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Hybrid iterative algorithm for finite families of countable Bregman quasiLipschitz mappings with applications in Banach spaces
Journal of Inequalities and Applications volume 2015, Article number: 210 (2015)
Abstract
The purpose of this paper is to introduce and consider a new hybrid shrinking projection method for finding a common element of the set EP of solutions of a generalized equilibrium problem, the common fixed point set F of finite uniformly closed families of countable Bregman quasiLipschitz mappings in reflexive Banach spaces. It is proved that under appropriate conditions, the sequence generated by the hybrid shrinking projection method converges strongly to some point in \(\mathit{EP} \cap F\). Relative examples are given. Strong convergence theorems are proved. The application for Bregman asymptotically quasinonexpansive mappings is also given. The main innovative points in this paper are as follows: (1) the notion of the uniformly closed family of countable Bregman quasiLipschitz mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable Bregman quasiLipschitz mappings are given in classical Banach spaces \(l^{2}\) and \(L^{2}\); (3) the application for Bregman asymptotically quasinonexpansive mappings is also given; (4) because the main theorems do not need the boundedness of the domain of mappings, so a corresponding technique for the proof is given. This new results improve and extend the previously known ones in the literature.
Introduction
Let C be a nonempty subset of a real Banach space and T be a mapping from C into itself. We denote by \(F(T)\) the set of fixed points of T. Recall that T is said to be asymptotically nonexpansive [1] if there exists a sequence \(\{k_{n}\}\subset [1,+\infty)\) with \(\lim_{n\rightarrow\infty}k_{n}=1\) such that
In the framework of Hilbert spaces, Takahashi et al. [2] have introduced a new hybrid iterative scheme called a shrinking projection method for nonexpansive mappings. It is an advantage of projection methods that the strong convergence of iterative sequences is guaranteed without any compact assumption. Moreover, Schu [3] has introduced a modified Mann iteration to approximate fixed points of asymptotically nonexpansive mappings in uniformly convex Banach spaces. Motivated by [2, 3], Inchan [4] has introduced a new hybrid iterative scheme by using the shrinking projection method with the modified Mann iteration for asymptotically nonexpansive mappings. The mapping T is said to be asymptotically nonexpansive in the intermediate sense (cf. [5]) if
If \(F(T)\) is nonempty and (1.1) holds for all \(x \in C\) and \(y \in F(T)\), then T is said to be asymptotically quasinonexpansive in the intermediate sense. It is worth mentioning that the class of asymptotically nonexpansive mappings in the intermediate sense contains properly the class of asymptotically nonexpansive mappings since the mappings in the intermediate sense are not Lipschitz continuous in general.
Recently, many authors have studied further new hybrid iterative schemes in the framework of real Banach spaces; for instance, see [6–8]. Qin and Wang [9] have introduced a new class of mappings which are asymptotically quasinonexpansive with respect to the Lyapunov functional (cf. [10]) in the intermediate sense. By using the shrinking projection method, Hao [11] has proved a strong convergence theorem for an asymptotically quasinonexpansive mapping with respect to the Lyapunov functional in the intermediate sense.
In 1967, Bregman [12] discovered an elegant and effective technique for using of the socalled Bregman distance function (see Section 2) in the process of designing and analyzing feasibility and optimization algorithms. This opened a growing area of research in which Bregman’s technique is applied in various ways in order to design and analyze not only iterative algorithms for solving feasibility and optimization problems, but also algorithms for solving variational inequalities, for approximating equilibria, and for computing fixed points of nonlinear mappings.
Many authors have studied iterative methods for approximating fixed points of mappings of nonexpansive type with respect to the Bregman distance; see [13–17]. In [18], the authors has introduced a new class of nonlinear mappings which is an extension of asymptotically quasinonexpansive mappings with respect to the Bregman distance in the intermediate sense and has proved the strong convergence theorems for asymptotically quasinonexpansive mappings with respect to Bregman distances in the intermediate sense by using the shrinking projection method.
The purpose of this paper is to introduce and consider a new hybrid shrinking projection method for finding a common element of the set EP of solutions of a generalized equilibrium problem, the common fixed point set F of finite uniformly closed families of countable Bregman quasiLipschitz mappings in reflexive Banach spaces. It is proved that under appropriate conditions, the sequence generated by the hybrid shrinking projection method converges strongly to some point in \(\mathit{EP} \cap F\). Relative examples are given. Strong convergence theorems are proved. The application for Bregman asymptotically quasinonexpansive mappings is also given. The main innovative points in this paper are as follows: (1) the notion of the uniformly closed family of countable Bregman quasiLipschitz mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable Bregman quasiLipschitz mappings are given in classical Banach spaces \(l^{2}\) and \(L^{2}\); (3) the application for Bregman asymptotically quasinonexpansive mappings is also given; (4) because the main theorems do not need the boundedness of the domain of mappings, so a corresponding technique for the proof is given. This new results improve and extend the previously known ones in the literature.
Preliminaries
Throughout this paper, we assume that E is a real reflexive Banach space with the dual space of \(E^{*}\) and \(\langle\cdot,\cdot\rangle\) the pairing between E and \(E^{*}\).
Let \(f: E\rightarrow(\infty, +\infty]\) be a function. The effective domain of f is defined by
When \(\operatorname{dom} f\neq\emptyset\), we say that f is proper. We denote by \(\operatorname{int}\operatorname{dom} f\) the interior of the effective domain of f. We denote by ranf the range of f.
The function f is said to be strongly coercive if
Given a proper and convex function \(f :E \rightarrow (\infty,+\infty]\), the subdifferential of f is a mapping \(\partial f: E\rightarrow E^{*}\) defined by
for all \(x \in E\).
The Fenchel conjugate function of f is the convex function \(f^{*}: E \rightarrow(\infty,+\infty)\) defined by
We know that \(x^{*} \in\partial f(x)\) if and only if
for all \(x \in E\) (see [18]).
Proposition 2.1
([19])
Let \(f :E \rightarrow(\infty,+\infty]\) be a proper, convex, and lower semicontinuous function. Then the following conditions are equivalent:

(i)
\(\operatorname{ran} \partial f =E^{*}\) and \(\partial f^{*} =(\partial f)^{1}\) is bounded on bounded subsets of \(E^{*}\);

(ii)
f is strongly coercive.
Let \(f : E\rightarrow(\infty, +\infty]\) be a convex function and \(x\in \operatorname{int}\operatorname{dom} f\). For any \(y \in E\), we define the righthand derivative of f at x in the direction y by
The function f is said to be Gâteaux differentiable at x if the limit (2.1) exists for any y. In this case, the gradient of f at x is the function \(\nabla f(x) : E\rightarrow E^{*}\) defined by \(\langle\nabla f(x),y \rangle= f^{\circ}(x, y)\) for all \(y \in E\). The function f is said to be Gâteaux differentiable if it is Gâteaux differentiable at each \(x \in \operatorname{int}\operatorname{dom} f\). If the limit (2.1) is attained uniformly in \(\y\=1\), then the function f is said to be Fréchet differentiable at x. The function f is said to be uniformly Fréchet differentiable on a subset C of E if the limit (2.1) is attained uniformly for \(x \in C\) and \(\y\ = 1\). We know that if f is uniformly Fréchet differentiable on bounded subsets of E, then f is uniformly continuous on bounded subsets of E (cf. [19]). We will need the following results.
Proposition 2.2
([20])
If a function \(f : E\rightarrow R\) is convex, uniformly Fréchet differentiable, and bounded on bounded subsets of E, then ∇f is uniformly continuous on bounded subsets of E from the strong topology of E to the strong topology of \(E^{*}\).
Proposition 2.3
([20])
Let \(f : E\rightarrow R\) be a convex function which is bounded on bounded subsets of E. Then the following assertions are equivalent:

(i)
f is strongly coercive and uniformly convex on bounded subsets of E;

(ii)
\(f^{*}\) is Fréchet differentiable and \(\nabla f^{*}\) is uniformly normtonorm continuous on bounded subsets of \(\operatorname{dom} f^{*} = E^{*}\).
A function \(f :E\rightarrow(\infty,+\infty]\) is said to be admissible if it is proper, convex, and lower semicontinuous on E and Gâteaux differentiable on \(\operatorname{int}\operatorname{dom} f\). Under these conditions we know that f is continuous in \(\operatorname{int}\operatorname{dom} f\), ∂f is singlevalued and \(\partial f =\nabla f\); see [17, 21]. An admissible function \(f :E\rightarrow (\infty,+\infty]\) is called Legendre (cf. [17]) if it satisfies the following two conditions:

(L1)
the interior of the domain of f, \(\operatorname{int}\operatorname{dom} f\), is nonempty, f is Gâteaux differentiable, and \(\operatorname{dom} \nabla f = \operatorname{int}\operatorname{dom} f\);

(L2)
the interior of the domain of \(f^{*}\), \(\operatorname{int}\operatorname{dom} f^{*}\) is nonempty, \(f^{*}\) is Gâteaux differentiable, and \(\operatorname{dom} \nabla f^{*} = \operatorname{int}\operatorname{dom} f^{*}\).
Let f be a Legendre function on E. Since E is reflexive, we always have \(\nabla f = (\nabla f^{*})^{1}\). This fact, when combined with conditions (L1) and (L2), implies the following equalities:
Conditions (L1) and (L2) imply that the functions f and \(f^{*}\) are strictly convex on the interior of their respective domains. In [22], author gave an example of the Legendre function.
Let \(f : E \rightarrow(\infty, +\infty]\) be a convex function on E which is Gâteaux differentiable on \(\operatorname{int}\operatorname{dom} f\). The bifunction \(D_{f}: \operatorname{dom} f\times \operatorname{int}\operatorname{dom} f\rightarrow[0,+\infty)\) given by
is called the Bregman distance with respect to f (cf. [23]). In general, the Bregman distance is not a metric since it is not symmetric and does not satisfy the triangle inequality. However, it has the following important property, which is called the three point identity (cf. [24]): for any \(x \in\operatorname{dom} f\) and \(y, z \in \operatorname{int}\operatorname{dom} f\),
With a Legendre function \(f : E \rightarrow(\infty, +\infty]\), we associate the bifunction \(W_{f} : \operatorname{dom} f^{*}\times \operatorname{dom} f \rightarrow[0, +\infty)\) defined by
Proposition 2.4
([14])
Let \(f : E \rightarrow (\infty, +\infty]\) be a Legendre function such that \(\nabla f^{*}\) is bounded on bounded subsets of \(\operatorname{int}\operatorname{dom} f^{*}\). Let \(x\in \operatorname{int}\operatorname{dom} f\). If the sequence \(\{D_{f}(x,x_{n})\}\) is bounded, then the sequence \(\{x_{n}\}\) is also bounded.
Proposition 2.5
([14])
Let \(f : E \rightarrow (\infty, +\infty]\) be a Legendre function. Then the following statements hold:

(i)
the function \(W^{f}(\cdot,x)\) is convex for all \(x\in \operatorname{dom} f\);

(ii)
\(W^{f}(\nabla f(x),y)=D_{f}(y,x)\) for all \(x\in \operatorname{int}\operatorname{dom} f\) and \(y\in \operatorname{dom} f\).
Let \(f : E \rightarrow(\infty, +\infty]\) be a convex function on E which is Gâteaux differentiable on \(\operatorname{int}\operatorname{dom} f\). The function f is said to be totally convex at a point \(x \in \operatorname{int}\operatorname{dom} f\) if its modulus of total convexity at x, \(v_{f}(x,\cdot):[0,+\infty)\rightarrow[0,+\infty]\), defined by
is positive whenever \(t >0\). The function f is said to be totally convex when it is totally convex at every point of \(\operatorname{int}\operatorname{dom} f\). The function f is said to be totally convex on bounded sets if, for any nonempty bounded set \(B \subset E\), the modulus of total convexity of f on B, \(v_{f}(B, t)\) is positive for any \(t > 0\), where \(v_{f}(B,\cdot): [0,+\infty)\rightarrow[0,+\infty]\) is defined by
We remark in passing that f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets; see [25, 26].
Proposition 2.6
([25])
Let \(f : E \rightarrow(\infty, +\infty]\) be a convex function whose domain contains at least two points. If f is lower semicontinuous, then f is totally convex on bounded sets if and only if f is uniformly convex on bounded sets.
Proposition 2.7
([27])
Let \(f : E\rightarrow R\) be a totally convex function. If \(x \in E\) and the sequence \(\{D_{f}(x_{n},x)\}\) is bounded, then the sequence \(\{x_{n}\}\) is also bounded.
Let \(f: E\rightarrow[0,+\infty)\) be a convex function on E which is Gâteaux differentiable on \(\operatorname{int}\operatorname{dom} f\). The function f is said to be sequentially consistent (cf. [26]) if for any two sequences \(\{x_{n}\}\) and \(\{y_{n}\}\) in \(\operatorname{int}\operatorname{dom} f\) and domf, respectively, such that the first one is bounded,
Proposition 2.8
([28])
A function \(f : E\rightarrow[0,+\infty)\) is totally convex on bounded subsets of E if and only if it is sequentially consistent.
Let C be a nonempty, closed, and convex subset of E. Let \(f : E\rightarrow(\infty,+\infty]\) be a convex function on E which is Gâteaux differentiable on \(\operatorname{int}\operatorname{dom} f\). The Bregman projection \(\operatorname{proj}_{C}^{f}(x)\) with respect to f (cf. [28]) of \(x \in \operatorname{int}\operatorname{dom} f\) onto C is the minimizer over C of the functional \(D_{f} (\cdot, x): \rightarrow[0,+\infty]\), that is,
Let E be a Banach space with dual \(E^{*}\). We denote by J the normalized duality mapping from E to \(2^{E^{*}}\) defined by
where \(\langle\cdot,\cdot\rangle\) denotes the generalized duality pairing. It is well known that if E is smooth, then J is singlevalued.
Proposition 2.9
([29])
Let \(f : E\rightarrow R\) be an admissible, strongly coercive, and strictly convex function. Let C be a nonempty, closed, and convex subset of domf. Then \(\operatorname{proj}_{C}^{f}(x)\) exists uniquely for all \(x \in \operatorname{int}\operatorname{dom} f\).
Let \(f(x)=\frac{1}{2}\x\^{2}\).

(i)
If E is a Hilbert space, then the Bregman projection is reduced to the metric projection onto C.

(ii)
If E is a smooth Banach space, then the Bregman projection is reduced to the generalized projection \(\Pi_{C}(x)\) which is defined by
$$\Pi_{C}(x)=\operatorname{argmin} \bigl\{ \phi(y,x): y\in C\bigr\} , $$where ϕ is the Lyapunov functional (cf. [10]) defined by
$$\phi(y,x)=\y\^{2}2\langle y, Jx\rangle+\x\^{2} $$for all \(y,x \in E\).
Proposition 2.10
([26])
Let \(f : E\rightarrow (\infty, +\infty]\) be a totally convex function. Let C be a nonempty, closed, and convex subset of \(\operatorname{int}\operatorname{dom} f\) and \(x\in \operatorname{int}\operatorname{dom} f\). If \(x^{*} \in C\), then the following statements are equivalent:

(i)
The vector \(x^{*}\) is the Bregman projection of x onto C.

