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 Open Access
A modified Mann iterative scheme by generalized fprojection for a countable family of relatively quasinonexpansive mappings and a system of generalized mixed equilibrium problems
 Siwaporn Saewan^{1} and
 Poom Kumam^{1, 2}Email author
https://doi.org/10.1186/168718122011104
© Saewan and Kumam; licensee Springer. 2011
 Received: 23 July 2011
 Accepted: 21 December 2011
 Published: 21 December 2011
Abstract
The purpose of this paper is to introduce a new hybrid projection method based on modified Mann iterative scheme by the generalized fprojection operator for a countable family of relatively quasinonexpansive mappings and the solutions of the system of generalized mixed equilibrium problems. Furthermore, we prove the strong convergence theorem for a countable family of relatively quasinonexpansive mappings in a uniformly convex and uniform smooth Banach space. Finally, we also apply our results to the problem of finding zeros of $\mathcal{B}$monotone mappings and maximal monotone operators. The results presented in this paper generalize and improve some wellknown results in the literature.
2000 Mathematics Subject Classification: 47H05; 47H09; 47H10.
Keywords
 The generalized fprojection operator
 relatively quasinonexpansive mapping
 $\mathcal{B}$monotone mappings
 maximal monotone operator
 system of generalized mixed equilibrium problems
1 Introduction
The theory of equilibrium problems, the development of an efficient and implementable iterative algorithm, is interesting and important. This theory combines theoretical and algorithmic advances with novel domain of applications. Analysis of these problems requires a blend of techniques from convex analysis, functional analysis, and numerical analysis.
Equilibrium problems theory provides us with a natural, novel, and unified framework for studying a wide class of problems arising in economics, finance, transportation, network, and structural analysis, image reconstruction, ecology, elasticity and optimization, and it has been extended and generalized in many directions. The ideas and techniques of this theory are being used in a variety of diverse areas and proved to be productive and innovative. In particular, generalized mixed equilibrium problem and equilibrium problems are related to the problem of finding fixed points of nonlinear mappings.
The generalized mixed equilibrium problems include fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and the equilibrium problems as special cases. Moreover, the above formulation (1.5) was shown in [1] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, optimization problems, vector equilibrium problems, and Nash equilibria in noncooperative games. In other words, the GMEP(θ, A, φ), MEP(θ, φ) and EP(θ) are an unifying model for several problems arising in physics, engineering, science, optimization, economics, etc. Many authors studied and constructed some solution methods to solve the GMEP(θ, A, φ), MEP(θ, φ), EP(θ) [[1–16], and references therein].
A point x ∈ C is a fixed point of T provided Tx = x. Denote by F(T) the set of fixed points of T, that is, F(T) = {x ∈ C : Tx = x}.
The existence and uniqueness of the operator Π_{ C } follow from the properties of the functional ϕ(y, x) and strict monotonicity of the mapping J [17–21]. It is well known that the metric projection operator plays an important role in nonlinear functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, and complementarity problems, etc. [17, 22]. In 1994, Alber [23] introduced and studied the generalized projections from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces. Moreover, Alber [17] presented some applications of the generalized projections to approximately solve variational inequalities and von Neumann intersection problem in Banach spaces. In 2005, Li [22] extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solve the variational inequality in Banach spaces. Later, Wu and Huang [24] introduced a new generalized fprojection operator in Banach spaces. They extended the definition of the generalized projection operators introduced by Abler [23] and proved some properties of the generalized fprojection operator. In 2009, Fan et al. [25] presented some basic results for the generalized fprojection operator and discussed the existence of solutions and approximation of the solutions for generalized variational inequalities in noncompact subsets of Banach spaces.
 (1)
G(ξ, ϖ) is convex and continuous with respect to ϖ when ξ is fixed;
 (2)
G(ξ, ϖ) is convex and lower semicontinuous with respect to ξ when ϖ is fixed.
Observe that, if f(x) = 0, then the generalized fprojection operator (1.12) reduces to the generalized projection operator (1.9).
