# An Implicit Algorithm for Finding a Fixed Point of aQ-Nonexpansive Mapping in Locally Convex Spaces

Authors: E. Soori; M.R. Omidi; A.P. Farajzadeh; Yuanheng Wang
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this paper, the notion of theq-duality mappings in locally convex spaces is introduced. An implicit method for finding a fixed point of a Q-nonexpansive mapping is provided. Finally, the convergence of the proposed implicit scheme is investigated. Some examples in order to illustrate of the main results are presented.


---

## Body

## 1. Introduction

LetC be a nonempty closed and convex subset of a Banach space E and E∗ be the dual space of E. Let .,. denote the pairing between E and E∗. The normalized duality mapping J:E⟶E∗ is defined by(1)Jx=f∈E∗:x,f=x2=f2,for all x∈E. It is well known that if J is a single-valued mapping, then it is norm-weak∗ continuous (see, for more details, Theorem 2.6.10 in [1]). In this paper, the duality mapping of locally convex space is studied and denoted by Jq for the seminorm q.Suppose thatQ is a family of seminorms on a locally convex space E which determines the topology of E that will be denoted by τQ. Let C be a nonempty closed and convex subset of E. A mapping T:C⟶C is called Q-nonexpansive if qTx−Ty≤qx−y, for all x,y∈C and q∈Q. Also, the function f:E⟶E is Q-contraction if there exists β∈0,1 such that qfx−fy≤βqx−y, for all x,y∈C and q∈Q.The methods of iterations, which find a number of real applications, are popular for problems in sciences and engineering; see, e.g., [2–6]. Recently, various iterative algorithms have been investigated for fixed points of the nonlinear mapping, especially, for the mappings of nonexpansive type; see, e.g., [7–11] and the references therein.Also, the general implicit algorithm for finding a fixed point of aQ-nonexpansive mapping is given by(2)zn=εnfzn+1−εnTzn,n=1,2,…,where f is a Q-contraction and T is a Q-nonexpansive mapping. On the contrary, it is proven that there exists a sunny Q-nonexpansive retraction P of C onto FixT and x∈C such that the the sequence zn converges to Px with respect to the topology τQ.Moreover, some notions and concepts in locally convex spaces are recalled. The Hahn–Banach theorem and Banach contraction principle will be generalized to locally convex spaces. The results of this article can be viewed as implicit and locally convex space version of some results given in [1, 12–17].


## 2. Methods and Preliminaries

Suppose thatQ is a family of seminorms on a locally convex space E which determines the topology of E will be denoted by τQ.LetD⊆B⊆E and P be a retraction of B onto D, that is, Px=x, for each x∈D. Then, P is said to be sunny if, for each x∈B and t≥0 with Px+tx−Px∈B, we have(3)PPx+tx−Px=Px.A subsetD of B is said to be sunny Q-nonexpansive retract of B if there exists a sunny Q-nonexpansive retraction P of B onto D. We know that if E is a smooth Banach space and P a retraction of B onto D, then P is sunny nonexpansive if and only if(4)x−Px,Jz−Px≤0,for each x∈B and z∈D. For more details, see [18]. In the sequel, we will prove inequality (4) for a real locally convex space.Now, we recall the following definitions:(1)
The locally convex topologyτQ is separated if and only if the family of seminorms Q possesses the following property: for each x∈E\0, there exists q∈Q such that qx≠0 (see [19]) or equivalently(5)∩q∈Qx∈E:qx=0=0.(2)
LetE be a locally convex topological vector space over ℝ or ℂ. If U⊂E, the polar of U is denoted by(6)U∘=f∈E∗:fx≤1,∀x∈U,see [20].(3)
SupposeG is a Hausdorff topological group, and xα is a net in G defined on a directed set D. xα is left Cauchy when the following happens: for every neighborhood B of the identity e of G, there exists an α0∈D such that(7)∀β,γ∈D:β≻α0,γ≻α0⟹xβ−1xγ∈B,see [20].Suppose thatQ is a family of seminorms on a locally convex space X which determines the topology of X and q∈Q. Let Y be a subset of X and(8)qY∗f=supfy:y∈Y,qy≤1.Also,(9)q∗f=supfx:x∈X,qx≤1,for every linear functional f on X. Notice that if, for each x∈X, qx≠0, and f∈X∗, then(10)x,f≤qxq∗f,for more details, see [12].We will make use of the following theorems.Theorem 1 (see [12]).