(ii)
The vector \(x^{*}\) is the unique solution z of the variational inequality
$$\bigl\langle zy, \nabla f(x)\nabla f(z)\bigr\rangle \geq0, \quad \forall y \in C. $$ 
(iii)
The vector \(x^{*}\) is the unique solution z of the inequality
$$D_{f}(y,z)+D_{f}(z,x)\leq D_{f}(y,x), \quad \forall y \in C. $$
In this paper, we present the following definition.
Definition 2.11
Let C be a nonempty, closed, and convex subset of E and \(f : E\rightarrow(\infty, +\infty]\) be an admissible function. Let T be a mapping from C into itself with a nonempty fixed point set \(F(T)\). The mapping T is said to be Bregman quasiLipschitz if there exists a constant \(L\geq1\) such that
The mapping T is said to be Bregman quasinonexpansive if
Bregman quasiLipschitz mappings are a more generalized class than the class of Bregman quasimappings. On the other hand, this class also contains the relatively quasiLipschitz mappings and quasiLipschitz mappings. Therefore, Bregman quasiLipschitz mappings are very important in the nonlinear analysis and fixed point theory and applications.
Definition 2.12
Let C be a nonempty, closed, and convex subset of E. Let \(\{T_{n}\}\) be sequence of mappings from C into itself with a nonempty common fixed point set \(F=\bigcap_{n=1}^{\infty}F(T_{n})\). \(\{T_{n}\}\) is said to be uniformly closed if for any convergent sequence \(\{z_{n}\} \subset C\) such that \(\T_{n}z_{n}z_{n}\\rightarrow0\) as \(n\rightarrow\infty\), the limit of \(\{z_{n}\}\) belongs to F.
In Section 4, we will give two examples of a uniformly closed family of countable Bregman quasiLipschitz mappings.
Let E be a real Banach space with the dual \(E^{*}\) and C be a nonempty closed convex subset of E. Let \(A:C\rightarrow E^{*}\) be a nonlinear mapping and \(F:C\times C\rightarrow R\) be a bifunction. Then consider the following generalized equilibrium problem of finding \(u \in C\) such that:
The set of solutions of (2.3) is denoted by EP, i.e.,
Whenever \(E=H\) a Hilbert space, problem (2.3) was introduced and studied by Takahashi and Takahashi [30].
Whenever \(A\equiv0\), problem (2.3) is equivalent to finding \(u\in C\) such that
which is called the equilibrium problem. The set of its solutions is denoted by \(\mathit{EP}(F )\).
Whenever \(F\equiv0\), problem (1.1) is equivalent to finding \(u\in C\) such that
which is called the variational inequality of Browder type. The set of its solutions is denoted by \(\mathit{VI}(C, A)\).
Problem (2.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games and others; see, e.g., [31, 32].
In order to solve the equilibrium problem, let us assume that \(F : C \times C \rightarrow(\infty,+\infty)\) satisfies the following conditions [33]:

(A1)
\(F(x,x)=0\) for all \(x\in C\),

(A2)
F is monotone, i.e., \(F(x,y)+F(y,x)\leq0\), for all \(x,y\in C\),

(A3)
for all \(x,y,z\in C\), \(\limsup_{t\downarrow0}F(tz+(1t)x,y)\leq F(x,y)\),

(A4)
for all \(x\in C\), \(F(x,\cdot)\) is convex and lower semicontinuous.
For \(r>0\), we define a mapping \(K_{r} : E \rightarrow C\) as follows:
for all \(x \in E\). The following two lemmas were proved in [14].
Lemma 2.13
Let E be a reflexive Banach space and let \(f : E \rightarrow R\) be a Legendre function. Let C be a nonempty, closed, and convex subset of E and let \(F : C \times C \rightarrow R\) be a bifunction satisfying (A1)(A4). For \(r> 0\), let \(T_{r} : E \rightarrow C\) be the mapping defined by (2.5). Then \(\operatorname{dom} T_{r} =E\).
Lemma 2.14
Let E be a reflexive Banach space and let \(f : E \rightarrow R\) be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E and let \(F : C \times C \rightarrow R\) be a bifunction satisfying (A1)(A4). For \(r > 0\), let \(T_{r} : E \rightarrow C\) be the mapping defined by (2.5). Then the following statements hold:

(i)
\(T_{r}\) is singlevalued.

(ii)
\(T_{r}\) is a firmly nonexpansivetype mapping, i.e., for all \(x,y\in E\),
$$\bigl\langle T_{r}xT_{r}y, \nabla f(T_{r}x) \nabla f(T_{r}y) \bigr\rangle \leq\bigl\langle T_{r}xT_{r}y, \nabla f(x)\nabla f(y) \bigr\rangle . $$ 
(iii)
\(F(T_{r})=\hat{F}(T_{r})=\mathit{EP}(F)\).

