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Common solutions of equilibrium and fixed point problems
Journal of Inequalities and Applications volume 2013, Article number: 425 (2013)
Abstract
In this paper, common solutions of equilibrium and fixed point problems are investigated. Convergence theorems of common solutions are established in a uniformly smooth and strictly convex Banach space.
MSC:47H09, 47H10, 47J25.
1 Introduction and preliminaries
Let E be a real Banach space. Let {U}_{E}=\{x\in E:\parallel x\parallel =1\} be the unit sphere of E. E is said to be smooth iff {lim}_{t\to 0}\frac{\parallel x+ty\parallel \parallel x\parallel}{t} exists for each x,y\in {U}_{E}. It is also said to be uniformly smooth iff the above limit is attained uniformly for x,y\in {U}_{E}. E is said to be strictly convex iff \parallel \frac{x+y}{2}\parallel <1 for all x,y\in E with \parallel x\parallel =\parallel y\parallel =1 and x\ne y. It is said to be uniformly convex iff {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}{y}_{n}\parallel =0 for any two sequences \{{x}_{n}\} and \{{y}_{n}\} in E such that \parallel {x}_{n}\parallel =\parallel {y}_{n}\parallel =1 and {lim}_{n\to \mathrm{\infty}}\parallel \frac{{x}_{n}+{y}_{n}}{2}\parallel =1.
Recall that the normalized duality mapping J from E to {2}^{{E}^{\ast}} is defined by
where \u3008\cdot ,\cdot \u3009 denotes the generalized duality pairing. It is well known that if E is uniformly smooth, then J is uniformly normtonorm continuous on each bounded subset of E. It is also well known that E is (uniformly) smooth if and only if {E}^{\ast} is (uniformly) convex.
In what follows, we use ⇀ and → to stand for the weak and strong convergence, respectively. Recall that E enjoys the KadecKlee property iff for any sequence \{{x}_{n}\}\subset E, and x\in E with {x}_{n}\rightharpoonup x, and \parallel {x}_{n}\parallel \to \parallel x\parallel, then \parallel {x}_{n}x\parallel \to 0 as n\to \mathrm{\infty}. It is well known that if E is a uniformly convex Banach space, then E enjoys the KadecKlee property.
Let E be a smooth Banach space. Consider the functional defined by
Observe that, in a Hilbert space H, the equality is reduced to \varphi (x,y)={\parallel xy\parallel}^{2}, x,y\in H. As we all know, if C is a nonempty closed convex subset of a Hilbert space H and {P}_{C}:H\to C is the metric projection of H onto C, then {P}_{C} is nonexpansive. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [1] recently introduced a generalized projection operator {\mathrm{\Pi}}_{C} in a Banach space E, which is an analogue of the metric projection {P}_{C} in Hilbert spaces. Recall that the generalized projection {\mathrm{\Pi}}_{C}:E\to C is a map that assigns to an arbitrary point x\in E the minimum point of the functional \varphi (x,y), that is, {\mathrm{\Pi}}_{C}x=\overline{x}, where \overline{x} is the solution to the minimization problem \varphi (\overline{x},x)={min}_{y\in C}\varphi (y,x). Existence and uniqueness of the operator {\mathrm{\Pi}}_{C} follows from the properties of the functional \varphi (x,y) and strict monotonicity of the mapping J. If E is a reflexive, strictly convex and smooth Banach space, then \varphi (x,y)=0 if and only if x=y; for more details, see [1] and the references therein. In Hilbert spaces, {\mathrm{\Pi}}_{C}={P}_{C}. It is obvious from the definition of a function ϕ that
and
Let C be a nonempty subset of E, and let T:C\to C be a mapping. In this paper, we use F(T) to stand for the fixed point set of T. T is said to be closed iff for any sequence \{{x}_{n}\}\subset C such that {lim}_{n\to \mathrm{\infty}}{x}_{n}={x}_{0} and {lim}_{n\to \mathrm{\infty}}T{x}_{n}={y}_{0}, then T{x}_{0}={y}_{0}. T is said to be asymptotically regular on C iff for any bounded subset K of C,
Recall that a point p in C is said to be an asymptotic fixed point of T iff C contains a sequence \{{x}_{n}\} which converges weakly to p such that {lim}_{n\to \mathrm{\infty}}\parallel {x}_{n}T{x}_{n}\parallel =0. The set of asymptotic fixed points of T will be denoted by \tilde{F}(T). T is said to be relatively nonexpansive iff
T is said to be relatively asymptotically nonexpansive iff
where \{{\mu}_{n}\}\subset [0,\mathrm{\infty}) is a sequence such that {\mu}_{n}\to 0 as n\to \mathrm{\infty}.
