EJ-MATHEJ-MATHEJ-MATHEuropean Journal of Mathematics and Statistics2736-5484European Open ScienceUK30910.24018/ejmath.2024.5.5.309Research ArticleIterative Procedure for Finite Family of Total Asymptotically Nonexpansive Maps (TAN)Iterative Procedure for Finite Family of Total Asymptotically Nonexpansive Maps (TAN)Nnubia et al.NnubiaAgatha Chizoba1ac.nnubia@unizik.edu.ngAkabuikeNkiruka Maria-Assumpta2MooreChika1Department of Mathematics, Nnamdi Azikiwe University Awka, NigeriaDepartment of Mathematics and Computer, Federal Polytechnic Oko, NigeriaCorresponding Author: e-mail: ac.nnubia@unizik.edu.ng

Conflict of Interest: Authors declare that they do not have any conflict of interest.

In this paper, CQ Algorithms for iterative approximation of a common fixed point of a finite family of nonlinear maps were introduced and sufficient conditions for the strong convergence of this process to a common fixed point of the family of Total asymptotically Nonexpansive maps (TAN) were proved.

Let H be a normed space, K be a nonempty closed convex subset of H and T:K→K be a map. The mapping T is called asymptotically nonexpansive mapping if and only if there exists a sequence {μn}n≥1⊂[0,+∞), with limn→∞μn=0 such that for all x.y∈K,‖Tnx−Tny‖≤(1+μn)‖x−y‖∀n∈N

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalisation of nonexpansive mappings. As further generalisation of class of nonexpansive mappings, Alber et al. [2] introduced the class of total asymptotically nonexpansive mappings, where a mapping T:K⟶K is called total asymptotically nonexpansive (TAN) if and only if there exist two sequences {μn}n≥1,{ηn}n≥1⊂[0,+∞), with limn→∞μn=0=limn→∞ηn and nondecreasing continuous function ϕ:[0,+∞)⟶[0,+∞) with ϕ(0)=0 such that for all x,y∈K,‖Tnx−Tny‖≤‖x−y‖+μnϕ(‖x−y‖)+ηnn≥1

In Ofoedu and Nnubia [3], an example to show that the class of asymptotically nonexpansive mappings is properly contained in the class of total asymptotically nonexpansive mappings was given. The class of asymptotically nonexpansive type mappings includes the class of mappings which are asymptotically nonexpansive in the intermediate sense. These classes of mappings had been studied extensively by several authors (see e.g., [4]–[9]).

A map T is said to satisfies condition B if there exists f:[0,∞)→[0,∞) strictly increasing, continuous, f(0)=0,f(r)>0∀r>0 such that for all x∈D(T),‖x−Tx‖≥f(d(x,F)) where F=F(T)={x∈D(T):x=Tx} and d(x,F)=inf{‖x−y‖:y∈F}.

Lemma 1.1Takahashi [10]

Let {μn},{βn},{γn} be sequences of nonnegative numbers satisfying the conditons: ∑n≥0∞βn=∞,βn⟶0 as n⟶∞ and γn=0(βn). Suppose that
μn+12≤μn2−βnψ(μn+1)+γn;n=1,2,...where ψ:[0,1)→[0,1) is a strictly increasing function with ψ(0)=0. Then μn→0 as n→∞.

Lemma 1.2Moore and Nnoli [11]

Let K be closed convex nonempty subset of a real Hilbert Space H. Let {xn} be a sequence in H, y∈H and z=PKy be such that ωw(xn)⊆K and ‖xn−y‖≤‖y−z‖∀n≥1, then {xn} converges strongly to z.ωw(xn)={z∈H:∃{xnj(z)}⊂{xn}∋xnj(z)→wzasj→∞}

Lemma 1.3Ofoedu and Nnubia [9, p. 703]

Let E be a reflexive Banach space with weakly continuous normalized duality mapping. Let K be a closed convex subset of E and T:K→K a uniformly continuous total asymptotically nonexpansive mapping with bounded orbits. Then I − T is demiclosed at zero.

Proposition 1.1Ofoedu and Nnubia [9, p. 704]

Let H be a real Hilbert space, let K be a nonempty closed convex subset of H and let Ti:K⟶K,wherei∈I={1,2,...,m}, be m uniformly continuous total asymptotically nonexpansive mappings from K into itself with sequences {μn,i}n≥1,{ηn,i}n≥1⊂[0,+∞) such that limn→∞μn,i=0=limn→∞ηn,i and with function ϕi:[0,+∞)⟶[0,+∞) satisfying ϕi(t)≤M0t∀t>M1 for some constantsM0,M1>0. Let μn=maxi∈I{μn,i} and ηn=maxi∈I{ηn,i} and, ϕ(t)=maxi∈I{ϕi(t)}∀t∈[0,∞). Suppose that F(T)=⋂i=1mF(Ti), then F(T) is closed and convex.

