Abstract
The purpose of this paper is to introduce and consider a new hybrid shrinking projection algorithm for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of a system of variational inequality problems, the set of solutions of a system of optimization problems, the common fixed point set of a uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings in reflexive Banach spaces. Strong convergence theorems have been proved under the appropriate conditions. The main innovative points in this paper are as follows: (1) the notion of the uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings are given in classical Banach spaces \(l^{2}\) and \(L^{2}\); (3) the hybrid shrinking projection method presented in this paper modified some mistakes in the recent result of Ugwunnadi et al. (Fixed Point Theory Appl. 2014:231, 2014). These new results improve and extend the previously known ones in the literature.
Similar content being viewed by others
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
It is well known that T is said to be nonexpansive if
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 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 author 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 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.
Recently, Zegeye and Shahzad [19] have proved a strong convergence theorem for the common fixed point of a finite family of right Bregman strongly nonexpansive mappings in a reflexive Banach space. Alghamdi et al. [20] proved a strong convergence theorem for the common fixed point of a finite family of quasi-Bregman nonexpansive mappings. Pang et al. [21] proved weak convergence theorems for Bregman relatively nonexpansive mappings. Shahzad and Zegeye [22] proved a strong convergence theorem for multivalued Bregman relatively nonexpansive mappings, while Zegeye and Shahzad [23] proved a strong convergence theorem for a finite family of Bregman weak relatively nonexpansive mappings.
Motivated and inspired by the above works, in 2015 Ugwunnadi et al. [24] proved a new strong convergence theorem for a finite family of closed quasi-Bregman strictly pseudocontractive mappings and a system of equilibrium problems in a real reflexive Banach space.
The purpose of this paper is to introduce and consider a new hybrid shrinking projection algorithm for finding a common element of the set of solutions of a system of equilibrium problems, the set of solutions of a system of variational inequality problems, the set of solutions of a system of optimization problems, the common fixed point set of a uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings in reflexive Banach spaces. Strong convergence theorems have been proved under the appropriate conditions. The main innovative points in this paper are as follows: (1) the notion of uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings is presented and the useful conclusions are given; (2) the relative examples of the uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings are given in classical Banach spaces \(l^{2}\) and \(L^{2}\); (3) the hybrid shrinking projection method presented in this paper modified some mistakes in the recent result of Ugwunnadi et al. [24]. These 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 \(\langle\cdot,\cdot\rangle\) is 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
([25])
Let \(f :E \rightarrow(-\infty,+\infty]\) be a proper, convex, and lower semi-continuous 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
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. [25, 26]). We will need the following results.
Proposition 2.2
([27])
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
([27])
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 semi-continuous 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, 22]. 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 [23], authors 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. [28]). 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. [29]): 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 [26, 27].
Proposition 2.6
([30])
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
([32])
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. [31]) 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
([24])
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. [23]) 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 single-valued.
Proposition 2.9
([33])
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
([31])
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 recent years, the following notions have been presented by some authors.
A point \(p \in C\) is said to be asymptotic fixed point of a map T if there exists a sequence \(\{x_{n}\}\) in C which converges weakly to p such that \(\lim_{n\rightarrow\infty}\|x_{n}-Tx_{n}\|= 0\). We denote by \(\widehat{F}(T)\) the set of asymptotic fixed points of T. A point \(p \in C\) is said to be strong asymptotic fixed point [34] of a mapping T if there exists a sequence \(\{x_{n}\}\) in C which converges strongly to p such that \(\lim_{n\rightarrow\infty}\|x_{n}-Tx_{n}\|= 0\). We denote by \(\widetilde{F}(T)\) the set of strong asymptotic fixed points of T. Let \(f : E \rightarrow R\), a mapping \(T : C \rightarrow C\) is said to be Bregman relatively nonexpansive [17] if \(F(T)=\widehat{F}(T)\) and \(D_{f}(p, T(x))\leq D_{f}(p, x)\) for all \(x \in C\) and \(p \in F(T)\). The mapping \(T : C \rightarrow C\) is said to be Bregman weak relatively nonexpansive if \(F(T)=\widetilde{F}(T)\) and \(D_{f}(p, T(x))\leq D_{f}(p, x)\) for all \(x \in C\) and \(p \in F(T)\). The mapping \(T : C \rightarrow C\) is said to be quasi-Bregman relatively nonexpansive [24] if \(F(T)\neq\emptyset\) and \(D_{f}(p, T(x))\leq D_{f}(p, x)\) for all \(x \in C\) and \(p \in F(T)\). In [24] quasi-Bregman relatively nonexpansive is called left quasi-Bregman relatively nonexpansive. A mapping \(T : C \rightarrow C\) is said to be right quasi-Bregman relatively nonexpansive [24] if \(F(T)\neq\emptyset\) and \(D_{f}( T(x),p)\leq D_{f}(x, p)\) for all \(x \in C\) and \(p \in F(T)\).