Remark 1.1 The class of relatively asymptotically nonexpansive mappings which is an extension of the class of relatively nonexpansive mappings was first considered in [2] and [3].
Recall that T is said to be quasiϕnonexpansive iff
Recall that T is said to be asymptotically quasiϕnonexpansive iff there exists a sequence \{{\mu}_{n}\}\subset [0,\mathrm{\infty}) with {\mu}_{n}\to 0 as n\to \mathrm{\infty} such that
Remark 1.2 The class of asymptotically quasiϕnonexpansive mappings, which is an extension of the class of quasiϕnonexpansive mappings, was considered in [4, 5]; see also [6].
Remark 1.3 The class of quasiϕnonexpansive mappings and the class of asymptotically quasiϕnonexpansive mappings are more general than the class of relatively nonexpansive mappings and the class of relatively asymptotically nonexpansive mappings. Quasiϕnonexpansive mappings and asymptotically quasiϕnonexpansive mappings do not require the restriction F(T)=\tilde{F}(T).
Remark 1.4 The class of quasiϕnonexpansive mappings and the class of asymptotically quasiϕnonexpansive mappings are generalizations of the class of quasinonexpansive mappings and the class of asymptotically quasinonexpansive mappings in Banach spaces.
Recall that T is said to be generalized asymptotically quasiϕnonexpansive iff F(T)\ne \mathrm{\varnothing}, and there exist two nonnegative sequences \{{\mu}_{n}\}\subset [0,\mathrm{\infty}) with {\mu}_{n}\to 0, and \{{\xi}_{n}\}\subset [0,\mathrm{\infty}) with {\xi}_{n}\to 0 as n\to \mathrm{\infty} such that
Remark 1.5 The class of generalized asymptotically quasiϕnonexpansive mappings [7] is a generalization of the class of generalized asymptotically quasinonexpansive mappings in the framework of Banach spaces which was introduced by Agarwal et al. [8].
Let F be a bifunction from C\times C to ℝ, where ℝ denotes the set of real numbers. Recall the following equilibrium problem. Find p\in C such that F(p,y)\ge 0, \mathrm{\forall}y\in C. We use \mathit{EP}(F) to denote the solution set of the equilibrium problem. Given a mapping Q:C\to {E}^{\ast}, let
Then p\in \mathit{EP}(F) if and only if p is a solution of the following variational inequality. Find p such that
Numerous problems in physics, optimization and economics reduce to finding a solution of the equilibrium problem; see [9–36] and the related references therein. In [25], Kim studied a sequence \{{x}_{n}\} which is generated in the following manner:
where {M}_{n}=sup\{\varphi (z,{x}_{n}):z\in \mathcal{F}\} for each n\ge 1, \{{\alpha}_{n}\} is a real sequence in [0,1], \{{r}_{n}\} is a real sequence in [a,\mathrm{\infty}), where a is some positive real number. In a uniformly smooth and strictly convex Banach space, which also enjoys the KadecKlee property, the author obtained a strong convergence theorem; for more details, see [25] and the references therein.
In this paper, motivated by the above result, we consider the projection algorithm for treating solutions of the equilibrium problem and fixed points of generalized asymptotically quasiϕnonexpansive mappings. A strong convergence theorem is established in a Banach space. The results presented this paper mainly improve the corresponding results announced in Qin Cho and Kang [5] and Kim [25].
In order to prove our main results, we need the following lemmas.
Lemma 1.6 [36]
Let E be a smooth and uniformly convex Banach space, and let r>0. Then there exists a strictly increasing, continuous and convex function g:[0,2r]\to R such that g(0)=0 and
for all x,y\in {B}_{r}=\{x\in E:\parallel x\parallel \le r\} and t\in [0,1].
