International Journal of Analysis and Applications Volume 18, Number 1 (2020), 71-84 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-18-2020-71 STRONG CONVERGENCE THEOREM FOR FINITE FAMILY OF GENERALISED ASYMPTOTICALLY NONEXPANSIVE MAPS AGATHA CHIZOBA NNUBIA1, SHEILA AMINA BISHOP2,∗ 1Department of Mathematics, Nnamdi Azikiwe University, Awka, Nigeria 2Department of Mathematics, Convenant University, Ota, Nigeria ∗Corresponding author: sheilabishop95@yahoo.com; sheila.bishop@covenantuniersity.edu.ng Abstract. Let K be a nonexpansive retract of a uniformly convex Banach space X with retraction P and Ti:1··· ,m : K −→ X a finite family of uniformly continuous generalised asymptotically nonexpansive maps with a nonempty common fixed point set F. We provided and proved sufficient conditions for the strong convergence of a sequence of successive approximations generated by an m-step algorithm to a point of F . 1. Introduction Let K be a nonempty subset of a normed linear space E. K is said to be (sequentially) compact if every closed bounded sequence in K has a subsequence that converges in K. K is said to be boundedly compact if every bounded subset of K is compact. Finite dimensional spaces are boundedly compact. Given a subset S of K, we shall denote by co(S) and ccl(S) the convex hull and the closed convex hull of S respectively. If K is boundedly compact convex and S is bounded, then co(S) and hence ccl(S) are compact convex subsets of K. A map T : K → E is said to be semi-compact if for any bounded sequence {xn} ⊂ K such that Received 2019-06-14; accepted 2019-07-18; published 2020-01-02. 2010 Mathematics Subject Classification. 47H09, 47H10. Key words and phrases. common fixed point set; generalised asymptotically nonexpansive maps; strong convergence; uni- formly convex Banach space. c©2020 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 71 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-18-2020-71 Int. J. Anal. Appl. 18 (1) (2020) 72 ‖xn − Txn‖ → 0 as n → ∞ there exists a subsequence {xnj} of {xn} such that xnj converges strongly to some x∗ ∈ K as j → ∞. The map T is said to be demi-compact at z ∈ E if for any bounded sequence {xn} ⊂ K such that ‖xn − Txn‖ → z as n → ∞ there exist a subsequence {xnj} of{xn} and a point p ∈ K such that xnj converges strongly to p as j →∞. (Observe that if T is additionally continuous, then p−Tp = z). A nonlinear map T : K → E is said to be completely continuous if it maps bounded sets into relatively compact sets. A mapping T : K −→ E is called nonexpansive if and only if for all x,y ∈ K, we have that ‖Tx−Ty‖≤‖x−y‖. (1.1) K is called a nonexpansive retract of E if there exists a nonexpansive map P : E −→ K which is onto and such that P2 = I. The map P is called the nonexpansive retraction of E onto K. Let K be a nonempty subset of a real normed space E. Let P : E −→ K be a nonexpansive retraction of E onto K. A nonself map T : K −→ E is called asymptotically nonexpansive mapping if and only if there exists a sequence {µn}n≥1 ⊂ [0, +∞), with lim n→∞ µn = 0 such that for all x,y ∈ K, ‖T(PKT)n−1x−T(PKT)n−1y‖≤ (1 + µn)‖x−y‖ ∀ n ∈ N (1.2) where PK : X −→ K is nonexpansive retraction of E onto K. T is called generalised