We introduce the class of (α,β)-hybrid mappings relative
to a Bregman distance Df in a Banach space, and then we study the fixed
point and weak convergence problem for such mappings.

1. Introduction

Let C be a nonempty subset of a Hilbert space H. A mapping T:C→H is said to be

nonexpansive if ∥Tx-Ty∥≤∥x-y∥,forallx,y∈C;

nonspreading if 2∥Tx-Ty∥2≤∥Tx-y∥2+∥Ty-x∥2,forallx,y∈C, cf. [1, 2];

hybrid if 3∥Tx-Ty∥2≤∥x-y∥2+∥Tx-y∥2+∥Ty-x∥2,forallx,y∈C, cf. [3].

Recently, Kocourek et al. [4] introduced a new class of nonlinear mappings in a Hilbert space containing the classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings and established some fixed point and ergodic theorems for mappings in this new class. For α,β∈ℝ, they call a mapping T:C→H

(α,β)-hybrid if
α‖Tx-Ty‖2+(1-α)‖x-Ty‖2≤β‖Tx-y‖2+(1-β)‖x-y‖2

for all x,y∈C.

Obviously, T is nonexpansive if and only if it is (1,0)-hybrid; T is nonspreading if and only if it is (2,1)-hybrid; T is hybrid if and only if it is (3/2,1/2)-hybrid.

Motivated by the above works, we extend the concept of (α,β)-hybrid from Hilbert spaces to Banach spaces. For a nonempty subset C of a Banach space X, a Gâteaux differentiable convex function f:X→(-∞,∞] and α,β∈ℝ, a mapping T:C→X is said to be

(α,β)-hybrid relative to Df if there are α,β∈ℝ such that
αDf(Tx,Ty)+(1-α)Df(x,Ty)≤βDf(Tx,y)+(1-β)Df(x,y)

for all x,y∈C, where Df is the Bregman distance associated with f. Sections 3 and 4 are devoted to investigation of the fixed point and weak convergence problem for such type of mappings, respectively. Our fixed point theorem extends that of [4].

2. Preliminaries

In what follows, X will be a real Banach space with topological dual X*, and f:X→(-∞,∞] will be a convex function. 𝒟 denotes the domain of f, that is, D={x∈X:f(x)<∞},
and 𝒟∘ denotes the algebraic interior of 𝒟, that is, the subset of 𝒟 consisting of all those points x∈𝒟 such that, for any y∈X∖{x}, there is z in the open segment (x,y) with [x,z]⊆𝒟. The topological interior of 𝒟, denoted by Int(𝒟), is contained in 𝒟∘. f is said to be proper provided that 𝒟≠∅. f is called lower semicontinuous (l.s.c.) at x∈X if f(x)≤liminfy→xf(y). f is strictly convex if
f(αx+(1-α)y)<αf(x)+(1-α)f(y)
for all x,y∈X and α∈(0,1).

The function f:X→(-∞,∞] is said to be Gâteaux differentiable at x∈X if there is f′(x)∈X* such that limt→0f(x+ty)-f(x)t=〈y,f′(x)〉
for all y∈X.

The Bregman distance Df associated with a proper convex function f is the function Df:𝒟×𝒟→[0,∞] defined byDf(y,x)=f(y)-f(x)+f∘(x,x-y),
where f∘(x,x-y)=limt→0+(f(x+t(x-y))-f(x))/t. Df(y,x) is finite valued if and only if x∈𝒟∘, compare Proposition 1.1.2 (iv) of [5]. When f is Gâteaux differentiable on 𝒟, (2.4) becomes
Df(y,x)=f(y)-f(x)-〈y-x,f′(x)〉,
and then the modulus of total convexity is the function νf:𝒟∘×[0,∞)→[0,∞] defined by νf(x,t)=inf{Df(y,x):y∈D,‖y-x‖=t}.
It is known thatνf(x,ct)≥cνf(x,t)
for all t≥0 and c≥1, compare Proposition 1.2.2 (ii) of [5]. By definition it follows thatDf(y,x)≥νf(x,‖y-x‖).

The modulus of uniform convexity of f is the function δf:[0,∞)→[0,∞] defined by δf(t)=inf{f(x)+f(y)-2f(x+y2):x,y∈D,‖x-y‖≥t}.

