# On Best Proximity Point Theorems and Fixed Point Theorems forp-Cyclic Hybrid Self-Mappings in Banach Spaces

Authors: M. De la Sen
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper relies on the study of fixed points and best proximity points of a class of so-called generalized point-dependent(K,λ)-hybrid p-cyclic self-mappings relative to a Bregman distance Df, associated with a Gâteaux differentiable proper strictly convex function f in a smooth Banach space, where the real functions λ and K quantify the point-to-point hybrid and nonexpansive (or contractive) characteristics of the Bregman distance for points associated with the iterations through the cyclic self-mapping. Weak convergence results to weak cluster points are obtained for certain average sequences constructed with the iterates of the cyclic hybrid self-mappings.


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## Body

## 1. Introduction and Preliminaries

The following objects are considered through the paper.(1)
The Hilbert spaceH on  the field X (in particular, R or C) is endowed with the inner product 〈x,y〉 which maps H×H to X, for all x,y∈H which maps H×H to X, where (X,∥∥) is a Banach space when endowed with a norm ∥∥ induced by the inner product and defined by ∥x∥=〈x,x〉1/2, for all x∈H. It is wellknown that all Hilbert spaces are uniformly convex Banach spaces and that Banach spaces are always reflexive.(2)
Thep(≥2)-cyclic self-mapping T:A→A with A:=⋃i∈p-Ai is subject to Ap+1≡Ap, where Ai(≠⌀)⊂H are p subsets of H, for all i∈p-={1,2,…,p}, that is, a self-mapping satisfying T(Ai)⊆Ai+1, foralli∈p-(3)
The functionf:D(≡domf)⊂X→(-∞,∞] is a proper convex function which is Gâteaux differentiable in the topological interior of the convex set D; intD,  that is, D:={x∈X:f(x)<∞}≠⌀ and convex since f is proper with
(1)f(αx+(1-α)y)≤αf(x)+(1-α)f(y),∀x,y∈D,∀α∈[0,1],since f is convex,  and for each x∈D, there is x*=f(x)∈X* (the topological dual of X) such that
(2)∃limt→0f(x+ty)-f(x)t=〈y,f′(x)〉,∀y∈D,
since f  is Gâteaux differentiable in intD where f′(x) denotes the Gâteaux derivative of f at x if x∈intD. On the other hand, f is said to be strictly convex if
(3)f(αx+(1-α)y)<αf(x)+(1-α)f(y),∀x,y(≠x)∈D,∀α∈(0,1).(4)
The Bregman distance (or Bregman divergence)Df associated with the proper convex function f  Df:D×D→(-∞,∞], where R0+:={z∈R:z≥0}=R+∪{0}, is defined by
(4)Df(y,x)=f(y)-f(x)-〈y-x,f′(x)〉,∀x,y∈D,provided that it is Gâteaux differentiable everywhere in intD. If f is not Gâteaux differentiable at x∈intD, then (4) is replaced by
(5)Df(y,x)=f(y)-f(x)+f0(x,x-y),
wheref0(x,x-y):=limt→0+((f(x+t(x-y))-f(x))/t)  and  Df(y,x) is finite if and only if x∈D0⊂D, the algebraic interior of D defined by
(6)D0:={x∈D:∃z∈(x,y),[x,z]⊆D;∀y∈X∖{x}}.
The topological interior of D is int(D):={x∈D:x∈fr(D)}⊂D0, where fr(D) is the boundary of D. It is well known that the Bregman distance does not satisfy either the symmetry property or the triangle inequality which are required for standard distances while they are always nonnegative because of the convexity of the function f:D(≡domf)⊂X→(-∞,∞]. The Bregman distance between sets B,C⊂H⊂X is defined as Df(B,C):=infx∈B,y∈CDf(x,y). If Ai∈A⊂H for i∈p-, then Dfi:=Df(Ai,Ai+1)=infx∈Ai,y∈Ai+1Df(x,y). Through the paper, sequences {xn}n∈N0≡{Tnx}n∈N0 with N0=N∪{0} are simply denoted by {Tnx} for the sake of notation simplicity.Fixed points and best proximity points of cyclic self-mappingsT:⋃i∈p-Ai→⋃i∈p-Ai in uniformly convex Banach spaces (X,∥∥) have been widely studied along the last decades for the cases when the involved sets intersect or not. See, for instance, [1–3] and references therein.  In parallel, interesting results have been obtained for both nonspreading, nonexpansive, and hybrid maps in Hilbert spaces including also to focus the related problems  via iterative methods  supported by fixed point theory and the use of more general mappings such as nonspreading and pseudocontractive mappings. See, for instance, recent background [4–7] and references therein. Let C be a nonempty subset of a Hilbert space H. On the other hand, it has to be pointed out that the characterization of several classes of iterative computations by invoking results of fixed point theory has received much attention in the background literature. See, for instance, [8–11] and references therein. In [12–18], the existence of fixed points of mappings T:C→H is discussed when T:C→H is:(1.1)
nonexpansive; that is,∥Tx-Ty∥≤∥x-y∥, for all x,y∈C,(1.2)
nonspreading; that is,∥Tx-Ty∥2≤∥x-y∥2+2〈x-Tx,y-Ty〉, for all x,y∈C,(1.3)
λ-hybrid [17]; that is, ∥Tx-Ty∥2≤∥x-y∥2+λ〈x-Tx,y-Ty〉, for all x,y∈C. If λ=1, then  T:C→H is referred to as hybrid [14, 15], and if λ=λ(y) and (1.3) is changed to,(1.4)
Df(Tx,Ty)≤Df(x,y)+λ(y)〈x-Tx,f′(y)-f′(Ty)〉, for all x,y∈C,wheref:D(≡domf)⊂X→(-∞,∞] is a Gâteaux differentiable convex function, then T:C→H is referred to as being point-dependent λ-hybrid relative to the Bregman distance Df, [16]. A well-known result is that a nonspreading mapping, and then a nonexpansive one,  on a nonempty closed convex subset C of a Hilbert space H has a fixed point if and only it has a bounded sequence on such a subset [18]. The result has been later on extended to λ-hybrid mappings, [17] and to point-dependent λ-hybrid ones [16]. As pointed out in [16], what follows directly from the previous definitions, T:C→H is nonexpansive if and only if it is 0-hybrid while it is nonspreading if and only if it is 2-hybrid; T is hybrid if and only if it is 1-hybrid.This paper is focused on the study of fixed points and best proximity points of a class of generalized point-dependent(K,λ)-hybrid p(≥2)-cyclic self-mappings T:⋃i∈p-Ai→⋃i∈p-Ai, relative to a Bregman distance Df in a smooth Banach space, where λ:⋃i∈p-Ai→R is a point-dependent real function in (1.4) quantifying the “hybrideness” of the p(≥2) cyclic self-mapping and K=K(y):⋃i∈p-Ai→[0,1] is added as a weighting factor in the first right-hand-side term of (1.4).  Such a function is defined through a point-dependent product of the particular point p-functions while quantifies either the “nonexpansiveness” or the “contractiveness” of the Bregman distance for points associated with the iterates of the cyclic self-mapping in each of the sets Ai×Ai+1∪Ai+1×Ai for i∈p-={1,2,…,p}, where Ai(i∈p-) are nonempty closed and convex. Thus, the generalization of the hybrid map studied in this paper has two main characteristics, namely, (a) a weighting point-dependent term is introduced in the contractive condition; (b) the hybrid self-mapping is a cyclic self-mappings.  Precise definitions and meaning of those functions are given in Definition 2 of Section 2 which are then used to get the main results obtained in the paper.  In most of the results obtained in this paper, the Bregman distance Df is defined associated with a Gâteaux differentiable proper strictly convex function f whose domain includes the union of the p subsets Ai(i∈p-) of the (K,λ)-hybrid p(≥2)-cyclic self-mapping which are not assumed, in general, to intersect.  Weak convergence results to weak cluster points of certain average sequences built with the iterates of the cyclic hybrid self-mappings are also obtained. In particular, such weak cluster points are proven to be also fixed points of the composite self-mappings on the sets Ai(i∈p-), even if such sets do not intersect, while they are simultaneously best proximity points of the point-dependent (K,λ)-hybrid p(≥2)-cyclic self-mapping relative to Df.


## 2. Some Fixed Point Theorems for Cyclic Hybrid Self-Mappings on the Union of Intersecting Subsets

The Bregman distance is not properly a distance, since it does not satisfy symmetry and the triangle inequality, but it is always nonnegative and leads to the following interesting result towards its use in applications of fixed point theory.Lemma 1.
Iff:D×D→(-∞,∞] is a proper strictly convex function being Gâteaux differentiable in intD, then
(7)Df(x,x)=0,∀x∈intD,(8)Df(y,x)>0,∀x,y(≠x)∈intD,(9)Df(y,x)+Df(x,y)=〈x-y,f′(x)-f′(y)〉≥0,∀x,y∈intD,(10)Df(y,x)-Df(x,y)=2(f(y)-f(x))-〈y-x,f′(x)+f′(y)〉,∀x,y∈intD.Proof.