Proposition 1.2Nnubia and Bishop [6, p. 74]

Let K be a nonempty subset of a real normed space E and Ti:K⟶K,wherei∈I={1,2,...,m}, be m total asymptotically nonexpansive mappings, then there exist sequences {μn}n≥1,{ηn}n≥1⊂[0,+∞), with limn→∞μn=0=limn→∞ηn and nondecreasing continuous function ϕ:[0,+∞)⟶[0,+∞) with ϕ(0)=0 such that for all x,y∈K,‖Tinx−Tiny‖≤‖x−y‖+μnϕ(‖x−y‖)+ηn;n≥1,∀i∈I.

Main Result

Proposition 2.1Result

Suppose that there exist c>0,k>0 constants such that ϕ(t)≤ct∀t≥k, then T is total asymptotically nonexpansive if ∃νn=μnc and γn=μnc0+ηn such that
‖Tnx−Tny‖≤(1+νn)‖x−y‖+γn

Proof

Suppose T is total asymptotically nonexpansive, that is, let T be such that
‖Tnx−Tny‖≤‖x−y‖+μnϕ(‖x−y‖)+ηnn≥1

Since ϕ is continuous, it follows that ϕ attains its maximum (say c0) on the interval [0,k]; moreover, ϕ(t)≤ct whenever t>k. Thus,
ϕ(t)≤c0+ct∀t∈[0,+∞).

So, we have,
‖Tnx−Tny‖≤‖x−y‖+μn(c0+c‖x−y‖)+ηnn≥1=(1+μnc)‖x−y‖+μnc0+ηn=(1+νn)‖x−y‖+γnwhere νn=μnc and γn=μnc0+ηn Thus completing the proof.

Theorem 2.1 Let H, K, Ti, and F be as in Propostion 1.1, then {xn}n≥1 generated iteratively by:
yn=(1−αn)xn+αnTi(n)m(n)xn;i(n)≡nmodm∀n∈Z;m(n)=1+[nm]Kn={z∈K:‖yn−z‖2≤‖xn−z‖2−αn(1−αn)‖xn−Ti(n)m(n)xn‖2+σn}Qn={z∈K:⟨xn−z,xo−xn⟩≥0}xn+1=PKn∩Qnx0.converges to PFx0 where σn=αn(km(n),i(n)2−1)(diam.K)2+αn(2km(n),i(n)diam.K+vm(n),i(n))vm(n),(n) and {αn}⊂[a,b]⊂(0,1).

Proof

Let x∗∈F.
‖yn−x∗‖2=‖(1−αn)(xn−x∗)+αn(Ti(n)m(n)xn−x∗)‖2≤(1−αn)‖xn−x∗‖2+αn‖Ti(n)m(n)xn−x∗‖2−αn(1−αn)‖xn−Ti(n)m(n)xn‖2≤(1−αn)‖xn−x∗‖2+αn(1+ki(n)m(n))2‖xn−x∗‖2+(2(1+ki(n)m(n))‖xn−x∗‖+vi(n)m(n))vi(n)m(n)−αn(1−αn)‖xn−Ti(n)m(n)xn‖2=(1+αn[1+ki(n)m(n)−1])‖xn−x∗‖2+(2(1+ki(n)m(n))‖xn−x∗‖+vi(n)m(n))vi(n)m(n)−αn(1−αn)‖xn−Ti(n)m(n)xn‖2≤‖xn−x∗‖2−αn(1−αn)‖xn−Ti(n)m(n)xn‖2+αn[(1+ki(n)m(n))2−1](diam.K)2+αn[2(1+ki(n)m(n))(diam.K)+vi(n)m(n)]vi(n)m(n)=‖xn−x∗‖2−αn(1−αn)‖xn−Ti(n)m(n)xn‖2+σn.

So that x∗∈Kn∀n. Hence, F⊂Kn∀n. For n=0,Qo=K.F⊂Qo.

Let F⊂Qv, we show that F⊂Qv+1. Now, xv+1 is the projection of xo onto Kv∩Qv. then (i) ⟨xv+1−z,xo−xv+1⟩≥0.∀z∈Kv∩Qv.