Lemma 1.7 [1]
Let C be a nonempty closed convex subset of a smooth Banach space E and x\in E. Then {x}_{0}={\mathrm{\Pi}}_{C}x if and only if
Lemma 1.8 [1]
Let E be a reflexive, strictly convex and smooth Banach space, let C be a nonempty closed convex subset of E and x\in E. Then
Let C be a closed convex subset of a smooth, strictly convex and reflexive Banach space E. Let F be a bifunction from C\times C to ℝ satisfying (A1)(A4). Let r>0 and x\in E. Then there exists z\in C such that F(z,y)+\frac{1}{r}\u3008yz,JzJx\u3009\ge 0, \mathrm{\forall}y\in C. Define a mapping {T}_{r}:E\to C by
Then the following conclusions hold:

(1)
{S}_{r} is a singlevalued firmly nonexpansivetype mapping, i.e., for all x,y\in E,
\u3008{S}_{r}x{S}_{r}y,J{S}_{r}xJ{S}_{r}y\u3009\le \u3008{S}_{r}x{S}_{r}y,JxJy\u3009; 
(2)
F({S}_{r})=\mathit{EP}(F) is closed and convex;

(3)
{S}_{r} is quasiϕnonexpansive;

(4)
\varphi (q,{S}_{r}x)+\varphi ({S}_{r}x,x)\le \varphi (q,x), \mathrm{\forall}q\in F({S}_{r}).
Lemma 1.10 [7]
Let E be a uniformly smooth and strictly convex Banach space which also enjoys the KadecKlee property, and let C be a nonempty closed and convex subset of E. Let T:C\to C be a generalized asymptotically quasiϕnonexpansive mapping. Then F(T) is closed and convex.
2 Main results
Theorem 2.1 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the KadecKlee property, and let C be a nonempty closed and convex subset of E. Let Δ be an index set. Let {F}_{i} be a bifunction from C\times C to ℝ satisfying (A1)(A4) for every i\in \mathrm{\Delta}. Let T:C\to C be a generalized asymptotically quasiϕnonexpansive mapping. Assume that T is closed asymptotically regular on C and \mathrm{\Omega}:=F(T)\cap {\bigcap}_{i\in \mathrm{\Delta}}EF({F}_{i}) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where {M}_{n}=sup\{\varphi (z,{x}_{n}):z\in \mathrm{\Omega}\}, \{{\alpha}_{n}\} is a real number sequence in (0,1) such that {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})>0, \{{r}_{n,i}\} is a real number sequence in [{a}_{i},\mathrm{\infty}), where \{{a}_{i}\} is a positive real number sequence. Then the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{\mathrm{\Omega}}{x}_{0}, where {\mathrm{\Pi}}_{\mathrm{\Omega}} is the generalized projection from E onto Ω.
Proof In view of Lemmas 1.9 and 1.10, we find that the common solution set Ω is closed and convex. Next, we show that {C}_{n} is closed and convex. It suffices to show, for any fixed but arbitrary i\in \mathrm{\Delta}, that {C}_{n,i} is closed and convex. This can be proved by induction on n. It is obvious that {C}_{1,i}=C is closed and convex. Assume that {C}_{j,i} is closed and convex for some j\ge 1. We next prove that {C}_{j+1,i} is closed and convex for the same j. This completes the proof that {C}_{n} is closed and convex. It is clear that {C}_{j+1,i} is closed. We only prove the convexity. Indeed, \mathrm{\forall}a,b\in {C}_{j+1,i}, we see that a,b\in {C}_{j,i}, and
and
Notice that the two inequalities above are equivalent to the following inequalities, respectively:
and
These imply that
Since {C}_{j,i} is convex, we see that ta+(1t)b\in {C}_{j,i}. Notice that the above inequality is equivalent to
This proves that {C}_{j+1,i} is convex. This completes that {C}_{n} is closed and convex.
Next, we prove that \mathrm{\Omega}\subset {C}_{n}. It suffices to claim that \mathrm{\Omega}\subset {C}_{n,i} for every i\in \mathrm{\Delta}. Note that \mathrm{\Omega}\subset {C}_{1,i}=C. Suppose that \mathrm{\Omega}\subset {C}_{j,i} for some j and for every i\in \mathrm{\Delta}. Then, for \mathrm{\forall}w\in \mathrm{\Omega}\subset {C}_{j,i}, we have
This shows that w\in {C}_{j+1,i}. This implies that \mathrm{\Omega}\subset {C}_{n} for every n\ge 1.
