Hybrid iterative algorithm for finite families of countable Bregman quasi-Lipschitz map**s with applications in Banach spaces

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 quasi-Lipschitz map**s 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 quasi-nonexpansive map**s is also given. The main innovative points in this paper are as follows: (1) the notion of the uniformly closed family of countable Bregman quasi-Lipschitz map**s is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable Bregman quasi-Lipschitz map**s are given in classical Banach spaces $l^{2}$ and $L^{2}$; (3) the application for Bregman asymptotically quasi-nonexpansive map**s is also given; (4) because the main theorems do not need the boundedness of the domain of map**s, so a corresponding technique for the proof is given. This new results improve and extend the previously known ones in the literature.

1 Introduction

Let C be a nonempty subset of a real Banach space and T be a map** 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

$$\bigl\Vert T^{n}x-T^{n}y\bigr\Vert \leq k_{n} \|x-y\|, \quad \forall x, y \in C, n\geq1. $$

In the framework of Hilbert spaces, Takahashi et al. [2] have introduced a new hybrid iterative scheme called a shrinking projection method for nonexpansive map**s. 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 map**s 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 map**s. The map** T is said to be asymptotically nonexpansive in the intermediate sense (cf. [5]) if

$$ \limsup_{n\rightarrow\infty} \sup_{x,y \in C}\bigl( \bigl\Vert T^{n}x-T^{n}y\bigr\Vert -\|x-y\|\bigr)\leq0. $$

(1.1)

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 quasi-nonexpansive in the intermediate sense. It is worth mentioning that the class of asymptotically nonexpansive map**s in the intermediate sense contains properly the class of asymptotically nonexpansive map**s since the map**s 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 map**s which are asymptotically quasi-nonexpansive 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 quasi-nonexpansive map** with respect to the Lyapunov functional in the intermediate sense.

In 1967, Bregman [12] discovered an elegant and effective technique for using of the so-called 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 map**s.

Many authors have studied iterative methods for approximating fixed points of map**s of nonexpansive type with respect to the Bregman distance; see [13–17]. In [18], the authors has introduced a new class of nonlinear map**s which is an extension of asymptotically quasi-nonexpansive map**s with respect to the Bregman distance in the intermediate sense and has proved the strong convergence theorems for asymptotically quasi-nonexpansive map**s 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 quasi-Lipschitz map**s 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 quasi-nonexpansive map**s is also given. The main innovative points in this paper are as follows: (1) the notion of the uniformly closed family of countable Bregman quasi-Lipschitz map**s is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable Bregman quasi-Lipschitz map**s are given in classical Banach spaces $l^{2}$ and $L^{2}$; (3) the application for Bregman asymptotically quasi-nonexpansive map**s is also given; (4) because the main theorems do not need the boundedness of the domain of map**s, so a corresponding technique for the proof is given. This new results improve and extend the previously known ones in the literature.

2 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

$$\operatorname{dom} f:=\bigl\{ x \in E: f(x)< +\infty\bigr\} . $$

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

$$\lim_{\|x\|\rightarrow\infty}\frac{f(x)}{\|x\|}=+\infty. $$

Given a proper and convex function $f :E \rightarrow (-\infty,+\infty]$, the subdifferential of f is a map** $\partial f: E\rightarrow E^{*}$ defined by

$$\partial f(x)=\bigl\{ x^{*} \in E^{*}: f(y)\geq f(x)+\bigl\langle x^{*}, y-x\bigr\rangle , \forall y \in E\bigr\} $$

for all $x \in E$.

The Fenchel conjugate function of f is the convex function $f^{*}: E \rightarrow(-\infty,+\infty)$ defined by

$$f^{*}\bigl(x^{*}\bigr)=\sup\bigl\{ \bigl\langle x^{*},x\bigr\rangle -f(x), x \in E\bigr\} . $$

We know that $x^{*} \in\partial f(x)$ if and only if

$$f(x)+f^{*}\bigl(x^{*}\bigr)=\bigl\langle x^{*},x\bigr\rangle $$

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 right-hand derivative of f at x in the direction y by

$$ f^{\circ}(x,y)=\lim_{t\downarrow0}\frac{f(x+ty)-f(x)}{t}. $$

(2.1)