(iv)
\(\mathit{EP}(F)\) is closed and convex.

(v)
\(D_{f}(p, T_{r}x)+D_{f}(T_{r}x,x)\leq D_{f}(p,x)\), \(\forall p\in \mathit{EP}(F)\), \(\forall x \in E\).
Lemma 2.15
Let E be a reflexive Banach space and let \(f : E \rightarrow R\) be a convex, continuous, and strongly coercive function which is bounded on bounded subsets and uniformly convex on bounded subsets of E. Let C be a nonempty, closed, and convex subset of E and let \(F : C \times C \rightarrow R\) be a bifunction satisfying (A1)(A4). Let \(A: C\rightarrow E^{*}\) be a monotone mapping, i.e.,
For \(r > 0\), let \(K_{r} : E \rightarrow C\) be the mapping defined by
Then the following statements hold:

(i)
\(K_{r}\) is singlevalued.

(ii)
\(K_{r}\) is a firmly nonexpansivetype mapping, i.e., for all \(x,y\in E\),
$$\bigl\langle K_{r}xK_{r}y, \nabla f(K_{r}x) \nabla f(K_{r}y) \bigr\rangle \leq\bigl\langle K_{r}xK_{r}y, \nabla f(x)\nabla f(y) \bigr\rangle . $$ 
(iii)
\(F(K_{r})=\hat{F}(K_{r})=\mathit{EP}\).

(iv)
EP is closed and convex.

