Let E be a real Banach space, and C be a nonempty, closed and convex subset of E. Let g:C×C→ℝ be a bifunction, the Equilibrium Problem with respect to g denoted by EP(g) is define as finding a point z∈C such that

(1.1)(z,y)≥0, ∀y∈C.

The EP(g) was shown by Blum and Oettli [7] to cover several other optimization problems such as monotone inclusion problems, saddle point problems, minimization problems, variational inequality problems and Nash equilibria in non-cooperative games. In addition, there are many other important problems, for example, the complementarity problem and fixed point problems, which can be written in the form of EP(g) (1.1). Thus, the EP(g) is a unifying model for several problems arising in physics, engineering, science, optimization, economics etc.

In the last two decades, the existence of solutions of the EP(g) have been mentioned in many papers, see for instance [7, 11, 13, 26, 36, 39, 40 ], and several iterative methods have been proposed for solving EP(g) and related optimization problems, see for instance [1, 2, 4, 14, 15, 17, 18, 19, 28, 29, 32, 30, 31, 38, 41, 42] and reference therein. In solving the EP(g) (1.1) it is necessary to assume that the bifunction g satisfies the following assumptions:

• (A1) g(x,x) =0 for all x ∈ C;

• (A2) g is monotone, that is g(x,y)+g(y,x) ≤ 0 for all x,y ∈ C;

• (A3) For all x,y, z∈ C

  limsupt↓0+g(tz+(1−t)x,y)≤g(x,y);

• (A4) For all x ∈ C, g(x,⋅) is convex and lower semicontinuous.

Definition 1.1. Let f:E→(−∞,+∞] be a convex and Gâteaux differentiable function. The function Df:dom f×int(dom f)→[0,+∞) defined by

Df(y,x)=f(y)−f(x)−⟨∇f(x),y−x⟩

is called a Bregman distance with respect to f.

From the definition, we know that the following properties are satisfied (see [6]):

(i) The three points identity, for any x∈dom f and y,z∈int(dom f)

(1.2)Df(x,y)+Df(y,z)−Df(x,z)=⟨∇f(z)−∇f(y),x−y⟩;

(ii) Four point identity, for any x,w∈dom f and y,z∈int(dom f)

(1.3)Df(x,y)−Df(x,z)−Df(w,y)+Df(w,z)=⟨∇f(z)−∇f(y),x−w⟩.

Definition 1.2. Let C be a nonempty closed convex subset of int(domf) and T:C→C be a mapping. A point x ∈ C is called a fixed point of T if Tx =x. We denote the set of all fixed points of T by F(T). The mapping T:C→C is called

(a) Bregman nonexpansive [33] if

Df(Tx,Ty)≤Df(x,y) ∀x,y∈C;

Bregman nonspreading [23] if

Df(Tx,Ty)+Df(Ty,Tx)≤Df(Tx,y)+Df(Ty,x),  ∀x,y∈C,

(c) (α,β,γ,δ)-generalized Bregman nonspreading [3, 16] if there exist α,β,γ,δ∈ℝ such that

αDf(Tx,Ty)+(1−α)Df(x,Ty)+γ{Df(Ty,Tx)−Df(Ty,x)}      ≤βDf(Tx,y)+(1−β)Df(x,y)+δ{Df(y,Tx)−Df(y,x)},                  ∀x,y∈C.

for all x,y∈C.

Next, we introduce a n-generalized Bregman nonspreading mapping in Banach spaces.

Definition 1.3. Let f:E→ℝ∪{+∞} be a convex and Gâteaux differentiable function and C be a nonempty closed convex subset of int(domf). A mapping T:C→C is called a n-generalized Bregman nonspreading mapping if there exist αi,βi,γi,δi∈ℝ (i=1,2,…,n) such that

(1.4)∑k=1nαkDf(Tn+1−kx,Ty)+(1−∑k=1nαk)Df(x,Ty)+∑k=1nγkDf(Ty,Tn+1−kx)−Df(Ty,x)≤∑k=1nβkDf(Tn+1−kx,y)+(1−∑k=1nβk)Df(x,y)+∑k=1nδkDf(y,Tn+1−kx)−Df(y,x),

for all x,y∈C.

Remark 1.4. From Definition 1.3,

(a) when n=2, (1.4) becomes

α1Df(T2x,Ty)+α2Df(Tx,Ty)+(1−α1−α2)Df(x,Ty)+γ1(Df(Ty,T2x)−Df(Ty,x))+γ2(Df(Ty,Tx)−Df(Ty,x))≤β1Df(T2x,y)+β2Df(Tx,y)+(1−β1−β2)Df(x,y)+δ1(Df(y,T2x)−Df(y,x))+δ2(Df(y,Tx)−Df(y,x)),

which is called 2-generalized Bregman nonspreading in the sense of [44], where f(x)=12||x||2.