By using (4) for Df(y,x) and defining Df(x,y)=f(x)-f(y)-〈x-y,f′(y)〉, for all x,y∈intD by interchanging x and y in the definition of Df(y,x) in (4),
(11)Df(y,x)+Df(x,y)=〈x,f′(x)〉+〈y,f′(y)〉-〈x,f′(y)〉>-〈y,f′(x)〉=〈x,f′(x)-f′(y)〉+〈y,f′(y)-f′(x)〉,
which leads to (9) since Df(y,x)≥0, for all x,y∈intD, [16, 17], if f:D×D→(-∞,∞] is proper strictly convex, and the fact that Df(x,y)≥0, for all x,y∈intD.
Equation (7) follows from (9) for x=y leading to 2Df(x,x)=0. To prove (8), take x,y(≠x)∈intD and proceed by contradiction using (4) by assuming that Df(y,x)=0 for such x,y(≠x)∈intD so that
(12)0=Df(y,x)=f(y)-f(x)-〈y-x,f′(x)〉=f(y)-f(x)+〈x-y,f′(x)-f′(y)〉+〈x-y,f′(y)〉>f(y)-f(x)-〈y-x,f′(y)〉=Df(y,x),
which contradicts Df(y,x)=0. Then, Df(y,x)>0, and, hence, (8) follows
(13)Df(y,x)-Df(x,y)=f(y)-f(x)-〈y-x,f′(x)〉-f(x)+f(y)+〈x-y,f′(y)〉=2(f(y)-f(x))+〈x-y,f′(y)+f′(x)〉,∀x,y∈intD,
And, hence, (10) via (7) and (9).The following definition is then used.Definition 2.
IfD∩Ai≠⌀, for all i∈p-, and f:D(≡domf)⊂X→(-∞,∞] is a proper convex function which is Gâteaux differentiable in intD, then the p-cyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai, where A:=⋃i∈p-Ai⊆intD⊂H and Ai≠⌀, for all i∈p-, is said to be a generalizedcontractive point-dependent (K,λ)-hybrid p(≥2)-cyclic self-mapping relative to Df if
(14)Df(Tx,Ty)≤Ki(y)Df(x,y)+λ(y)〈x-Tx,f′(y)-f′(Ty)〉,∀x∈Ai,∀y∈Ai+1,∀i∈p-,
for some given functions λ:⋃i∈p-Ai→R and Ki:Ai+1→(0,ai] with ai∈R+, for all i∈p-, where K:⋃i∈p-Ai+1→(0,1] defined by K(y)=∏j=ii+p-1[Kj(Tj-iy)] for any y∈Ai+1, for all i∈p-.
If, furthermore,K:⋃i∈p-Ai→(0,1), for all i∈p-, then  T:⋃i∈p-Ai→⋃i∈p-Ai is said to be a generalized point-dependent (K,λ)-hybrid p(≥2)-cyclic self-mapping relative to Df.Ifp=1, it is possible to characterize T:A1→A1 as a  trivial 1-cyclic  self-mapping with A1=A2 which does not need to be specifically referred to as 1-cyclic.AlthoughKi:Ai+1→(0,ai] depends on i∈p-, the whole K:⋃i∈p-Ai→(0,1) does not depend on i∈p- so that the cyclic self-mapping is referred to as generalizedpoint-dependent (K,λ)-hybrid in the definition.The following concepts are useful.f:D(≡domf)⊂X→(-∞,∞] is said to be totally convex if the modulus of total convexity vf:D0×[0,∞)→[0,∞]; that is, vf(x,t)=inf{Df(x,y):y∈D,∥y-x∥=t} is positive for t>0.f:D(≡domf)⊂X→(-∞,∞] is said to be uniformly convex if the modulus of uniform convexity δf:[0,∞)→[0,∞]; that is, δf(t)=inf{f(x)+f(y)-2f((x+y)/2):x,y∈D,∥y-x∥≥t} is positive for t>0. It holds that vf(x,t)≥δf(t), for all x∈D [16]. The following result holds.Theorem 3.
Assume that(1)
f:D(≡domf)⊂X→(-∞,∞] is a lower-semicontinuous proper strictly totally convex function which is Gâteaux differentiable in intD;(2)
Ai(≠⌀)⊆intD⊂H, for all i∈p-, are bounded, closed, and convex  subsets of H which intersect and T:⋃i∈p-Ai→⋃i∈p-Ai is a generalized point-dependent (K,λ)-hybrid p(≥2)-cyclic self-mapping relative to Df for some given functions λ:⋃i∈p-Ai→Λ⊂R and K:⋃i∈p-Ai→(0,1), defined by K(y)=∏j=ii+p-1[Kj(Tj-iy)] for any y∈Ai+1, for all i∈p-, and some functions Ki:Ai+1→(0,ai], for all i∈p-, with Λ being bounded;(3)
there is a convergent sequence{Tnx}  to some z∈⋂i∈p-Ai for some x∈⋃i∈p-Ai.
Then,z=Tz is the unique fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai to which all sequences {Tnx} converge for any x∈⋃i∈p-Ai, for all i∈p-.Proof.
The recursive use of (14) yields
(15)Df(T2x,T2y)≤Ki+1(Ty)Df(Tx,Ty)+λ(Ty)×〈Tx-T2x,f′(Ty)-f′(T2y)〉≤Ki+1(Ty)[〈x-Tx,f′(y)-f′(Ty)〉Ki(y)Df(x,y)+λ(y)×〈x-Tx,f′(y)-f′(Ty)〉]+λ(Ty)〈Tx-T2x,f′(Ty)-f′(T2y)〉,∀x∈Ai,∀y∈Ai+1,∀i∈p-,(16)Df(Tpx,Tpy)≤Ki+p-1(Tp-1y)Df(Tp-1x,Tp-1y)+λ(Tp-1y)〈Tp-1x-Tpx,f′(Tp-1y)-f′(Tpy)〉≤[∏j=1pKp-j+1(Tp-j+1-iy)]Df(x,y)+∑k=1p(∏j=k+1p[Kp-j+i(Tp-j+1)y])λ(Tk-1y)×〈Tk-1x-Tkx,f′(Tk-1y)-f′(Tky)〉,∀x∈Ai,∀y∈Ai+1,∀i∈p-,
with Tpx∈Ai+p, Tpy∈Ai+1+p with Ai+p=Ai, Ki+p=Ki, for all i∈p-, where T0 is the identity mapping on ⋃i∈p-Ai. Now, define K^(y):=[∏j=1pKp-j+1(Tp-j+1-iy)] so that one gets
(17)Df(Tnpx,Tnpy)≤K^n(y)Df(x,y)+∑k=1np(∏j=k+1np[Knp-j+i(Tnp-j+1)y])λ(Tk-1y)×〈Tk-1x-Tkx,f′(Tk-1y)-f′(Tky)〉≤K^n(y)Df(x,y)+∑k=1(n-1)p(∏j=k+1np[Knp-j+i(Tnp-j+1)y])λ(Tk-1y)×〈Tk-1x-Tkx,f′(Tk-1y)-f′(Tky)〉+∑k=(n-1)pnp(∏j=k+1np[Knp-j+i(Tnp-j+1)y])λ(Tk-1y)×〈Tk-1x-Tkx,f′(Tk-1y)-f′(Tky)〉≤K^n(y)Df(x,y)+(1-K^(n-1)p+1(y^)1-K^(y)+Mnp)×(max1≤j≤np[λ(Tj-1y)×〈Tj-1x-Tjx,f′(Tj-1y)-f′(Tjy)〉]max1≤j≤np),
since K(y)=[Kj(Tj-iy)]<1, for all y∈Ai+1, since T:⋃i∈p-Ai→⋃i∈p-Ai is a generalized contractive point-dependent (K,λ)-hybrid p(≥2)-cyclic self-mapping relative to Df, implies that K^(y)<1, for all y∈⋃i∈p-Ai, since K(y)=∏j=ii+p-1[Kj(Tj-iy)]<1 for any y∈Ai+1, for all i∈p- (then for any y∈⋃i∈p-Ai), where
(18)Mnp≥Mnp0:=∑k=(n-1)pnp(∏j=k+1np[Knp-j+i(Tnp-j+1)y])×λ(Tk-1y)×〈Tk-1x-Tkx,f′(Tk-1y)-f′(Tky)〉,Df(Tmnpx,Tmnpy)≤K^m(y)Df(Tnpx,Tnpy)+(1-K^(m-1)p+1(y^)1-K^(y)+Mnmp)×(maxnp+1≤j≤nmp[λ(Tj-1y)〈Tj-1x-Tjx,f′(Tj-1y)-f′(Tjy)〉]maxnp+1≤j≤nmp),
where K^n(y):=K^(y)·K^(Tpy)⋯K^(Tnpy)< 1, since K(y)=∏j=ii+p-1[Kj(Tj-iy)]<1, for all y∈⋃i∈p-Ai, so that
(19)0≤limsupn,m→∞Df(Tmnpx,Tmnpy)≤(11-K^(y)+limsupn,m→∞Mnmp)×limsupn,m→∞(maxnp+1≤j≤nmp×[λ(Tj-1y)〈Tj-1x-Tjx,f′(Tj-1y)-f′(Tjy)〉]maxnp+1≤j≤nmp)=0,
since λ:⋃i∈p-Ai→Λ⊂R is bounded, f:D⊂X→(-∞,∞] is lower-semicontinuous then with all subgradients in any bounded subsets of intD being bounded,  and {Tjx} and {Tj-1x-Tjx}, for all x∈⋃i∈p-Ai, for all i∈p-,  converge so that they are Cauchy sequences being then bounded, for all x∈⋃i∈p-Ai, for all i∈p-, where z∈⋂i∈p-Ai, since ⋂i∈p-Ai is nonempty and closed, is some fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai. As a result, ∃limn→∞Df(Tnpx,Tnpy)=limn→∞Df(Tnx,Tny)=0, for all x∈Ai, for all y∈Ai+1, for all i∈p-. From a basic property of Bregman distance, Tny→z, Tnx→z as n→∞, for all x∈Ai, for all y∈Ai+1, for all i∈p-, if f:D(≡domf)⊂X→(-∞,∞] is sequentially consistent. But, since ⋃i∈p-Ai is closed, f:D∣⋃i∈p-Ai→⋃i∈p-Ai is sequentially consistent if and only if it is totally convex [19]. Thus, {Tny} converges also to z for any x∈Ai and y∈Ai+1, for all i∈p-, so that z=Tz is a fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai. Assume not and proceed by contradiction so as then obtaining Df(Tnx,Tnz)→0; Df(z,Tnz)→0 as n→∞ from a basic property of Bregman distance. Thus, [Df(z,Tnz)-f(z)+f(Tnz)]→0 as n→∞ since 〈z-Tnz,f′(Tnz)〉→0 as n→∞. As a result, f(Tnz)→f(z), Tnz→z=Tz as n→∞ from the continuity of f:D(≡domf)⊂X→(-∞,∞], and z is a fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai. Now, take any y1∈Aj so that y=Ti+1-jy1∈Ai+1, then, Df(Tnx,Tn+i+1-jy)→Df(z,Ti+1-jz)=Df(z,z)=0 since z is a fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai and f:D(≡domf)⊂X→(-∞,∞] is a proper strictly totally convex function. As a result, {Tny} converges to z, for all y∈Ai+1.