The function f is called uniformly convex if δf(t)>0 for all t>0. If f is uniformly convex then for any ε>0 there is δ>0 such thatf(x+y2)≤f(x)2+f(y)2-δ
for all x,y∈𝒟 with ∥x-y∥≥ε.

Note that for y∈𝒟 and x∈𝒟∘, we have f(x)+f(y)-2f(x+y2)=f(y)-f(x)-f(x+((y-x)/2))-f(x)1/2≤f(y)-f(x)-f∘(x,y-x)≤Df(y,x),
where the first inequality follows from the fact that the function t→(f(x+tz)-f(x))/t is nondecreasing on (0,∞). Thereforeνf(x,t)≥δf(t),
whenever x∈𝒟∘ and t≥0. For other properties of the Bregman distance Df, we refer readers to [5].

The normalized duality mapping J from X to 2X* is defined by Jx={x*∈X*:〈x,x*〉=‖x‖2=‖x*‖2}
for all x∈X.

When f(x)=∥x∥2 in a smooth Banach space, it is known that f′(x)=2J(x) for x∈X, compare Corollaries 1.2.7 and 1.4.5 of [6]. Hence we have
Df(y,x)=‖y‖2-‖x‖2-〈y-x,f′(x)〉=‖y‖2-‖x‖2-2〈y-x,Jx〉=‖y‖2+‖x‖2-2〈y,Jx〉.
Moreover, as the normalized duality mapping J in a Hilbert space H is the identity operator, we have
Df(y,x)=‖y‖2+‖x‖2-2〈y,x〉=‖y-x‖2.
Thus, in case f(x)=∥x∥2 in a Hilbert space, (1.5) coincides with (1.4). However, in general they are different as the following example shows.

Example 2.1.

Let f(x)=|x| for x∈ℝ. f is a continuous convex function with
Df(y,x)={0,ifx>0,y≥0orx<0,y≤0,-2y,ifx≥0,y≤0,2y,ifx≤0,y≥0.
Let C=[0,1] and define T:C→C by
T(x)={x,ifx∈(0,1],12,ifx=0.
If T were (α,β)-hybrid for some α,β∈ℝ, then we would have
α‖Tx-T0‖2+(1-α)‖x-T0‖2≤β‖Tx-0‖2+(1-β)‖x-0‖2,∀x∈(0,1].
Since
(2.18)⟺α|x-12|2+(1-α)|x-12|2≤β|x|2+(1-β)|x|2,∀x∈(0,1]⟺|x-12|2≤|x|2,∀x∈(0,1]⟺14≤x,∀x∈(0,1],
we see that T is not (α,β)-hybrid for any α,β∈ℝ. But some simple computations show that T is (1,0)-hybrid relative to Df.

A function g:X→(-∞,∞] is said to be subdifferentiable at a point x∈X if there exists a linear functional x*∈X* such that g(y)-g(x)≥〈y-x,x*〉,∀y∈X.
We call such x* the subgradient of g at x. The set of all subgradients of g at x is denoted by ∂g(x), and the mapping ∂g:X→2X* is called the subdifferential of g. For a l.s.c. convex function f,∂f is bounded on bounded subsets of Int(𝒟) if and only if f is bounded on bounded subsets there, compare Proposition 1.1.11 of [5]. A proper convex l.s.c. function f is Gâteaux differentiable at x∈Int(𝒟) if and only if it has a unique subgradient at x; in such case ∂f(x)=f′(x), compare Corollary 1.2.7 of [6].

The following lemma will be quoted in the sequel.

Lemma 2.2 (see Proposition 1.1.9 of [<xref ref-type="bibr" rid="B1">5</xref>]).

If a proper convex function f:X→(-∞,∞] is Gâteaux differentiable on Int(𝒟) in a Banach space X, then the following statements are equivalent.

The function f is strictly convex on Int(𝒟).

For any two distinct points x,y∈Int(𝒟), one has Df(y,x)>0.

For any two distinct points x,y∈Int(𝒟), one has

〈x-y,f′(x)-f′(y)〉>0.

Throughout this paper, F(T) will denote the set of all fixed points of a mapping T.