(b) When n=1, then (1.4) becomes

α1Df(Tx,Ty)+(1−α1)Df(x,Ty)+γ1(Df(Ty,Tx)−Df(Ty,x))     ≤β1Df(Tx,y)+(1−β1)Df(x,y)+δ1(Df(y,Tx)−Df(y,x)),

which is the generalized Bregman nonspreading mapping in the sense of [3, 16]. Note that, the 2-generalized Bregman nonspreading mapping reduces to the generalized Bregman nonspreading mapping if α1=β1=γ1=δ1=0.

(c) The class of generalized Bregman nonspreading mapping reduces to Bregman nonspreading [23] if α1=β1=γ1=1 and δ1=0.

(d) The class of generalized Bregman nonspreading mapping reduces to Bregman nonexpansive [35] if α1=1 and β1=γ1=δ1=0.

We now present an example of Bregman nonspreading mapping which is not nonspreading in the usual Hilbert space setting.

Example 1.5. Let E=ℝ with the usual metric. Let f:E→ℝ be defined by f(x)=x10 for all x∈ℝ and T:[0,0.85]→[0,0.85] be defined by Tx=x2. We first show that T is not nonspreading, i.e.,

||Tx−Ty||2≤||x−y||2+2⟨x−Tx,y−Ty⟩ ∀x,y∈C, 

does not hold. Taking x = 0.5 and y = 0.85, then

||Tx−Ty||2=(x2−y2)2=[(0.5)2−(0.85)2]2=0.22325625,

while

||x−y||2+2⟨x−Tx,y−Ty⟩=(x−y)2+2(x−x2)(y−y2)          =(0.5−0.85)2+2(0.5−0.52)(0.85−0.852)          =0.18625.

Hence, T is not nonspreading. Put

h(x,y)=Df(Tx,Ty)+Df(Ty,Tx)−Df(Tx,y)−Df(Ty,x). 

By simple calculations, we obtain

Df(Tx,Ty)=x20+9y20−10x2y18,Df(Ty,Tx)=y20+9x20−10x18y2,Df(Tx,y)=x20+9y10−10y2x9,Df(Ty,x)=y20+9x10−10x9y2.

Then

h(x,y)=9y10(y10−1)+9x10(x10−1)−10x2y9(y9−1)−10x9y2(x9−1)  ≤0,

for all x,y∈[0,0.85]. Thus T is Bregman nonspreading.

We further give an example of 2-generalized Bregman nonspreading mapping which is not necessarily 1-generalized Bregman nonspreading.

Example 1.6. Let E=ℝ and f(x)=x22 then the associated Bregman distance is given by

Df(x,y)=f(x)−f(y)−⟨x−y,∇f(y)⟩   =12x2−12y2−(x−y)(y)   12(x−y)2, ∀x,y∈ℝ.

Define T:[0,2]→[0,2] by

(1.5)Tx=0, if x∈[0,2),1, if x=2.

It is easy to see that F(T)={0}. Let

h(x,y)=α1Df(T2x,Ty)+α2Df(Tx,Ty)+(1−α1−α2)Df(x,Ty)   +γ1(Df(Ty,T2x)−Df(Ty,x))+γ2(Df(Ty,Tx)−Df(Ty,x))   −β1Df(T2x,y)−β2Df(Tx,y)−(1−β1−β2)Df(x,y)   −δ1(Df(y,T2x)−Df(y,x))−δ2(Df(y,Tx)−Df(y,x)),

for all x,y∈[0,2]. We consider the following possible cases.

Case I: Suppose x=y=2, then Tx=Ty=1 and T2x=0. Thus

Df(Tx,Ty)=Df(Ty,Tx)=Df(x,y)=Df(y,x)=0,Df(x,Ty)=Df(Ty,x)=Df(Tx,y)=Df(y,Tx)=12,Df(T2x,Ty)=Df(Ty,T2x)=12,Df(T2x,y)=Df(y,T2x)=2.

Hence

h(x,y)=12−12(α2+γ2+β2+δ2)−2(β1+δ1). 

Case II: Suppose x = 2 and y∈[0,2), then Tx = 1 and Ty = T^{2x = 0.} Thus

Df(Tx,Ty)=Df(Ty,Tx)=12,Df(x,Ty)=Df(Ty,x)=2,Df(Tx,y)=Df(y,Tx)=12(y−1)2,Df(x,y)=Df(y,x)=12(y−2)2,Df(T2x,y)=Df(y,T2x)=y22,Df(T2x,Ty)=Df(Ty,T2x)=0.

Hence

h(x,y)=−12(y2−4y)−2(α1+γ1)−32(α2+γ2)    −2(y−2)(β2+δ1)−12(2y−3)(β2+δ2).

Case III: Suppose x,y∈[0,2) then Tx=Ty=T2x=0. Thus

Df(Tx,Ty)=Df(Ty,Tx)=Df(T2x,Ty)=Df(Ty,T2x)=0,Df(x,y)=Df(y,x)=12(x−y)2,Df(x,Ty)=Df(Ty,x)=x22,Df(Tx,y)=Df(y,Tx)=Df(T2x,y)=Df(y,T2x)=y22.