It is now proven thatz∈⋂i∈p-Ai is the unique fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai. Assume not so that there is z1(≠z)=Tz1∈⋂i∈p-Ai. Then, Df(Tnx,Tnz1)→0 as  n→∞ from (17) for y=z1 since T:⋃i∈p-Ai→⋃i∈p-Ai is a generalized point-dependent (K,λ)-hybrid p(≥2)-cyclic self-mapping relative to Df with λ:⋃i∈p-Ai→Λ⊂R and K:⋃i∈p-Ai→(0,1) so that {Tnx}→z1=Tnz1, since Df(x,y)>0 if x,y(≠x)∈⋂i∈p-Ai=int(⋂i∈p-Ai) since f:D(≡domf)⊂X→(-∞,∞] is proper and totally strictly convex, and since Tnx→z, and z∈⋂i∈p-Ai. Since ⋂i∈p-Ai is closed and  convex,  it turns out  that z is the unique fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai.
Note that the result also holds for any for ally1∈⋃i∈p-Ai since y∈⋃j(≠i)∈p-Aj maps to y1=Tkiy∈Ai+1 for some nonnegative integer ki≤p-1 through the self-mapping  T:⋃i∈p-Ai→⋃i∈p-Ai so that Df(Tnx,Tny1)=Df(Tnx,Tn+kiy)→Df(z,Tkiz)=Df(z,z)=0 as n→∞ since z∈⋂i∈p-Ai is the unique fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai and {Tny} converges to z for any y∈⋃i∈p-Ai.The subsequent result  directly extends Theorem3 to  the  p-composite self-mappings Tip:⋃j∈p-Aj∣Ai→Ai, foralli∈p-, defined  as Tpx=Tj(Tp-jx); for all x∈⋃i∈p-Ai, subject to i=p-j-k, for all i,j∈p-. The subsets Ai⊂X,  i∈p- are not required to intersect since the restricted composite mappings as defined earlier are self-mappings on nonempty, closed, and convex sets.Corollary 4.
Assume that(1)
fi:D(≡domf)⊂X→(-∞,∞] is a proper strictly totally convex function which is lower-semicontinuous and Gâteaux differentiable in intD, and, furthermore, it is bounded on any bounded subsets of intD;(2)
Ai(≠⌀)⊆intD⊂H is bounded and closed, for all i∈p-, T:⋃i∈p-Ai→⋃i∈p-Ai is a p-cyclic self-mapping so that Tip:⋃j∈p-Aj∣Ai→Ai  for some i∈p- is a generalized point-dependent (K,λi)-hybrid p(≥2)-cyclic self-mapping relative to Df for some given functions λi:⋃i∈p-Ai→Λ⊂R  and K:⋃i∈p-Ai→(0,1)  for some i∈p- defined by K(y)=∏j=ii+p-1[Kj(Tj-iy)] for any y∈Ai+1, for all i∈p-, and Ki:Ai+1→(0,ai] for some ai∈R+, for all i∈p-, where A:=⋃i∈p-Ai⊂H, Λ being bounded and Ai being, furthermore, convex for the given i∈p-;(3)
there is a convergent sequence{Tinpx}  to some zi∈Ai for some x∈Ai and i∈p-.
Then,zi=Tzi is a unique fixed point of Tip:⋃j∈p-Aj∣Ai→Ai to which all sequences {Tinpx} converge for any x∈Ai for i∈p-.
Also, if conditions (1)–(3) are satisfied with all the subsetsAi, for all i∈p-, being nonempty, closed, and convex for some proper strictly convex function f≡fi:D(≡domf)⊂X→(-∞,∞] which is Gâteaux differentiable in intD,  then zi=Tzi, for all i∈p-, is a unique fixed point of Tip:⋃j∈p-Aj∣Ai→Ai, for all i∈p-, to which all sequences {Tinpx} converge for any x∈Ai, for all i∈p-. The p unique fixed points of each generalized point-dependent (K,λi)-hybrid 1-cyclic composite self-mappings Tip:⋃j∈p-Aj∣Ai→Ai, for all i∈p-, fulfil the relations zp-i=Tjzk for i=p-j-k, for all i∈p-1¯, for all j∈p-.Outline of Proof. Note that D∩⋂i∈p-(Ai)≠⌀. Equation (14) is now extended to Tip:⋃j∈p-Aj∣Ai→Ai for the given i∈p- leading to
(20)Df(Tipx,Tipy)≤Ki(y)Df(x,y)+λi(y)〈x-Tipx,f′(y)-f′(Tipy)〉,∀x,y∈Ai,i∈p-,
since Tip is a trivial 1-cyclic self-mapping on Ai for i∈p-. The previous relation leads recursively to. 						(21)Df(Tinpx,Tinpy)≤K^n(y)Df(x,y)+(11-K^(y)+M^inp)×(max1≤j≤n[λ(Ti(j-1)py)〈Ti(j-1)px-Tijpx,f′(Ti(j-1)py)-f′(Tijpy)〉]max1≤j≤n),
with Tinpx,Tinpy∈Ai, forally∈Ai+1, for the given i∈p- with K(y)<1, where K(y) is independent of the particular Tip:⋃j∈p-Aj∣Ai→Ai for i∈p-. One gets by using very close arguments to those used in the proof of Theorem 3 that ∃limn,m→∞Df(Tinmpx,Tinmpy)=0. Then, {Tinmpx} converges to some zi∈Ai which is proven to be a unique fixed point in the nonempty, closed, and convex set Ai for i∈p-. The remaining of the proof is similar to that of Theorem 3. The last part of the result follows by applying its first part to each of the p generalized point-dependent (K,λ)-hybrid 1-cyclic composite self-mappings Tip:⋃j∈p-Aj∣Ai→Ai relative to Df, for all i∈p-.Remark 5.