Suppose thatQ is a family of seminorms on a real locally convex space X which determines the topology of X and q∈Q is a continuous seminorm and Y is a vector subspace of X such that(11)Y∩x∈X:qx=0=0.Letf be a real linear functional on Y such that qY∗f<∞. Then, there exists a continuous linear functional h on X that extends f such that(12)qY∗f=q∗h.Theorem 2 (see [12]).
Suppose thatQ is a family of seminorms on a real locally convex space X which determines the topology of X and q∈Q a nonzero continuous seminorm. Let x0 be a point in X. Then, there exists a continuous linear functional f on X such that q∗f=1 and fx0=qx0.Next, we bring the following known results for easy reference.Theorem 3 (see Proposition 2.5.2 in [1]).
LetX be a topological space and f:X⟶−∞,+∞ a function. Then, the following statements are equivalent: (a) f is lower semicontinuous; (b) for each α∈ℝ, the level set x∈X:fx≤α is closed.Theorem 4 (see Theorem 3.26 (Banach–Alaoglu) in [20]).
Suppose thatX is a locally convex space and U is a neighborhood of zero in X. Then, U∘, the polar set of U, is weak∗ compact.For general topological spaces, we know the useful property as follows.Theorem 5 (see Theorem 3.3.18 in [21]).
For a topological spaceX,τ, the following are equivalent:(i)
X is compact(ii)
Each net inX has a convergent subnet


## 3. Main Result

First, we define our notation of theq-duality mapping as follows.Suppose thatQ is a family of seminorms on a real locally convex space X which determines the topology of X, q∈Q is a continuous seminorm, and X∗ is the dual space of X. A multivalued mapping Jq:X⟶2X∗, defined by(13)Jqx=j∈X∗:x,j=qx2=q∗j2,is called q-duality mapping. Obviously, Jq−x=−Jqx and Jqx≠∅. Indeed, let x∈X; if qx=0, then j=0∈Jqx, and if qx≠0, by applying Theorem 2, there exists a linear functional f∈X∗ such that q∗f=1 and x,f=qx. Putting j:=qxf, then(14)x,j=x,qxf=qxx,f=qx2,q∗j=supjy:y∈X,qy≤1=supqxfy:y∈X,qy≤1=qxsupfy:y∈X,qy≤1=qxq∗f=qx,and then, j=qxf∈Jqx.Now, we present the following result that extends Proposition 4.1.1 in [1].Lemma 1.
LetQ be a family of seminorms on a Hausdorff and complete locally convex space E which determines the topology of E, ϕq:E⟶−∞,∞ a bounded below and lower semicontinuous function, for each q∈Q, and xn a sequence in E such that qxn−xn+1≤ϕqxn−ϕqxn+1, for all n∈ℕ0=ℕ∪0 and q∈Q. Then, xn converges to a point v∈E, and for each q∈Q,(15)qxn−v≤ϕqxn−ϕqv,for all n∈ℕ0.Proof.
Sinceqxn−xn+1≤ϕqxn−ϕqxn+1, for each n∈ℕ0 and q∈Q, then ϕqxn is a decreasing sequence for each q∈Q. Moreover,(16)∑n=0mqxn−xn+1=qx0−x1+qx1−x2+⋯+qxm−xm+1≤ϕqx0−ϕqxm+1≤ϕqx0−infn∈ℕ0ϕqxn,for m∈ℕ0. Letting m⟶∞, we have(17)∑n=0∞qxn−xn+1<∞,and then, limnqxn−xn+1=0, for each q∈Q. This implies that xn is a left Cauchy sequence in E. Since E is Hausdorff and complete, there exists a unique v∈E such that limn⟶∞xn=v. Let m,n∈ℕ0 with m>n. Then,(18)qxn−xm≤∑i=nm−1qxi−xi+1≤ϕqxn−ϕqxm,for each q∈Q. Since ϕq is lower semicontinuous and q∈Q is continuous (Theorem 1.4 in [19]), letting m⟶∞, we conclude that(19)qxn−v≤ϕqxn−limm⟶∞ϕqxm≤ϕqxn−ϕqv,for all n∈ℕ0 and q∈Q.Now, we state an extension of Banach contraction principle to locally convex spaces, and we call it BanachQ-contraction principle.Theorem 6 (BanachQ-contraction principle).
LetQ be a family of seminorms on a separated and complete locally convex space E which determines the topology of E and T:E⟶E a Q-contraction mapping with Lipschitz constant k∈0,1. Then,(a)
T has a unique fixed pointv∈E.(b)
For arbitraryx0∈E, the Picard iteration process defined by(20)xn+1=Txn,n∈ℕ0,converges tov.(c)
qxn−v≤kn/1−kqx0−x1, for all n∈ℕ0 and q∈Q.Proof.