Hence

h(x,y)=(1−α1−α2)x22−x22(γ1+γ2)−y22(β1+β2)    −12(1−β1−β2)(x−y)2−δ1xy−x22−δ2xy−y22.

Choosing suitable choices of α1,α2,β1,β2,γ1,γ,δ1,δ2∈ℝ, for instance, α1=α2=β1=β2=γ1=γ2=1 and δ1=δ2=−1, we see that h(x,y)≤0 for all the cases. Hence, T is 2-generalized Bregman nonspreading. However, in this case, T is not 1-generalized Bregman nonspreading (since α1≠0,β1≠0,γ1≠0,δ1≠0).

In 2010, by making use of the Bregman projection, Reich and Sabach [33] studied some approximation methods for finding common zeros of maximal monotone operators in reflexive Banach spaces. They also studied some approximation techniques for finding common solutions of finitely many Bregman nonexpansive operators, see [35]. In the same sense, Kassay et al. [20] studied the approximation of solutions of system of variational inequalities in reflexive Banach spaces. It is worth noting that extension of many theory from Hilbert space to general Banach space suffer some difficulties because many of the useful techniques employed in Hilbert space (for instance the inner product and the nonexpansiveness of resolvent operators) are no longer valid in Banach spaces setting.

Motivated by the works given in [21, 35, 46], we prove some properties of the n-generalized Bregman nonspreading mappings in reflexive Banach space. Further, we introduce a hybrid method for finding a common solution of countable family of equilibrium problem and finite family of fixed points of n-generalized Bregman nonspreading mapping in reflexive Banach space. We also discuss some applications and numerical example to demonstrate the applicability of our iterative algorithm and result. The method and results present in this paper generalized and unify many previously known related results, see for instance [21, 22, 35, 45, 46].

In this section, we recall some definitions and preliminary results which will be used in the sequel. We denote the strong convergence (resp. weak convergence) of a sequence {xn}⊂E to a point x ∈ E by xn→x (resp. xn⇀x).

Let E be a real reflexive Banach space with the dual space E^* and C a nonempty closed convex subset of E. Throughout this paper, we shall assume that the mapping f:E→ℝ∪{+∞} is proper, convex and lower semi-continuous and also denote the domain of f by domf, where dom f={x∈E:f(x)<∞}. Let x∈ int(domf), the subdifferential of f at x is the convex set defined by

∂f(x)={x*∈E*:f(x)+⟨x*,y−x⟩≤f(y), ∀y∈E} 

and the Frénchet conjugate of f is the function f*:E*→(−∞,+∞] defined by

f*(y*)=sup{⟨y*,x⟩−f(x):x∈E}.

Let x ∈ int(domf), for any y ∈ E, the directional derivative of f at x is defined by

(2.1)fo(x,y):=limh→0f(x+hy)−f(x)h.

If the limit in (2.1) exists as h→0 for each y, then the function f is said to be Ga^teaux differentiable at x. In this case, the gradient of f at x is the linear function ∇f(x), which is defined by ⟨∇f(x),y⟩:=fo(x,y) for all y ∈ E. The function f is said to be Ga^teaux differentiable if it is Ga^teaux differentiable at each x ∈ int(dom f). When the limit as h→0 in (2.1) is attained uniformly for any y ∈ E with ||y||=1, we say that f is Fréchet differentiable at x. It is well known that f is Ga^teaux (resp. Fréchet) differentiable at x ∈ int(dom f) if and only if the gradient ∇ f is norm-to-weak^* (resp. norm-to-norm) continuous at x (see [6]).

Let E be a reflexive Banach space. The function f is called Legendre if and only if it satisfies the following two conditions:

(L1) f is Gâteaux differentiable, int(dom f)≠∅ and dom ∇f = int(dom f),

(L2) f^* is Gâteaux differentiable, int(dom f*)≠∅ and dom ∇f* = int(dom f^*).

Since E is reflexive, we know that (∇f)−1=∇f*, this together with conditions (L1) and (L2) implies that

ran∇f=dom∇f*=int(domf*),

and

ran∇f*=dom∇f=int(domf).

The notion of Legendre function in infinite dimensional spaces was first introduced by Bauschke, Borwein and Combettes in [6]. By their definition, the conditions (L1) and (L2) also yield that f and f^* are Gâteaux differentiable and strictly convex in the interior of their respective domains. It follows that f is Legendre if and only if f^* is Legendre (see [6], Corollary 5.5, p. 634).

One important and interesting example of Legendre function is 1p||⋅||p (1<p<∞) when E is a smooth and strictly convex Banach space. In this case, the gradient ∇f of f coincide with the generalized duality mapping of E. More examples of Legendre functions can be found in [5, 6]. In the rest of this paper, we always assume that f:E→ℝ∪{+∞} is a Legendre function.

Definition 2.1. Let f:E→(−∞,+∞] be a convex and Gâteaux differentiable function. The Bregman projection of x∈int(domf) onto the nonempty, closed and convex subset C⊂domf is the necessarily unique vector ProjCf(x)∈C satisfying

Df(ProjCf(x),x)=inf{Df(y,x):y∈C}. 

