1 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
$$\bigl\Vert T^{n}x-T^{n}y\bigr\Vert \leq k_{n} \|x-y\|, \quad \forall x, y \in C, n\geq1. $$
In the framework of Hilbert spaces, Takahashi
et al. [
2] have introduced a new hybrid iterative scheme called a shrinking projection method for nonexpansive 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
$$ \limsup_{n\rightarrow\infty} \sup_{x,y \in C}\bigl( \bigl\Vert T^{n}x-T^{n}y\bigr\Vert -\|x-y\|\bigr)\leq0. $$
(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 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 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 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 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 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 quasi-nonexpansive mappings with respect to the Bregman distance in the intermediate sense and has proved the strong convergence theorems for asymptotically quasi-nonexpansive 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 quasi-Lipschitz 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 quasi-nonexpansive 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 quasi-Lipschitz mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable Bregman quasi-Lipschitz mappings are given in classical Banach spaces \(l^{2}\) and \(L^{2}\); (3) the application for Bregman asymptotically quasi-nonexpansive 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.
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 ran
f 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 mapping
\(\partial f: E\rightarrow E^{*}\) 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]).
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.
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
$$W^{f}(w,x)=f(x)-\langle w,x \rangle+f^{*}(w). $$
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].
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 dom
f, respectively, such that the first one is bounded,
$$\lim_{n\rightarrow\infty} D_{f}(y_{n},x_{n})=0 \quad \Rightarrow \quad \lim_{n\rightarrow\infty}\|y_{n}-x_{n} \|=0. $$
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 mapping 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.
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
\(y,x \in E\).
In this paper, we present the following definition.
Bregman quasi-Lipschitz mappings are a more generalized class than the class of Bregman quasi-mappings. On the other hand, this class also contains the relatively quasi-Lipschitz mappings and quasi-Lipschitz mappings. Therefore, Bregman quasi-Lipschitz mappings are very important in the nonlinear analysis and fixed point theory and applications.
In Section
4, we will give two examples of a uniformly closed family of countable Bregman quasi-Lipschitz 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:
$$ F(u,y)+\langle Au,y-u \rangle\geq0, \quad \forall y\in C. $$
(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 mapping
\(K_{r} : E \rightarrow C\) 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].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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 quasi-Lipschitz mappings. MC performed all the steps of the proofs in this research for the finite families of countable Bregman quasi-Lipschitz mappings. JB performed the application to the Bregman asymptotically quasi-nonexpansive mappings. All authors read and approved the final manuscript.