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 norm-to-norm 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 single-valued 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:

$$\operatorname{ran} \nabla f = \operatorname{dom} f^{*} = \operatorname{int} \operatorname{dom} f^{*} \quad \mbox{and} \quad \operatorname{ran} \nabla f^{*} = \operatorname{dom} f = \operatorname{int}\operatorname{dom} f . $$

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

$$D_{f}(x,y)=f(x)-f(y)-\bigl\langle x-y, \nabla f(y)\bigr\rangle $$

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 $y, z \in \operatorname{int}\operatorname{dom} f$,

$$ D_{f}(x,y)+D_{f}(y,z)-D_{f}(x,z)=\bigl\langle x-y, \nabla f(z)-\nabla f(y)\bigr\rangle . $$

(2.2)

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

$$W^{f}(w,x)=f(x)-\langle w,x \rangle+f^{*}(w). $$

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

$$v_{f}(x,t)=\inf\bigl\{ D_{f}(y,x): y\in \operatorname{dom} f, \|y-x\|=t\bigr\} , $$

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

$$v_{f}(B,t)=\inf\bigl\{ v_{f}(x,t): x\in B \cap \operatorname{int}\operatorname{dom} f\bigr\} . $$

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 semi-continuous, 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,

$$\lim_{n\rightarrow\infty} D_{f}(y_{n},x_{n})=0 \quad \Rightarrow \quad \lim_{n\rightarrow\infty}\|y_{n}-x_{n} \|=0. $$

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,

$$\operatorname{proj}_{C}^{f}(x)=\operatorname{argmin}\bigl\{ D_{f}(y,x): y\in C\bigr\} . $$

Let E be a Banach space with dual $E^{*}$. We denote by J the normalized duality map** from E to $2^{E^{*}}$ defined by

$$Jx=\bigl\{ f\in E^{*}:\langle x,f\rangle=\|x\|^{2}=\|f\|^{2} \bigr\} , $$

where $\langle\cdot,\cdot\rangle$ denotes the generalized duality pairing. It is well known that if E is smooth, then J is single-valued.

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

  $$\bigl\langle z-y, \nabla f(x)-\nabla f(z)\bigr\rangle \geq0, \quad \forall y \in C. $$

• (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 map** from C into itself with a nonempty fixed point set $F(T)$. The map** T is said to be Bregman quasi-Lipschitz if there exists a constant $L\geq1$ such that

$$D_{f}(p,Tx)\leq L D_{f}(p,x), \quad \forall p\in F(T), \forall x \in C. $$

The map** T is said to be Bregman quasi-nonexpansive if

$$D_{f}(p,Tx)\leq D_{f}(p,x), \quad \forall p\in F(T), \forall x \in C. $$

Bregman quasi-Lipschitz map**s are a more generalized class than the class of Bregman quasi-map**s. On the other hand, this class also contains the relatively quasi-Lipschitz map**s and quasi-Lipschitz map**s. Therefore, Bregman quasi-Lipschitz map**s 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 map**s 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 quasi-Lipschitz map**s.

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 map** and $F:C\times C\rightarrow R$ be a bifunction. Then consider the following generalized equilibrium problem of finding $u \in C$ such that:

$$ F(u,y)+\langle Au,y-u \rangle\geq0, \quad \forall y\in C. $$

(2.3)

The set of solutions of (2.3) is denoted by EP, i.e.,

$$\mathit{EP}=\bigl\{ u\in C: F(u,y)+\langle Au,y-u \rangle\geq0, \forall y\in C \bigr\} . $$

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

$$ F(u, y)\geq0, \quad \forall y\in C, $$

(2.4)

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

$$\langle Au,y-u \rangle\geq0,\quad \forall y\in C, $$

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 $x,y,z\in C$, $\limsup_{t\downarrow0}F(tz+(1-t)x,y)\leq F(x,y)$,

• (A4) for all $x\in C$, $F(x,\cdot)$ is convex and lower semi-continuous.

For $r>0$, we define a map** $K_{r} : E \rightarrow C$ as follows:

$$ T_{r}(x)=\biggl\{ z\in C:F(z,y)+\frac{1}{r}\bigl\langle y-z, \nabla f(z)-\nabla f(x) \bigr\rangle \geq0, \forall y\in C\biggr\} $$

(2.5)

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 map** 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 map** defined by (2.5). Then the following statements hold:

• (i) $T_{r}$ is single-valued.