Remark 2.2.

1. If E is a Hilbert space and f(x)=12||x||2, then the Bregman projection ProjCf(x) is reduced to the metric projection of x onto C.

2. If E is smooth and strictly convex and f(x)=1p||x||p (1<p<∞), then the Bregman projection ProjCf(x) reduces to the generalized projection ΠC(x), which is defined by

Dp(ΠC(x),x):=inf{Dp(z,x):z∈C}.

It is known from [10] that z = ProjCf(x) if and only if

⟨∇f(x)−∇f(z),y−z⟩≤0  for all y∈C.

We also have

Df(y,ProjCf(x))+Df(ProjCf(x),x)≤Df(y,x)  for all x∈E, y∈C.

Similar to the metric projection in Hilbert space, the Bregman projection also has a variational characterization which is given below.

Lemma 2.3. [33] (Characterization of Bregman Projection)) Let f be totally convex on int(domf). Let C be a nonempty, closed and convex subset of int(domf) and x ∈ int(domf), if ω ∈ C, then the following conditions are equivalent:

(i)the vector ω is the Bregman projection of x onto C, with respect to f,

(ii) the vector ω is the unique solution of the variational inequality

(2.2)⟨∇f(x)−∇f(z),z−y⟩≥0  ∀y∈C,

(iii) the vector ω is the unique solution of the inequality

Df(y,z)+Df(z,x)≤Df(y,x)  ∀y∈C.

Definition 2.4. Let f:E→(−∞,+∞] be a convex and Ga^teaux differentiable function. The function f is called:

(i) totally convex at x if its modulus of totally convexity at x ∈ int(domf), that is, the bifunction v_f: int(domf)×[0,+∞)→[0,+∞) defined by

(2.3)vf(x,t):=inf{Df(y,x):y∈domf,||y−x||=t}

is positive for any t>0,

(ii) totally convex if it is totally convex at every point x ∈ int(dom f),

(iii) totally convex on bounded subset B of E, if v_f(B,t) is positive for any nonempty bounded subset B, where the function v_f: int(dom f)×[0,+∞)→[0,+∞] is defined by

(2.4)vf(B,t):=inf{vf(x,t):x∈B∩int(domf)}, t>0.

(iv) cofinite if domf*=E*,

(v) coercive if lim||x||→+∞(f(x)||x||)=+∞,

(vi) sequentially consistent if for any two sequences {xn} and {yn} in E such that {xn} is bounded,

(2.5)limn→∞Df(yn,xn)=0⇒limn→∞||yn−xn||=0.

For further details and examples on totally convex functions see [8, 9, 10].

Lemma 2.5. ([9]) The function f:E→ℝ is totally convex on bounded subsets if and only if it is sequentially consistent.

Lemma 2.6. ([34]) Let f:E→ℝ be a Ga^teaux differentiable and totally convex function. If x0∈E and the sequence {Df(x0,xn)} is bounded, then the sequence {xn} is also bounded.

Lemma 2.7. ([10]) Let f:E→(−∞,+∞] be a convex function whose domain contains at-least two points. Then the following statements holds:

(i) f is sequentially consistent if and only if it is totally convex on bounded subsets.

(ii) If f is lower semicontinuous, then f is sequential consistent if and only if it is uniformly convex on bounded subsets.

(iii) If f is uniformly strictly convex on bounded subsets, then it is sequentially consistent and the converse implication holds when f is lower semicontinuous, Fréchet differentiable on its domain, and the Fréchet derivative ∇f is uniformly continuous on bounded subsets.

Lemma 2.8. ([33]) If f:E→ℝ is 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^*.

Let f:E→ℝ be a convex Legendre and Gâteaux differentiable function. The function Vf:E×E*→[0,∞) associated with f defined by

Vf(x,x*)=f(x)−⟨x*,x⟩+f*(x*),  ∀x∈E,x*∈E*.

Then, V_f is non-negative and Vf(x,x*)=Df(x,∇f*(x*)) for all x∈E and x*∈E*. More so, by the subdifferential inequality,

Vf(x,x*)+⟨y*,∇f*(x*)−x⟩≤Vf(x,x*+y*)

for all x∈E and x*,y*∈E* (see [24]). In addition, if f:E→(−∞,+∞] is a proper lower semicontinuous function, then f*:E*→(−∞,+∞] is a proper weak^* lower semicontinuous and convex function. Hence, V_f is convex in the second variable. Thus, for all z ∈ E

(2.6)Df(z,∇f*(∑ i=1Nti∇f(xi)))≤∑ i=1NtiDf(z,xi),

where {xi}⊂E and {ti}⊂(0,1) with ∑ i=1Nti=1.