3. Fixed Point Theorems

In this section, we apply Lemma 2.2 to study the fixed point problem for mappings satisfying (1.5).

Theorem 3.1.

Let X be a reflexive Banach space and let f:X→(-∞,∞] be a l.s.c. strictly convex function so that it is Gâteaux differentiable on Int(𝒟) and is bounded on bounded subsets of Int(𝒟). Suppose C⊆Int(𝒟) is a nonempty closed convex subset of X and T:C→C is (α,β)-hybrid relative to Df. Then the following two statements are equivalent.

There is a point x∈C such that {Tnx}n∈ℕ is bounded.

F(T)≠∅.

Proof.

If F(T)≠∅, then {Tnv}n∈ℕ={v} is bounded for any v∈F(T). Now assume that {Tnx}n∈ℕ is bounded for some x∈C, and for any n∈ℕ, let Sn(x)=(1/n)∑k=0n-1Tkx. Then {Snx}n∈ℕ is bounded, and so, in view of X being reflexive, it has a subsequence {Snix}i∈ℕ so that Snix converges weakly to some v∈C as ni→∞. Since T is (α,β)-hybrid relative to Df, we have, for any y∈C and k∈ℕ∪{0},
0≤βDf(Tk+1x,y)+(1-β)Df(Tkx,y)-[αDf(Tk+1x,Ty)+(1-α)Df(Tkx,Ty)].
Rewrite αDf(Tk+1x,Ty)+(1-α)Df(Tkx,Ty) as
αDf(Tk+1x,Ty)+(1-α)Df(Tkx,Ty)=α[f(Tk+1x)-f(Ty)-〈Tk+1x-Ty,f′(Ty)〉]+(1-α)[f(Tkx)-f(Ty)-〈Tkx-Ty,f′(Ty)〉]=αf(Tk+1x)+(1-α)f(Tkx)-f(Ty)-〈α(Tk+1x-Ty)+(1-α)(Tkx-Ty),f′(Ty)〉=αf(Tk+1x)+(1-α)f(Tkx)-f(Ty)-〈αTk+1x+(1-α)Tkx-Ty,f′(Ty)〉.
Similarly, we also have
βDf(Tk+1x,y)+(1-β)Df(Tkx,y)=βf(Tk+1x)+(1-β)f(Tkx)-f(y)-〈βTk+1x+(1-β)Tkx-y,f′(y)〉.
Consequently, we obtain from (3.1) that
0≤(β-α)[f(Tk+1x)-f(Tkx)]+[f(Ty)-f(y)]+〈Tkx+α(Tk+1x-Tkx)-Ty,f′(Ty)〉-〈Tkx+β(Tk+1x-Tkx)-y,f′(y)〉.
Summing up these inequalities with respect to k=0,1,…,n-1, we get
0≤(β-α)[f(Tnx)-f(x)]+n[f(Ty)-f(y)]+〈nSnx+α(Tnx-x)-nTy,f′(Ty)〉-〈nSnx+β(Tnx-x)-ny,f′(y)〉.
Dividing the above inequality by n, we have
0≤β-αn[f(Tnx)-f(x)]+f(Ty)-f(y)+〈Snx+αn(Tnx-x)-Ty,f′(Ty)〉-〈Snx+βn(Tnx-x)-y,f′(y)〉.
Replacing n by ni and letting ni→∞, we obtain from the fact that {f(Tnx)}n∈ℕ is bounded that
0≤f(Ty)-f(y)+〈v-Ty,f′(Ty)〉-〈v-y,f′(y)〉.
Putting y=v in (3.7), we get
0≤f(Tv)-f(v)+〈v-Tv,f′(Tv)〉,
that is,
0≤-Df(v,Tv),
from which follows that Df(v,Tv)=0. Therefore Tv=v by Lemma 2.2.

Since the function f(x)=∥x∥2 in a Hilbert space H satisfies all the requirements of Theorem 3.1, the corollary below follows immediately.

Corollary 3.2 (see [<xref ref-type="bibr" rid="B5">4</xref>]).

Let C be a nonempty closed convex subset of Hilbert space H and Suppose T:C→C is (α,β)-hybrid. Then the following two statements are equivalent.

There exists x∈C such that {Tn(x)}n∈ℕ is bounded.

T has a fixed point.