(a)
For eachq∈Q, let ϕq:E⟶ℝ+ be a function defined by(21)ϕqx=qx−Tx1−k,for eachx∈E. Since q is continuous, ϕq is also a continuous function. Also, T is a Q-contraction mapping; then,(22)qTx−T2x≤kqx−Tx,for eachx∈E and q∈Q, which conclude that(23)qx−Tx−kqx−Tx≤qx−Tx−qTx−T2x.Hence,(24)qx−Tx≤11−kqx−Tx−qTx−T2x≤ϕqx−ϕqTx,for eachq∈Q. Let x be an arbitrary and fixed element in E. We define the sequence xn in E by xn=Tnx (n∈ℕ). By using (24), we have(25)qxn−xn+1≤ϕqxn−ϕqxn+1,for eachn∈ℕ and q∈Q. Since E is Hausdorff, by applying Lemma 1, there exists an element v∈E such that(26)limn⟶∞xn=v,qxn−v≤ϕqxn,for eachn∈ℕ0 and q∈Q. Notice that T is continuous (page 3 in [19]) and xn+1=Txn; therefore, v=Tv. Let z∈E be another fixed point for T and qv−z>0 for some q∈Q. Then,(27)0<qv−z=qTv−Tz≤kqv−z<qv−z,which is a contradiction. Therefore,qv−z=0, for each q∈Q. Since E is separated, we have v=z. Hence, T has unique fixed point v∈E.(b)
This assertion follows from part (a).(c)
By applying (22), we have ϕqxn≤knϕqx0, for each q∈Q. This implies from (26) that qxn−v≤knϕqx0, for each q∈Q.Next, we prove a generalization of Proposition 2.4.7 in [1] as follows.Lemma 2.
LetE be a locally convex space. Then, for x,y∈E and for each q∈Q with qx≠0, the following statements are equivalent:(a)
qx≤qx+ty, for all t>0 with qx+ty≠0(b)
There existsjq∈Jqx such that y,jq≥0Proof.
(a)
⟹ (b). Let t>0 and ft∈Jqx+ty, and we define gt=ft/q∗ft. It is clear that q∗gt=1 and gt∈1/q∗ftJqx+ty; then,(28)qx≤qx+ty=x+ty,ftq∗ft=x+ty,gt=x,gt+ty,gt≤qx+ty,gt.PuttingU=x∈E:qx≤1, it is clear that(29)gtt⊂U∘=f∈E∗:fx≤1,∀x∈U,from Theorem4, U∘ is weak∗ compact; then, by applying Theorem 5, the net gt has a limit point g∈E∗ such that q∗g≤1. From the above equations qx≤x+ty,gt, let y=0; by limiting, we have qx≤x,g. Also, since t>0, by using (28) and limiting, we have y,g≥0. Therefore,(30)qx≤x,g≤qxq∗g≤qx,which conclude thatx,g=qx and q∗g=1. Set jq=gqx; then, jqx=gxqx=qx2 and(31)q∗jq=supjqz:z∈X,qz≤1=supqxgz:z∈X,qz≤1=qxsupgz:z∈X,qz≤1=qxq∗g=qx.Hence,jq∈Jqx and y,jq=y,gqx=qxy,g≥0.(b)
⟹ (a). Suppose, for x,y∈X and q∈Q with qx≠0, there exists jq∈Jqx such that y,jq≥0. We know that(32)x+ty,jq≤qx+tyq∗jq=qx+tyqx.
Hence, fort>0 with qx+ty≠0, we have(33)qx2=x,jq≤x,jq+ty,jq=x+ty,jq≤qx+tyqx,which implies that qx≤qx+ty. This completes the proof.The next theorem is significant in the sequel, and it extends Proposition 2.10.21 in [1].Theorem 7.
LetE be a separated locally convex space, C a nonempty convex subset of E, and D a nonempty subset of C. Also, let Jq:E⟶E∗ be a single valued, for every q∈Q and P a retraction of C onto D. Then, for each q∈Q, the following statements are equivalent:(a)
x−Px,Jqy−Px≤0,x∈C,y∈D(b)
P is sunny Q-nonexpansiveProof.