Let E be a Banach space and let Br:={z∈E:||z||≤r} for all r>0. Then, a function f:E→ℝ is said to be uniformly convex on bounded subsets of E if ρr(t)>0 for all t≥ 0, where ρr:[0,+∞)→[0,∞]is defined by

(2.7)ρr(t)=infx,y∈Br,||x−y||=t,α∈(0,1)αf(x)+(1−α)f(y)−f(αx+(1−α)y)α(1−α).

The function ρ_r is called the gauge of uniform convexity of f. More so, the function f:E→(−∞,+∞] is called totally coercive if

lim||x||→+∞(f(x)||x||)=+∞.

Lemma 2.9. ([27]) Let r > 0 be a constant and let f:E→ℝ be a continuous uniformly convex function on bounded subsets of E. Then

(2.8)f(∑ k=0∞αkxk)≤∑ k=0∞αkf(xk)−αiαjρr*(||xi−xj||),

for all i,j∈ℕ∪0, xk∈Br, αk∈(0,1) and k∈ℕ∪0 with ∑ k=0∞αk=1, where ρr* is the gauge of uniform convexity of f.

Let l∞ be the Banach lattice of bounded real sequences with the supremum norm. It is well known that there exists a bounded linear functional µ on l∞ such that the following three conditions hold:

(i) if {tn} in l∞ and tn≥0 for every n∈ℕ, then μ({tn})≥0,

(ii) if tn=1 for every n∈ℕ, then μ({tn})=1,

(iii) μ({tn+1})=μ({tn}) for all {tn} in l∞.

Here, {tn+1} denotes the sequence (t2,t3,…,tn,tn+1,…) in l∞. Such a functional µ is called a Banach limit and the value of µ at {tn} in l∞ is denoted by μntn. Therefore, condition (3) means μntn=μntn+1. If µ satisfies conditions (1) and (2), we call µ a mean on l∞ (see, for example, [43] for more details).

Lemma 2.10. ([12]) Let C be a nonempty, closed and convex subset of a real reflexive Banach space E. Let f:E→ℝ be strictly convex, continuous, strongly coercive, Gâteaux differentiable, locally bounded and local uniformly convex on E. Let T:C→C be a mapping and {xn} be a bounded sequence of C and µ be a mean on l∞. Supposet that

μnDf(xn,Ty)≤μnDf(xn,y) ∀y∈C.

Then, T has a fixed point in C.

Let T be a mapping from C into itself. A point x ∈ C is said to be an asymptotic fixed point of T if there exists a sequence {xn} in C which converges weakly to p and limn→∞||xn−Txn||=0. We denote the set of all asymptotic fixed points of T by F^(T).

Recall that a mapping T:C→C is said to be Bregman quasi-nonexpansive [27] if F(T)≠∅ and

Df(p,Tx)≤Df(p,x) ∀x∈C,p∈F(T).

A mapping T:C→C is to be Bregman relatively nonexpansive [27] if the following conditions are satisfied:

(i) F(T) is nonempty;

(ii) Df(p,Tv)≤Df(p,v), ∀p∈F(T), v∈C;

(iii) F^(T)=F(T).

Lemma 2.11. ([37]) Let C be a nonempty, closed and convex subset of a real reflexive Banach space E and let f:E→ℝ be a strictly convex and Gâteaux differentiable function. Let g:C×C→ℝ be a bifunction satisfying conditions (A1)-(A4). For all λ>0 be any given number and x ∈ E, there exists z ∈ C such that

(2.9)g(z,y)+1r⟨∇(z)−∇(x),y−z⟩≥0, ∀ y∈C.

Define the resolvent mapping Tr:E→2C as follows

(2.10)Resλ,gf(x)={z∈C:g(z,y)+1r⟨∇f(z)−∇f(x),y−z⟩≥0, ∀ y∈C},

then, Resλ,gf has the following properties:

(i) Resλ,gf is single-valued;

(ii) Resλ,gf is a firmly nonexpansive mapping, that is;

⟨Resλ,gfz−Resλ,gfy,∇f(Resλ,gfz)−∇f(Resλ,gfy)⟩≤⟨Resλ,gfz−Resλ,gfy,∇f(z)−∇f(y)⟩

∀z,y∈E;

(iii) F(Resλ,gf)=EP(g);

(iv) EP(g) is closed and convex.

It is easy to see that the resolvent operator satisfies the following inequality: for all r > 0, u ∈ EP(g) and x ∈ E, then

(2.11)Df(x,Resλ,gfx)+Df(Resλ,gfx,u)≤Df(x,u).

In this section, we present the existence and some properties of fixed points of n-generalized Bregman nonspreading mapping in a reflexive Banach space. This result extend the corresponding results of [45] and [25] to reflexive Banach space.