We now show that the fixed point set F(T) is closed and convex under the assumptions of Theorem 3.1.

A mapping T:C→X is said to be quasi-nonexpansive with respect to Df if F(T)≠∅ and Df(v,Tx)≤Df(v,x) for all x∈C and all v∈F(T).

The following lemma is shown in Huang et al. [7].

Lemma 3.3.

Let f:X→(-∞,∞] be a proper strictly convex function on a Banach space X so that it is Gâteaux differentiable on Int(𝒟), and let C⊆Int(𝒟) be a nonempty closed convex subset of X. If T:C→C is quasi-nonexpansive with respect to Df, then F(T) is a closed convex subset.

Proposition 3.4.

Let f:X→(-∞,∞] be a proper strictly convex function on a reflexive Banach space X so that it is Gâteaux differentiable on Int(𝒟) and is bounded on bounded subsets of Int(𝒟), and let C⊆Int(𝒟) be a nonempty closed convex subset of X. Suppose T:C→C is (α,β)-hybrid relative to Df and has a point x0∈C such that {Tn(x0)}n∈ℕ is bounded. Then T is quasi-nonexpansive with respect to Df, and therefore F(T) is a nonempty closed convex subset of C.

Proof.

In view of Theorem 3.1, F(T)≠∅. Now, for any v∈F(T) and any y∈C, as T is (α,β)-hybrid relative to Df, we have
Df(v,Ty)=αDf(v,Ty)+(1-α)Df(v,Ty)=αDf(Tv,Ty)+(1-α)Df(v,Ty)≤βDf(Tv,y)+(1-β)Df(v,y)=βDf(v,y)+(1-β)Df(v,y)=Df(v,y),
so T is quasi-nonexpansive with respect to Df, and hence F(T) is a nonempty closed convex subset of C by Lemma 3.3.

For the remainder of this section, we establish a common fixed point theorem for a commutative family of (α,β)-hybrid mappings relative to Df.

Lemma 3.5.

Let X be a reflexive Banach space and let f:X→(-∞,∞] be a l.s.c. strictly convex function so that it is Gâteaux differentiable on Int(𝒟) and is bounded on bounded subsets of Int(𝒟). Suppose C⊆Int(𝒟) is a nonempty bounded closed convex subset of X and {T1,T2,…,TN} is a commutative finite family of (α,β)-hybrid mappings relative to Df from C into itself. Then {T1,T2,…,TN} has a common fixed point.

Proof.

We prove this lemma by induction with respect to N. To begin with, we deal with the case that N=2. By Proposition 3.4, we see that F(T1) and F(T2) are nonempty bounded closed convex subsets of X. Moreover, F(T1) is T2-invariant. Indeed, for any v∈F(T1), it follows from T1T2=T2T1 that T1T2v=T2T1v=T2v, which shows that T2v∈F(T1). Consequently, the restriction of T2 to F(T1) is (α,β)-hybrid relative to Df, and so by Theorem 3.1, T2 has a fixed point u∈F(T1), that is, u∈F(T1)∩F(T2).

By induction hypothesis, assume that for some n≥2, E=∩k=1nF(Tk) is nonempty. Then E is a nonempty closed convex subset of X, and the restriction of Tn+1 to E is a (α,β)-hybrid mapping relative to Df from E into itself. By Theorem 3.1, Tn+1 has a fixed point in X. This shows that E∩F(Tn+1)≠∅, that is, ∩k=1n+1F(Tk)≠∅, completing the proof.

Here we would like to remark that in the above lemma, the assumption f is bounded on bounded subsets of Int(𝒟) can be dropped by checking the proof of Theorem 3.1 and noting that we have assumed C is bounded.

Theorem 3.6.

Let X be a reflexive Banach space and let f:X→(-∞,∞] be a l.s.c. strictly convex function so that it is Gâteaux differentiable on Int(𝒟). Suppose C⊆Int(𝒟) is a nonempty bounded closed convex subset of X and {Ti}i∈I is a commutative family of (α,β)-hybrid mappings relative to Df from C into itself. Then {Ti}i∈I has a common fixed point.

Proof.