On the other hand, it follows from Lemma 1.8 that
This shows that the sequence \varphi ({x}_{n},{x}_{0}) is bounded. In view of (1.2), we see that the sequence \{{x}_{n}\} is also bounded. Since the space is reflexive, we may, without loss of generality, assume that {x}_{n}\rightharpoonup p\in {C}_{n}. Note that \varphi ({x}_{n},{x}_{0})\le \varphi (p,{x}_{0}). It follows that
This implies that
Hence, we have \parallel {x}_{n}\parallel \to \parallel p\parallel as n\to \mathrm{\infty}. In view of the KadecKlee property of E, we obtain that {x}_{n}\to p as n\to \mathrm{\infty}.
Next, we show that p\in F(T). By the construction of {C}_{n}, we have that {C}_{n+1}\subset {C}_{n} and {x}_{n+1}={\mathrm{\Pi}}_{{C}_{n+1}}{x}_{0}\in {C}_{n}. It follows that
Letting n\to \mathrm{\infty}, we obtain that \varphi ({x}_{n+1},{x}_{n})\to 0. In view of {x}_{n+1}\in {C}_{n+1}, we see that
It follows that
From (1.2), we see that {lim}_{n\to \mathrm{\infty}}\parallel {u}_{n,i}\parallel =\parallel p\parallel. It follows that {lim}_{n\to \mathrm{\infty}}\parallel J{u}_{n,i}\parallel =\parallel Jp\parallel. This implies that \{J{u}_{n,i}\} is bounded. Note that E is reflexive and {E}^{\ast} is also reflexive. We may assume that J{u}_{n,i}\rightharpoonup {x}^{\ast ,i}\in {E}^{\ast}. In view of the reflexivity of E, we see that J(E)={E}^{\ast}. This shows that there exists an {x}^{i}\in E such that J{x}^{i}={x}^{\ast ,i}. It follows that
Taking {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} on the both sides of the equality above yields that
That is, p={x}^{i}, which in turn implies that {x}^{\ast ,i}=Jp. It follows that J{u}_{n,i}\rightharpoonup Jp\in {E}^{\ast}. Since {E}^{\ast} enjoys the KadecKlee property, we obtain that J{u}_{n,i}Jp\to 0 as n\to \mathrm{\infty}. Note that {J}^{1}:{E}^{\ast}\to E is demicontinuous. It follows that {u}_{n,i}\rightharpoonup p. Since E enjoys the KadecKlee property, we obtain that {u}_{n,i}\to p as n\to \mathrm{\infty}. Note that
It follows that
Since J is uniformly normtonorm continuous on any bounded sets, we have
Let r={sup}_{n\ge 0}\{\parallel {x}_{n}\parallel ,\parallel {T}^{n}{x}_{n}\parallel \}. Since E is uniformly smooth, we know that {E}^{\ast} is uniformly convex. In view of Lemma 1.6, we see that
It follows that
Notice that
It follows from (2.1) and (2.2) that \varphi (w,{x}_{n})\varphi (w,{u}_{n,i})\to 0 as n\to \mathrm{\infty}. In view of {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})>0, we see that {lim}_{n\to \mathrm{\infty}}g(\parallel J{x}_{n}J{T}^{n}{x}_{n}\parallel )=0. It follows from the property of g that
Since {x}_{n}\to p as n\to \mathrm{\infty} and J:E\to {E}^{\ast} is demicontinuous, we obtain that J{x}_{n}\rightharpoonup Jp\in {E}^{\ast}. Note that
This implies that \parallel J{x}_{n}\parallel \to \parallel Jp\parallel as n\to \mathrm{\infty}. Since {E}^{\ast} enjoys the KadecKlee property, we see that
Notice that
It follows from (2.3) and (2.4) that
Note that {J}^{1}:{E}^{\ast}\to E is demicontinuous. It follows that {T}^{n}{x}_{n}\rightharpoonup p. On the other hand, we have
In view of (2.5), we obtain that \parallel {T}^{n}{x}_{n}\parallel \to \parallel p\parallel as n\to \mathrm{\infty}. Since E enjoys the KadecKlee property, we obtain that
Note that
It follows from the asymptotic regularity of T and (2.6) that
That is, T{T}^{n}{x}_{n}p\to 0 as n\to \mathrm{\infty}. It follows from the closedness of T that Tp=p.