(v)
\(D_{f}(p, K_{r}x)+D_{f}(K_{r}x,x)\leq D_{f}(p,x)\), \(\forall p\in \mathit{EP}(F)\), \(\forall x \in E\).
Proof
Let
It is easy to show that, \(G(x,y)\) satisfies conditions (A1)(A4). Replacing \(F(x,y)\) by \(G(x,y)\) in Lemma 2.14, we can get the conclusions. □
Main results
Theorem 3.1
Let \(f : E \rightarrow (\infty,+\infty]\) be a Legendre function which is totally convex on bounded subsets of E. Suppose that \(\nabla f^{*}\) is bounded on bounded subsets of \(\operatorname{int}\operatorname{dom} f^{*}\). Let C be a nonempty, closed, and convex subset of \(\operatorname{int}\operatorname{dom} f\). Let \(\{T_{n}\} : C \rightarrow C\) be a uniformly closed family of countable Bregman quasiLipschitz mappings with the condition \(\lim_{n\rightarrow\infty}L_{n}=1\), where
Let F be a common fixed point set of \(\{T_{n}\}\). Then F is closed and convex.
Proof
Firstly, we prove that F is closed. Let \(\{p_{n}\}\subset F\), \(p_{n}\rightarrow p\) as \(n\rightarrow\infty\), then \(\T_{n}p_{n}p_{n}\=0\rightarrow0\) as \(n\rightarrow\infty\). Since \(\{T_{n}\}\) is uniformly closed, we know that \(p\in F\). Hence F is closed.
Next we prove that F is convex. Let \(p_{1},p_{2} \in F\), \(p=tp_{1}+(1t)p_{2}\), where \(t \in(0,1)\). We prove that \(p \in F\). By the three point identity (2.2), we know that
This implies
for \(i=1,2\). Combining (3.1) and (3.2) yields
for \(i =1,2\). Multiplying t and \((1t)\) on both sides of (3.3) with \(i = 1\) and \(i = 2\), respectively, yields
where
This implies that \(\{D_{f}(p, T_{n}p)\}\) is bounded. By Propositions 2.4 and 2.8, we see that the sequence \(\{T_{n}p_{n}\}\) is bounded and \(\pT_{n}p\\rightarrow0\) as \(n\rightarrow\infty\). Since \(p\rightarrow p\), and \(\{T_{n}\}\) is uniformly closed, then \(p\in F\). Therefore F is convex. This completes the proof. □
Next we will prove the main strong convergence theorem for the finite families of countable Bregman quasiLipschitz mappings by using a new hybrid projection scheme. In this scheme, we will use some detailed technology.
Theorem 3.2
Let \(f:E\rightarrow(\infty, +\infty]\) be a Legendre function which is bounded, strongly coercive, uniformly Fréchet differentiable and totally convex on bounded subsets on E. Let C be a nonempty, closed, and convex subset of \(\operatorname{int} \operatorname{dom} f\). Let \(\{T^{(i)}_{n}\}_{n=1}^{\infty} : C \rightarrow C\) be N uniformly closed families of countable Bregman quasiLipschitz mappings with the condition \(\lim_{n\rightarrow\infty}L^{(i)}_{n}=1\) for \(i=1,2,3, \ldots,N\). Let \(F=\bigcap_{n=1}^{\infty}\bigcap_{i=1}^{N} F(T^{(i)}_{n})\) and \(F\cap \mathit{EP}\) be nonempty. Let \(\{x_{n}\}\) be a sequence of C generated by
where
and \(\{\alpha_{n}\}\) is a sequence satisfying \(\limsup_{n\rightarrow\infty} \alpha_{n} <1\). Then \(\{x_{n}\}\) converges to \(q= P_{F\cap \mathit{EP}}^{f}x_{0}\).
Proof
We divide the proof into six steps.
Step 1. We show that \(C_{n}\) is closed and convex for all \(n\geq 1\). It is obvious that \(C_{i,1} = C\) is closed and convex. Suppose that \(C_{i,k}\) is closed and convex for some \(k\geq1\). We see for each \(i=1,2,3, \ldots,N\) that
and
is equivalent to
Therefore
It is easy to see that if \(z_{1}\), \(z_{2}\) satisfy (3.4), the element \(z=tz_{1}+(1t)z_{2}\) satisfies also (3.4) for all \(t \in(0,1)\), so that the set
is convex and closed, and hence \(C_{i,k+1}\) is closed and convex for all \(n\geq1\). Therefore \(C_{n+1}=\bigcap_{i=1}^{N}C_{i,n+1}\) is closed and convex.
Step 2. We show that \(F\cap \mathit{EP}\cap B(P_{F\cap \mathit{EP}}^{f}x_{0}, 1) \subset C_{n}\) for all \(n\geq1\). It is obvious that \(F\cap \mathit{EP} \cap B(P_{F\cap \mathit{EP}}^{f}x_{0}, 1) \subset C_{i,1}=C\) for all \(1\leq i \leq N\). Suppose that \(F\cap \mathit{EP} \cap B(P_{F\cap \mathit{EP}}^{f}x_{0}, 1) \subset C_{n}\) for some \(n\geq1\). Let \(p\in F \cap \mathit{EP} \cap B(P_{F\cap \mathit{EP}}^{f}x_{0}, 1)\). By Proposition 2.7, we have
On the other hand, by Lemma 2.15, we have \(p=K_{r}(p)\) and
that is,
Combining (3.5) and (3.6) we know that \(p \in C_{i,n+1}\) for all \(1\leq i \leq N\), which implies that \(F\cap \mathit{EP}\cap B(P_{F\cap \mathit{EP}}^{f}x_{0}, 1) \subset C_{i,n+1}\). Therefore \(F\cap \mathit{EP}\cap B(P_{F\cap \mathit{EP}}^{f}x_{0}, 1) \subset C_{n+1}\). By induction we know that \(F\cap \mathit{EP}\cap B(P_{F\cap \mathit{EP}}^{f}x_{0}, 1) \subset C_{n}\) for all \(n\geq1\).
Step 3. We show that \(\{x_{n}\}\) converges to a point \(p \in C\).
Since \(x_{n}=P^{f}_{C_{n}}x_{0}\) and \(C_{n+1}\subset C_{n}\), then we get
Therefore \(\{D_{f}(x_{n},x_{0})\}\) is nondecreasing. On the other hand, by Proposition 2.10, we have