For the generalized fprojection operator, Wu and Hung [24] proved the following basic properties:
 (1)
${\pi}_{C}^{f}\varpi $, is a nonempty closed convex subset of C for all ϖ ∈ E*;
 (2)if E is smooth, then for all ϖ ∈ E*, $x\in {\pi}_{C}^{f}\varpi $if and only if$\u3008xy,\varpi Jx\u3009+\rho f\left(y\right)\rho f\left(x\right)\ge 0,\phantom{\rule{1em}{0ex}}\forall y\in C;$
 (3)
if E is strictly convex and f : C → ℝ ∪ {+∞} is positive homogeneous (i.e., f(tx) = tf(x) for all t > 0 such that tx ∈ C where x ∈ C), then ${\pi}_{C}^{f}\varpi $is singlevalued mapping.
Recently, Fan et al. [25] show that the condition f is positive homogeneous which appeared in [[25], Lemma 2.1 (iii)] can be removed.
Lemma 1.3. [25]Let E be a real reflexive Banach space with its dual E* and C a nonempty closed convex subset of E. If E is strictly convex, then ${\pi}_{C}^{f}\varpi $is single valued.
Now we consider the second generalized f projection operator in Banach space [26].
Next, we give the following example [27] of metric projection, generalized projection operator and generalized fprojection operator do not coincide.
Example 1.5. Let X = ℝ^{3} be provided with the norm $\left\right\left({x}_{1},{x}_{2},{x}_{3}\right)\left\right\phantom{\rule{2.77695pt}{0ex}}=\sqrt{\left({x}_{1}^{2}+{x}_{2}^{2}\right)}+\sqrt{{x}_{2}^{2}+{x}_{3}^{2}}$.
Recall that a point p in C is said to be an asymptotic fixed point of T [28] if C contains a sequence {x_{ n }} which converges weakly to p such that lim_{n→∞}x_{ n }  Tx_{ n } = 0. The set of asymptotic fixed points of T will be denoted by $\hat{F}\left(T\right)$. A mapping T from C into itself is said to be relatively nonexpansive mapping [29–31] if
(R1) F(T) is nonempty;
(R2) ϕ(p, Tx) ≤ ϕ(p, x) for all x ∈ C and p ∈ F(T);
(R3) $\hat{F}\left(T\right)=F\left(T\right)$.
A mapping T is said to be relatively quasinonexpansive (or quasiϕnonexpansive) if the conditions (R1) and (R2) are satisfied. The asymptotic behavior of a relatively nonexpansive mapping was studied in [32–34]. The class of relatively quasinonexpansive mappings is more general than the class of relatively nonexpansive mappings [11, 32–35] which requires the strong restriction: $F\left(T\right)=\hat{F}\left(T\right)$. In order to explain this better, we give the following example [36] of relatively quasinonexpansive mappings which is not relatively nonexpansive mapping. It is clearly by the definition of relatively quasinonexpansive mapping T is equivalent to F(T) ≠ ∅, and G(p, JTx) ≤ G(p, Jx) for all x ∈ C and p ∈ F(T).
Example 1.6. Let E be any smooth Banach space and let x_{0} ≠ 0 be any element of E.
Then T is a relatively quasinonexpansive mapping but not a relatively nonexpansive mapping. Actually, T above fails to have the condition (R 3).
Next, we give some examples which are closed quasiϕnonexpansive [[4], Examples 2.3 and 2.4].
Example 1.7. Let E be a uniformly smooth and strictly convex Banach space and A ⊂ E × E* be a maximal monotone mapping such that its zero set A^{ 1}0 ≠ ∅. Then, J_{ r } = (J + rA)^{ 1}J J is a closed quasiϕnonexpansive mapping from E onto D(A) and F(J_{ r }) = A^{ 1}0.
Proof By Matsushita and Takahashi [[35], Theorem 4.3], we see that J_{ r } is relatively nonexpansive mapping from E onto D(A) and F(J_{ r }) = A^{ 1}0. Therefore, J_{ r } is quasiϕnonexpansive mapping from E onto D(A) and F (J_{ r }) = A^{ 1}0. On the other hand, we can obtain the closedness of J_{ r } easily from the continuity of the mapping J and the maximal monotonicity of A; see [35] for more details. □
Example 1.8. Let C be the generalized projection from a smooth, strictly convex, and reflexive Banach space E onto a nonempty closed convex subset C of E. Then, C is a closed quasiϕnonexpansive mapping from E onto C with F(Π_{ C }) = C.
They proved that if the sequence {α_{ n }} bounded above from one, then {x_{ n }} defined by (1.15) converges strongly to P_{F(T)}x.