Since C is a nonempty bounded closed convex subset of the reflexive Banach space X, it is weakly compact. By Proposition 3.4, each F(Ti) is a nonempty weakly compact subset of C. Therefore, the conclusion follows once we note that {F(Ti)}i∈I has the finite intersection property by Lemma 3.5.

4. Weak Convergence Theorems

In this section, we discuss the demiclosedness and the weak convergence problem of (α,β)-hybrid mappings relative to Df. We denote the weak convergence and strong convergence of a sequence {xn} to v in a Banach space X by xn⇀v and xn→v, respectively. For a nonempty closed convex subset C of a Banach space X, a mapping T:C→X is demiclosed if for any sequence {xn} in C with xn⇀v and xn-Txn→0, one has Tv=v.

The following Opial-like inequality for the Bregman distance is proved in [7]. For the Opial’s inequality we refer readers to Lemma 1 of [8].

Lemma 4.1.

Suppose f:X→(-∞,∞] is a proper strictly convex function so that it is Gâteaux differentiable on Int(𝒟) in a Banach space X and {xn}n∈ℕ is a sequence in 𝒟 such that xn⇀v for some v∈Int(𝒟). Then
liminfn→∞Df(xn,v)<liminfn→∞Df(xn,y),∀y∈Int(D)withy≠v.

Proposition 4.2.

Suppose f:X→(-∞,∞] is a strictly convex function so that it is Gâteaux differentiable on Int(𝒟) and is bounded on bounded subsets of Int(𝒟) in a Banach space X, and suppose C is a closed convex subset of Int(𝒟). If T:C→X is (α,β)-hybrid relative to Df, then T is demiclosed.

Proof.

Let {xn} be any sequence in C with xn⇀v and xn-Txn→0. We have to show that Tv=v. Since f is bounded, by Proposition 1.1.11 of [5] there exists a constant M>0 such that
max{sup{‖f′(xn)‖:n∈N},‖f′(Tv)‖,‖f′(v)‖}≤M.
Note that, for α∈ℝ,
αDf(Txn,Tv)+(1-α)Df(xn,Tv)=α[f(Txn)-f(Tv)-〈Txn-Tv,f′(Tv)〉]+(1-α)[f(xn)-f(Tv)-〈xn-Tv,f′(Tv)〉]=α[f(Txn)-f(Tv)+f(Tv)-f(xn)]-〈Txn-Tv-xn+Tv,f′(Tv)〉+f(xn)-f(Tv)-〈xn-Tv,f′(Tv)〉=α[f(Txn)-f(xn)-〈Txn-xn,f′(Tv)〉]+Df(xn,Tv).
Similarly, for β∈ℝ, we have
βDf(Txn,v)+(1-β)Df(xn,v)=β[f(Txn)-f(xn)-〈Txn-xn,f′(v)〉]+Df(xn,v).
Thus we obtain from the (α,β)-hybrid of T that
α[f(Txn)-f(xn)-〈Txn-xn,f′(Tv)〉]+Df(xn,Tv)≤β[f(Txn)-f(xn)-〈Txn-xn,f′(v)〉]+Df(xn,v),
which implies that
Df(xn,Tv)≤Df(xn,v)+(α-β)(f(xn)-f(Txn))+〈Txn-xn,αf′(Tv)-βf′(v)〉≤Df(xn,v)+(α-β)〈xn-Txn,f′(xn)〉+〈Txn-xn,αf′(Tv)-βf′(v)〉≤Df(xn,v)+3M(|α|+|β|)‖xn-Txn‖.
Consequently, if Tv≠v, then Lemma 4.1 implies that
liminfn→∞Df(xn,v)<liminfn→∞Df(xn,Tv)=liminfn→∞[Df(xn,v)+3M(|α|+|β|)‖xn-Txn‖]=liminfn→∞Df(xn,v),
a contradiction. This completes the proof.

A mapping T:C→C is said to be asymptotically regular if, for any x∈C, the sequence {Tn+1x-Tnx} tends to zero as n→∞.

Theorem 4.3.

Suppose the following conditions hold.

f:X→(-∞,∞] is l.s.c. uniformly convex function so that it is Gâteaux differentiable on Int(𝒟) and is bounded on bounded subsets of Int(𝒟) in a reflexive Banach space X.

C⊆Int(𝒟) is a closed convex subset of X.