Since, F⊂Kv∩Qv, then ⟨xv+1−x∗,xo−xv+1⟩≥0∀x∗∈F. So, F⊂Qv+1 and hence F⊂Qn∀n≥0.

Now,
‖xo−PQnxo‖≤‖xo−y‖∀y∈Qn

Hence, ∀n‖xn−xo‖=‖xo−PQnxo‖≤‖xo−y‖∀y∈Qn.and since F⊂Qn, then ‖xn−xo‖≤‖xo−x∗‖∀x∗∈F.

In particular, ‖xn−xo‖≤‖xo−x∗‖;x∗=PFxo.

Since, xn+1∈Qn,⟨xn+1−xn,xn−xo⟩≥0.

So,
‖xn+1−xn‖2=‖xn+1−xo−(xn−xo)‖2=‖xn+1−xo∗‖2−‖xn−xo‖2−2⟨xn+1−xn,xn−xo⟩≤‖xn+1−xo‖2−‖xn−xo‖2

Then, ‖xn−xo‖≤‖xn+1−xo‖.

Since {‖xn−xo‖} is bounded, then limn→∞‖xn−xo‖ exists.

Thus,
limn→∞‖xn+1−xn‖=0.

Observe that by (7)limn→∞‖xn+i−xn‖=0=limn→∞‖xn−i−xn‖∀i∈{1,...,m}.

Now, ∀n>m, we have n=(n−m)(modm) and since n=(m(n)−1)m+i(n), we obtain n−m=(m(n)−1)N+i(n)−m=(m(n−m)−1)+i(n−m), so that n−m=[(m(n)−1)−1]m+i(n)=(m(n−m)−1)m+i(n−m). Hence, m(n)−1=m(n−m) and i(n)=i(n−m). Similarly, m(n+1−m)=m(n+1)−1 and i(n + 1 − N) = i(n + 1). Using this we obtain
‖xn−Ti(n+1)xn+1‖≤‖xn−Ti(n+1)m(n+1)xn+1‖+‖Ti(n+1)m(n+1)xn+1−Ti(n+1)xn+1‖≤‖xn−xn+1‖+‖xn+1−Ti(n+1)m(n+1)xn+1‖+‖Ti(n+1)m(n+1)xn+1−Ti(n+1)xn+1‖but,
‖Ti(n+1)m(n+1)−1xn+1−xn+1‖≤‖Ti(n+1)m(n+1)−1xn+1−Ti(n+1)m(n+1)−1m(n+1)−1n+1−m‖+‖Ti(n+1)m(n+1)−1xn+1−m−xn+1−m‖+‖xn+1−m−xn+1‖≤(2+km(n+1)−1)‖xn+1−xn+1−m‖+‖Ti(n+1−m)m(n+1−mxn+1−m−xn+1−m‖+vm(n+1)−1so that by hypothesis, (9), (8) and the uniform countinuity of Tii∈I we have:
limn→∞‖xn−Ti(n+1)xn+1‖=0

So that using (7) and (11) we have,
limn→∞‖xn+1−Ti(n+1)xn+1‖=0.

Now, let k∈I be arbitrarily choosen, then,
‖xn−Ti(n+k)xn‖≤‖xn−xn+k‖+‖xn+k−Ti(n+k)xn+k‖+‖Ti(n+k)xn+k−Ti(n+k)xn‖

By uniform continuity of Ti∀i∈I and from (8) and (13) we have that
limn→∞‖xn−Ti(n+k)xn‖=0∀k∈I={1,...,m}

Observe that ∀k∈I={1,..,m}∃ηk∈Isuch thati(n)+ηk≡kmodm, put in another way, ∀k∈I∃ik∈Isuch thati(n+k)≡imodN Hence,
limn→∞‖xn−Tixn‖=0;∀i∈{1,...,m}.

Since (1−Ti) is demiclosed at 0 ∈H∀i.{xn} is bounded and H is reflexive,

so, ∃z∈K and {xnj}⊂{xn} such that, xnj→wz as j→∞. Since, xnj−Tixnj→0 as j→∞∀i then z∈F(Ti)∀i and so z∈F=⋂i=1mF(Ti)

Let q∈ωw(xn) arbitrary. Then ∃{xnr}⊂{xn}∋xnr→wq and xnr−Tixnr→0 as r→∞∀i. So that since 1−Ti is demiclosed at 0∀i,q∈F. Hence, ωw(xn)⊆F. Moreover, ‖xn−xo‖≤‖xo−x∗‖∀n≥0 where x∗=PFxo. Then by the lemma 1.2 {xn} converges strongly to x∗=PFxo (that is the common fixed point nearest to x0).