In 2007, Aoyama et al. [[42], Lemma 3.1] introduced {T_{ n }} is a sequence of nonexpansive mappings of C into itself with ${\cap}_{n=1}^{\infty}F\left({T}_{n}\right)\ne \varnothing $ satisfy the following condition: if for each bounded subset B of C, ${\sum}_{n=1}^{\infty}\mathrm{sup}\left\{\left\right{T}_{n+1}z{T}_{n}z\left\right\phantom{\rule{2.77695pt}{0ex}}:z\in B<\infty \right\}.$ Assume that if the mapping T : C → C defined by Tx = lim_{n→∞}T_{ n }x for all x ∈ C, then lim_{n→∞}sup{Tz  T_{ n }z : z ∈ C} = 0. They proved that the sequence {T_{ n }} converges strongly to some point of C for all x ∈ C.
where 0 ≤ α_{ n } ≤ a < 1 for all n ∈ ∞ and {T_{ n }} is a sequence of nonexpansive mappings of C into itself such that ${\cap}_{n=1}^{\infty}F\left({T}_{n}\right)\ne \varnothing $. They proved that if {T_{ n }} satisfies some appropriate conditions, then {x_{ n }} converges strongly to ${P}_{{\cap}_{n=1}^{\infty}F\left({T}_{n}\right)}{x}_{0}$.
They proved that {x_{ n }} converges strongly to Π_{F(T)}x_{0}, where Π_{F(T)}is the generalized projection from C onto F(T). Plubtieng and Ungchittrakool [44] introduced and proved the processes for finding a common fixed point of a countable family of relatively nonexpansive mappings in a Banach space. They proved the strong convergence theorems for a common fixed point of a countable family of relatively nonexpansive mappings {T_{ n }} provided that {T_{ n }} satisfies the following condition:

if for each bounded subset D of C, there exists a continuous increasing and convex function h : ℝ^{+} → ℝ^{+}. such that h(0) = 0 and limk,_{l→∞}sup_{z∈D}h(T_{ k }z  T_{ l }z) = 0.
where ${T}_{{r}_{i,n}}^{{F}_{i}}$, i = 1, 2, 3, ..., m defined in Lemma 2.8. Then, they proved that under certain appropriate conditions imposed on {α_{ n }}, and {r_{n,i}}, the sequence {x_{ n }} converges strongly to ${\mathrm{\Pi}}_{{C}_{n+1}}{x}_{0}$.
They obtained a strong convergence theorem for finding an element in the fixed point set of T. The results of Li et al. [26] extended and improved on the results of Matsushita and Takahashi [35].
where T_{ n } = T_{ n }(mod N ). He proved that the sequence {x_{ n }} converges strongly to ${\mathrm{\Pi}}_{{C}_{n+1}}{x}_{0}$ under certain appropriate conditions.
It follows directly from the definitions above that if T_{ n } ≡ T and T is closed, then {T_{ n }} satisfies (*)condition [46]. Next, we give the following example:
for all n ≥ 0 and for each x ∈ ℝ. Hence, ${\bigcap}_{n=1}^{\infty}F\left({T}_{n}\right)=F\left({T}_{n}\right)=\left\{0\right\}$ and ϕ(0, T_{ n }x) = 0  T_{ n }x ≤ 0  x = ϕ(0, x), ∀x ∈ ℝ. Then, T is a relatively quasinonexpansive mapping but not a relatively nonexpansive mapping. Moreover, for each bounded sequence z_{ n } ∈ E, we observe that ${T}_{n}{z}_{n}=\frac{1}{n}\to 0$ as n → ∞, and hence z = lim_{n→∞}z_{ n } = lim_{n→∞}T_{ n }z_{ n } = 0 as n → ∞; this implies that z = 0 ∈ F(T_{ n }). Therefore, T_{ n } is a relatively quasinonexpansive mapping and satisfies the (*)condition.
In 2010, Shehu [47] introduced a new iterative scheme by hybrid methods and proved strong convergence theorem for approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings which is also a solution to a system of generalized mixed equilibrium problems in a uniformly convex real Banach space which is also uniformly smooth using the properties of generalized fprojection operator.
The following questions naturally arise in connection with the above results using the (*)condition:
Question 1: Can the Mann algorithms (1.20) of [45] still be valid for an infinite family of relatively quasinonexpansive mappings?
Question 2: Can an iterative scheme (1.19) to solve a system of generalized mixed equilibrium problems?