Next, we show that p\in {\bigcap}_{i\in \mathrm{\Delta}}EF({F}_{i}). Notice that \varphi (w,{y}_{n})\le \varphi (w,{x}_{n})+{\mu}_{n}{M}_{n}+{\xi}_{n}. In view of {u}_{n,i}={S}_{{r}_{n,i}}{y}_{n}, we find from Lemma 1.8 that
This in turn implies that
It follows from (1.2) that \parallel {u}_{n,i}\parallel \parallel {y}_{n}\parallel \to 0 as n\to \mathrm{\infty}. In view of {u}_{n,i}\to p as n\to \mathrm{\infty}, we arrive at {lim}_{n\to \mathrm{\infty}}\parallel {y}_{n}\parallel =\parallel p\parallel. It follows that {lim}_{n\to \mathrm{\infty}}\parallel J{y}_{n}\parallel =\parallel Jp\parallel. Since {E}^{\ast} is reflexive, we may assume that J{y}_{n}\rightharpoonup {f}^{\ast}\in {E}^{\ast}. In view of J(E)={E}^{\ast}, we see that there exists f\in E such that Jf={f}^{\ast}. It follows that
Taking {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}} on the both sides of the equality above yields that
That is, p=f, which in turn implies that {f}^{\ast}=Jp. It follows that J{y}_{n}\rightharpoonup Jp\in {E}^{\ast}. Since {E}^{\ast} enjoys the KadecKlee property, we obtain that J{y}_{n}Jp\to 0 as n\to \mathrm{\infty}. Note that {J}^{1}:{E}^{\ast}\to E is demicontinuous. It follows that {y}_{n}\rightharpoonup p. Since E enjoys the KadecKlee property, we obtain that {y}_{n}\to p as n\to \mathrm{\infty}. Notice that \parallel {u}_{n,i}{y}_{n}\parallel \le \parallel {u}_{n,i}p\parallel +\parallel p{y}_{n}\parallel. It follows that
Since J is uniformly normtonorm continuous on any bounded sets, we have
From the assumption {r}_{n,i}\ge {a}_{i}, we see that
Notice that
It follows from condition (A2) that
By taking the limit as n\to \mathrm{\infty} in the above inequality, from condition (A4) we obtain that
For 0<{t}_{i}<1 and y\in C, define {y}_{{t}_{i}}={t}_{i}y+(1{t}_{i})p. It follows that {y}_{t,i}\in C, which yields that {F}_{i}({y}_{t,i},p)\le 0. It follows from conditions (A1) and (A4) that
That is,
Letting {t}_{i}\downarrow 0, we find from condition (A3) that {F}_{i}(p,y)\ge 0, \mathrm{\forall}y\in C. This implies that p\in \mathit{EP}({F}_{i}). This completes the proof that p\in \mathrm{\Omega}.
Finally, we prove that p={\mathrm{\Pi}}_{\mathrm{\Omega}}{x}_{0}. From {x}_{n}={\mathrm{\Pi}}_{{C}_{n}}{x}_{0}, we see that
In view of \mathrm{\Omega}\subset {C}_{n}, we find that
Letting n\to \mathrm{\infty} in the above inequality, we see that
In view of Lemma 1.7, we can obtain that p={\mathrm{\Pi}}_{\mathrm{\Omega}}{x}_{0}. This completes the proof. □
If T is asymptotically quasiϕnonexpansive, then we find from Theorem 2.1 the following result.
Corollary 2.2 Let E be a uniformly smooth and strictly convex Banach space which also enjoys the KadecKlee property, and let C be a nonempty closed and convex subset of E. Let Δ be an index set. Let {F}_{i} be a bifunction from C\times C to ℝ satisfying (A1)(A4) for every i\in \mathrm{\Delta}. Let T:C\to C be an asymptotically quasiϕnonexpansive mapping. Assume that T is closed asymptotically regular on C and \mathrm{\Omega}:=F(T)\cap {\bigcap}_{i\in \mathrm{\Delta}}EF({F}_{i}) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where {M}_{n}=sup\{\varphi (z,{x}_{n}):z\in \mathrm{\Omega}\}, \{{\alpha}_{n}\} is a real number sequence in (0,1) such that {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})>0, \{{r}_{n,i}\} is a real number sequence in [{a}_{i},\mathrm{\infty}), where \{{a}_{i}\} is a positive real number sequence. Then the sequence \{{x}_{n}\} converges strongly to {\mathrm{\Pi}}_{\mathrm{\Omega}}{x}_{0}, where {\mathrm{\Pi}}_{\mathrm{\Omega}} is the generalized projection from E onto Ω.