• (ii) $T_{r}$ is a firmly nonexpansive-type map**, i.e., for all $x,y\in E$,

  $$\bigl\langle T_{r}x-T_{r}y, \nabla f(T_{r}x)- \nabla f(T_{r}y) \bigr\rangle \leq\bigl\langle T_{r}x-T_{r}y, \nabla f(x)-\nabla f(y) \bigr\rangle . $$

• (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 map**, i.e.,

$$\langle Ax-Ay, x-y\rangle\geq0, \quad \forall x,y \in C. $$

For $r > 0$, let $K_{r} : E \rightarrow C$ be the map** defined by

$$K_{r}(x)=\biggl\{ z\in C:F(z,y)+\langle Az, y-z\rangle+ \frac{1}{r}\bigl\langle y-z,\nabla f(z)-\nabla f(x) \bigr\rangle \geq0, \forall y\in C\biggr\} . $$

Then the following statements hold:

• (i) $K_{r}$ is single-valued.

• (ii) $K_{r}$ is a firmly nonexpansive-type map**, i.e., for all $x,y\in E$,

  $$\bigl\langle K_{r}x-K_{r}y, \nabla f(K_{r}x)- \nabla f(K_{r}y) \bigr\rangle \leq\bigl\langle K_{r}x-K_{r}y, \nabla f(x)-\nabla f(y) \bigr\rangle . $$

• (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

$$G(x,y)=F(x,y)+\langle Ax,y-x\rangle, \quad \forall x,y \in C. $$

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. □

3 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 quasi-Lipschitz map**s with the condition $\lim_{n\rightarrow\infty}L_{n}=1$, where

$$ D_{f}(p,T_{n}x)\leq L_{n} D_{f}(p,x), \quad \forall p\in F, \forall x \in C. $$

(3.1)

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 $p_{1},p_{2} \in F$, $p=tp_{1}+(1-t)p_{2}$, where $t \in(0,1)$. We prove that $p \in F$. By the three point identity (2.2), we know that

$$D_{f}(x,y)=D_{f}(x,z)+D_{f}(z,y)+ \bigl\langle x-z, \nabla f(z)-\nabla f(y)\bigr\rangle . $$

This implies

$$ D_{f}(p_{i},T_{n}p)=D_{f}(p_{i},p)+D_{f}(p,T_{n}p)+ \bigl\langle p_{i}-p, \nabla f(p)-\nabla f(T_{n}p)\bigr\rangle $$

(3.2)

for $i=1,2$. Combining (3.1) and (3.2) yields

$$ D_{f}(p,T_{n}p)\leq(L_{n}-1)D_{f}(p_{i},p)- \bigl\langle p_{i}-p, \nabla f(p)-\nabla f(T_{n}p)\bigr\rangle $$

(3.3)

for $i =1,2$. Multiplying t and $(1-t)$ on both sides of (3.3) with $i = 1$ and $i = 2$, respectively, yields

$$\lim_{n\rightarrow\infty} D_{f}(p,T_{n}p)\leq \lim _{n\rightarrow\infty} \bigl(\xi_{n}- \bigl\langle tp_{1}+(1-t)p_{2}-p, \nabla f(p)-\nabla f(T_{n}p) \bigr\rangle \bigr)=0, $$

where

$$\xi_{n}=(L_{n}-1)\bigl[tD_{f}(p_{1},p)+(1-t)D_{f}(p_{2},p) \bigr]. $$

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 $\|p-T_{n}p\|\rightarrow0$ 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 quasi-Lipschitz map**s 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 quasi-Lipschitz map**s 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

$$\textstyle\begin{cases} x_{0} \in \operatorname{int} \operatorname{dom} f,\quad \textit{arbitrarily}, \\ y_{i,n} = \nabla f^{*}(\alpha_{n}\nabla f(x_{n})+(1-\alpha_{n}) \nabla f(T^{(i)}_{n}x_{n})),\quad i=1,2,3, \ldots, N, \\ F(u_{i,n},y)+\langle Au_{i,n},y-u_{i,n}\rangle+\frac{1}{r_{n}}\langle\nabla f(u_{i,n})- \nabla f(y_{i,n}), y-u_{i,n}\rangle\geq0, \quad \forall y \in C, \\ C_{i,n+1}=\{z\in C_{n}: D_{f}(z,u_{i,n})\leq D_{f}(z,y_{i,n})\leq D_{f}(z, x_{n})+\xi_{n}\}, \quad n\geq1, \\ C_{i,1}=C, \qquad C_{n+1}=\bigcap_{i=1}^{N}C_{i,n+1}, \\ x_{n}=P_{C_{n}}^{f}x_{0}, \end{cases} $$