Iff:D∣⋃i∈p-Ai⊂X→(-∞,∞] is totally convex if it is a continuous strictly convex function which is Gâteaux differentiable in intD, dimX<∞ and D∣⋃i∈p-Ai is closed, [20]. In view of this result, Theorem 3 and Corollary 4 are still valid if the condition of its strict total convexity of f:D∣⋃i∈p-Ai→(-∞,∞] is replaced by its continuity and its strict convexity if the Banach space is finite dimensional. Since vf(x,t)≥δf(t)>0, for all t∈R+, it turns out that if f:D∣⋃i∈p-Ai→(-∞,∞] is uniformly convex, then it is totally convex. Therefore, Theorem 3 and Corollary 4 still hold if the condition of strict total convexity is replaced with the sufficient one of strict uniform convexity. Note that if a convex function f is totally convex then it is sequentially consistent in the sense that Df(xn,yn)→0 as n→∞ if ∥xn-yn∥→0 as n→∞ for any sequences {xn} and {yn} in D.Some results on weak cluster points of average sequences built with the iterated sequences  generated from hybrid cyclic self-mappings{Tnx} relative to a Bregman distance Df, for x∈Ai and some i∈p-, are investigated in the following results related to the fixed points of {Tnx}.Theorem 6.
Assume that(1)
X is a reflexive space and f:D⊂X→(-∞,∞] is a lower-semicontinuous strictly convex function, so that it is Gâteaux differentiable in int(D), and it is bounded on any bounded subsets of intD;(2)
ap-cyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai is given defining a composite self-mapping Tp:⋃i∈p-Ai→⋃i∈p-Ai with Ai(≠⌀)⊆intD⊂H being bounded, convex, and closed, for all i∈p-, so that its restricted composite mapping Tip:⋃j∈p-Aj∣Ai→Ai to Ai, for some  given i∈p-,  is generalized point-dependent (1,λi)-hybrid relative to Df for some λi:Ai→R  and the given i∈p-.Define the sequence{Sn(i)x}≡{(1/n)∑k=0n-1Tkpx} for x∈Ai, where Ti0≡T0p is the identity mapping on Ai so that T0px=x, for all x∈Ai, and assume that {Tnx} is bounded for x∈Ai. Then, the following properties hold.(i)
Every weak cluster point of{Sn(i)x} for x∈Ai is a fixed point νi∈Ai of Tip:⋃j∈p-Aj∣Ai→Ai of Tip:⋃j∈p-Aj∣Ai→Ai for the given i∈p-. Under the conditions of Theorem 3, there is a unique fixed point of Tip:⋃j∈p-Aj∣Ai→Ai which coincides with the unique cluster point of {Sn(i)x}.(ii)
Define sequences{Sn(i,j)x}≡{(1/n)∑k=0n-1Tkp+jx} for any integer 1≤j≤p-1 and x∈Ai where Ai are bounded, closed, and convex, for all i∈p-. Thus, {Sn(i,j)x} converges weakly to νi+j=Tjνi∈Ai+j for x∈Ai, where νi∈Ai is a fixed point of Tip:⋃j∈p-Aj∣Ai→Ai and a weak cluster point of {Sn(i)x} for x∈Ai and νi+j∈Ai+j (1≤j≤p-1) is both a fixed point of Ti+jp:⋃j∈p-Aj∣Ai+j→Ai+j and a weak cluster point of {Sn(i,j)x} for x∈Ai. Furthermore, νi+j=Tjνi if  T:⋃i∈p-Ai→⋃i∈p-Ai  is continuous.Proof.
Using (14) with Tp:⋃i∈p-Ai→⋃i∈p-Ai being a generalized point-dependent (K,λ)-hybrid p(≥2)-cyclic self-mapping relative to Df for λ:⋃i∈p-Ai→R, with K(y)=∏j=ii+p-1[Kj(Tj-iy)]=1, for all y∈⋃i∈p-Ai, yields
(22)Df(Tkpx,y)-Df(T(k+1)px,Tpy)+λ(y)〈Tkpx-T(k+1)px,f′(y)-f′(Tpy)〉=f(Tkpx)-f(T(k+1)px)+f(Tpy)-f(y)-〈Tkpx-y,f′(y)〉+〈T(k+1)px-Tpy,f′(Tpy)〉+λ(y)〈Tkpx-T(k+1)px,f′(y)-f′(Tpy)〉≥0,∀k∈N0=N∪{0}.
Summing up from k=0 to k=n-1 and taking n→∞ yields
(23)f(x)-f(Tnpx)n+f(Tpy)-f(y)+λ(y)〈x-Tnpxn,f′(y)-f′(Tpy)〉-〈1n∑k=0n-1Tkpx-y,f′(y)〉+〈1n∑k=0n-1T(k+1)px-Tpy,f′(Tpy)〉=f(x)-f(Tnpx)n+f(Tpy)-f(y)+λ(y)〈x-Tnpxn,f′(y)-f′(Tpy)〉-〈1n∑k=0n-1Tkpx-y,f′(y)〉+〈1n∑k=0n-1Tkpx+Tnpxn-Tpy,f′(Tpy)〉⟶f(Tpvi)-f(vi)+〈vi-Tpy,f′(Tpy)〉+〈y-vi,f′(vi)〉(≥0)asn⟶∞,∀y∈Ai,
since {Tnx} is bounded for x∈Ai, its subsequence {Tnpx} is then bounded for x∈Ai, and {f(Tnpx)} is also bounded on the bounded subset ⋃i∈p-Ai of intD. Then, {(x-Tnpx)/n} converges to zero since {x-Tnpx} is bounded for x∈Ai. Since f:D⊂X→(-∞,∞] is lower-semicontinuous, the set of subgradients is bounded in all bounded subsets Ai⊆intD, for all i∈p-. As a result, {Sn(i)x}≡{(1/n)∑k=0n-1Tkpx} is bounded for x∈Ai and has a subsequence {Sni(i)x} being weakly convergent to some vi∈Ai for x∈Ai since X is reflexive. One gets by taking y=vi that
(24)-Df(vi,Tipvi)=f(Tipvi)-f(vi)+〈vi-Tipvi,f′(Tipvi)〉≥0.
Hence, it follows from Lemma 1 that vi=Tipvi is a fixed point νi∈Ai of  Tip:⋃j∈p-Aj∣Ai→Ai for the given i∈p-. Now, consider {Sni(i,j)x},  x∈Ai, for any integers 1≤j≤p-1 so that Tjx∈Ai+j for x∈Ai. Hence, Property (i) follows with the uniqueness and the coincidence of the weak cluster point of {Sn(i)x} and  fixed point of Tip:⋃j∈p-Aj∣Ai→Ai under Theorem 3. The previous reasoning remains valid so that {Sni(i,j)x} is weakly convergent to some νi+j∈Ai+j since {Tn+jx}∈Ai+j is bounded for x∈Ai (since {Tnx} is bounded for x∈Ai and j is finite), its subsequence {Tnp+kx} for x∈Ai is also bounded so that {(Tjx-Tnp+jx)/n} converges to zero since {Tjx-Tnp+jx} in Ai+j is bounded for x∈Ai, 1≤j≤p-1. Now, take y=νi+j so that Df(νi+j,Ti+jpνi+j)=0, and then νi+j=Ti+jpνi+j, 1≤j≤p-1, is both a fixed point of Ti+jp:⋃j∈p-Aj∣Ai→Ai+j and a weak cluster point of {Sni(i,j)x}. Now, take x∈Ai so that Tjx∈Ai+j. Then, Tj(Tnpx)=Tnp+jx=Tnp(Tjx)→νi+j as n→∞ for x∈Ai and, in addition, νi+j=Tjνi if  T:⋃i∈p-Ai→⋃i∈p-Ai  is continuous. Hence, Property (ii) follows.


## 3. Extensions for Generalized Point-Dependent Cyclic Hybrid Self-Mappings on Nonintersecting Subsets: Weak Convergence to Weak Cluster Points of a Class of Sequences

Some of the results of Section2 are now generalized to the case when the subsets of the cyclic mapping do not intersect T:⋃i∈p-Ai→⋃i∈p-Ai, in general, by taking advantage of the fact that best proximity points of such a self-mapping are fixed points of the restricted composite mapping Tip:⋃j∈p-Aj∣Ai→Ai for i∈p-. Weak convergence of averaging sequences to weak cluster points and their links with the best proximity points in the various subsets of the p-cyclic self-mappings is discussed. Firstly, the following result follows from a close proof to that of Theorem 6 which is omitted.Theorem 7.
LetX be a reflexive space, and let f:D⊂X→(-∞,∞] be a lower-semicontinuous strictly convex function so that it is Gâteaux differentiable in int(D) and it is bounded on any bounded subsets of intD. Consider the generalized point-dependent (p≥1)-cyclic hybrid self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai being (1,λi) relative to Df for some λi:Ai→R such that Ai(≠⌀)⊆intD⊂H are all bounded, convex, closed, and with nonempty intersection. Define the sequence {Snx}≡{(1/n)∑k=0n-1Tkx} for x∈⋃i∈p-Ai, where T0 is the identity mapping on ⋃i∈p-Ai, and assume that {Tnx} is bounded for x∈⋃i∈p-Ai. Then, the following properties hold.(i)
Every weak cluster point of{Sn(i)x} for x∈Ai is a fixed point νi∈Ai of T:⋃i∈p-Ai→⋃i∈p-Ai.(ii)
Define the sequence{Snx}≡{(1/n)∑k=0n-1Tkx} for x∈⋃i∈p-Ai which is bounded, closed, and convex, for all i∈p- and any integer 1≤j≤p-1. Thus, {Snx} converges weakly to the fixed point ν=Tν∈⋂i∈p-Ai of T:⋃i∈p-Ai→⋃i∈p-Ai for x∈⋃i∈p-Ai which is also a weak cluster point of {Snx}.Remark 8.