asymptotically nonexpansive mapping if and only if there exist a sequences {µn}n≥1,{ηn}n≥1 ⊂ [0, +∞), with lim n→∞ µn = 1 and lim n→∞ ηn = 0 such that for all x,y ∈ D(T), ‖T(PKT)n−1x−T(PKT)n−1y‖≤ µn‖x−y‖ + ηn n ≥ 1. (1.3) Goebel and Kirk [3] introduced the class of asymptotically nonexpansive mappings as a generalisation of nonexpansive mappings, Zegeye and Shahzad [13] introduced the class of generalised asymptotically nonexpansive mappings as a generalization of asymptotically nonexpansive maps. As further generali- sation, Alber, Chidume and Zegeye [1] introduced the class of total asymptotically nonexpansive map- pings, where T : K −→ H is called total asymptotically nonexpansive if and only if there exist two se- quences {µn}n≥1,{ηn}n≥1 ⊂ [0, +∞), with lim n→∞ µn = 0 = lim n→∞ ηn and nondecreasing continuous function φ : [0, +∞) −→ [0, +∞) with φ(0) = 0 such that for all x,y ∈ K, ‖T(PKT)n−1x−T(PKT)n−1y‖≤‖x−y‖ + µnφ(‖x−y‖) + ηn n ≥ 1. (1.4) Ofoedu and Nnubia [8] gave an example to show that the class of asymptotically nonexpansive mappings is a proper subset of the class of total asymptotically nonexpansive mappings. The class of total asymptotically nonexpansive type mappings includes the class of mappings which are asymptotically nonexpansive in the intermediate sense. These classes of mappings had been studied by several authors (see e.g., [3], [5], [11], [14]). Int. J. Anal. Appl. 18 (1) (2020) 73 2. Preliminaries We shall make use of the following result: A Banach space E is said to satisfy Opial’s condition if for each sequence {xn}n≥1 ∈ E which converges weakly to a point z ∈ E, we have that lim inf n→∞ ‖xn − z‖ < lim inf n→∞ ‖xn − y‖, ∀y ∈ E, such that y 6= z. It is well known that every Hilbert space satisfies Opial’s condition(see e.g., [9]). A map T is said to satisfy condition B if there exists f : [0,∞) → [0,∞) strictly increasing, continuous, f(0) = 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 2.1. [2] Let H be a real Hilbert space. Then for all x,y ∈ H the following inequality holds. ‖x + y‖2 ≤‖x‖2 + 2〈y,x + y〉. Lemma 2.2. [2] For any x,y,z in a real Hilbert space H and a real number λ ∈ [0, 1], ‖λx + (1 −λ)y −z‖2 = λ‖x−z‖2 + (1 −λ)‖y −z‖2 −λ(1 −λ)‖x−y‖2. Lemma 2.3. [12] Let {xn} be sequence of nonegative real numbers satisfying the following relation: xn+1 ≤ xn −αnxn + δn, n ≥ n0, where {αn}n≥1 ⊂ (0, 1) and {δn}n≥1 ⊂ R satisfying the following conditions: ∞∑ n=0 αn = ∞, lim n→∞ αn = 0 and lim sup n→∞ δn ≤ 0, then lim n→∞ xn = 0. Lemma 2.4. Let {µn},{νn} and{ηn} be nonnegative sequences such that∑ n≥0 νn < ∞ , ∑ n≥0 ηn < ∞ and µn+1 ≤ (1 + νn)µn + ηn. Then lim n→∞ µn exists. The nearest point projection PK : H −→ K defined from H onto K is the function which assign to each x ∈ H its nearest point denoted by PKx ∈ K. Thus PKx is the unique point in K such that ‖x−PKx‖≤‖x−y‖ for all y ∈ K and we have the following Lemmas. Lemma 2.5. [12]. Let K be a closed convex nonempty subset of a real Hilbert space H. Let x ∈ H, then z = PKx if and only if 〈x−z,y −z〉≤ 0 ∀ y ∈ K. Lemma 2.6. Let K be a closed convex nonempty subset of a Banach space E and let Ti∈I : K −→ K where i ∈ I = {1, 2, ...,m}. be finite family of continous nonlinear maps in K such