where

$$\begin{aligned}& \xi_{n}=(L_{n}-1)\sup_{p\in F\cap \mathit{EP} \cap B(P_{F\cap \mathit{EP}}^{f}x_{0}, 1)}D_{f}(p,x_{0}), \\& B(x,1)=\bigl\{ y \in E: D_{f}(y,x) \leq1\bigr\} , \\& L_{n}=\max\bigl\{ L^{(1)}_{n}, L^{(2)}_{n}, L^{(3)}_{n}, \ldots, L^{(N)}_{n} \bigr\} \end{aligned}$$

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

$$C_{i,k+1}=\bigl\{ z \in C: D_{f}(z,u_{i,k})\leq D_{f}(z,y_{i,k})\leq D_{f}(z, x_{k})+ \xi_{k}\bigr\} \cap C_{i,k} $$

and

$$D_{f}(z,u_{i,k})\leq D_{f}(z,y_{i,k})\leq D_{f}(z, x_{k})+\xi_{k} $$

is equivalent to

$$ \textstyle\begin{cases} \langle\nabla f(x_{k})-\nabla f(y_{i,k}),z \rangle\leq\langle f^{*}( \nabla f(x_{k}))-f^{*}(\nabla f(y_{i,k})) \rangle+\xi_{k}, \\ \langle\nabla f(y_{i,k})-\nabla f(u_{i,k}),z \rangle\leq\langle f^{*}( \nabla f(y_{i,k}))-f^{*}(\nabla f(u_{i,k})) \rangle. \end{cases} $$

(3.4)

Therefore

$$C_{i,k+1}=\bigl\{ z \in C: z \text{ satisfies (\ref{eq3.4})}\bigr\} \cap C_{i,k}. $$

It is easy to see that if $z_{1}$, $z_{2}$ satisfy (3.4), the element $z=tz_{1}+(1-t)z_{2}$ satisfies also (3.4) for all $t \in(0,1)$, so that the set

$$\bigl\{ z \in C: z \text{ satisfies (\ref{eq3.4})}\bigr\} $$

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

$$\begin{aligned} D_{f}(p,y_{i,n})&=D_{f}\bigl(p, \nabla f^{*}\bigl( \alpha_{n}\nabla f(x_{n})+(1-\alpha_{n}) \nabla f \bigl(T^{(i)}_{n}x_{n}\bigr)\bigr)\bigr) \\ &=W^{f}\bigl(p, \nabla f^{*}\bigl(\alpha_{n}\nabla f(x_{n})+(1-\alpha_{n}) \nabla f\bigl(T^{(i)}_{n}x_{n} \bigr)\bigr)\bigr) \\ &\leq\alpha_{n} W^{f}\bigl(p, \nabla f(x_{n}) \bigr)+(1-\alpha_{n}) W^{f}\bigl(p,\nabla f \bigl(T^{(i)}_{n}x_{n}\bigr)\bigr) \\ &= \alpha_{n} D_{f}(p, x_{n})+(1- \alpha_{n}) D_{f}\bigl(p,T^{(i)}_{n}x_{n} \bigr) \\ &\leq\alpha_{n} D_{f}(p, x_{n})+(1- \alpha_{n}) D_{f}(p,x_{n})+\xi_{n} \\ & \leq D_{f}(p,x_{n})+\xi_{n}. \end{aligned}$$

(3.5)

On the other hand, by Lemma 2.15, we have $p=K_{r}(p)$ and

$$D_{f}(p, K_{r}y_{i,n})+D_{f}(K_{n}y_{i,n},y_{i,n}) \leq D_{f}(p,y_{i,n}), $$

that is,

$$ D_{f}(p, u_{i,n})+D_{f}(K_{n}y_{i,n},y_{i,n}) \leq D_{f}(p,y_{i,n}). $$

(3.6)

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

$$ D_{f}(x_{n},x_{0})\leq D_{f}(x_{n+1},x_{0}) \quad \mbox{for all } n\geq 1. $$

(3.7)

Therefore $\{D_{f}(x_{n},x_{0})\}$ is nondecreasing. On the other hand, by Proposition 2.10, we have

$$D_{f}(x_{n},x_{0})=D_{f} \bigl(P_{C_{n}}^{f}x_{0},x_{0}\bigr)\leq D_{f}(p,x_{0})-D_{f}(p,x_{n})\leq D_{f}(p,x_{0}) $$