In [24], authors presented the definition of quasi-Bregman strictly pseudocontractive mapping. In this paper, we extend this definition to the quasi-Bregman pseudocontractive mapping as follows.
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 quasi-Bregman k-pseudocontractive if there exists a constant \(k\in[0,+\infty)\) such that
If \(k\in[0,1)\), the mapping T is said to be quasi-Bregman strictly pseudocontractive. If \(k=1\), the mapping T is said to be quasi-Bregman pseudocontractive. The mapping T is said to be Bregman quasi-nonexpansive if
In this paper, we will use the following definition.
Definition 2.12
([34])
Let C be a nonempty, closed, and convex subset of E. Let \(\{T_{n}\}\) be a 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.
The next lemmas have been proved in [24], which is useful for the results of [24], but in this paper we do not use Lemma 2.13 and Lemma 2.15.
Lemma 2.13
([24])
Let \(f : E \rightarrow R\) be a Legendre function which is uniformly Fréchet differentiable and bounded on subsets of E, let C be a nonempty, closed, and convex subset of E, and let \(T : C \rightarrow C\) be a quasi-Bregman strictly pseudocontractive mapping with respect to f. Then, for any \(x\in C\), \(p\in F(T)\) and \(k\in[0, 1)\), the following holds:
Lemma 2.14
([24])
Let \(f : E \rightarrow R\) be a Legendre function which is uniformly Fréchet differentiable on bounded subsets of E, let C be a nonempty, closed, and convex subset of E, and let \(T : C \rightarrow C\) be a quasi-Bregman strictly pseudocontractive mapping with respect to f. Then \(F(T)\) is closed and convex.
Lemma 2.15
([24])
Let E be a real reflexive Banach space, \(f : E \rightarrow(-\infty, +\infty]\) be a proper lower semi-continuous function, then \(f^{*} : E^{*} \rightarrow(-\infty, +\infty]\) is a proper weak ∗ lower semi-continuous and convex function. Thus, for all \(z \in E\), we have
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 \(A\equiv0\), \(\varphi(x)\equiv0\), 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 \(F\equiv0\), \(\varphi(x)\equiv0\), problem (2.3) 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 \(\operatorname{VI}(C, A)\).
Whenever \(F\equiv0\), \(A\equiv0\), problem (2.3) is equivalent to finding \(u\in C\) such that
which is called the convex optimization problem. The set of its solutions is denoted by \(\operatorname{MIN}(\varphi)\).
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 for finding an element \(x \in C\) such that
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:
for all \(x \in E\). The following two lemmas were proved in [14].
Lemma 2.16
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 2.17
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 single-valued.
-
(ii)
\(T_{r}\) is a firmly nonexpansive-type mapping, 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)
\(F(T_{r})=\widehat{F}(T_{r})=\mathit{EP}(F)\).
-
(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 2.18
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.,
Let \(\varphi(x): C\rightarrow R\) be a convex lower semi-continuous functional. For \(r > 0\), let \(K_{r} : E \rightarrow C\) be the mapping defined by
where
Then the following statements hold:
-
(i)
\(K_{r}\) is single-valued.
-
(ii)
\(K_{r}\) is a firmly nonexpansive-type mapping, 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)
\(F(K_{r})=\widehat{F}(K_{r})=\mathit{EP}\).