The results of Theorems6 and 7 are extendable without difficulty to the weak cluster points of other related sequences to the considered ones.
(
1
) Define sequences {Sn(j)x}≡{(1/n)∑k=0n-1Tk+jx}, x∈⋃i∈p-Ai, for any given finite non-negative integer j under all the hypotheses of Theorem 7. With this notation, the sequence considered in such a corollary is {Snx}≡{Sn(0)x}. Direct calculation yields (Sn(j)x-Snx)=(1/n)∑k=0j-1(Tk+nx-Tkx)→0 for x∈⋃i∈p-Ai as n→∞ since {Tk+nx-Tkx}, and then  {∑k=0j-1(Tk+nx-Tkx)}, is bounded. Then, Sn(j)x→ν weakly which is the same fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai in ⋂i∈p-Ai which is a weak cluster point of  {Sn(j)x} for x∈⋃i∈p-Ai for  any finite non-negative integer j.
(
2
) Consider all the hypotheses of Theorem 7 and now define sequences {Sn[j]x}≡{(1/n)∑k=0n+j-1Tkx}, x∈⋃i∈p-Ai, for any given finite non-negative integer j. With this notation, the sequence considered in the corollary is {Snx}≡{Sn[0]x}. Direct calculation yields (Sn[j]x-Snx)=(1/n)∑k=0j-1(Tk+nx)→0 weakly for x∈⋃i∈p-Ai as n→∞ since {Tk+nx}, and then  {∑k=0j-1(Tk+nx)}, is bounded. Then, Sn[j]x→ν weakly which is the same fixed point of T:⋃i∈p-Ai→⋃i∈p-Ai in ⋂i∈p-Ai which is a weak cluster point of  {Sn[j]x} for x∈⋃i∈p-Ai and for any finite non-negative integer j.
(
3
) Now consider the hypotheses of Theorem 6. It turns out that the sequence {Sn(i,j)x}≡{(1/n)∑k=0n-1Tkp+jx} for x∈Ai satisfies for any integer 1≤j≤p-1,
(25)Sn(i,j)x=1n+1n+1n(∑k=0n-1Tkp+jx)+Tjn(∑k=0n-1Tkpx-∑k=0nTkpx)=Tj(n+1nSn+1(i)x-1nTnpx),Sn(i,j)x=1n+1n+1n(∑k=0n-1Tkp(Tjx))+1n(∑k=0n-1Tkp(Tjx)-∑k=0nTkpx)=(n+1nSn+1(i+j)xi+j-1nTnpxi+j)⟶νi+j≡Ti+jpνi+j(∈Ai+k)
weakly as n→∞, where xi+j(=Tjx)∈Ai+j since x∈Ai, {Tnx} is bounded,  and {Sn+1(i,j)z}≡(1/n)∑k=0n-1Tkpz=(1/n)∑k=0n-1(Ti+jp)kz for z∈Ai+j and 1≤j≤p-1. Thus, νi+j is a fixed point of Ti+jp:⋃j∈p-Aj∣Ai+j→Ai+j which is also a weak cluster point of the sequences {Sn+1(i,j)z} for 1≤j≤p-1. However, it is not guaranteed that νi+j=Tjνi=Tipvi=Ti+jpvi+j without additional hypotheses on T:⋃i∈p-Ai→⋃i∈p-Ai such as its continuity, or at least that of the composite mapping Tj:Ai→Ai+j allowing to equalize the function of the limit with the limit of the function at such a fixed point.
(
4
) Now, define {Sn[i,j]x}≡{(1/n)∑k=0n+j-1Tkpx} for x∈Ai. Note that for x∈Ai, ∃νi∈Ai,
(26)Sn[i,j]x=1n∑k=0n-1Tkpx+1n∑k=0j-1T(n+k)px=Sn(i)x+1n∑k=0j-1T(n+k)px⟶νi(=Tipvi)
weakly as n→∞ sincej is finite, which is a fixed point in Ai of  the composite mapping Tip and a weak cluster point of {Sn[i,j]x} for finite j.Note that Theorem6 are supported by boundedness  constraints for the sequences of iterates obtained through the cyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai which is generalized point-dependent with respect to some convex function.  The results of identification of weak cluster points of some average sequences with fixed points of the cyclic self-mapping or its composite mappings do not guarantee uniqueness of fixed points and weak cluster points because the cyclic self-mapping is not restricted to be contractive. By incorporating some background contractive-type conditions for the cyclic self-mapping, the previous results can be extended to include uniqueness of fixed points as follows.Theorem 9.
Assume that.(1)
Assumption 1 of Theorem6 holds with the restriction of (X,∥∥) to be a uniformly convex Banach space;(2)
Assumption 2 of Theorem6 holds, and, furthermore, all the p-cyclic composite mappings with restricted domain Tip:⋃j∈p-Aj∣Ai→Ai; for all i∈p- are either contractive or Meir-Keeler contractions.Then, the following properties hold.(i)
Theorem6(i)-(ii) holds. Furthermore, each of the mappings Tip:⋃j∈p-Aj∣Ai→Ai has a unique fixed point νi∈Ai which are also best proximity points of T:⋃i∈p-Ai→⋃i∈p-Ai in Ai so that νi+j=Tjνi; for all j∈p-i¯, for all i∈p-.(ii)
If, in addition,⋂i∈p-Ai≠⌀, then, there is a unique fixed point ν∈⋂i∈p-Ai of T:⋃i∈p-Ai→⋃i∈p-Ai and Tip:⋃j∈p-Aj∣Ai→Ai, for all i∈p-.Proof.
Note that uniformly convex Banach spaces(X,∥∥) are also reflexive spaces required by Theorem 6. Each mapping Tip:⋃j∈p-Aj∣Ai→Ai has a unique fixed point νi∈Ai, for all i∈p-, irrespective of ⋂i∈p-Ai being empty or not if Tip:⋃j∈p-Aj∣Ai→Ai is either a cyclic contraction or a Meir-Keeler contraction [1–3],  since Ai(i∈p-) are non-empty, closed, and convex, and (X,∥∥) is a uniformly convex Banach space so that each νi∈Ai; for all i∈p- is a best proximity point in Ai of T:⋃i∈p-Ai→⋃i∈p-Ai. It follows from the hypothesis that there is a unique weak cluster point of {Sn(i)x} for x∈Ai which is the unique fixed point of Tip:⋃j∈p-Aj∣Ai→Ai, for all i∈p-, and also the unique best proximity point of T:⋃i∈p-Ai→⋃i∈p-Ai in Ai for i∈p-.
It is now proven that if⋂i∈p-Ai≠⌀ then (⋂i∈p-Ai)∋ν=νi∈Ai, for all i∈p-. Take some x∈Ai∩Aj for some i,j(≠i)∈p-. Thus, Tpnx→νi(∈Ai) and Tpnx→νj as n→∞ since νi is the unique fixed point of Tip:⋃j∈p-Aj∣Ai→Ai and νj(∈Aj) is the unique fixed point of Tjp:⋃i∈p-Ai∣Aj→Aj. Then, ν=νi=Tkν, for all i∈p-; for all k∈N, Sn(i)x→ν weakly as n→∞, for all i∈p-, and ν∈⋂i∈p-Ai is the unique weak cluster point of {Sn(i)x}, for all i∈p-.Theorem9 can be also extended “mutatis-mutandis” to the convergence of weak cluster points of the alternative sequences discussed in Remark 8.  It is now proven that the sets of fixed points of the restricted composite mapping Tip:⋃j∈p-Aj∣Ai→Ai, some i∈p-, are convex if such mappings  are quasi-nonexpansive with respect to Df in the sense that it has (at least) a fixed point in Ai and Df(v,Tipx)≤Df(v,x), for all x∈Ai, and f:D⊂X→(-∞,∞] is a proper strictly convex function,  [16]. The concept of quasi-nonexpansive mapping is addressed in the subsequent result to discuss the   topology of fixed points and best proximity points of  composite mappings of cyclic self-mappings.Theorem 10.