Proposition 3.1. Let E be a real reflexive Banach space and f:E→ℝ be a strictly convex and Gâteaux differentiable function. Let C⊂int(domf) be a nonempty, closed and convex set and T:C→C be a n-generalized Bregman nonspreading mapping. Then, the following are equivalent

(i) F(T) is nonempty;

(ii) {Tmz} is bounded for some z ∈ C and m∈ℕ.

proof First we show that (i) implies (ii). Suppose F(T)≠∅, then {Tmz}={z} for z∈F(T). So {Tmz} is bounded. Next, we show that (ii) implies (i). Let {Tmz} be bounded for some z ∈ C. Since T is n-Bregman generalized nonspreading, then there exist αi,βi,γi,δi∈ℝ for i=1,2,…,n,

such that

(3.1)∑k=1nαkDf(Tn+1−kx,Ty)+(1−∑k=1nαk)Df(x,Ty)+∑k=1nγk{Df(Ty,Tn+1−kx)−Df(Ty,x)}≤∑k=1nβkDf(Tn+1−kx,y)+(1−∑k=1nβk)Df(x,y)+∑k=1nδk{Df(y,Tn+1−kx)−Df(y,x)},

for all x,y ∈ C. Replacing x by T^m-1z in (3.1), we have that for any y,z ∈ C,

(3.2)∑k=1nαkDf(Tn+1−kTm−1z,Ty)+(1−∑k=1nαk)Df(Tm−1z,Ty)+∑k=1nγk{Df(Ty,Tn+1−kTm−1z)−Df(Ty,Tm−1z)}≤∑k=1nβkDf(Tn+1−kTm−1z,y)+(1−∑k=1nβk)Df(Tm−1z,y)+∑k=1nδk{Df(y,Tn+1−kTm−1z)−Df(y,Tm−1z)}.

Since {Tmz} is bounded, we can apply Banach limit µ to both sides of (3.2), then we have

μm(∑k=1nαkDf(Tm+n−kz,Ty)+(1−∑k=1nαk)Df(Tm−1z,Ty)+∑k=1nγk{Df(Ty,Tm+n−kz)−Df(Ty,Tm−1z)})≤μm(∑k=1nβkDf(Tm+n−kz,y)+(1−∑k=1nβk)Df(Tm−1z,y)+∑k=1nδk{Df(y,Tm+n−kz)−Df(y,Tm−1z)}).

Thus, we obtain

(3.3)∑k=1nαkμmDf(Tm+n−kz,Ty)+(1−∑k=1nαk)μmDf(Tm−1z,Ty)+∑k=1nγk{μmDf(Ty,Tm+n−kz)−μmDf(Ty,Tm−1z)}≤∑k=1nβkμmDf(Tm+n−kz,y)+(1−∑k=1nβk)μmDf(Tm−1z,y)+∑k=1nδk{μmDf(y,Tm+n−kz)−μmDf(y,Tm−1z)}.

Then

∑k=1nαkμmDf(Tmz,Ty)+(1−∑k=1nαk)μmDf(Tmz,Ty)+∑k=1nγk{μmDf(Ty,Tmz)−μmDf(Ty,Tmz)}≤∑k=1nβkμmDf(Tmz,y)+(1−∑k=1nβk)μmDf(Tmz,y)+∑k=1nδk{μmDf(y,Tmz)−μmDf(y,Tmz)}.

Hence

μmDf(Tmz,Ty)≤μmDf(Tmz,y). 

Therefore by Lemma 2.10, T has a fixed point in C. This completes the proof.

The following results follow as direct consequences of Theorem 3.1.

Corollary 3.2. Let C be a nonempty, closed and convex subset of a smooth, strictly convex Banach space E, let p be a real number such that 1<p<+∞ and let f be a function defined by f(x)=1p||x||p and T:C→C be a n-generalized Bregman nonspreading mapping. Then, the following assertions are equivalent:

(i) F(T) is nonempty;

(ii) {Tmz} is bounded for some z ∈ C.

Corollary 3.3. Let C be a nonempty bounded closed convex subset of a real reflexive Banach space E and f:E→ℝ be a strictly convex and Gâteaux differentiable function. Let T:C→C be a n-generalized Bregman nonspreading mapping. Then, T has a fixed point.

Remark 3.4. Corollary 3.2 is a generalization of the corresponding result in Theorem 3.2 of [45], where the equivalence between the two assertions was shown for p=2.

We now show another important property of the fixed points of n-generalized Bregman nonspreading mapping.

Proposition 3.5. Let C be a nonempty, closed and convex subset of a real reflexive Banach space E and f:E→ℝ be a strictly convex and Gâteaux differentiable function. Let T:C→C be a n-generalized Bregman nonspreading mapping such that F(T)≠∅. Then F(T) is closed and convex.

Proof. Let u∈F(T), then putting u=x∈F(T) in (1.4), we have

∑k=1nαkDf(u,Ty)+(1−∑k=1nαk)Df(u,Ty)+∑k=1nγk{Df(Ty,u)−Df(Ty,u)}≤∑k=1nβkDf(u,y)+(1−∑k=1nβk)Df(u,y)+∑k=1nδk{Df(y,u)−Df(y,u)},

which implies that

(3.4)Df(u,Ty)≤Df(u,y),  ∀u∈F(T),y∈C.

This means that T is quasi-Bregman nonexpansive. Now let {xn}⊂F(T) such that xn→p. Then

Df(p,Tp)=limn→∞Df(xn,Tp)≤Df(xn,p)=Df(p,p)=0. 