Question 3: Can the Mann algorithms (1.20) be extended to more generalized fprojection operator?
The purpose of this paper is to solve the above questions. We introduce a new hybrid iterative scheme of the generalized fprojection operator for finding a common element of the fixed point set for a countable family of relatively quasinonexpansive mappings and the set of solutions of the system of generalized mixed equilibrium problem in a uniformly convex and uniformly smooth Banach space by using the (*)condition. Furthermore, we show that our new iterative scheme converges strongly to a common element of the aforementioned sets. Our results extend and improve the recent result of Li et al. [26], Matsushita and Takahashi [35], Takahashi et al. [43], Nakajo and Takahashi [41] and Shehu [45] and others.
2 Preliminaries
A Banach space E is said to be strictly convex if $\left\right\frac{x+y}{2}\left\right\phantom{\rule{2.77695pt}{0ex}}<1$ for all x, y ∈ E with x = y = 1 and x ≠ y. Let U = {x ∈ E : x = 1} be the unit sphere of E. Then a Banach space E is said to be smooth if the limit $\underset{t\to 0}{lim}\frac{\left\rightx+ty\left\right\left\rightx\left\right}{t}$ exists for each x, y ∈ U. It is also said to be uniformly smooth if the limit exists uniformly in x, y ∈ U. Let E be a Banach space. The modulus of smoothness of E is the function ρ_{ E } : [0, ∞] → [0, ∞] defined by ${\rho}_{E}\left(t\right)=sup\left\{\frac{\left\rightx+y\left\right+\left\rightxy\left\right}{2}1:\phantom{\rule{2.77695pt}{0ex}}\left\rightx\left\right\phantom{\rule{2.77695pt}{0ex}}=1,\left\righty\left\right\le t\right\}$. The modulus of convexity of E is the function δ_{ E } : [0, 2] → [0, 1] defined by ${\delta}_{E}\left(\epsilon \right)=inf\left\{1\left\right\frac{x+y}{2}\left\right\phantom{\rule{2.77695pt}{0ex}}:x,y\in E,\left\rightx\left\right\phantom{\rule{2.77695pt}{0ex}}=\phantom{\rule{2.77695pt}{0ex}}\left\righty\left\right\phantom{\rule{2.77695pt}{0ex}}=1,\left\rightxy\left\right\phantom{\rule{2.77695pt}{0ex}}\ge \epsilon \right\}.$ The normalized duality mapping $J:E\to {2}^{E*}$ is defined by J(x) = {x* ∈ E* : 〈x, x*〉 = x^{2}, x* = x}. If E is a Hilbert space, then J = I, where I is the identity mapping.
It is also known that if E is uniformly smooth, then J is uniformly normtonorm continuous on each bounded subset of E.
Remark 2.1. If E is a reflexive, strictly convex and smooth Banach space, then for x, y ∈ E, ϕ(x, y) = 0 if and only if x = y. It is sufficient to show that if ϕ(x, y) = 0 then x = y. From (1.8), we have x = y. This implies that 〈x, Jy〉 = x^{2} = Jy^{2}. From the definition of J, one has Jx = Jy. Therefore, we have x = y; see [19, 21] for more details.
We also need the following lemmas for the proof of our main results:
Lemma 2.2. [20]Let E be a uniformly convex and smooth Banach space and let {x_{ n }} and {y_{ n }} be two sequences of E. If ϕ(x_{ n }, y_{ n }) → 0 and either {x_{ n }} or {y_{ n }} is bounded, then x_{ n }  y_{ n } → 0.
 1.
${\mathrm{\Pi}}_{C}^{f}x$is nonempty closed convex subset of C for all x ∈ E;
 2.for all x ∈ E, $\widehat{x}\in {\mathrm{\Pi}}_{C}^{f}x$ if and only if$\u3008\widehat{x}y,JxJ\widehat{x}\u3009+\rho f\left(y\right)\rho f\left(\widehat{x}\right)\ge 0,\phantom{\rule{1em}{0ex}}\forall y\in C;$
 3.
if E is strictly convex, then ${\mathrm{\Pi}}_{C}^{f}$is a singlevalued mapping.
Remark 2.6. Let E be a uniformly convex and uniformly smooth Banach space and f(x) = 0 for all x ∈ E; then Lemma 2.5 reduces to the property of the generalized projection operator considered by Alber [17].