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

$$ \lim_{n\rightarrow\infty}D_{f}(x_{n}, x_{0})=d. $$

(3.8)

From Proposition 2.10, we have, for any positive integer m, that

$$\begin{aligned} \begin{aligned} D_{f}(x_{n+m},x_{n})&=D_{f} \bigl(x_{n+m},P_{C_{n}}^{f}x_{0}\bigr)\leq D_{f}(x_{n+m},x_{0})-D_{f} \bigl(P_{C_{n}}^{f}x_{0},x_{0}\bigr) \\ &=D_{f}(x_{n+m},x_{0})-D_{f}(x_{n},x_{0}) \end{aligned} \end{aligned}$$

for all $n\geq1$. This together with (3.8) implies that

$$\lim_{n\rightarrow\infty}D_{f}(x_{n+m}, x_{n})=0 $$

holds uniformly for all m. Therefore, we get that

$$\lim_{n\rightarrow\infty}\|x_{n+m}- x_{n}\|=0 $$

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

$$D_{f}(x_{n+1}, u_{i,n})\leq D_{f}(x_{n+1},y_{i,n}) \leq D_{f}(x_{n+1}, x_{n})+\xi_{n} \rightarrow0 $$

as $n\rightarrow\infty$. By Proposition 2.8, we obtain that

$$ \lim_{n\rightarrow\infty}\|x_{n+1}-y_{i,n}\|=0, \lim _{n\rightarrow\infty}\|x_{n+1}-u_{i,n}\|=0. $$

(3.9)

From

$$y_{i,n} = \nabla f^{*}\bigl(\alpha_{n}\nabla f(x_{n})+(1-\alpha_{n}) \nabla f\bigl(T^{(i)}_{n}x_{n} \bigr)\bigr), $$

we get

$$\nabla f(y_{i,n}) =\alpha_{n}\nabla f(x_{n})+(1- \alpha_{n}) \nabla f\bigl(T^{(i)}_{n}x_{n} \bigr), $$

which implies that

$$\nabla f(y_{i,n})-\nabla f(x_{n}) =(1-\alpha_{n}) \bigl( \nabla f\bigl(T^{(i)}_{n}x_{n}\bigr)-\nabla f(x_{n})\bigr). $$

By Proposition 2.2, we have

$$\lim_{n\rightarrow\infty} \bigl\Vert \nabla f(y_{i,n})-\nabla f(x_{n})\bigr\Vert =0 , $$

so that

$$\lim_{n\rightarrow\infty} \bigl\Vert \nabla f\bigl(T^{(i)}_{n}x_{n} \bigr)-\nabla f(x_{n})\bigr\Vert =0 . $$

By Propositions 2.3 and 2.8, $\nabla f^{*}$ is uniformly continuous on bounded subsets of E and thus

$$\lim_{n\rightarrow\infty} \bigl\Vert T^{(i)}_{n}x_{n}-x_{n} \bigr\Vert =0 . $$

Since $\{T^{(i)}_{n}\}$ is an asymptotically countable family of Bregman weak relatively nonexpansive map**s 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 norm-to-norm 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

$$\lim_{n\rightarrow\infty}\frac{\|\nabla f(u_{i,n})-\nabla f(y_{i,n})\|}{r_{n}}=0. $$

By the definition of $u_{n}:=K_{r_{n}}y_{n}$, we have

$$G(u_{i,n},y)+\frac{1}{r_{n}}\bigl\langle y-u_{i,n},\nabla f(u_{i,n})-\nabla f(y_{i,n}) \bigr\rangle \geq0,\quad \forall y \in C, $$

where

$$G(u_{i,n},y)=F(u_{i,n},y)+\langle Au_{i,n},y-u_{i,n} \rangle. $$

We have from (A2) that

$$\frac{1}{r_{n}}\bigl\langle y-u_{i,n},\nabla f(u_{i,n})- \nabla f(y_{i,n}) \bigr\rangle \geq-G(u_{i,n},y)\geq G(y,u_{i,n}),\quad \forall y\in C. $$