Hence, p∈F(T). Therefore F(T) is closed.

Next, we show that F(T) is convex. For any x,y∈F(T) and λ∈(0,1), let z=λx+(1−λ)y. Then

(3.5)Df(z,Tz)=f(z)−f(Tz)−⟨∇f(Tz),z−Tz⟩    =f(z)−f(Tz)−⟨∇f(Tz),λx+(1−λ)y−Tz⟩    =f(z)+λDf(x,Tz)+(1−λ)Df(y,Tz)−λf(x)−(1−λ)f(y)    ≤f(z)+λDf(x,z)+(1−λ)Df(y,z)−λf(x)−(1−λ)f(y)    =f(z)−f(z)−⟨∇f(z),λx+(1−λ)y−z⟩    =f(z)−f(z)−⟨∇f(z),z−z⟩    =0.

Hence, z = Tz. Therefore, F(T) is convex.

Using Corollary 3.3 and Proposition 3.5, we prove the following common fixed point theorem for a commutative family of n-generalized Bregman nonspreading mapping in a reflexive Banach space.

Theorem 3.6. Let f:E→ℝ be a strictly convex and Gâteaux differentiable function, C be a nonempty bounded closed convex subset of a real reflexive Banach space E and let {Tα}α∈I be a commutative family of n-generalized Bregman nonspreading mappings from C into itself. Then {Tα}α∈I has a common fixed point.

Proof. By Theorem 3.5, we know that F(T_α) is a closed convex subset of C. Since E is reflexive and C is a bounded closed and convex subset, C is weakly compact. To show that ∩α∈IF(Tα) is nonempty, it is sufficient to show that {F(Tα)}α∈I has a nonempty finite intersection property.

Now, let {T1,T2,…,TN} be a commutative finite family of n-generalized Bregman nonspreading mapping from C into itself. We prove by induction that {T1,T2,…,TN} has a common fixed point. To do this, we start by showing the case for N=2. By Corollary 3.3 and Theorem 3.5, F(T₁) is nonempty, bounded, closed and convex. Let u∈F(T1), since T1T2=T2T1, then we have T1T2u=T2T1u=T2u. This implies that T2u∈F(T1). Hence, F(T₁) is T₂-invariant. Thus, the restriction of T₂ to F(T₁) is a n-generalized Bregman nonspreading self mapping. By Corollary (3.3), T₂ has a fixed point in F(T₁), that is, we have z∈F(T1) such that T2z=z. Hence, z∈F(T1)∩F(T2).

Suppose that for some N ≥ 2, Γ=∩k=1NF(Tk) is nonempty. Then Γ is a nonempty, bounded, closed and convex subset of C and the restriction of TN+1 to Γ is a n-generalized Bregman nonspreading self mapping. By Corollary 3.3, TN+1 has a fixed point in Γ. This implies that Γ∩F(TN+1) is nonempty. Hence, ∩k=1N+1F(Tk) is nonempty. This completes the proof.

The following result will be used in the sequel.

Proposition 3.7. Let E be a real reflexive Banach space and let C be a nonempty, closed and convex subset of E. Let f:E→ℝ be a strongly coercive Legendre function which is bounded, uniformly Fréchet differentiable and totally convex on bounded subsets of E. Let T:C→C be a n-generalized Bregman nonspreading mapping. Then, for any x,y∈C, αi,βi,γi,δi∈ℝ, for i=1,2,…,n, we have

(3.6)0≤∑k=1n(βk−αk)(Df(Tn+1−kx,Ty)−Df(x,Ty))+Df(Ty,y)+⟨∇f(Ty)−∇f(y),∑k=1nβk(Tn+1−kx−x)+x−Ty⟩+∑k=1nδk{Df(y,Tn+1−kx)−Df(y,x)}−∑k=1nγk{Df(Ty,Tn+1−kx)−Df(Ty,x)}.

Proof. From the definition of n-generalized Bregman nonspreading mapping, we have

(3.7)∑k=1nαkDf(Tn+1−kx,Ty)+(1−∑k=1nαk)Df(x,Ty)+∑k=1nγk{Df(Ty,Tn+1−kx)−Df(Ty,x)}≤∑k=1nβkDf(Tn+1−kx,y)+(1−∑k=1nβk)Df(x,y)+∑k=1nδk{Df(y,Tn+1−kx)−Df(y,x)},

for all x,y∈C. This implies that

0≤∑k=1nβkDf(Tn+1−kx,y)+(1−∑k=1nβk)Df(x,y)+∑k=1nδk{Df(y,Tn+1−kx)−Df(y,x)}−∑k=1nαkDf(Tn+1−kx,Ty)−(1−∑k=1nαk)Df(x,Ty)−∑k=1nγk{Df(Ty,Tn+1−kx)−Df(Ty,x)}.