Lemma 2.7. [4]Let E be a real uniformly smooth and strictly convex Banach space, and C be a nonempty closed convex subset of E. Let T : C → C be a closed and relatively quasinonexpansive mapping. Then F(T) is a closed and convex subset of C.
For solving the equilibrium problem for a bifunction θ : C × C → ℝ, let us assume that θ satisfies the following conditions:
(A1) θ(x, x) = 0 for all x ∈ C;
(A2) θ is monotone, i.e., θ(x, y) + θ(y, x) ≤ 0 for all x, y ∈ C;
(A4) for each x ∈ C, y ↦ θ(x, y) is convex and lower semicontinuous.
Then, θ satisfies (A1)(A4). The following result is in Blum and Oettli [1].
Motivated by Combettes and Hirstoaga [3] in a Hilbert space and Takahashi and Zembayashi [12] in a Banach space, Zhang [16] obtain the following lemma:
 (1)
${T}_{r}^{F}$ is singlevalued;
 (2)
${T}_{r}^{F}$is firmly nonexpansive, i.e., for all x, y ∈ E, $\u3008{T}_{r}^{F}x{T}_{r}^{F}y,J{T}_{r}^{F}xJ{T}_{r}^{F}y\u3009\le \u3008{T}_{r}^{F}x{T}_{r}^{F}y,JxJy\u3009$;
 (3)
$F\left({T}_{r}^{F}\right)=\hat{F}\left({T}_{r}^{F}\right)=\mathsf{\text{GMEP}}\left(\theta ,A,\phi \right)$;
 (4)
GMEP(θ, A, φ) is closed and convex;
 (5)
$\varphi \left(p,{T}_{r}^{F}z\right)+\varphi \left({T}_{r}^{F}z,z\right)\le \varphi \left(p,z\right),$, $\forall p\in F\left({T}_{r}^{F}\right)$and z ∈ E.
3 Main results
In this section, by using the (*)condition, we prove the new convergence theorems for finding a common fixed points of a countable family of relatively quasinonexpansive mappings, in a uniformly convex and uniformly smooth Banach space.
where J is the duality mapping on E, {α_{ n }} is a sequence in [0, 1] and ${\left\{{r}_{j,n}\right\}}_{n=1}^{\infty}\subset \left[d,\infty \right)$for some d > 0 (j = 1, 2, ..., m). If lim inf_{n→∞}(1  α_{ n }) > 0, then {x_{ n }} converges strongly to $p\in \ud509$, where $p={\mathrm{\Pi}}_{\ud509}^{f}{x}_{0}$.
Proof We split the proof into five steps.
Step 1: We first show that C_{ n } is closed and convex for each n ∈ ℕ.
So, C_{n+1}is closed and convex. This implies that ${\mathrm{\Pi}}_{{C}_{n+1}}^{f}{x}_{0}$ is well defined.
Step 2 : We show that $\ud509\subset {C}_{n}$ for all n ∈ ℕ.
This shows that q ∈ C_{n+1}which implies that $\ud509\subset {C}_{n+1}$ and hence, $\ud509\subset {C}_{n}$ for all n ∈ ℕ.
Step 3 : We show that {x_{ n }} is a Cauchy sequence in C and lim_{n→∞}G(x_{ n }, Jx_{0}) exist.
Taking m, n → ∞, we have ϕ(x_{ m }, x_{ n }) → 0. From Lemma 2.2, we get x_{ n }  x_{ m } → 0. Hence, {x_{ n }} is a Cauchy sequence and by the completeness of E and the closedness of C, we can assume that there exists p ∈ C such that x_{ n } → p ∈ C as n → ∞.
 (a)We show that $p\in {\cap}_{n=1}^{\infty}F\left({T}_{n}\right)$. Since ϕ(x_{ m }, x_{ n }) → 0 as m, n → ∞, we obtain in particular that ϕ(x_{n+1}, x_{ n }) → 0 as n → ∞. By Lemma 2.2, we have$\underset{n\to \infty}{lim}\left\right{x}_{n+1}{x}_{n}\left\right\phantom{\rule{2.77695pt}{0ex}}=0.$(3.5)
 (b)
We show that $p\in {\cap}_{j=1}^{m}\mathsf{\text{GMEP}}\left({\theta}_{j},{A}_{j},{\phi}_{j}\right)$.