Remark 2.3 Since the index set Δ is arbitrary, Corollary 2.2 is an improvement of the corresponding results in Kim [25].
Remark 2.4 Corollary 2.2 also improves the corresponding results in Qin et al. [5] in the following aspects:

(a)
from a uniformly smooth and uniformly convex space to a uniformly smooth and strictly convex Banach space which also enjoys the KadecKlee property;

(b)
from a single bifunction to a family of bifunctions;

(c)
from a quasiϕnonexpansive mapping to an asymptotically quasiϕnonexpansive mapping.
In the framework of Hilbert spaces, the theorem is reduced to the following.
Corollary 2.5 Let E be a Hilbert space, and let C be a nonempty closed and convex subset of E. Let Δ be an index set. Let {F}_{i} be a bifunction from C\times C to ℝ satisfying (A1)(A4) for every i\in \mathrm{\Delta}. Let T:C\to C be a generalized asymptotically quasinonexpansive mapping. Assume that T is closed asymptotically regular on C and \mathrm{\Omega}:=F(T)\cap {\bigcap}_{i\in \mathrm{\Delta}}EF({F}_{i}) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where {M}_{n}=sup\{{\parallel z{x}_{n}\parallel}^{2}:z\in \mathrm{\Omega}\}, \{{\alpha}_{n}\} is a real number sequence in (0,1) such that {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})>0, \{{r}_{n,i}\} is a real number sequence in [{a}_{i},\mathrm{\infty}), where \{{a}_{i}\} is a positive real number sequence. Then the sequence \{{x}_{n}\} converges strongly to {Proj}_{\mathrm{\Omega}}{x}_{0}, where {Proj}_{\mathrm{\Omega}} is the metric projection from E onto Ω.
Proof In the framework of Hilbert spaces, we find that \varphi (x,y)={\parallel xy\parallel}^{2}, J is reduced to the identity mapping and the generalized projection {\mathrm{\Pi}}_{C} is reduced to the metric projection {Proj}_{C}. This completes the proof. □
For a single bifunction, we also have the following.
Corollary 2.6 Let E be a Hilbert space, and let C be a nonempty closed and convex subset of E. Let F be a bifunction from C\times C to ℝ satisfying (A1)(A4). Let T:C\to C be a generalized asymptotically quasinonexpansive mapping. Assume that T is closed asymptotically regular on C and \mathrm{\Omega}:=F(T)\cap EF(F) is nonempty and bounded. Let \{{x}_{n}\} be a sequence generated in the following manner:
where {M}_{n}=sup\{{\parallel z{x}_{n}\parallel}^{2}:z\in \mathrm{\Omega}\}, \{{\alpha}_{n}\} is a real number sequence in (0,1) such that {lim\hspace{0.17em}inf}_{n\to \mathrm{\infty}}{\alpha}_{n}(1{\alpha}_{n})>0, \{{r}_{n,i}\} is a real number sequence in [a,\mathrm{\infty}), where a is a positive real number. Then the sequence \{{x}_{n}\} converges strongly to {Proj}_{\mathrm{\Omega}}{x}_{0}, where {Proj}_{\mathrm{\Omega}} is the metric projection from E onto Ω.
Proof In the framework of Hilbert spaces, we find that \varphi (x,y)={\parallel xy\parallel}^{2}, J is reduced to the identity mapping, and the generalized projection {\mathrm{\Pi}}_{C} is reduced to the metric projection {Proj}_{C}. In view of Corollary 2.5, we may immediately conclude the desired results. □
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Zhang, QN. Common solutions of equilibrium and fixed point problems. J Inequal Appl 2013, 425 (2013). https://doi.org/10.1186/1029242X2013425
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DOI: https://doi.org/10.1186/1029242X2013425