(a)
⟹ (b). First, we show that P is sunny. For x∈C and t>0, set xt:=Px+tx−Px. Since C is convex, xt∈C, for each t∈0,1. Hence, from (a), we have(34)x−Px,JqPx−Pxt≥0,xt−Pxt,JqPxt−Px≥0.Becausext−Px=tx−Px and tx−Px,JqPx−Pxt≥0, we have(35)xt−Px,JqPx−Pxt≥0.Combining (34) and (35), we have(36)qPx−Pxt2=Px−xt+xt−Pxt,JqPx−Pxt=−xt−Px,JqPx−Pxt+xt−Pxt,JqPx−Pxt≤0.Then,qPx−Pxt=0, for each q∈Q. Thus, from the fact that E is separated, Px=Pxt. This means that P is sunny. Now, we show that P is Q-nonexpansive. For x,z∈C and q∈Q, we have from (a) that x−Px,JqPx−Pz≥0 and z−Pz,JqPz−Px≥0. Hence,(37)x−z−Px−Pz,JqPx−Pz≥0,x−z,JqPx−Pz=x−Px+Px−Pz+Pz−z,JqPx−Pz=x−Px,JqPx−Pz−z−Pz,JqPx−Pz+Px−Pz,JqPx−PzgeqPx−Pz,JqPx−Pz=q2Px−Pz.(notice thatJqPx−Pz is single valued). On the contrary,(38)x−z,JqPx−Pz≤qx−zq∗JqPx−Pz=qx−zqPx−Pz.Therefore, be applying (37) and (38), we have(39)qPx−Pz≤qx−z,it follows thatP is Q-nonexpansive.(b)
⟹ (a). Conversely, suppose that P is the sunny Q-nonexpansive retraction from C onto D, x∈C and y∈D. Then, Px∈D and Py=y. Since C is convex, Px+tx−Px∈C, for 0<t≤1. Also, P is sunny; then,(40)Px=PPx+tx−Px,and therefore,(41)qy−Px=qPy−Px=qPy−PPx+tx−Px≤qy−Px+tx−Px=qy−Px+tPx−x.
Hence, from Lemma2, there exists jq∈Jqy−Px such that(42)Px−x,jqy−Px≥0,x∈C,y∈D.
SinceJq:E⟶E∗ is single valued, we can say that(43)x−Px,Jqy−Px≤0,x∈C,y∈D.Next, as a direct consequence of Theorem 6.5.3 in [18], we have the following result. To apply Theorem 6.5.3 in [18] in Theorems 8, 9, 11, and 12 and Corollary 1, it is considered that infqx>0, for each q∈Q, when x is in K (28) and (34), C0, FixT, and C, respectively.Theorem 8.
LetQ be a family of seminorms on a locally convex space E which determines the topology of E and K be a convex subset of E, x∈E and x0∈K. Then, for each q∈Q, the following are equivalent:(1)
qx0−x=infqx−y:y∈K.(2)
There exists anLq∈E∗ such that Lqy≤qy, for each y∈E, q∗Lq=1, and(44)infLqy−x:y∈K=qx0−x.(3)
There exists anLq∈E∗ such that Lqy≤qy, for each y∈E, q∗Lq=1, and(45)infLqy:y∈K=Lqx0,Lqx0−x=qx0−x.The next theorem is similar to Corollary 6.5.5 in [18].Theorem 9.
LetQ be a family of seminorms on a real locally convex space E which determines the topology of E and K be a convex subset of E, x∈E and x0∈K. Then, for each q∈Q, the following statements are equivalent:(1)
qx0−x=infqx−y:y∈K(2)
There exists anfq∈Jqx−x0 such that fqx0−y≥0, for every y∈KProof.
(1)
1⟹2. Let q∈Q and qx0−x=infqx−y:y∈K. Then, from Theorem 8, there exists an Lq∈E∗ such that Lqy≤qy, for each y∈E, q∗Lq=1, and(46)infLqy−x:y∈K=qx0−x.Setfq=−qx0−xLq. It is easy to see that Lqx0−x=qx0−x; therefore,(47)fqx−x0=−qx0−xLqx−x0=q2x0−x=q∗fq2.Then,fq∈Jqx−x0. Hence, from (46) and (47), we have(48)fqx0−y=fqx0−x+x−y=−q2x0−x−qx0−xLqx−y=−q2x0−x+qx0−xLy−x≥−q2x0−x+q2x0−x=0.2⟹1. Since fq∈Jqx−x0, then(49)fqx−x0=q2x0−x=q∗fq2.
Also,fqx0−y≥0, for every y∈K; therefore,(50)q2x0−x=fqx−x0=fqx−y+y−x0≤qx−yq∗fq+fqy−x0≤qx−yq∗fq=qx−yqx0−x,for every y∈K; hence, we have(51)qx0−x≤qx−y,for all y∈K. This means that(52)qx0−x=infqx−y:y∈K.Corollary 1.