Since $y\mapsto f(x,y)+\langle Ax,y-x \rangle$ is convex and lower semi-continuous, letting $n\rightarrow\infty$ in the last inequality, from (A4) we have

$$G(y, p)\leq0, \quad \forall y\in C. $$

For t, with $0< t<1$, and $y\in C$, let $y_{t}=ty+(1-t)p$. 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

$$0=G(y_{t},y_{t})\leq tG(y_{t},y)+(1-t)G(y_{t}, p)\leq tG(y_{t},y). $$

Dividing by t, we have

$$G(y_{t},y)\geq0, \quad \forall y\in C. $$

Letting $t\rightarrow0$, from (A3) we can get

$$G(p, y)\geq0,\quad \forall y\in C. $$

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

$$ D_{f}\bigl(p, P_{F\cap \mathit{EP}}^{f}x_{0} \bigr)+D_{f}\bigl(P_{F\cap \mathit{EP}}^{f}x_{0}, x_{0}\bigr)\leq D_{f}(p, x_{0}). $$

(3.10)

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

$$ D_{f}\bigl(P_{F\cap \mathit{EP}}^{f}x_{0}, x_{n+1}\bigr)+D_{f}(x_{n+1}, x_{0})\leq D_{f}\bigl(P_{F\cap \mathit{EP}}^{f}x_{0}, x_{0}\bigr). $$

(3.11)

By the definition of $D_{f}(x,y)$, we know that

$$ \lim_{n\rightarrow\infty}D_{f}(x_{n+1}, x_{0})=D_{f}(p,x_{0}). $$

(3.12)

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 map** from C into itself with a nonempty fixed point set $F(T)$. The map** T is said to be Lyapunov quasi-Lipschitz if there exists a constant $L\geq1$ such that

$$\phi(p,Tx)\leq L \phi(p,x), \quad \forall p\in F(T), \forall x \in C. $$

The map** T is said to be Lyapunov quasi-nonexpansive if

$$\phi(p,Tx)\leq \phi(p,x),\quad \forall p\in F(T), \forall x \in C. $$

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 quasi-Lipschitz map**s 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

$$\textstyle\begin{cases} x_{0} \in \operatorname{int} \operatorname{dom} f,\quad \textit{arbitrarily}, \\ y_{i,n} = J^{-1}(\alpha_{n} Jx_{n}+(1-\alpha_{n}) JT^{(i)}_{n}x_{n}),\quad i=1,2,3, \ldots, N, \\ F(u_{i,n},y)+\langle Au_{i,n},y-u_{i,n}\rangle+\frac{1}{r_{n}}\langle J(u_{i,n})- J(y_{i,n}), y-u_{i,n}\rangle\geq0,\quad \forall y \in C, \\ C_{i,n+1}=\{z\in C_{n}: \phi(z,u_{i,n})\leq \phi(z,y_{i,n})\leq\phi(z, x_{n})+\xi_{n}\},\quad n\geq1, \\ C_{i,1}=C, \qquad C_{n+1}=\bigcap_{i=1}^{N}C_{i,n+1}, \\ x_{n}=P_{C_{n}}^{f}x_{0}, \end{cases} $$

where

$$\begin{aligned}& \xi_{n}=(L_{n}-1)\sup_{p\in F\cap \mathit{EP} \cap B(P_{F\cap \mathit{EP}}^{f}x_{0}, 1)} \phi(p,x_{0}), \\& B(x,1)=\bigl\{ y \in E: \phi(y,x) \leq1\bigr\} , \\& L_{n}=\max\bigl\{ L^{(1)}_{n}, L^{(2)}_{n}, L^{(3)}_{n}, \ldots, L^{(N)}_{n} \bigr\} \end{aligned}$$

and $\{\alpha_{n}\}$ is a sequence satisfying $\limsup_{n\rightarrow\infty} \alpha_{n} <1$. Then $\{x_{n}\}$ converges to $q= P_{F\cap \mathit{EP}}^{f}x_{0}$.