-
(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 2.17, we can get the conclusions. □
From [36] we have the following conclusion.
Theorem 2.19
Let E be a p-uniformly convex Banach space with \(p \geq2\). Then for all \(x,y \in E\), \(j(x)\in J_{p}(x)\), \(j(y)\in J_{p}(y)\),
where \(J_{p}\) is the generalized duality mapping from E into \(E^{*}\) and \(1/c\) is the p-uniformly convexity constant of E.
From Theorem 2.19, we know that the generalized duality mapping \(J_{p}: E\rightarrow E^{*}\) is a monotone operator. It is well known that if E is also smooth and 2-uniformly convex, the normalized duality mapping \(J=J_{2}: E\rightarrow E^{*}\) is a single-valued monotone operator.
3 Main results
We now prove the following theorem.
Theorem 3.1
Let C be a nonempty, closed, and convex subset of a real reflexive Banach space E and \(f : E \rightarrow R\) be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable, and totally convex on a bounded subset of E. Let \(\{F_{j}\}_{j=1}^{m}\) be finite bifunctions from \(C \times C\) to R satisfying (A1)-(A4) and let \(\{A_{j}\}_{j=1}^{m}: C\rightarrow E^{*}\) be finite monotone mappings, i.e.,
Let \(\{\varphi_{j} (x)\}_{j=1}^{m}: C\rightarrow R\) be finite convex lower semi-continuous functionals. Let \(\{T_{n}\}_{n=1}^{\infty}\) be a uniformly closed family of countable quasi-Bregman strictly pseudocontractive mappings from C into itself with uniformly \(k \in[0,1)\) such that \(F=\bigcap_{j=1}^{m}\mathit{EP}_{j} \cap (\bigcap_{n=1}^{\infty}F(T_{n}))\) is nonempty. For given \(x_{0} \in C\), let \(\{T_{n}\}_{n=1}^{\infty}\) be a sequence generated by
where
for \(j=1,2,3, \ldots, m\), and \(\{\alpha_{n}\}\), \(\{\beta_{j,n}\}\) are sequences satisfying \(\limsup_{n\rightarrow\infty} \alpha_{n} <1\), \(\{r_{n}\}\) is a sequence satisfying \(\liminf_{n\rightarrow\infty} r_{n} >0\). Then \(\{x_{n}\}\) converges to \(q= P_{F}^{f}x_{0}\).
Proof
We divide the proof into six steps.
Step 1. We show that \(C_{n}\) is closed and convex for all \(n\geq 1\). Let
then
Since \(C_{1}=C\) is closed and convex, it is sufficient to prove that the sets \(D_{n}\), \(E_{j,n}\) are closed and convex for all \(n\geq1\). We show that \(D_{n}\) is closed and convex for all \(n\geq1\). We rewrite \(D_{n}\) as follows:
From the above expression, we know that \(D_{n}\) is closed and convex for all \(n\geq1\).
Next we show that \(E_{j,n}\) is closed and convex for all \(n\geq1\), \(j=1,2,3,\ldots,m\). We rewrite \(E_{j,n}\) as follows:
From the above expression, we know that \(E_{j,n}\) is closed and convex for all \(n\geq1\), \(j=1,2,3,\ldots,m\). Therefore \(C_{n}\) is closed and convex for all \(n\geq1\).
Step 2. We show that \(F \subset C_{n}\) for all \(n\geq1\). Note that \(F \subset C_{1} = C\). Suppose \(F \subset C_{n}\) for \(n\geq1\), then for all \(p \in F \subset C_{n}\), since \(u_{j,n}=K^{(j)}_{r}(y_{n})\) for all \(n\geq1\), \(j=1,2,3, \ldots,m \), from Lemma 2.18, we have
where
Since
from (3.1) and (3.2) we know that \(p\in C_{n+1}\), which implies \(F \subset C_{n}\) 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 (3.3) 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 (3.4) 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 \(\bigcap_{n=1}^{\infty}F(T_{n})\).