LetQ be a family of seminorms on a locally convex space E which determines the topology of E, C a nonempty closed convex subset of E, C0⊆C, and P be a sunny Q-nonexpansive retraction of C onto C0. Let Jq:E⟶E∗ be single-valued duality mapping for each q∈Q; then,(53)x−Px,Jqy−Px≤0,for each x∈C and y∈C0.Proof.
Letx∈C and y∈C0. Set xt=Px+tx−Px, for each 0≤t≤1; then, xt∈C and(54)qPx−y=qPxt−Py≤qxt−y,for each q∈Q. Now, from Theorem 9, there exists fq∈Jqy−Px such that Px−xt,fq=fqPx−xt≥0. Since Jq is single valued, we can say that(55)Px−xt,Jqy−Px≥0,and since Px−xt=tPx−x, then(56)x−Px,Jqy−Px≤0.We now extend the result of Theorem 2.6.10 in [1] as follows.Theorem 10.
LetE be a locally convex space and Jq:E⟶E∗ be a single-valued duality mapping. Then, Jq is continuous from τQ to weak∗ topology, for each q∈Q.Proof.
Letq∈Q be fixed and arbitrary, xαα∈I⊂E, and x∈E such that xα⟶x in τQ, and we prove that Jqxα⟶Jqx in the weak∗ topology. First, assume that qx=0; then, Jqx=0 (from (13)). It is clear that q∗Jqxα=qxα, also q∈Q is continuous Theorem 1.4 in [19]; therefore,(57)limαq∗Jqxα=limαqxα=0,and hence, limαy,Jqxα=0, for every y∈E that qy≤1 (from (9)). On the contrary, if qy>1, then qy/qy=1; therefore, limαy/qy,Jqxα=0. So, we conclude that limαy,Jqxα=0, for every y∈E. Hence, Jqxα⟶0 in the weak∗ topology. Second, assume that qx≠0. Since(58)Jqxα=j∈X∗:xα,j=qxα2=q∗j2,and Jq is single-valued set fαq:=Jqxα. Then,(59)xα,fαq=qxαq∗fαq,qxα=q∗fαq.
Sincexαα is bounded with respect to τQ, then there exists Nq>0 such that qxα≤Nq, for each α∈I. Now, it is shown that fαqα∈I is bounded in E∗.
One can see thatE∗ is a locally convex space and the family of seminorms py:y∈E defines the weak∗ topology on it, where pyx∗=y,x∗, for each x∗∈E (Chapter 5, Definition 1.1 in [22]). Hence,(60)pyfαq=y,fαq≤qyq∗fαq=qyqxα≤qyNq,when qy≠0. Also, if qy=0,(61)pyfαq=y,fαq≤supfαqy:y∈E,qy≤1=q∗fαq=qxα≤Nq,for each α∈I. By applying (60) and (61), there exists Mq,y>0 related to each y∈E and q∈Q, such that pyfαq≤Mq,y, for each α∈I. This means that fαqα∈I is bounded in the weak∗ topology of E∗. Putting Uq=z∈E:qz<1, we have 1/Mq,yfαqα∈I⊂Uq∘, for each y∈E, where Uq∘ is the polar of Uq and it is weak∗ compact (Theorem 4). Then, from Theorem 5, there exists a subnet 1/Mq,yfαβqβ of 1/Mq,yfαα such that 1/Mq,yfαβq⟶f∈Uq∘ in the weak∗ topology, equivalently(62)x,Mq,yf−fαβq⟶0.
We know that the functionq∗ on E∗ is lower semicontinuous in the weak∗ topology. Indeed, if gββ⊂f∈E∗:q∗f≤α such that gβ⟶g in the weak∗ topology, then(63)y,gβ−g=pygβ−g⟶0,for each y∈E; therefore, y,gβ⟶y,g and q∗g≤α, i.e.,(64)g∈f∈E∗:q∗f≤α.
So, by applying Theorem3, q∗ is lower semicontinuous in the weak∗ topology on E∗. Hence, we have(65)q∗f≤liminfβq∗1Mq,yfαβq=1Mq,yliminfβqxαβ=1Mq,yqx.
Sincexαβ⟶x in τQ and xαβ⟶x in weak topology, then(66)x−xαβ,fαβq⟶0.
Therefore, using (62) and (66) implies that(67)x,Mq,yf−qxαβ2=x,Mq,yf−xαβ,fαβq≤x,Mq,yf−fαβq+x−xαβ,fαβq⟶0.
On the contrary,qx2xαβ⟶qx2x. Thus,(68)x,Mq,yf=qx2.