4 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 map**s $\{T_{n}\}: C\rightarrow C$ as follows:

$$T_{n}(x)= \textstyle\begin{cases} \frac{n+1}{n}x_{n} & \mbox{if }x=x_{n}\ (\exists n\geq 1), \\ -x & \mbox{if } x\neq x_{n}\ (\forall n\geq1). \end{cases} $$

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 quasi-Lipschitz map**s 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

$$D_{f}(x,y)=\phi(x,y)=\|x\|^{2}-2\langle x,Jy\rangle+\|y \|^{2} $$

for all $x,y \in C$ and

$$D_{f}(0, T_{n}x)=\|T_{n}x\|^{2}= \textstyle\begin{cases} \frac{n+1}{n}\|x_{n}\|^{2} &\mbox{if } x=x_{n} , \\ \|x\|^{2}&\mbox{if } x\neq x_{n} . \end{cases} $$

Therefore

$$D_{f}(0, T_{n}x)\leq\frac{n+1}{n} \|x\|^{2}= \frac{n+1}{n} D_{f}(0, x) $$

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

$$\begin{aligned}& l^{2}=\Biggl\{ \xi=(\xi_{1}, \xi_{2}, \xi_{3}, \ldots, \xi_{n}, \ldots): \sum _{n=1}^{\infty}|x_{n}|^{2}< \infty \Biggr\} , \\& \|\xi\|=\Biggl(\sum_{n=1}^{\infty}| \xi_{n}|^{2}\Biggr)^{\frac{1}{2}}, \quad \forall \xi \in l^{2}, \\& \langle\xi, \eta\rangle=\sum_{n=1}^{\infty} \xi_{n}\eta_{n}, \quad \forall \xi=(\xi_{1}, \xi_{2},\xi_{3},\ldots,\xi_{n},\ldots), \eta=( \eta_{1},\eta_{2},\eta_{3},\ldots, \eta_{n},\ldots)\in l^{2}. \end{aligned}$$

Let $\{x_{n}\}\subset E$ be a sequence defined by

$$\begin{aligned}& x_{0}= (1, 0, 0,0,\ldots), \\& x_{1}= (1, 1, 0,0,\ldots), \\& x_{2}= (1, 0, 1, 0,0,\ldots), \\& x_{3}= (1, 0, 0, 1, 0,0,\ldots), \\& \ldots, \\& x_{n}= (\xi_{n,1}, \xi_{n,2}, \xi_{n,3}, \ldots, \xi_{n,k}, \ldots), \\& \ldots, \end{aligned}$$

where

$$\xi_{n,k}= \textstyle\begin{cases} 1 &\mbox{if }k=1, n+1 , \\ 0 &\mbox{if }k\neq1, k\neq n+1 \end{cases} $$

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 map**s $T_{n}: E\rightarrow E$ as follows:

$$T_{n}(x)= \textstyle\begin{cases} \frac{n+1}{n}x_{n} & \mbox{if } x=x_{n}, \\ -x & \mbox{if } x\neq x_{n} \end{cases} $$

for all $n\geq0$. By using Conclusions 4.1 and 4.2, $\{T_{n}\}$ is a uniformly closed family of countable Bregman quasi-Lipschitz map**s 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

$$x_{n}=1-\frac{1}{2^{n}},\quad n=1,2,3, \ldots . $$

Define a sequence of functions in $L^{p}[0,1]$ by the following expression:

$$f_{n}(x)= \textstyle\begin{cases} \frac{2}{x_{n+1}-x_{n}} &\mbox{if } x_{n}\leq x < \frac{x_{n+1}+x_{n}}{2} , \\ \frac{-2}{x_{n+1}-x_{n}}&\mbox{if } \frac{x_{n+1}+x_{n}}{2} \leq x < x_{n+1}, \\ 0&\mbox{otherwise} \end{cases} $$

for all $n\geq1$. Firstly, we can see, for any $x \in[0,1]$, that

$$ \int^{x}_{0}f_{n}(t)\, dt\rightarrow0= \int^{x}_{0}f_{0}(t)\, dt, $$

(4.1)

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

$$\begin{aligned} \|f_{n}-f_{m}\|&=\biggl(\int^{1}_{0} \bigl\vert f_{n}(x)-f_{m}(x)\bigr\vert ^{p}\, dx\biggr)^{\frac{1}{p}} \\ & = \biggl(\int^{x_{n+1}}_{x_{n}}\bigl\vert f_{n}(x)-f_{m}(x)\bigr\vert ^{p}\, dx+\int ^{x_{m+1}}_{x_{m}}\bigl\vert f_{n}(x)-f_{m}(x) \bigr\vert ^{p}\, dx\biggr)^{\frac{1}{p}} \\ & = \biggl(\int^{x_{n+1}}_{x_{n}}\bigl\vert f_{n}(x)\bigr\vert ^{p}\, dx+\int^{x_{m+1}}_{x_{m}} \bigl\vert f_{m}(x)\bigr\vert ^{p}\, dx \biggr)^{\frac{1}{p}} \\ &= \biggl(\biggl(\frac{2}{x_{n+1}-x_{n}}\biggr)^{p} (x_{n+1}-x_{n})+ \biggl(\frac{2}{x_{m+1}-x_{m}}\biggr)^{p}(x_{m+1}-x_{m}) \biggr)^{\frac{1}{p}} \\ &= \biggl(\frac{2^{p}}{(x_{n+1}-x_{n} )^{p-1}} +\frac{2^{p}}{(x_{m+1}-x_{m} )^{p-1}}\biggr)^{\frac{1}{p}} \\ & \geq \bigl(2^{p} +2^{p}\bigr)^{\frac{1}{p}}>0. \end{aligned}$$