Letf:D⊂X→(-∞,∞] be a proper strictly convex function on the Banach space (X,∥∥) so that it is Gâteaux differentiable in intD, and consider the restricted composite mapping Tip:⋃j∈p-Aj∣Ai→Ai for some given i∈p- built from the p-cyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai so that Ai is nonempty, convex, and closed. Assume that Aj⊆intD, for all j∈p-, and that  the composite mapping Tip:⋃j∈p-Aj∣Ai→Ai is quasi-nonexpansive with respect to Df for the given i∈p-.Then, the following properties hold.(i)
The set of fixed pointsF(Tip) of Tip:⋃j∈p-Aj∣Ai→Ai is a  closed and convex subset of Ai for the given i∈p-.(ii)
Assume, in addition, thatAi, for all  i∈p-, are nonempty convex closed subsets of H subject to ⋃i∈p-Ai⊆intD, and assume also that Tip:⋃j∈p-Aj∣Ai→Ai are quasi-nonexpansive with respect to Df, for all  i∈p-. Then, the set of best proximity points in Ai of the p-cyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai coincides with F(Tip), and it is then a closed and convex subset of Ai, for all  i∈p-. Furthermore, if ⋂i∈p-Ai≠⌀, then F(T)=clF(T)⊆⋂i∈p-F(Tip)⊆⋂i∈p-Ai which is then nonempty, closed, and convex.Proof.
Takex∈clF(Tip)⊆Ai⊆⋃i∈p-Ai⊆intD and {xn}⊆clF(Tip) so that {xn}→x as n→∞. Note that F(Tip) and clF(Tip) are nonempty sets since Tip:⋃j∈p-Aj∣Ai→Ai is quasi-nonexpansive with respect to Df then possessing at least a fixed point. By the continuity of Df(·,Tipx) and that of Df(x,·), the strict convexity of f:D⊂X→(-∞,∞], and the assumption Df(v,Tipx)≤Df(v,x), one has
(27)Df(x,Tipx)=limn→∞Df(xn,Tipx)≤limn→∞Df(xn,x)≤Df(x,x)=0
and x=Tipx, from the strict convexity of f:D⊂X→(-∞,∞] and Lemma 1, which is in F(Tip) which is then a closed subset of Ai as a  result. Now, it is proven that is convex. Following the steps of a parallel result proven in [16] for noncyclic self-mappings, take x,y(≠x)∈F(Tip) and consider for some arbitrary real constant α∈[0,1] a point z=αx+(1-α)y which is in Ai since such a set is convex. Since Tip:⋃j∈p-Aj∣Ai→Ai is quasi-nonexpansive with respect to Df leading to
(28)Df(x,Tipz)≤Df(x,z)=f(x)-f(z)-〈x-zf′(z)〉,Df(y,Tipz)≤Df(y,z)=f(y)-f(z)-〈y-zf′(z)〉,
and,  since q=αq+(1-α)q and f(q)=αf(q)+(1-α)f(q) for any q∈Ai, that
(29)Df(z,Tipz)=f(z)-f(Tipz)-〈z-Tipz,f′(Tipz)〉=f(z)+[αf(x)-f(Tipz)-〈x-Tipz,f′(Tipz)〉]+(1-α)[f(y)-f(Tipz)-〈y-Tipz,f′(Tipz)〉]-[αf(x)+(1-α)f(y)]=f(z)+αDf(x,Tip)+(1-α)Df(y,z)-[αf(x)+(1-α)f(y)]≤f(z)+αDf(x,z)+(1-α)Df(y,z)-[αf(x)+(1-α)f(y)]=-〈αx+(1-α)y-(αz+(1-α))z,f′(z)〉=-〈0,f′(z)〉=0,
which implies from Lemma 1 that Tz=z=αx+(1-α)y for any x,y(≠x)∈F(Tip) sincef:D⊂X→(-∞,∞] is strictly convex and F(Tip)=clF(Tip)⊆(intD∩Ai). Thus, F(Tip) is a convex subset of Ai. Hence, Property (ii) follows. The first part of property (ii) is a direct consequence of property (i) if the p composite self-mappings on all the sets Ai are quasi-nonexpansive with respect to Df since the respective sets of fixed points are the best proximity points of the p-cyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai in each of the sets Ai, for all i∈p-. If, furthermore, the sets Ai, for all i∈p-, have a nonempty intersection, then its set of fixed points coincides with the intersection of the sets of best proximity points of the composite mappings Tip:⋃j∈p-Aj∣Ai→Ai which are all identical for i∈p-. The proof is trivial. Take any x∈⋂i∈p-Ai(≠⌀). Then, the sequence of iterates obtained through the composite Tip:⋃j∈p-Aj∣Ai→Ai converges to some z∈⋂i∈p-(Ai∩F(Tip)). This implies that ⋂i∈p-F(Tip)≠⌀, closed and convex from property (i). Thus, F(T)(≠⌀)⊆⋂i∈p-F(Tip). Then, the set of fixed points of T:⋃i∈p-Ai→⋃i∈p-Ai is F(T)≡clF(T)⊆⋂i∈p-(Ai∩F(Tip)). Property (ii) has been proven.Concerning that Theorem10(ii), note that the set inclusion F(T)=⋂i∈p-F(Tip) does not guarantee, in general, that the identity F(T)⊆⋂i∈p-F(Tip) is not guaranteed for the case when ⋂i∈p-Ai≠⌀ except for cases under extra conditions such as the contractiveness of the composite mappings built from the p-cyclic one leading, for instance, to the uniqueness of the fixed point of the cyclic self-mapping. See, for instance, Theorem 3 and Corollary 4.It is direct to give sufficient conditions for the restricted composite mappings of thep-cyclic self-mapping  to be quasi-nonexpansive under the  relevant conditions of Theorem 3, Corollary 4, Theorem 6, and Theorem 7 (see Proposition 3.5 of [16]) for noncyclic self-mappings, as follows.Theorem 11.
Assume that.(1)
X is a reflexive space and f:D⊂X→(-∞,∞] is a proper strictly convex function, so that it is Gâteaux differentiable in int(D), and it is bounded on any bounded subsets of intD.(2)
Ap-cyclic self-mapping T:⋃i∈p-Ai→⋃i∈p-Ai is given defining a composite self-mapping Tp:⋃i∈p-Ai→⋃i∈p-Ai with the subsets Ai(≠⌀)⊆intD⊂H being bounded, closed, and convex, for all i∈p-.(3)
The restricted composite mapping toAi for some given i∈p-, that is,  Tip:⋃j∈p-Aj∣Ai→Ai, is generalized point-dependent (1,λi)-hybrid relative to Df for some function λi:Ai→R and the given i∈p- which possesses a bounded sequence {Tpnx}⊂Ai for some point x∈Ai.
Then, the restricted composite mappingTip:⋃j∈p-Aj∣Ai→Ai is quasi-nonexpansive with respect to Df so that F(Tip) is a nonempty closed convex subset of Ai.Remark 12.
The well-known concepts of nonexpansive,  nonspreading, hybrid, and contractive cyclic self-mappings [12–16, 19]  are useful in the context of particular cases of interest of (14) within the given framework for generalized nonexpansive p-cyclic self-mappings relative to Df.(1)
If (14) holds 0≤Ki(y)≤1 and λi(y)=0, for all y∈Ai+1, for each x∈Ai, for all i∈p-, then T:⋃i∈p-Ai→⋃i∈p-Ai is said to be a generalized nonexpansive p-cyclic self-mapping relative to Df.(2)
If (14) holds 0≤Ki(y)≤1 and λi(y)=2, for all y∈Ai+1, for each x∈Ai, for all i∈p-, then T:⋃i∈p-Ai→⋃i∈p-Ai is said to be a generalized nonspreading p-cyclic self-mapping relative to Df.(3)
If (14) holds, for all x∈Ai,  for  all  y∈Ai+1,  for  all  i∈p- with 0≤Ki(y)≤1 and λi(y)=1, for all i∈p-, then T:⋃i∈p-Ai→⋃i∈p-Ai is said to be a generalized nonexpansive (1,1)-hybrid p-cyclic self-mapping relative to Df.(4)
If (14) holds, for all x∈Ai, for all y∈Ai+1, for all i∈p- with 0≤Ki(y)≤1 and λi(y)≠1 for some y∈Ai+1, for all i∈p-, then T:⋃i∈p-Ai→⋃i∈p-Ai is said to be a generalized nonexpansive (point-dependent if some λi(y) for some i∈p- is not constant) (1,λ)-hybrid p-cyclic self-mapping relative to Df.(5)
If (14) holds, for all x∈Ai,forally∈Ai+1,foralli∈p- with 0≤Ki(y)≤1 and λi(y)≠0 for some y∈Ai+1; foralli∈p- then T:⋃i∈p-Ai→⋃i∈p-Ai is said to be a generalized nonexpansive (point-dependent if some λi(y) for some i∈p- is not constant) p-cyclic self-mapping relative to Df;(6)
If (14) holds, for all x∈Ai, for all y∈Ai+1, for all i∈p- with 0≤Ki(y)≤K<1 and λi(y)≠1 for some y∈Ai+1, for all i∈p-, then T:⋃i∈p-Ai→⋃i∈p-Ai is said to be a generalized contractive (point-dependent if some λi(y) for some i∈p- is not constant) (K,λ)-hybrid p-cyclic self-mapping relative to Df.(7)
If (14) holds 0≤Ki(y)≤K<1 and λi(y)=0, for all y∈Ai+1 for each x∈Ai, for all i∈p-, then T:⋃i∈p-Ai→⋃i∈p-Ai is said to be a generalized K-contractive p-cyclic self-mapping relative to Df.