Sinceqx≠0, using (65), we have(69)qx2=x,Mq,yf≤q∗Mq,yfqx≤qx2,and therefore,(70)x,Mq,yf=qx2,qx=q∗Mq,yf.
This means thatMq,yf=Jqx. Finally, since fαq=Jqxα, from (62), we conclude that Jqxα⟶Jqx. This completes the proof.In the next theorem, we prove an existence theorem of a sunnyQ-nonexpansive retract.Theorem 11.
LetQ be a family of seminorms on a real separated and complete locally convex space E which determines the topology of E, C a nonempty closed convex and bounded subset of E such that every sequence in C has a convergent subsequence, and T:C⟶C be a Q-nonexpansive mapping such that FixT≠∅, the fixed points set of T. Let Jq:E⟶E∗ be single valued for every q∈Q. Then, FixT is a sunny Q-nonexpansive retract of C and the sunny Q-nonexpansive retraction of C onto FixT is unique.Proof.
Letx∈C be fixed; then, there exists a sequence zn in C such that(71)zn=1nx+1−1nTzn,n∈ℕ.
For this means, we define the functionKn:C⟶C, with(72)Knt=1nx+1−1nTt,for each n∈ℕ and t∈C. It is clear that Kn is Q-nonexpansive because(73)qKnt1−Knt2≤1−1nqTt1−Tt2≤qt1−t2,for every t1,t2∈C. Then, by applying Theorem 6, there exists a unique point zn in C such that Knzn=zn, for each n∈ℕ; therefore,(74)zn=1nx+1−1nTzn,n∈ℕ.
SinceC is bounded in τQ, there exists Mq>0 such that qy≤Mq, for each y∈C and q∈Q. Also,(75)qTzn−zn=1nqTzn−x≤2nMq,and then,(76)limnqTzn−zn=0,for each q∈Q. Next, we show that the sequence zn converges to an element of FixT. In the other words, we show that the limit set of zn (denoted by Szn) is a subset of FixT. First, for each z∈FixT and n∈ℕ, we have(77)zn−x,Jqzn−z≤0because Tzn−x=nTzn−zn and(78)Tzn−Tz,Jqzn−z≤qTzn−Tzq∗Jqzn−z≤qzn−zqzn−z=q2zn−z.
Also, sinceJq is single valued, from definition (13), we have(79)z−zn,Jqzn−z=−zn−z,Jqzn−z=−q2zn−z.
Therefore,(80)zn−x,Jqzn−z=zn−Tzn+Tzn−x,Jqzn−z=zn−Tzn+nTzn−zn,Jqzn−z=n−1Tzn−zn,Jqzn−z=n−1Tzn−Tz,Jqzn−z+n−1z−zn,Jqzn−z≤n−1qTzn−Tzqzn−z−qzn−z2≤n−1qzn−z2−qzn−z2=0.
From our assumption,zn has a subsequence converges to a point in C. Let zni and znj be subsequences of zn such that zni and znj converge to y and z, respectively. Then, (76) implies that(81)qy−Ty≤qy−zni+qzni−Tzni+qTzni−Ty≤2qy−zni+qzni−Tzni⟶0,for each q∈Q. Since E is separated, Ty=y; hence, y∈FixT and similarly z∈FixT.
Now,zni⟶y and z∈FixT, and we claim that(82)y−x,Jqy−z=limi⟶∞zni−x,Jqzni−z.
Indeed, from the fact thatJq is single valued and continuous from τQ to weak∗ topology, we have(83)zni−x,Jqzni−z−y−x,Jqy−z=zni−x,Jqzni−z−y−x,Jqzni−z+y−x,Jqzni−z−y−x,Jqy−z≤zni−y,Jqzni−z+y−x,Jqzni−z−Jqy−z≤qzni−yqzni−z+y−x,Jqzni−z−Jqy−z≤2qzni−yMq+y−x,Jqzni−z−Jqy−z⟶0,and hence, by using (77),(84)y−x,Jqy−z=limi⟶∞zni−x,Jqzni−z≤0.
Similarly,z−x,Jqz−y≤0 since Jqy−z=−Jqz−y; then,(85)qy−z2=y−z,Jqy−z=y−x,Jqy−z+x−z,Jqy−z=y−x,Jqy−z+z−x,Jqz−y≤0,that is, qy−z=0, for each q∈Q, and since E is separated, y=z. Thus, zn converges to an element of FixT. Therefore, a mapping P:C⟶C can be defined by Px=limnzn. Then, since zn⟶Px, by using (84) and for each z∈FixT,(86)x−Px,Jqz−Px=Px−x,JqPx−z,(87)=limn⟶∞zn−x,Jqzn−z≤0.