Since \(x_{n+1}\in C_{n+1}\), we have from the definition of \(C_{n+1}\) that
which implies that \(\lim_{n\rightarrow\infty} D_{f}(x_{n+1},y_{n})=0\). Since f is totally convex on bounded subsets of E, f is sequentially consistent (see [35]). It follows that
From the uniform continuity of ∇f, we have
Since
we obtain that
Since f is strongly coercive and uniformly convex on bounded subsets of E, \(f^{*}\) is uniformly Fréchet differentiable on bounded sets. Moreover, \(f^{*}\) is bounded on bounded sets, and from (3.6) we obtain
Since \(\{T_{n}\}\) is uniformly closed and \(x_{n}\rightarrow p\), we have \(p \in\bigcap_{n=1}^{\infty}F(T_{n})\).
Step 5. We show that the limit of \(\{x_{n}\}\) belongs to \(\mathit{EP}_{j}\) for all \(j=1,2,3, \ldots ,m\).
We have proved that \(x_{n}\rightarrow p\) as \(n\rightarrow\infty\). Now let us show that \(p\in \mathit{EP}_{j}\) for any \(j=1,2,3, \ldots ,m\). Since \(x_{n+1}\in C_{n+1}\), we have from the definition of \(C_{n+1}\) that
Since \(\lim_{n\rightarrow\infty} D_{f}(x_{n+1},y_{n})=0\), we have
Since f is totally convex on bounded subsets of E, f is sequentially consistent (see [35]). It follows that
This together with (3.5) implies that
Since ∇f is uniformly norm-to-norm continuous on bounded subsets of E, from (3.5) we have \(\lim_{n\rightarrow\infty}\|\nabla f(u_{j,n})-\nabla f(y_{n})\|=0\). From \(\liminf_{n\rightarrow\infty}r_{n}>0\) it follows that
By the definition of \(u_{j,n}:=K^{(j)}_{r_{n}}y_{n}\), we have
where
We have from (A2) that
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
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 \(G_{j}(y_{t}, p)\leq0\). So, from (A1) we have
Dividing by t, we have
Letting \(t\rightarrow0\), from (A3) we can get
So, \(p\in \mathit{EP}_{j}\) for all \(j=1,2,3,\ldots,m\).
Step 6. Finally, we prove that \(p= P_{F}^{f}x_{0}\), from Proposition 2.10 we have
On the other hand, since \(x_{n}= P_{C_{n}}^{f}x_{0}\) and \(F \subset C_{n}\) for all n, also from Proposition 2.10, we have
By the definition of \(D_{f}(x,y)\), we know that
Combining (3.7), (3.8), and (3.9), we know that \(D_{f}(p,x_{0})=D_{f}(P_{F}^{f}x_{0}, x_{0})\). Therefore, it follows from the uniqueness of \(P_{F}^{f}x_{0}\) that \(p= P_{F}^{f}x_{0}\). This completes the proof. □
Remark 3.2
Theorem 3.1 includes the following three special cases.
(1) Take \(T_{n}\equiv I\), \(\varphi(x)\equiv0\), \(F(x,y)\equiv0\), where I denotes the identity operator, then the iterative sequence \(\{x_{n}\}\) converges strongly to a solution of the system of variational inequalities
In this case, the iterative sequence \(\{x_{n}\}\) is defined by
(2) Take \(T_{n}\equiv I\), \(\varphi(x)\equiv0\), \(A \equiv0\), where I denotes the identity operator, then the iterative sequence \(\{x_{n}\}\) converges strongly to a solution of the system of equilibrium problems
In this case, the iterative sequence \(\{x_{n}\}\) is defined by
(3) Take \(T_{n}\equiv I\), \(F(x,y)\equiv0\), \(A \equiv0\), where I denotes the identity operator, then the iterative sequence \(\{x_{n}\}\) converges strongly to a solution of the system of convex optimization problems
In this case, the iterative sequence \(\{x_{n}\}\) is defined by
4 Examples
Let E be a Hilbert 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 quasi-Bregman \((2n+1)\)-pseudocontractive mappings.
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 quasi-Bregman \((2n+1)\)-pseudocontractive mappings.
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 quasi-Bregman \((2n+1)\)-pseudocontractive mappings.
