k2
≤ αn kxn − pk2 + (1 − αn )kn kun − pk2
− (1 − αn )(1 − βn )(βn − κ)kun − p − (S n un − S n p)k2
≤ (kn − 1)kxn − pk2 − (1 − αn )kn kun − xn k2 + kxn − pk2 .
This implies that
(1 − αn )kn kun − xn k2 ≤ (kn − 1)kxn − pk2 − kyn − pk2 + kxn − pk2
≤ (kxn − pk + kyn − pk)kxn − yn k + (kn − 1)kxn − pk2 .
By use of (2.3), we find that limn→∞ kun − xn k = 0. Since {xn } is bounded, there exists a subsequence
{xni } of {xn } such that xni * q. It follows uni * q. Using the restriction imposed on {rn }, we may assume
there exists a positive real number r such that rn ≥ r. Hence,
kun − xn k
= 0.
n→∞
rn
lim
(2.6)
Next, we show that q ∈ F (S). Note that
kxn − βn xn + (1 − βn )S n xn k ≤ kxn − βn un + (1 − βn )S n un k + βn kun − xn k + (1 − βn )kS n un − S n xn k
≤ kxn − βn un + (1 − βn )S n un k + Lkun − xn k.
D. Li, J. Zhao, J. Nonlinear Sci. Appl. 9 (2016), 757–765
763
It follows from (2.4) that
lim kxn − βn xn + (1 − βn )S n xn k = 0.
(2.7)
n→∞
Since
kS n xn − xn k ≤ kS n xn − βn xn + (1 − βn )S n xn k + k βn xn + (1 − βn )S n xn − xn k
≤ βn kS n xn − xn k + k βn xn + (1 − βn )S n xn − xn k,
one sees from the restriction imposed on {βn } and (2.7) that
lim kS n xn − xn k = 0.
(2.8)
n→∞
Since S is Lipschitz continuous, one has
kxn − Sxn k ≤ kxn − xn+1 k + kxn+1 − S n+1 xn+1 k + kS n+1 xn+1 − S n+1 xn k + kS n+1 xn − Sxn k
≤ (1 + L)kxn − xn+1 k + kxn+1 − S n+1 xn+1 k + LkS n xn − xn k.
It follows from (2.1) and (2.8) that
lim kSxn − xn k = 0.
n→∞
Using Lemma 1.1, we obtain that q ∈ F (S).
Next, we prove q ∈ GM EP (B, A, ϕ) = EP (G). By use of (B-1), one has
uni −xni
i
rni
1
rn hu − un , un − xn i
≥ G(u, un ).
≥ G(u, uni ). Using assumption (B-3), we get from (2.6)
Replacing n by ni , we arrive at hu − uni ,
that G(u, q) ≤ 0, ∀u ∈ C. For any t with 0 < t ≤ 1 and u ∈ C, let ut = tu + (1 − t)q. Since u ∈ C and
q ∈ C, we have ut ∈ C and hence G(ut , q) ≤ 0. It follows that
0 = G(ut , ut ) ≤ tG(ut , u) + (1 − t)G(ut , q) ≤ tG(ut , u),
which yields that G(ut , u) ≥ 0, ∀u ∈ C. Letting t ↓ 0, we obtain from assumption (B-4) that
G(q, u) ≥ 0,
∀u ∈ C.
This implies that q ∈ EP (G) = GM EP (B, A, ϕ). This shows that q ∈ GM EP (B, A, ϕ) ∩ F (S). Since
x̄ = PGM EP (B,A,ϕ)∩F (S) x1 , we obtain that
kx1 − x̄k ≤ kx1 − qk ≤ lim inf kx1 − xni k ≤ lim sup kx1 − xni k ≤ kx1 − x̄k.
i→∞
i→∞
It follows that
lim kx1 − xni k = kx1 − qk = kx1 − x̄k.
i→∞
Hence, {xni } converges strongly to x̄. It follows that the sequence {xn } converges strongly to x̄ =
PGM EP (B,A,ϕ)∩F (S) x1 . This completes the proof.
Next, we give some subresults based on Theorem 2.1. First, we consider solutions of equilibrium problem
(1.5).
Corollary 2.2. Let C be a nonempty closed convex subset of H. Let B be a bifunction from C × C to R
which satisfies (B-1), (B-2), (B-3) and (B-4) and let S : C → C be a uniformly L-Lipschitz continuous and
asymptotically κ-strict pseudocontraction in the intermediate sense. Assume that EP (F ) is nonempty and
bounded. Let {αn } be a sequence in [0, 1]. Let {rn } be a real sequence in (0, ∞). Let {xn } be a sequence
generated in the following manner:


x1 ∈ H,





C1 = C,



F (u , u) + 1 hu − u , u − x i ≥ 0, ∀u ∈ C,
n
n n
n
rn

yn = αn xn + (1 − αn )un ,





Cn+1 = {w ∈ Cn : kyn − wk ≤ kxn − wk},



x
n ≥ 1.
n+1 = P rojCn+1 x1 ,
D. Li, J. Zhao, J. Nonlinear Sci. Appl. 9 (2016), 757–765
764
Assume that the control sequences {rn } and {αn } satisfy the following restrictions: lim inf rn > 0 and
n→∞
0 ≤ αn ≤ a < 1, where a is a real number. Then sequence {xn } converges strongly to x̄ = P rojEP (F ) x1 .
Proof. Putting ϕ = 0, A = 0 and S = I, we find from Theorem 2.1 the desired conclusion.
Next, we give a result on equilibrium problem (1.9).
Corollary 2.3. Let C be a nonempty closed convex subset of H. Let B be a bifunction from C × C to R
which satisfies (B-1), (B-2), (B-3) and (B-4) and A : C → H a continuous and monotone mapping. Assume
that EP (F, A) is nonempty and bounded. Let {αn } be a sequence in [0, 1]. Let {rn } be a real sequence in
(0, ∞). Let {xn } be a sequence generated in the following manner:


x1 ∈ H,




C1 = C,



F (u , u) + hAu , u − u i + 1 hu − u , u − x i ≥ 0, ∀u ∈ C,
n n
n
n
n
n
rn

y
=
α
x
+
(1
−
α
)u
,
n
n n
n n





C
=
{w
∈
C
:
ky
n+1
n
n − wk ≤ kxn − wk},



x
n ≥ 1.
n+1 = P rojCn+1 x1 ,
Assume that the control sequences {rn } and {αn } satisfy the following restrictions: lim inf rn > 0, 0 ≤ αn ≤
n→∞
a < 1, where a is a real number. Then sequence {xn } converges strongly to x̄ = P rojEP (F,A) x1 .
Finally, we give a result on fixed point of asymptotically κ-strict pseudocontraction in the intermediate
sense.
Corollary 2.4. Let C be a nonempty closed convex subset of H. Let S : C → C be a uniformly L-Lipschitz
continuous and asymptotically κ-strict pseudocontraction in the intermediate sense with a nonempty and
bounded fixed-point set. Let {αn } and {βn } be sequences in [0, 1]. Let {xn } be a sequence generated in the
following manner:


x1 ∈ C,





C1 = C,
yn = αn xn + (1 − αn ) βn xn + (1 − βn )S n xn ,



Cn+1 = {w ∈ Cn : kyn − wk2 ≤ (kn − 1) sup{kxn − wk2 : w ∈ F (S)} + µn + kxn − wk2 },




xn+1 = PCn+1 x1 , n ≥ 1.
Assume that the control sequences {αn } and {βn } satisfy the following restrictions: 0 ≤ αn ≤ a < 1 and
(b) 0 < κ ≤ βn ≤ b < 1, where a and b are two real numbers. Then sequence {xn } converges strongly to
x̄ = P rojF (S) x1 .
Acknowledgements:
The authors were grateful to the reviewers for useful suggestions which improved the contents of this
paper.
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