for all \(p\in F\subset C_{n}\) and for all \(n\geq1\). Therefore, \(D_{f}(x_{n},x_{0})\) is also bounded. This together with (3.7) implies that the limit of \(\{D_{f}(x_{n},x_{0})\}\) exists. Put
From Proposition 2.10, we have, for any positive integer m, that
for all \(n\geq1\). This together with (3.8) implies that
holds uniformly for all m. Therefore, we get that
holds uniformly for all m. Then \(\{x_{n}\}\) is a Cauchy sequence, therefore there exists a point \(p\in C\) such that \(x_{n}\rightarrow p\).
Step 4. We show that the limit of \(\{x_{n}\}\) belongs to F.
Since \(x_{n+1}\in C_{n+1}\), we have for all \(1\leq i \leq N\) that
as \(n\rightarrow\infty\). By Proposition 2.8, we obtain that
From
we get
which implies that
By Proposition 2.2, we have
so that
By Propositions 2.3 and 2.8, \(\nabla f^{*}\) is uniformly continuous on bounded subsets of E and thus
Since \(\{T^{(i)}_{n}\}\) is an asymptotically countable family of Bregman weak relatively nonexpansive mappings and \(x_{n}\rightarrow p\), so that \(p\in\bigcap_{n=1}^{\infty} F(T^{(i)}_{n})\) for each \(1\leq i \leq N\). Therefore \(p\in F=\bigcap_{n=1}^{\infty}\bigcap_{i=1}^{N} F(T^{(i)}_{n})\).
Step 5. We show that the limit of \(\{x_{n}\}\) belongs to EP.
We have proved that \(x_{n}\rightarrow p\) as \(n\rightarrow\infty\). Now let us show that \(p\in \mathit{EP}\). Since ∇f is uniformly normtonorm continuous on bounded subsets of E, from (3.9) we have \(\lim_{n\rightarrow\infty}\\nabla f(u_{i,n})\nabla f(y_{i,n})\=0\). From \(\liminf_{n\rightarrow\infty}r_{n}>0\) it follows that
By the definition of \(u_{n}:=K_{r_{n}}y_{n}\), we have
where
We have from (A2) that
Since \(y\mapsto f(x,y)+\langle Ax,yx \rangle\) is convex and lower semicontinuous, letting \(n\rightarrow\infty\) in the last inequality, from (A4) we have
For t, with \(0< t<1\), and \(y\in C\), let \(y_{t}=ty+(1t)p\). Since \(y\in C\) and \(p\in C\), then \(y_{t}\in C\) and hence \(G(y_{t}, p)\leq0\). So, from (A1) we have
Dividing by t, we have
Letting \(t\rightarrow0\), from (A3) we can get
So, \(p\in \mathit{EP}\).
Step 6. Finally, we prove that \(p= P_{F\cap \mathit{EP}}^{f}x_{0}\), from Proposition 2.10, we have
On the other hand, since \(x_{n}= P_{C_{n}}^{f}x_{0}\) and \(F\cdot \mathit{EP} \subset C_{n}\) for all n, also from Proposition 2.10, we have
By the definition of \(D_{f}(x,y)\), we know that
Combining (3.10), (3.11), and (3.12), we know that \(D_{f}(p,x_{0})=D_{f}(P_{F\cap \mathit{EP}}^{f}x_{0}, x_{0})\). Therefore, it follows from the uniqueness of \(P_{F\cap \mathit{EP}}^{f}x_{0}\) that \(p= P_{F\cap \mathit{EP}}^{f}x_{0}\). This completes the proof. □
Definition 3.3
Let C be a nonempty, closed, and convex subset of E. Let T be a mapping from C into itself with a nonempty fixed point set \(F(T)\). The mapping T is said to be Lyapunov quasiLipschitz if there exists a constant \(L\geq1\) such that
The mapping T is said to be Lyapunov quasinonexpansive if
If we choose \(f (x) =\frac{1}{2} \x\^{2}\) for all \(x \in E\), then Theorem 3.2 reduces to the following corollary.
Corollary 3.4
Let E be a smooth Banach space and C be a closed convex subset of E. Let \(\{T^{(i)}_{n}\}_{n=1}^{\infty} : C \rightarrow C\) be N uniformly closed families of countable Lyapunov quasiLipschitz mappings with the condition \(\lim_{n\rightarrow\infty}L^{(i)}_{n}=1\) for \(i=1,2,3, \ldots,N\). Let \(F=\bigcap_{n=1}^{\infty}\bigcap_{i=1}^{N} F(T^{(i)}_{n})\) and \(F\cap \mathit{EP}\) be nonempty. Let \(\{x_{n}\}\) be a sequence of C generated by
where
and \(\{\alpha_{n}\}\) is a sequence satisfying \(\limsup_{n\rightarrow\infty} \alpha_{n} <1\). Then \(\{x_{n}\}\) converges to \(q= P_{F\cap \mathit{EP}}^{f}x_{0}\).
Example
Let E be a smooth Banach space and C be a nonempty closed convex and balanced subset of E. Let \(\{x_{n}\}\) be a sequence in C such that \(\x_{n}\=r>0\), \(\{x_{n}\}\) converges weakly to \(x_{0}\neq0\) and \(\x_{n}x_{m}\\geq r>0\) for all \(n\neq m\). Define a countable family of mappings \(\{T_{n}\}: C\rightarrow C\) as follows:
Conclusion 4.1
\(\{T_{n}\}\) has a unique common fixed point 0, that is, \(F=\bigcap_{n=1}^{\infty}F(T_{n})=\{0\}\) for all \(n\geq0\).
Proof
The conclusion is obvious. □
Conclusion 4.2
\(\{T_{n}\}\) is a uniformly closed family of countable Bregman quasiLipschitz mappings with the condition \(\lim_{n\rightarrow\infty}L_{n}=\lim_{n\rightarrow\infty}\frac{n+1}{n}=1\).
Proof
Take \(f(x)=\frac{\x\^{2}}{2}\), then
for all \(x,y \in C\) and
Therefore
for all \(x \in C\). On the other hand, for any strong convergent sequence \(\{z_{n}\}\subset E\) such that \(z_{n}\rightarrow z_{0}\) and \(\z_{n}T_{n}z_{n}\\rightarrow0\) as \(n\rightarrow\infty\), it is easy to see that there exists a sufficiently large nature number N such that \(z_{n}\neq x_{m}\) for any \(n, m >N\). Then \(Tz_{n}=z_{n}\) for \(n>N\), it follows from \(\z_{n}T_{n}z_{n}\\rightarrow0\) that \(2z_{n}\rightarrow0\) and hence \(z_{n}\rightarrow z_{0}=0\). That is, \(z_{0} \in F\). □