5 The mistakes in the result of Ugwunnadi et al. [24]
In [24], from page 10, line −3 to page 11, line 2, there exists a mistake ratiocination as follows.
Mistake ratiocination 1
Since \(x_{ n+1} \in C_{n+1}\), it follows from (3.6), (3.7) that
which implies from (3.20), (3.18), (3.13), and (3.14) that
However, (3.6), (3.7) are the following:
for any \(w \in F\),
for any \(w \in F\).
In fact, the authors attempt taking \(w=x_{n+1}\) in (3.6) and (3.7) to get the (∗). This is an obvious mistake since (3.6) and (3.7) are right for only \(w \in F\), but \(x_{n+1}\) does not belong to F. Therefore, the definition of an iterative sequence \(\{x_{n}\}\) must be modified so that \(\lim_{n\rightarrow\infty} D_{f}(x_{n+1},x_{n})=0\) implies \(\lim_{n\rightarrow\infty} D_{f}(x_{n+1},y_{n})=0\).
In [24], page 12, line 3, there exists another mistake ratiocination as follows.
Mistake ratiocination 2
Also, since \(y_{n}\rightarrow p\) as \(n\rightarrow\infty\), we have from Lemma 2.3, for each \(j=1,2,3, \ldots,m\),
In fact, we are proving that \(p\in \mathit{EP}_{j}\) for any \(j=1,2,3,\ldots,m\), therefore, if we do not know whether \(p\in \mathit{EP}_{j}\), then the above inequalities are not right since if \(p\in \mathit{EP}_{j}\), the above inequalities are right. In this paper, we have overcome these shortcomings by modifying the iterative scheme.
References
Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35, 171-174 (1972)
Takahashi, W, Takeuchi, Y, Kubota, R: Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 341, 276-286 (2008)
Schu, J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 43, 153-159 (1991)
Inchan, I: Strong convergence theorems of modified Mann iteration methods for asymptotically nonexpansive mappings in Hilbert spaces. Int. J. Math. Anal. 2, 1135-1145 (2008)
Bruck, RE, Kuczumow, T, Reich, S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq. Math. 65, 169-179 (1993)
Hecai, Y, Aichao, L: Projection algorithms for treating asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense. J. Inequal. Appl. (2013). doi:10.1186/1029-242X-2013-265
Qing, Y: Some results on asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense. J. Fixed Point Theory 2012, Article ID 1 (2012)
Qin, X, Huang, S, Wang, T: On the convergence of hybrid projection algorithms for asymptotically quasi-ϕ-nonexpansive mappings. Comput. Math. Appl. 61, 851-859 (2011)
Qin, X, Wang, L: On asymptotically quasi-ϕ-nonexpansive mappings in the intermediate sense. Abstr. Appl. Anal. 2012, Article ID 636217 (2012). doi:10.1155/2012/636217
Alber, YI: Metric and generalized projection operators in Banach spaces: properties and applications. In: Kartsatos, AG (ed.) Theory and Applications of Nonlinear Operators of Accretive and Monotone Type. Lecture Notes Pure Appl. Math., vol. 178, pp. 15-50. Dekker, New York (1996)
Hao, Y: Some results on a modified Mann iterative scheme in a reflexive Banach space. Fixed Point Theory Appl. 2013, Article ID 227 (2013). doi:10.1186/1687-1812-2013-227
Bregman, LM: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200-217 (1967)
Martin-Marquez, V, Reich, S, Sabach, S: Iterative methods for approximating fixed points of Bregman nonexpansive operators. Discrete Contin. Dyn. Syst. 6, 1043-1063 (2013)
Reich, S, Sabach, S: Two strong convergence theorems for Bregman strongly nonexpansive operators in reflexive Banach spaces. Nonlinear Anal. 73, 122-135 (2010)
Reich, S, Sabach, S: Existence and approximation of fixed points of Bregman firmly nonexpansive mappings in reflexive Banach spaces. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optim. Appl., vol. 49, pp. 301-316. Springer, New York (2011)
Suantai, S, Cho, YJ, Cholamjiak, P: Halpern’s iteration for Bregman strongly nonexpansive mappings in reflexive Banach spaces. Comput. Math. Appl. 64, 489-499 (2012)