The various given results can be easily focused on these particular cases.


## 4. Examples

Dynamic systems are a very important tool to describe and design control systems in applications. Fixed point theory has been found useful to study their controllability and stability properties. See, for instance, [21–26] and references there in. Two examples are now given related to discrete dynamic systems in order  to illustrate the theoretical aspects  of this paper.Example 1.
Consider the scalar discrete dynamic system(30)xk+1=Txk:=akxk+ηk,∀k∈N,
for given initial condition x0 with ηk=ηk(x-k), x-k:={xj:j=0,1,…,k} being a state disturbance which can include combined effects of parametrical disturbances and unmodeled dynamics (roughly speaking, the neglected dynamic effects of describing a higher-order difference equation by the previous first-order one). The solution sequence is defined by the self-mapping T:clR→clR (clR=[-∞,+∞] being the extended real line including the infinity points) given by xk+1=(∏i=0k[ai])x0+∑i=0k(∏j=i+1k[aj])ηi, for all k∈N. It is assumed that a fixed point exists for some x0∈R; that is, a sequence {Txk} for some initial point x0 is bounded; That is,
(31)∃limk→∞[(∏i=0k[ai])x0+∑i=0k(∏j=i+1k[aj])ηi]=x*=x*(x0),
with |x*|<+∞. In particular, this holds for the unperturbed system with |ak|<1 and ηk≡0 which possess a unique globally asymptotically stable equilibrium x*=0 which is also a unique fixed point of the solution. If ak=1, then there is a stable constant solution xk=x0 for each initial condition  which is also a (nonunique) fixed point. In both cases, the mapping T:clR→clR is trivially nonexpansive and, in the first case, it is also contractive.  Note that the previous mapping is also a trivial cyclic self-mapping for p=1. Under a cyclic repetition of the sequence {ak} with
(32)a=∏i=qkq(k+1)-1[ai],b=∑i=0q-1(∏j=i+1q-1[aqk+j])ηqk+i,∀k∈N0,
for some q∈N  being constant with |a|<1 and |b|<+∞. In this case, we can describe the given difference equation equivalently  as
(33)x(k+1)q=Tqxk≔axkq+bq
for the same initial condition. Then, the composite self-mapping Tqxqk=x(k+1)q generating the subsequence {xqk}⊂{xk} of the solution has a unique fixed point xq*=b/(1-a) for any given x0. If, furthermore, there are finite limits aqk+i→ai*, ηqk+i→ηi*=(1-ai*)xq*=b(1-ai*)/(1-a) as k→∞, for alli∈q-1¯, then Txk→xq* as k→∞ which is a fixed point of T:clR→clR. If the second set of limits exists being all finite, but arbitrary, that is, the identities ηi*=(1-aqi)xq* do not all hold for i∈q-1¯, then  the solution sequence converges to a cycle {xq*=b/(1-a),xq+1*=a1*b/(1-a)+η1*,…,x2q-1*=aq-1*b/(1-a)+ηq-1*}. We now retake the example under the point if view of a point-dependent λ-hybrid map where f(x)=αx2+β(α>0,β≥0). The considered Banach space is (R,∥∥) which is a Hilbert space for the  inner product being the  Euclidean scalar product  and the norm is the Euclidean norm. Then,
(34)Df(yk,xk)=f(yk)-f(xk)-2α(yk-xk)xk=α[(yk2-xk2)-2(yk-xk)xk]=α(yk-xk)2,Df(yk+1,xk+1)=α(yk+1-xk+1)2=α[ak2(yk-xk)2+(ηyk-ηxk)2+2ak(yk-xk)(ηyk-ηxk)(ηyk-ηxk)2].
Condition (14) for T:clR→clR  to be point-dependent 1-cyclic λ-hybrid becomes in particular  for some real functions λ(y) and K(y)∈[0,1] for y∈R(35)α[ak2(yk-xk)2+(ηyk-ηxk)2+2ak(yk-xk)(ηyk-ηxk)(ηyk-ηxk)2]≤α[K(yk)(yk-xk)2+2λ(yk)(xk-xk+1)(yk-yk+1)(yk-xk)2]=α[((1-ak)yk-ηyk)K(yk)(yk-xk)2+2λ(yk)((1-ak)xk-ηxk)×((1-ak)yk-ηyk)]=α[-(1-ak)(ykηxk+xkηyk))K(yk)(yk-xk)2+2λ(yk)×((1-ak)2xkyk+ηxkηyk-(1-ak)(ykηxk+xkηyk))],∀k∈N0,
and, equivalently,
(36)(ak2-K(yk))(yk-xk)2≤(ηyk-ηxk)[2ak(yk-xk)-(ηyk-ηxk)]+2λ(yk)((1-ak)2xkyk+ηxkηyk-(1-ak)(ykηxk+xkηyk)),∀k∈N0.
Note that if xk=xk+1 or if yk=yk+1, equivalently, if ηxk=(1-ak)xk or if ηyk=(1-ak)yk, then  the previous  equivalent constraints (35)-(36) cannot be satisfied by a choice of some finite value of λ(yk) unless
(37)|ak(yk-xk)+(1-ak)yk-ηxk|≤K(yk)|yk-xk|ifyk=yk+1,|ak(yk-xk)+(1-ak)xk-ηyk|≤K(yk)|yk-xk|ifxk=xk+1,
so that any arbitrary value of λ(yk) would satisfy the inequalities. Note that both inequalities hold directly for any fixed points.
If the previous constraint (36) holds, subject to (37), for some real sequence {λ(yk)}, then any weak cluster point of {Sn(i)x}≡{(1/n)∑k=0n-1Tkx} is a fixed point of T:clR→clR according to Theorem 6. If there is a unique fixed point according to Theorem 3, then the unique fixed point of T:clR→clR and weak cluster point of {Sn(i)x}≡{(1/n)∑k=0n-1Tkx} coincide. The same property holds for the weak cluster point of the average sequences referred to in Remark 8. Note in particular the following.
(
1
) If ηxk=ηyk≡0 and |ak|≤K(yk)≤1, then the previous constraint leads to 0≤2λ(yk)(1-ak)2xkyk which guarantees that Theorem 6 holds for any real sequence {λ(yk)} satisfying λ(yk)=sign(xkyk) if |1-ak||yk-xk|xkyk≠0 and taking any arbitrary real value, otherwise for each k∈N0. Since ηxk=ηyk≡0, a choice of λ=λ(yk) independent of xk is as follows:
(38)λ=λ(yk)=λ0≥0ifmin(x0,y0)≥0,∀k∈N0λ=λ(yk)=λ0<0ifmin(x0,y0)<0orifsgn(x0)=-sgn(y0),∀k∈N0.
Thus, any cluster point of{Sn(i)x} is a fixed point of  T:clR→clR.  There is a unique such fixed point xq*=b/(1-a) if  |a|<1 and |b|<+∞, which is also a globally asymptotically stable equilibrium point of the solution, and there are finite limits aqk+i→ai*, ηqk+i→ηi*=(1-ai*)xq*=b(1-ai*)/(1-a) as k→∞ for some given q∈N and such a fixed point is also a fixed point of  the composite self-mapping Tq:clR→clR. If |ak|>1, then the constraint of point-dependent λ-hybrid self-mapping is satisfied for λ(yk)≥(ak2-K(yk))(yk-xk)2/(1-ak)2xkyk if xkyk≠0, for all k∈N0. If xkyk≥0 (i.e., both xk and yk have the same sign or one of them is zero) then T:clR→clR is λ-hybrid for λ=λ(yk)=λ0≥0, for all k∈N0. However, Theorem 6 is not applicable for cluster fixed points of the averaging sequence since there is no fixed point of the difference equation, in general.
(
2
) If ηk=ηxk=ηyk is not identically zero, then the constraint is satisfied if |ak|≤K(yk)≤1 and
(39)λ(yk)=sign[(1-ak)2xkyk+[ηk-(1-ak)(yk+xk)]ηk(1-ak)2]=sign(40){(1-ak)2(∏i=0k-1[ai])x0yk+∑i=0k-1(∏j=i+1k-1[aj])ηiyk+⌊+∑i=0k-1(∏j=i+1k-1[aj])ηi)ηk-(1-ak)(∑i=0k-1(∏j=i+1k-1[aj])ηiyk+(∏i=0k-1[ai])x0+∑i=0k-1(∏j=i+1k-1[aj])ηi)⌋ηk}
if [(1-ak)2xkyk+[ηk-(1-ak)(yk+xk)]ηk]|xk-yk|≠0 and taking any arbitrary real value, otherwise.