From the condition (A3) and letting t → 0, This implies that p ∈ GMEP(θ_{ j }, A_{ j }, φ_{ j }) for all j = 1, 2, 3, ..., m. Therefore, $p\in {\cap}_{j=1}^{m}\mathsf{\text{G}}MEP\left({\theta}_{j},{A}_{j},{\phi}_{j}\right)$. Hence, from (a) and (b), we obtain $p\in \ud509$.
From definition of ${\mathrm{\Pi}}_{\ud509}^{f}{x}_{0}$ and $p\in \ud509$, we can conclude that $v=p={\mathrm{\Pi}}_{\ud509}^{f}{x}_{0}$ and x_{ n } → p as n → ∞. This completes the proof. □
Setting T_{ n } ≡ T in Theorem 3.1, then we obtain the following result:
where J is the duality mapping on E, {α_{ n }} is a sequence in [0, 1] and ${\left\{{r}_{j,n}\right\}}_{n=1}^{\infty}\subset \left[d,\infty \right)$for some d > 0 (j = 1, 2, ..., m). If lim inf_{n→∞}(1  α_{ n }) > 0, then {x_{ n }} converges strongly to $p\in \ud509$, where $p={\mathrm{\Pi}}_{\ud509}^{f}{x}_{0}$.
Remark 3.3. Corollary 3.2 extends and improves the result of Li et al. [26].
Taking f(x) = 0 for all x ∈ E, we have G(ξ, Jx) = ϕ(ξ, x) and ${\mathrm{\Pi}}_{C}^{f}x={\mathrm{\Pi}}_{C}x$. By Theorem 3.1, then we obtain the following Corollaries:
where J is the duality mapping on E, {α_{ n }} is a sequence in [0, 1] and ${\left\{{r}_{j,n}\right\}}_{n=1}^{\infty}\subset \left[d,\infty \right)$for some d > 0 (j = 1, 2, ..., m). If lim inf_{n→∞}(1  α_{ n }) > 0, then {x_{ n }} converges strongly to $p\in \ud509$, where $p={\mathrm{\Pi}}_{\ud509}{x}_{0}$.
Remark 3.5. Corollary 3.4 extends and improves the result of Shehu [[45], Theorem 3.1] form finite family of relatively quasinonexpansive mappings to a countable family of relatively quasinonexpansive mappings.
4 Applications
4.1 A zero of $\mathcal{B}$monotone mappings
 1.
monotone if $\u3008\mathcal{B}x\mathcal{B}y,xy\u3009\ge 0$ for all x, y ∈ E;
 2.
strictly monotone if $\mathcal{B}$ monotone and $\u3008\mathcal{B}x\mathcal{B}y,xy\u3009=0$ if and only if x = y;
 3.
βLipschitz continuous if there exist a constant β ≥ 0 such that $\left\right\mathcal{B}x\mathcal{B}y\left\right\phantom{\rule{2.77695pt}{0ex}}\le \beta \left\rightxy\left\right$ for all x, y ∈ E.
 (i)
monotone if 〈x_{1}x_{2}, y_{1}y_{2}〉 ≥ 0 for each x_{ i } ∈ D(M) and y_{ i } ∈ Mx_{ i }, i = 1, 2;
 (ii)
rstrongly monotone if 〈x_{1}x_{2}, y_{1}y_{2}〉 ≥ rx_{1}x_{2} for each x_{ i } ∈ D(M) and y_{ i } ∈ Mx_{ i }, i = 1, 2;
 (iii)
maximal monotone if M is monotone and its graph $\mathfrak{G}(M)=\{(x,y):y\in Mx\}$ is not properly contained in the graph of any other monotone mapping;
 (iv)
general $\mathcal{B}$monotone if M is monotone and $\left(\mathcal{B}+\lambda M\right)E=E*$ holds for every λ > 0, where $\mathcal{B}$ is a mapping from E to E*.
We consider the problem of finding a point x* ∈ E satisfying 0 ∈ Mx*. We denote by M^{ 1}0 the set of all points x* ∈ E such that 0 ∈ Mx*, where M is maximal monotone operator from E to E*.
Lemma 4.1. [26]Let E be a Banach space with the dual space E*, $\mathcal{B}:E\to E*$be a strictly monotone mapping, and M : E → 2^{ E }* be a general $\mathcal{B}$monotone mapping. Then M is maximal monotone mapping.