Naraghirad, E, Yao, J-C: Bregman weak relatively nonexpansive mappings in Banach spaces. Fixed Point Theory Appl. 2013, Article ID 141 (2013)
Bauschke, HH, Borwein, JM, Combettes, PL: Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Commun. Contemp. Math. 3, 615-647 (2001)
Zegeye, H, Shahzad, N: Convergence theorems for right Bregman strongly nonexpansive mappings in reflexive Banach spaces. Abstr. Appl. Anal. 2014, Article ID 584395 (2014)
Alghamdi, MA, Shahzad, N, Zegeye, H: Strong convergence theorems for quasi-Bregman nonexpansive mappings in reflexive Banach spaces. J. Appl. Math. 2014, Article ID 580686 (2014)
Pang, CT, Naraghirad, E, Wen, CF: Weak convergence theorems for Bregman relatively nonexpansive mappings in Banach spaces. J. Appl. Math. 2014, Article ID 573075 (2014)
Shahzad, N, Zegeye, H: Convergence theorem for common fixed points of finite family of multivalued Bregman relatively nonexpansive mappings. Fixed Point Theory Appl. 2014, Article ID 152 (2014)
Zegeye, H, Shahzad, N: Strong convergence theorems for common fixed point of finite family of Bregman weak relatively nonexpansive mappings in reflexive Banach spaces. Sci. World J. 2014, Article ID 493450 (2014)
Ugwunnadi, GC, Ali, B, Idris, I, Minjibir, MS: Strong convergence theorem for quasi-Bregman strictly pseudocontractive mappings and equilibrium problems in Banach spaces. Fixed Point Theory Appl. 2014, Article ID 231 (2014)
Barbu, V, Precupanu, T: Convexity and Optimization in Banach Spaces. Springer, Dordrecht (2012)
Ambrosetti, A, Prodi, G: A Primer of Nonlinear Analysis. Cambridge University Press, Cambridge (1993)
Reich, S, Sabach, S: A strong convergence theorem for a proximal-type algorithm in reflexive Banach spaces. J. Nonlinear Convex Anal. 10, 471-485 (2009)
Censor, Y, Lent, A: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321-353 (1981)
Chen, G, Teboulle, M: Convergence analysis of a proximal-like minimization algorithm using Bregman functions. SIAM J. Optim. 3, 538-543 (1993)
Butnariu, D, Iusem, AN, Zalinescu, C: On uniform convexity, total convexity and convergence of the proximal points and outer Bregman projection algorithms in Banach spaces. J. Convex Anal. 10, 35-61 (2003)
Butnariu, D, Resmerita, E: Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces. Abstr. Appl. Anal. 2006, Article ID 84919 (2006)
Reich, S, Sabach, S: Two strong convergence theorems for a proximal method in reflexive Banach spaces. Numer. Funct. Anal. Optim. 31, 22-44 (2010)
Alber, Y, Butnariu, D: Convergence of Bregman projection methods for solving consistent convex feasibility problems in reflexive Banach spaces. J. Optim. Theory Appl. 92, 33-61 (1997)
Su, Y, Xu, H, Zhang, X: Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications. Nonlinear Anal. 73, 3890-3906 (2010)
Censor, Y, Lent, A: An iterative row-action method for interval convex programming. J. Optim. Theory Appl. 34, 321-353 (1981)
Su, Y, Xu, H: A duality fixed point theorem and applications. Fixed Point Theory 13(1), 259-265 (2012)
Acknowledgements
This project is supported by the National Natural Science Foundation of China under grant (11071279).
Author information
Authors and Affiliations
Corresponding author
Additional information
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 basic structure. YX performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Xu, Y., Su, Y. New hybrid shrinking projection algorithm for common fixed points of a family of countable quasi-Bregman strictly pseudocontractive mappings with equilibrium and variational inequality and optimization problems. Fixed Point Theory Appl 2015, 95 (2015). https://doi.org/10.1186/s13663-015-0347-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13663-015-0347-9