(
3
) In the general case, the constraint is satisfied if |ak|≤K(yk)≤1 with
(41)λ(yk)≥12((ηxk-ηyk)[2ak(yk-xk)-(ηyk-ηxk)]+(ak2-K(yk))(yk-xk)2)×((1-ak)2xkyk+ηxkηyk-(1-ak)(ykηxk+xkηyk))-1,∀k∈N0,
provided that the denominator of (41)  is nonzero or if so (37) hold so that λ(yk) may take an arbitrary value.
(
4
) Assume that ak∈[0,1], for all k∈N0, ak→1 as k→∞ with αk=1-ak≥0 and ηk satisfying ∑k=0∞αk=+∞, ηk≥0 and ∑k=0∞ηk<+∞ and that x0≥0. Thus, the difference equation can be described equivalently  by xk+1-xk=-αkxk+ηk, and summing up both sides from k=0 to k=n-1 yields since the solution sequence is non-negative for nonnegative initial conditions and since ∑k=0∞ηk<+∞:
(42)0≤xn=x0-∑k=0n-1αkxk+∑k=0n-1ηk≤x0-∑k=0n-1αkxk+C
for  some real constant C≥0. Thus, 0≤∑k=0n-1αkxk≤x0+C-∑k=0n-1ηk<+∞ which implies that liminfk→+∞xk=0,  limsupk→+∞xk<+∞, and supk→+∞xk<+∞. But if 0<x-=limsupk→+∞xk<+∞, then 0<x-=limsupn→∞xn≤x0+C+limsupn→∞(-∑k=0n-1αkxk)=-∞ which is a contradiction. Then, {xn} is bounded and ∃limn→∞xn=0. Thus, x*=0 is a fixed point of  T:clR→clR and the previous particular cases (1)–(3) can be applied for weak cluster points of the average sequences.
(
5
) Now, consider the r(≥2)-dimensional dynamic system xk+1=Txk:=Akxk+ηk, for all k∈N,  where Ak∈Rr×r, ηk∈Rr, for all k∈N0, and the convex function f(x)=(1/2)xTQx with Q=QT≻0 (i.e., a positive definite square r-matrix with the superscript T standing for transposes). The Bregman distance becomes Df(yk,xk)=(1/2)(ykTQyk-xkTQxk)-Q(yk-xk)Txk resulting in the point-dependent λ-hybrid constraint:
(43)(yk-xk)T(AkTAk-K(yk)Ir)(yk-xk)≤(ηyk-ηxk)T[2Ak(yk-xk)-(ηyk-ηxk)]+2λ(yk)(xkT(Ir-Ak)T(Ir-Ak)yk+ηxkTηyk-[ykT(Ir-Ak)ηxk+xkT(Ir-Ak)ηyk]),
where Ir is the rth identity matrix. A finite, in general nonunique, real sequence {λ(yk)} exists satisfying the previous constraint if for any k∈N0,
(44)(xkT(Ir-Ak)T(Ir-Ak)yk+ηxkTηyk-[ykT(Ir-Ak)ηxk+xkT(Ir-Ak)ηyk])=0⟹{(yk-xk)T(AkTAk-K(yk)Ir)(yk-xk)-(ηyk-ηxk)T[2Ak(yk-xk)-(ηyk-ηxk)]}≤0,
which is a generalization of (37) to the n-the dimensional case. Thus,(45a)λ(yk)≥((yk-xk)T(AkTAk-K(yk)Ir)(yk-xk)-(ηyk-ηxk)T[2Ak(yk-xk)-(ηyk-ηxk)](yk-xk)T)×(2(xkT(Ir-Ak)T(Ir-Ak)yk+ηxkTηyk-[ykT(Ir-Ak)ηxk+xkT(Ir-Ak)ηyk]))-1(45b)if(xkT(Ir-Ak)T(Ir-Ak)yk+ηxkTηyk-[ykT(Ir-Ak)ηxk+xkT(Ir-Ak)ηyk])≠0λ(yk)=λ0,otherwise.
The previous discussion for the particular scalar difference equation may be generalized for this case with the replacementak2→λmax(AkTAk)=∥Ak∥22≤K(yk)≤1, for all k∈N0,  where λmax(·) stands for the maximum (real) eigenvalue of the symmetric matrix (·) leading to the results for point-hybrid mappings to hold if there is a sequence {λ(yk)} satisfying the previous constraint (45a) and (45b).Example 2.
It is direct to extend Example1 to a 2-cyclic self-mapping  as follows. For instance, consider a scalar difference equation of the form
(46)xk+1=Txk:=akxk+uk+ηk,∀k∈N,
for a given initial condition x0≥0 where {uk} is a control sequence. Recursive computation for two consecutive samples yields
(47)xk+2=T2xk:=a^kxk+η^k=a^kxk+η^k0+uk+1,∀k∈N,
where x0≥0, a^k=ak+1ak; η^k=η^k0+uk+1=ak+1(ηk+uk)+ηk+1+uk+1. Define the sets Ai(i=1,2) as A1=R0+={z∈clR:z≥0}=-A2 so that A1∩A2={0}. If the control sequence {uk} is chosen as uk=Me-k(-1)k+1, for all k∈N0, for some constant M>0, then xk+1=Me-(k+1)(-1)k, sgnxk+1=-sgnxk, for all k∈N0, xk→0 as k→∞, and the sequences {x2k}⊂A1, {x2k+1}⊂A2 both converge to the unique fixed point x*=0 of both T:clR→clR  and T2:clR→clR. Now, suppose that the control sequence is changed to uk=max(Me-k(-1)k+1,εsgn((-1)k+1)), for all k∈N0 for some positive real constant ε, then {x2k}⊂A1, {x2k+1}⊂A2 and x2k→ε, x2k+1→-ε as k→∞ with Ai(i=1,2) being redefined as A1=R0+={z∈clR:∞≥z≥ε}=-A2. In this case, A1∩A2=⌀ and ±ε are the best proximity points of T:clR→clR in A1 and A2, respectively, while ε and (-ε) are also fixed points of T2:A1→A1 and T2:A2→A2, respectively.
Now, note thatT2kx0∈A1 and T2kx1(=T2k+1x0)∈A2 if x0∈A1 and T2kx0∈A2 and T2kx1(=T2k+1x0)∈A1 if x0∈A2. Thus, one gets
(48)x2k=(∏i=0k-1[a^2i])x0+∑i=0k-1(∏j=i+12k-2[a^2j])η^2i=(∏i=0k-1[a^2i])x0+∑i=0k-1(∏j=i+12k-2[a^2j])×(a2i+1(η2i+u2i)+η2i+1+u2i+1),x2k+1=a2kx2k+u2k+η2k,∀k∈N0,
with x2k∈A1 and x2k+1∈A2 if x0∈A1 and x2k∈A2 and x2k+1∈A1 if x0∈A2. The Bregman constraint for the composite self-mapping T2:A1→A1 to be λ(y)-hybrid relative to Df holds in a similar way as (36), subject to (37), by replacing the subscripts k→2k and the sequences {ak}→{a^2k}, {xk}→{x2k}⊂A1, {yk}→{y2k}⊂A1, {ηxk}→{η^x(2k)}, {ηyk}→{η^y(2k)} and “mutatis-mutandis” performed replacements for subscripts k→2k+1 for the composite self-mapping T2:A2→A2 for x0∈A1. If x0∈A2, then the modifications in the Bregman constraint (36), subject to (37), are referred to {xk}→{x2k}⊂A2, {yk}→{y2k}⊂A2.
Then, we have the following.(a)
IfA1=R0+={z∈clR:z≥0}=-A2 and the control sequence {uk} is chosen as uk=Me-k(-1)k+1; for all k∈N0, then {S^n(i)x}≡{(1/n)∑k=0n-1T2kx} has a unique weak cluster point {0} for any real x  which is the unique fixed point and best proximity point of T2k:A1→A1, T2k:A2→A2 and a fixed point of T:clR→clR  provided that the previous modified Bregman constraint (36), subject to (37), holds.(b)
Take a controluk=max(Me-k(-1)k+1,εsgn((-1)k+1)), for all k∈N0, and Ai(i=1,2) are redefined as A1=R0+={z∈clR:∞≥z≥ε}=-A2 for some positive real constant ε, then {x2k}⊂A1, {x2k+1}⊂A2, and x2k→ε, x2k+1→-ε as k→∞ if x0∈A1. Then, {S^n(i)x}≡{(1/n)∑k=0n-1T2kx} has a unique weak cluster point {ε} which is the unique fixed point of T2k:A1→A1 and the unique best proximity point in A1 of T:clR→clR  for any x0=x∈A1. Also,  {S^n(i)x}≡{(1/n)∑k=0n-1T2kx} has a unique weak cluster point {-ε} which is the unique fixed point of T2k:A2→A2 and the unique best proximity point in A2 of T:clR→clR  for any x0=x∈A2 provided that the mentioned modified Bregman constraint (36), subject to (37), holds.



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