Let

$$u_{n}(x)=f_{n}(x)+1, \quad \forall n\geq1. $$

It is obvious that $u_{n}$ converges weakly to $u_{0}(x)\equiv1$ and

$$ \|u_{n}-u_{m}\|=\|f_{n}-f_{m}\|\geq \bigl(2^{p} +2^{p}\bigr)^{\frac{1}{p}}>0,\quad \forall n\geq 1. $$

(4.2)

Define a map** $T: E\rightarrow E$ as follows:

$$T_{n}(x)= \textstyle\begin{cases} \frac{n+1}{n}u_{n} & \mbox{if } x=u_{n}\ (\exists n\geq 1) , \\ -x & \mbox{if } x\neq u_{n}\ (\forall n\geq1). \end{cases} $$

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 quasi-Lipschitz map**s with the condition $\lim_{n\rightarrow\infty}L_{n}=\lim_{n\rightarrow\infty}\frac{n+1}{n}=1$.

5 Application

The map** T is said to be Bregman asymptotically quasi-nonexpansive (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

$$D_{f}\bigl(p, T^{n}x\bigr)\leq k_{n} D_{f}(p,x), \quad \forall p \in F(T), \forall x \in C. $$

Every Bregman quasi-nonexpansive map** is Bregman asymptotically quasi-nonexpansive with $k_{n}\equiv1$. Let $S_{n}=T^{n}$ for all $n\geq1$, the above inequality becomes

$$D_{f}(p, S_{n}x)\leq k_{n} D_{f}(p,x), \quad \forall p \in F(T), \forall x \in C. $$

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

$$\bigl\Vert T^{n}x-T^{n}y\bigr\Vert \leq L \|x-y\|,\quad \forall x, y \in C $$

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

$$\bigl\Vert p-T^{n}p\bigr\Vert \leq\bigl\Vert p-T^{n}z_{n}\bigr\Vert +\bigl\Vert T^{n}z_{n}-T^{n}p \bigr\Vert \leq \bigl\Vert p-T^{n}z_{n}\bigr\Vert +L \|z_{n}-p\|\rightarrow0 $$

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 quasi-nonexpansive map**s.

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 quasi-nonexpansive map** 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

$$\textstyle\begin{cases} x_{0} \in \operatorname{int} \operatorname{dom} f,\quad \textit{arbitrarily}, \\ y_{i,n} = \nabla f^{*}(\alpha_{n}\nabla f(x_{n})+(1-\alpha_{n}) \nabla f(T^{n}_{i}x_{n})), \quad i=1,2,3, \ldots, N, \\ F(u_{i,n},y)+\langle Au_{i,n},y-u_{i,n}\rangle+\frac{1}{r_{n}}\langle\nabla f(u_{i,n})- \nabla f(y_{i,n}), y-u_{i,n}\rangle\geq0, \quad \forall y \in C, \\ C_{i,n+1}=\{z\in C_{n}: D_{f}(z,u_{i,n})\leq D_{f}(z,y_{i,n})\leq D_{f}(z, x_{n})+\xi_{n}\},\quad n\geq1, \\ C_{i,1}=C, \qquad C_{n+1}=\bigcap_{i=1}^{N}C_{i,n+1}, \\ x_{n}=P_{C_{n}}^{f}x_{0}, \end{cases} $$

where

$$\begin{aligned}& \xi_{n}=(k_{n}-1)\sup_{p\in F\cap \mathit{EP} \cap B(P_{F\cap \mathit{EP}}^{f}x_{0}, 1)}D_{f}(p,x_{0}), \\& B(x,1)=\bigl\{ y \in E: D_{f}(y,x) \leq1\bigr\} \end{aligned}$$

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. □