Remark 4.2. [26] Let E be a Banach space with the dual space E*, $\mathcal{B}:E\to E*$ be a strictly monotone mapping, and M : E → 2^{ E }* be a general $\mathcal{B}$monotone mapping. Then M is a maximal monotone mapping. Therefore, M^{ 1}0 = {z ∈ D(M) : 0 ∈ Mz} is closed and convex.
hold for all x and ξ in E, where $d=\sqrt{\left(\left\rightx{}^{2}+\left\xi \right{}^{2}\right)\u22152}$.
 1.
${\left(\mathcal{B}+\lambda M\right)}^{1}$ is single value;
 2.
if E is reflexive and M : E → 2^{ E }* a rstrongly monotone mapping, then ${\left(\mathcal{B}+\lambda M\right)}^{1}$is Lipschitz continuous with constant $\frac{1}{\lambda r}$, where r > 0.
Motivated by Li et al. [26] we obtain the following result:
where J is the duality mapping on E and {α_{ n }} is a sequence in [0, 1], and ${\left\{{r}_{j,n}\right\}}_{n=1}^{\infty}\subset \left[d,\infty \right)$for some d > 0 (j = 1, 2, ..., m). If lim inf_{n→∞}(1  α_{ n }) > 0, then {x_{ n }} converges strongly to $p\in \ud509$where $p={\mathrm{\Pi}}_{\ud509}^{f}{x}_{0}$.
it follows from (4.3) and (4.4) that $\varphi \left(p,{T}_{{\lambda}_{n}}q\right)\le \varphi \left(p,q\right)$ for all q ∈ E, $p\in F\left({T}_{{\lambda}_{n}}\right)$, n ≥ 1. Therefore, $\left\{{T}_{{\lambda}_{n}}\right\}$ is a family of relatively quasinonexpansive mapping. It follows from Theorem 3.1, so the desired conclusion follows. □
4.2 A zero point of maximal monotone operators
In this section, we apply our results to find zeros of maximal monotone operator. Such a problem contains numerous problems in optimization, economics, and physics. The following result is also well known.
Lemma 4.6. [50]Let E be a reflexive strictly convex and smooth Banach space and let M be a monotone operator from E to E*. Then M is maximal if and only if R(J + λM) = E* for all λ > 0.
Let E be a reflexive strictly convex and smooth Banach space, $\mathcal{B}=J$ and let M be a maximal monotone operator from E to E*. Using Lemma 4.6 and strict convexity of E, we obtain that for every λ > 0 and x ∈ E, there exists a unique x_{ λ } such that Jx ∈ (Jx_{ λ } + λMx_{ λ }). Then we can defined a singlevalued mapping J_{ λ } : E → D(M) by J_{ λ } = (J + λM)^{ 1}J and J_{ λ } is called the resolvent of M. We know that M^{ 1}0 = F(J_{ λ }) [21, 51].
where J is the duality mapping on E and {α_{ n }} is a sequence in [0, 1] and ${\left\{{r}_{j,n}\right\}}_{n=1}^{\infty}\subset \left[d,\infty \right)$for some d > 0 (j = 1, 2, ..., m). If lim inf_{n→∞}(1  α_{ n }) > 0, then {x_{ n }} converges strongly to $p\in \ud509$, where $p={\mathrm{\Pi}}_{\ud509}^{f}{x}_{0}$.
for all n ≥ 1. Therefore, $\left\{{J}_{{\lambda}_{n}}\right\}$ is a family of relatively quasinonexpansive mapping for all λ_{ n } > 0 with the common fixed point set ${\cap}_{n=1}^{\infty}F\left({J}_{{\lambda}_{n}}\right)={M}^{1}0$. Hence, it follows from Theorem 3.1, the desired conclusion follows: □
Declarations
Acknowledgements
The authors are greatly indebted to Professor Simeon Reich and the reviewers for their extremely constructive comments and valuable suggestions leading to the revised version. Ms. Siwaporn Saewan was supported by grant from under the program Strategic Scholarships for Frontier Research Network for the Join Ph.D. Program Thai Doctoral degree from the Office of the Higher Education Commission, Thailand and the King Mongkuts Diamond scholarship for the Ph.D. program at King Mongkuts University of Technology Thonburi (KMUTT; under NRUCSEC project no.54000267). Furthermore, this research was supported by the Center of Excellence in Mathematics, the Commission on Higher Education, Thailand (under the project no.RG153032).
Authors’ Affiliations
References
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