It follows from Theorem7 that P is a sunny Q-nonexpansive retraction of C onto FixT. Let R be another sunny Q-nonexpansive retraction of C onto FixT. Then, from Corollary 1,(88)x−Rx,Jqz−Rx≤0,for each x∈C and z∈FixT. Putting z=Rx in (86) and z=Px in (88), we have x−Px,JqRx−Px≤0 and x−Rx,JqPx−Rx≤0; then,(89)q2Rx−Px=Rx−Px,JqRx−Px≤0,for each q∈Q. Since E is separated, Rx=Px. This completes the proof.Proposition 1.
LetQ be a family of seminorms on a separated locally convex space E which determines the topology of E. Then,(90)qx2−qy2≥2x−y,j,for all x,y∈E and j∈Jqy such that qy≠0.Proof.
Letj∈Jqx, x∈E. Then,(91)qy2−qx2−2y−x,j=qx2+qy2−2y,j≥qx2+qy2−2qxqy≥qqx−qy2≥0.Theorem 12.
LetQ be a family of seminorms on a real separated and complete locally convex space E which determines the topology of E, C a nonempty closed convex and bounded subset of E such that every sequence in C has a convergent subsequence, and T:C⟶C a Q-nonexpansive mapping such that FixT≠∅. Let Jq:E⟶E∗ be single valued for each q∈Q and f be a Q-contraction on C; also, εn is a sequence in 0,1 such that limnεn=0. Then, there exists a unique x∈C and sunny Q-nonexpansive retraction P of C onto FixT such that the following sequence zn, generated by(92)zn=εnfzn+1−εnTzn,n∈ℕ,converges to Px.Proof.
Sincef is a Q-contraction, there exists 0≤β<1 such that(93)qfx−fy≤βqx−y,for each x,y∈E and q∈Q. We divide the proof into five steps.Step 1: the existence ofzn which satisfies (92).This follows immediately from the fact that, for everyn∈ℕ, the mapping Nn:C⟶C, given by(94)Nnx≔εnfx+1−εnTx,x∈C,is aQ-contraction. To see this, put βn=1+εnβ−1, 0≤βn<1,n∈ℕ. Then, we have(95)qNnx−Nny≤εnqfx−fy+1−εnqTx−Ty≤εnβqx−y+1−εnqx−y=1+εnβ−1qx−y=βnqx−y<qx−y.Therefore, by Theorem6, there exists a unique point zn∈C such that Nnzn=zn, that is,(96)zn=εnfzn+1−εnTzn.Step 2:limnqTzn−zn=0, for each q∈Q.SinceC is bounded in τQ, there exists Mq>0 such that qy≤Mq, for each y∈C and q∈Q.Then,(97)qTzn−zn=εnqTzn−fzn≤2Mqεn,limnqTzn−zn=0,for eachq∈Q.Step 3:Szn⊂FixT, where Szn denotes the set of τQ-limit points of subsequences of zn.Letz∈Szn and znk be a subsequence of zn such that znk⟶z. By using (97), we have(98)qTz−z≤qTz−Tznk+qTznk−znk+qznk−z≤2qznk−z+qTznk−znk⟶0,for eachq∈Q. Hence, qTz−z=0, for each q∈Q, and since E is separated, we conclude that z∈FixT.Step 4: there exists a unique sunnyQ-nonexpansive retraction P of C onto FixT and x∈C such that(99)K≔limsupnx−Px,Jqzn−Px≤0.We know from Theorem10 that Jq is continuous from τQ to weak∗ topology; then, by Theorem 11, there exists a unique sunny Q-nonexpansive retraction P of C onto FixT. Theorem 6 guarantees that fP has a unique fixed point x∈C. Since(100)K=limsupnx−Px,Jqzn−Px,by the definition oflimsup, there exists subsequence zni of zn such that(101)K=limix−Px,Jqzni−Px.On the contrary, every sequence inC has a convergent subsequence; then, there exists subsequence znik of zni such that znik⟶z and z∈C.By Step 3, we havez∈FixT. From Theorem 7, we have(102)x−Px,Jqz−Px≤0,and sinceJq is continuous, then(103)limkx−Px,Jqznik−Px=x−Px,Jqz−Px.Therefore, by applying (101)–(103), we have K≤0.Step 5:zn converges to Px in τQ.