Hence, from the three points identity (1.2), we have

0≤∑k=1nβk(Df(Tn+1−kx,Ty)+Df(Ty,y)+⟨∇f(Ty)−∇f(y),Tn+1−kx−Ty⟩)+(1−∑k=1nβk)(Df(x,Ty)+Df(Ty,y)+⟨∇f(Ty)−∇f(y),x−Ty⟩)−∑k=1nαkDf(Tn+1−kx,Ty)−(1−∑k=1nαk)Df(x,Ty)−∑k=1nγk{Df(Ty,Tn+1−kx)−Df(Ty,x)}+∑k=1nδk{Df(y,Tn+1−kx)−Df(y,x)}.

Therefore

0≤∑k=1n(βk−αk)(Df(Tn+1−kx,Ty)−Df(x,Ty))+Df(Ty,y)+⟨∇f(Ty)−∇f(y),∑k=1nβk(Tn+1−kx−x)+x−Ty⟩+∑k=1nδk{Df(y,Tn+1−kx)−Df(y,x)}−∑k=1nγk{Df(Ty,Tn+1−kx)−Df(Ty,x)}.

The following result is another important property which characterized the n-generalized Bregman nonspreading mapping.

Proposition 3.8. Let T:C→C be a n-generalized Bregman nonspreading mapping. Suppose F(T)≠∅, then T is Bregman relatively nonexpansive.

Proof. It is clear that

Df(p,Tx)≤Df(p,x) ∀p∈F(T),x∈C.

We show that F^(T)=F(T). It is easy to see that F(T)⊂F^(T). Now let p∈F^(T), that is, there exist a sequence {xn}⊂C such that xn⇀p and ||xn−Txn||→0. Since f is uniformly Frćhet differentiable on bounded subsets of E, then ∇f is uniformly continuous and thus

(3.8)limn→∞||f(xn)−f(Txn)||=limn→∞||∇f(xn)−∇f(Txn)||=0.

Putting x = x_n and y = q in Proposition 3.7, we have

(3.9)0≤∑k=1n(βk−αk)(Df(Tn+1−kxn,Tq)−Df(xn,Tq))+Df(Tq,q)+⟨∇f(Tq)−∇f(q),∑k=1nβk(Tn+1−kxn−xn)+xn−Tq⟩+∑k=1nδk{Df(q,Tn+1−kxn)−Df(q,xn)}−∑k=1nγk{Df(Tq,Tn+1−kxn)−Df(Tq,xn)}.

Observe that

(3.10)Df(Tn+1−kxn,Tq)−Df(xn,Tq)=f(Tn+1−kxn)−f(Tq)            −⟨∇f(Tq),Tn+1−kxn−Tq⟩            −f(xn)+f(Tq)+⟨∇f(Tq),xn−Tq⟩          =f(Tn+1−kxn)−f(xn)            +⟨∇f(Tq),xn−Tq⟩            −⟨∇f(Tq),Tn+1−kxn−Tq⟩          =f(Tn+1−kxn)−f(xn)            +⟨∇f(Tq),xn−Tn+1−kxn⟩.

Similarly

(3.11)Df(q,Tn+1−kxn)−Df(q,xn)=f(xn)−f(Tn+1−kxn)+⟨∇f(xn),Tn+1−kxn−xn⟩            +⟨∇f(xn)−∇f(Tn+1−kxn),q−xn⟩,

and

(3.12)Df(Tq,Tn+1−kxn)−Df(Tq,xn)=f(xn)−f(Tn+1−kxn)            +⟨∇f(xn),Tn+1−kxn−xn⟩            +⟨∇f(xn)−∇f(Tn+1−kxn),Tq−xn⟩.

Substituting (3.10), (3.11) and (3.12) into (3.9), we have

(3.13)0≤∑k=1n(βk−αk)(f(Tn+1−kxn)−f(xn)+⟨∇f(Tq),xn−Tn+1−kxn⟩)+Df(Tq,q)+⟨∇f(Tq)−∇f(q),∑k=1nβk(Tn+1−kxn−xn)+xn−Tq⟩+∑k=1nδk{f(xn)−f(Tn+1−kxn)+⟨∇f(xn),Tn+1−kxn−xn⟩+⟨∇f(xn)−∇f(Tn+1−kxn),q−xn⟩}−∑k=1nγk{f(xn)−f(Tn+1−kxn)+⟨∇f(xn),Tn+1−kxn−xn⟩+⟨∇f(xn)−∇f(Tn+1−kxn),Tq−xn⟩}.

Taking limit as n→∞ in (3.13) and using (3.8), we have

0 ≤Df(Tq,q)+⟨∇f(Tq)−∇f(q),q−Tq⟩. 

Using the four points identity (1.3), we have

0≤Df(Tq,q)+Df(Tq,Tq)−Df(Tq,q)−Df(q,Tq)+Df(q,q)=−Df(q,Tq).

Thus Df(q,Tq)≤0 and then Df(q,Tq)=0. Since f is strictly convex, we have q = Tq. Hence, q∈F(T). Therefore F^(T)⊂F(T). This thus implies that F^(T)=F(T).