T:C→C is (α,β)-hybrid relative to Df and is asymptotically regular with a bounded sequence {Tnx0}n∈ℕ for some x0∈C.

The mapping x→f′(x) for x∈X is weak-to-weak* continuous.

Then for any x∈C, {Tnx}n∈ℕ is weakly convergent to an element v∈F(T).Proof.

Let v∈F(T) and x∈C. If {Tnx}n∈ℕ is not bounded, then there is a subsequence {Tnix}i∈ℕ such that ∥v-Tnix∥≥1 for all i∈ℕ and ∥v-Tnix∥→∞ as i→∞. From (4.3.3), for any n∈ℕ∪{0} we have
Df(v,Tn+1x)=αDf(Tv,Tn+1x)+(1-α)Df(v,Tn+1x)≤βDf(Tv,Tnx)+(1-β)Df(v,Tnx)=Df(v,Tnx)≤Df(v,x),
which in conjunction with (2.7), (2.8), and (2.12) implies that
Df(v,x)≥Df(v,Tnix)≥νf(Tnix,‖v-Tnix‖)≥‖v-Tnix‖νf(Tnix,1)≥‖v-Tnix‖δf(1)⟶∞,asi⟶∞,
a contradiction. Therefore, for any x∈X, {Tnx}n∈ℕ is bounded, and so it has a subsequence {Tnjx}j∈ℕ which is weakly convergent to w for some w∈C. As Tnjx-Tnj+1x→0, it follows from the demiclosedness of T that w∈F(T). It remains to show that Tnx⇀w as n→∞. Let {Tnkx}k∈ℕ be any subsequence of {Tnx}n∈ℕ so that Tnkx⇀u for some u∈C. Then u∈F(T). Since both of {Df(w,Tnx)}n∈ℕ and {Df(u,Tnx)}n∈ℕ are decreasing, we have
limn→∞[Df(w,Tnx)-Df(u,Tnx)]=limn→∞[f(w)-f(u)-〈w-u,f′(Tnx)〉]=a
for some a∈ℝ. Particularly, from (4.3.4) we obtain
a=limnj→∞[f(w)-f(u)-〈w-u,f′(Tnjx)〉]=f(w)-f(u)-〈w-u,f′(w)〉,a=limnk→∞[f(w)-f(u)-〈w-u,f′(Tnkx)〉]=f(w)-f(u)-〈w-u,f′(u)〉.
Consequently, 〈w-u,f′(w)-f′(u)〉=0, and hence w=u by the strict convexity of f. This shows that Tnx⇀w for some w∈F(T).

5. Conclusion

In this paper, we have introduced the Bregman distance Df and a new class of mappings, (α,β)-hybrid mappings relative to Df in Banach spaces. We also have given and proved a necessary and sufficient condition for the existence of fixed points of the introduced mappings and some properties of the mappings. In fact, our result properly extends the Kocourek-Takahashi-Yao fixed point theorems for (α,β)-hybrid mappings in Hilbert spaces in 2010 [4]. Since the Kocourek-Takahashi-Yao fixed point theorems can be applied to study the nonexpansive mappings, the nonspreading mappings, and the hybrid mappings, our theorems in this paper are also good for these famous mappings in the field of fixed point theory.

According to [7], the fixed point theorems in this paper can be expected to discuss in a wider class of mappings, called point-dependent (α,β)-hybrid mappings relative to Df in Banach spaces. In a point-dependent (α,β)-hybrid mapping, the α and the β are not constant again but two functions from a nonempty subset of a Banach space to real numbers. Therefore, inequality (1.4) for point-dependent (α,β)-hybrid mappings becomes α(y)‖Tx-Ty‖2+(1-α(y))‖x-Ty‖2≤β(y)‖Tx-y‖2+(1-β(y))‖x-y‖2
for all x,y∈C.

In addition, Noor [9–11] provides algorithms to search the fixed points of nonexpansive mappings and then combines the result with general variational inequalities to study applied mathematical problems. We are motivated by that and expect to develop algorithms from the theorems of this paper to approach the fixed points of the introduced mappings. Through the combination of the fixed point theorems and the corresponding algorithms, the introduced mappings of this paper would be able to be applied to more fields of applied mathematics.

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