SincefPx=x, thus f−IPx=x−Px. Now, from Proposition 1 and our assumption, we have(104)εnβ−1qzn−Px2≥εnβqzn−Px+1−εnqzn−Px2−qzn−Px2≥εnqfzn−fPx+1−εnqTzn−Px2−qzn−Px2≥2εnfzn−fPx+1−εnTzn−Px−zn−Px,Jqzn−Px=−2εnf−IPx,Jqzn−Px=−2εnx−Px,Jqzn−Px,for each n∈ℕ. Hence,(105)qzn−Px2≤21−βx−Px,Jqzn−Px,for each q∈Q. Since Px∈FixT, from (99) and (105), we conclude that(106)limsupnqzn−Px2≤21−βlimsupnx−Px,Jqzn−Px≤0,for each q∈Q. That is, zn⟶Px in τQ.


## 4. Numerical Example

The following examples illustrate Theorem12. The first example is in the setting of a locally convex space that is not normable and the rest in finite dimensional spaces.Example 1.
Letε be an arbitrary positive number, E=C0,1, the set of all continuous complex valued functions, and εn=1/n. For each x∈0,1, the seminorm q on E is defined by qxf=fx, for each f∈E. Note that E is a locally convex space with the topology induced by Q. It can be proved by Urysohn’s Lemma that it is not normable.
In Theorem12, let C=Nx0,ε,0=f:qx0f=fx0<ε be convex neighborhood of 0 with x0∈0,1 and x0<ε/2. Suppose that I is the identity mapping on the complex numbers, and let T,F:E⟶E be defined by Tf. The following examples illustrate Theorem 12. The first example is in the setting of a locally convex space that is not normable and the rest in finite dimensional spaces.Example 2.
Letε be an arbitrary positive number and E=C0,1 be the set of all continuous complex valued functions and εn=1/n. For each x∈0,1, we define the seminorm qx on E by qxf=fx, for each f∈CC0,1. Note that E is a locally convex space with the topology induced by Q. It can be proved by Urysohn’s Lemma which is not normable.
In Theorem12, let C=Nx0,ε,0=f:qx0f=fx0<ε be convex neighborhood of 0 with x0∈0,1 and x0<ε/2. Suppose that I is the identity mapping on the complex numbers, and let T,F:E⟶E be defined by Tf=1/2f+I and Ff=1/2f, respectively. It is obvious that T is Q-nonexpansive and F is Q-contraction mapping on C. It can be verified that Pf=I is a sunny Q-nonexpansive retraction from C onto FixT=I; then, obviously, we have the sequence fnn∈ℕ=1−1/nIn∈ℕ generated by (92) which converges to PI=I with respect to the topology induced by Q.Example 3.
In Theorem12, let E=ℝ, εn=1/n, and qx=x be the only seminorm on ℝ, i.e., Q=.. Let C=0,2 and Tx=1/2x+1 be a Q-nonexpansive mapping and fx=1/2x be a Q-contraction mapping on 0,2. It is obvious Px=2 is a sunny Q-nonexpansive retraction from 0,2 onto FixT=2 and the sequence znn∈ℕ=2−2/nn∈ℕ generated by (92) converges to P1=2 and Ff=1/2f, respectively. It is obvious that T is Q-nonexpansive and F is Q-contraction mapping on C. It can be verified that Pf=I is a sunny Q-nonexpansive retraction from C onto FixT=I; then, obviously, we have the sequence fnn∈ℕ=1−1/nIn∈ℕ generated by (92) converges to PI=I with respect to the topology induced by Q.Example 4.
In Theorem12, let E=ℝ, εn=1/n, and qx=x be the only seminorm on ℝ, i.e., Q=.. Let C=0,2 and Tx=1/2x+1 be a Q-nonexpansive mapping and fx=1/2x be a Q-contraction mapping on 0,2. It is obvious Px=2 is a sunny Q-nonexpansive retraction from 0,2 onto FixT=2 and the sequence znn∈ℕ=2−2/nn∈ℕ, generated by (92), converges to P1=2.Example 5.
In Theorem12, assume E=ℝ2 and εn=1/n, and q1x,y=x and q2x,y=y are two seminorms on ℝ2, i.e., Q=q1,q2. Let C=0,2×0,2 and Tx,y=1/2x+1,1/3y+1 be a Q-nonexpansive mapping and fx,y=1/2x,1/3y be a Q-contraction mapping on 0,2×0,2 in Theorem 11. It is easy to check that Px,y=2,3/2 is a sunny Q-nonexpansive retraction from 0,2×0,2 onto FixT=2,3/2, and the sequence znn∈ℕ=xn,ynn∈ℕ=2−2/n,3/2−3/2nn∈ℕ generated by (92) converges to P1,1=2,3/2.



---

*Source: 9913909-2021-08-13.xml*