Example 4.3
Let \(E=l^{2}\), where
Let \(\{x_{n}\}\subset E\) be a sequence defined by
where
for all \(n\geq1\). It is well known that \(\x_{n}\=\sqrt{2}\), \(\forall n\geq1\) and \(\{x_{n}\}\) converges weakly to \(x_{0}\). Define a countable family of mappings \(T_{n}: E\rightarrow E\) as follows:
for all \(n\geq0\). By using Conclusions 4.1 and 4.2, \(\{T_{n}\}\) is a uniformly closed family of countable Bregman quasiLipschitz mappings with the condition \(\lim_{n\rightarrow\infty}L_{n}=\lim_{n\rightarrow\infty}\frac{n+1}{n}=1\).
Example 4.4
Let \(E=L^{p}[0,1]\) (\(1< p<+\infty\)) and
Define a sequence of functions in \(L^{p}[0,1]\) by the following expression:
for all \(n\geq1\). Firstly, we can see, for any \(x \in[0,1]\), that
where \(f_{0}(x)\equiv0\). It is well known that the above relation (4.1) is equivalent to \(\{f_{n}(x)\}\) converges weakly to \(f_{0}(x)\) in a uniformly smooth Banach space \(L^{p}[0,1]\) (\(1< p<+\infty\)). On the other hand, for any \(n\neq m\), we have
Let
It is obvious that \(u_{n}\) converges weakly to \(u_{0}(x)\equiv1\) and
Define a mapping \(T: E\rightarrow E\) as follows:
Since (4.2) holds, by using Conclusions 4.1 and 4.2, we know that \(\{ T_{n}\}\) is a uniformly closed family of countable Bregman quasiLipschitz mappings with the condition \(\lim_{n\rightarrow\infty}L_{n}=\lim_{n\rightarrow\infty}\frac{n+1}{n}=1\).
Application
The mapping T is said to be Bregman asymptotically quasinonexpansive (cf. [29]) if \(F(T)\neq\emptyset\) and there exists a sequence \(\{k_{n}\}\subset[1, +\infty)\) with \(\lim_{n\rightarrow\infty}k_{n}=1\) such that
Every Bregman quasinonexpansive mapping is Bregman asymptotically quasinonexpansive with \(k_{n}\equiv1\). Let \(S_{n}=T^{n}\) for all \(n\geq1\), the above inequality becomes
It is obvious that \(\bigcap_{n=1}^{\infty}F(S_{n})=\bigcap_{n=1}^{\infty}F(T^{n})=F(T)\).
Lemma 5.1
Assume that T is uniformly Lipschitz, that is, there exists a constant \(L\geq1\) such that
for all \(n\geq1\). Then \(\{S_{n}\}=\{T^{n}\}\) is uniformly closed.
Proof
Assume \(\z_{n}S_{n}z_{n}\ \rightarrow0\), \(z_{n}\rightarrow p\) as \(n\rightarrow\infty\), we have \(\z_{n}T^{n}z_{n}\ \rightarrow0\), therefore
as \(n\rightarrow\infty\). On the one hand, \(T^{n}p\rightarrow p\), on the other hand, \(T^{n+1}p\rightarrow Tp\), these imply that \(p=Tp\). Hence \(p \in\bigcap_{n=1}^{\infty}F(S_{n})\). This completes the proof. □
Next we give an application of Theorem 3.2 to find the fixed point of Bregman asymptotically quasinonexpansive mappings.
Theorem 5.2
Let \(f:E\rightarrow(\infty, +\infty]\) be a Legendre function which is bounded, strongly coercive, uniformly Fréchet differentiable, and totally convex on bounded subsets on E. Let C be a nonempty, closed, and convex subset of \(\operatorname{int} \operatorname{dom}f\). Let \(\{T_{i}\}_{i=1}^{N} : C \rightarrow C\) be an N uniformly Lipschitz Bregman asymptotically quasinonexpansive mapping with a nonempty common fixed point set \(F=\bigcap_{i=1}^{N}F(T_{i})\) and \(F\cap \mathit{EP}\) be nonempty. Let \(\{x_{n}\}\) be a sequence of C generated by
where
and \(\{\alpha_{n}\}\) is a sequence satisfying \(\limsup_{n\rightarrow\infty} \alpha_{n} <1\). Then \(\{x_{n}\}\) converges to \(q= P_{F\cap \mathit{EP}}^{f}x_{0}\).
Proof
Let \(S_{n}=T^{n}\) for all \(n\geq1\), by using Lemma 5.1 and Theorem 3.2 we can obtain the conclusion. □
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Acknowledgements
This project is supported by the National Natural Science Foundation of China under grant (11071279).
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The main idea of this paper was proposed by the corresponding author YS, and YS prepared the manuscript initially for one family of countable Bregman quasiLipschitz mappings. MC performed all the steps of the proofs in this research for the finite families of countable Bregman quasiLipschitz mappings. JB performed the application to the Bregman asymptotically quasinonexpansive mappings. All authors read and approved the final manuscript.
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Chen, M., Bi, J. & Su, Y. Hybrid iterative algorithm for finite families of countable Bregman quasiLipschitz mappings with applications in Banach spaces. J Inequal Appl 2015, 210 (2015). https://doi.org/10.1186/s1366001507313
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DOI: https://doi.org/10.1186/s1366001507313
MSC
 47H05
 47H09
 47H10
Keywords
 Bregman distance
 Bregman quasiLipschitz mapping
 generalized projection
 hybrid algorithm
 Bregman asymptotically quasinonexpansive mappings
 finite families
 equilibrium problem