Theorem 2.2 Let H, K, Ti, and F be as in Theorem 2.1, then {xn}n≥1 generated iteratively by
yn,0=xn;yn,i=(1−αn)xn+αnTinyn,i−1;i=1,...,mKn,i={z∈K:‖yn,i−z‖2≤‖xn−z‖2−αn(1−αn)ai∑j=0i−1‖xn−Ti−jnyn,i−j−1‖2+σn,i}Kn=⋂i=1mKn,iQn={z∈K:⟨xn−z,xo−xn⟩≥0}xn+1=PKn∩Qnx0.converges to PFx0 where σn,i=do∑j=1i[(kn,j2−1)+vn,j],i∈I,0<a≤αn≤b<1.

Proof

Let x∗∈F.
‖yn,i−x∗‖2≤(1−αn)‖xn−x∗‖2+αnkn,i2‖yn,i−1−x∗‖2+αn(2kn,i‖yn,i−1−x∗‖+vn,i)ηn,i−αn(1−αn)‖xn−Tinyn,i−1‖2

So,
‖yn,1−x∗‖2≤(1+αn(kn,12−1))‖xn−x∗‖2+αn(2kn,1‖xn−x∗‖+vn,1)vn,1−αn(1−αn)‖xn−T1nxn‖

So,
‖yn,i−x∗‖2≤(1+∑j=0i−1αnj+1Πt=0j−1kn,i−t2(kn,i−j2−1))‖xn−x∗‖2+∑j=0i−1αnj+1(2kn,i−j‖yn,i−j−1−x∗‖2+vn,i−j)vn,i−jΠt=0j−1kn,i−t2−∑j=0j−1αnj+1(1−αn)‖xn−Ti−jnyn,i−j−1‖2Πt=0j−1kn,i−t2≤(1+bqi∑j=1i(kn,j2−1))‖xn−x∗‖2+bqid1∑j=1ivn,j−ai(1−b)∑j=1i‖xn−Tjnyn,j−1‖2≤‖xn−x∗‖2−ai(1−b)∑j=1i‖xn−Tjnyn,j−1‖2+σn,iwhere do=bqimax{d1,(diamK)2} so, x∗∈Kn,i∀i and hence x∗∈⋂i=1mKn,i∀n. So, F⊂Kn,i∀n≥0,∀i and hence, F⊂Kn. Following the argument as in Theorem 2.1, we have that F⊂Qn∀n≥0 and
limn→∞‖xn+1−xn‖=0.

Now,
αn2‖xn−Tinyn,i−1‖2≤‖yn,i−xn+1‖2+2‖yn,i−xn+1‖.‖xn+1−xn‖+‖xn+1−xn‖2but xn+1∈Kn, so, ∀i∈I‖yn,i−xn+1‖2≤‖xn−xn+1‖2−αn(1−αn)ai∑j=1i‖xn−Tjnyn,j−1‖2+σn,i.so that ∀i∈Iαn2‖xn−Tinyn,i−1‖2≤‖xn+1−xn‖2+σn,i−αn(1−αn)ai∑j=1i‖xn−Tjnyn,j−1‖2+2‖yn,i−xn+1‖.‖xn+1−xn‖+‖xn+1−xn‖2and so ∀i∈Ia2‖xn−Tinyn,i−1‖2≤αn2‖xn−Tinyn,i−1‖2≤2‖xn+1−xn‖2+σn,i+2‖yn,i−xn+1‖.‖xn+1−xn‖

Hence, limn→∞‖xn−Tinyn,i−1‖=0∀i∈I and hence, limn→∞‖yn,i−xn‖=0∀i∈I‖xn−Tinxn‖≤‖xn−Tinyn,i−1‖+kn,i‖yn,i−1−xn‖+vn,i

Thus, limn→∞‖xn−Tin−1xn‖=0∀i∈I, so that by uniform continuity of Ti∀i∈I, limn→∞‖xn−Tixn‖=0. Now, since (1−Ti) is demiclosed at 0, the same argument in Theorem 2.1 completes the Proof.

Conclusion

CQ Algorithms for iterative approximation of a common fixed point of a finite family of nonlinear maps were introduced and sufficient conditions for the strong convergence of this process to a common fixed point of the family of Total asymptotically Nonexpansive maps were proved. Our iterative processes generalise some of the existing ones, our theorems improve, generalise and extend several known results and our method of proof is of independent interest.

Authors wish to acknowledge in advance the contributions of the reviewers.

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