that F = ⋂m i=1 F(Ti) 6= ∅ and let {xn}n≥1 be a sequence of successive approximation satisfying (1) lim n→∞ ‖xn −Tixn‖ = 0 ∀i ∈ I Int. J. Anal. Appl. 18 (1) (2020) 74 (2) ‖xn+1 −x∗‖≤ (1 + τn)‖xn −x∗‖ + νn where ∑ n≥0 νn < ∞ and ∑ n≥0 τn < ∞. Then, {xn} converges strongly to a common fixed point of Ti’s if and only if lim infn→∞d(xn,F) = 0. Proof Now, d(xn+1,F) ≤ (1 + τn)d(xn,F) + νn (2.1) hence lim infn→∞d(xn,F) = 0 ⇒ limn→∞d(xn,F). Also, ∀ k > 0 ‖xn+k+1 −x∗‖ ≤ Πkj=0(1 + τn+j)‖xn −x ∗‖ + k∑ j=0 νn+k−jΠ j−1 r=0(1 + τn+k−r) ≤ Πkj=0(1 + τn+j)(‖xn −x ∗‖ + k∑ j=0 νn+k−j) ≤ Q(‖xn −x∗‖ + k∑ j=0 νn+j) So that given any ε > 0 there exists an integer n0 > 0, such that for all n ≥ n0, d(xn,F) < ε4(Q+1) and νn+j < ε 4m(Q+1) ∀ j = 1, 2, ...,m. So ∃ x∗ ∈ F such that d(xn0,x∗) < ε 4(Q+1) that is, ‖xn0 − x∗‖ < ε 4(Q+1) Now, ‖xn+k −xn‖≤‖xn+k −x∗‖ + ‖xn −x∗‖ ≤ 2Q(‖xn0 −x ∗‖ + k∑ j=0 νn0+j) ≤ 2Q( ε 4(Q + 1) + m ε 4m(Q + 1) ) = 2Q( ε 2(Q + 1) < ε. So, {xn} is a Cauchy sequence in E and so it converges to some u∗ ∈ K. But, xn −Tixn → 0 as n →∞∀ i and Ti is continuous ∀ i. Hence, 0 = lim n→∞ (xn −Tixn) = lim n→∞ xn −Ti( lim n→∞ xn) = u ∗−Tiu∗. So that u∗ ∈ F. i.e u∗ = x∗ ∈ F. Hence, xn → x∗ as n →∞. This completes the proof since the other part is obvious. Lemma 2.7. Let K be a closed convex nonempty subset of a Banach space E and let Ti : K −→ K where i ∈ I = {1, 2, ...,m}. be finite family of continuous nonlinear maps in K such that F = ⋂m i=1 F(Ti) 6= ∅ and let {xn}n≥1 be a sequence of successive approximation satisfying (1) lim n→∞ ‖xn −Tixn‖ = 0 ∀i ∈ I (2) ‖xn+1 −x∗‖≤ (1 + τn)‖xn −x∗‖ + νn where ∑ n≥0 νn < ∞ and ∑ n≥0 τn < ∞ Then, {xn} converges strongly to a common fixed point of Ti’s if one of the Ti’s satisfy condition B. Int. J. Anal. Appl. 18 (1) (2020) 75 Proof Let Ti0 satisfy condition B. Then ∃f : [0,∞) → [0,∞) with f(0) = 0 and f(d(xn,F)) ≤‖xn −Ti0xn‖, hence lim n→∞ f(d(xn,F)) ≤ lim n→∞ ‖xn −Ti0xn‖ = 0. Therefore, lim n→∞ d(xn,F) = 0. Thus by Lemma 2.6, {xn} converges strongly to a common fixed point of Ti’s, this concludes the proof. Lemma 2.8. Let K be a closed convex nonempty subset of a Banach space E and let Ti : K −→ Kwhere i ∈ I = {1, 2, ...,m}. be finite family of continuous nonlinear maps in K such that F = ⋂N i=1 F(Ti) 6= ∅. Suppose the sequence {xn}n≥1 of successive approximation satisfies the following conditions (1) lim n→∞ ‖xn −x∗‖ exists withx∗ ∈ F, (2) lim n→∞ ‖xn −Tixn‖ = 0 ∀i ∈ I, (3) {xn}n≥1 has a convergent subsequence {xnj}n≥1. Then, {xn} converges strongly to a point of F. Proof Suppose that {xn} has a convergent subsequence {xnj} and let xnj → p as j →∞, since xn−Tixn → 0 as n →∞ for all i ∈{1, 2, ...,N}. It implies that xnj −Tixnj → 0 as j →∞ for all i ∈ I. Also, by continuity of Ti .Tixnj → Tip as j → ∞ for all i ∈ I . So, ‖p−Tip‖ = lim n→∞ ‖xnj −Tixnj‖ = 0,∀i which implies that p ∈ F. Now, lim n→∞ ‖xn −p‖ exists from our hypothesis and lim n→∞ ‖xnj −p‖ = 0, so lim n→∞ ‖xn −p‖ = 0. Thus, {xn} converges strongly to a point of F. Remark 2.1. The conditions for which {xn} has a convergent subsequence includes (1) Ti is completely continuous ∀i ∈{1, ...,N}. (2) Ti is demicompact ∀i ∈{1, ...,N}. (3) Ti is semicompact for some i ∈{1, ...,N}. (4) K is compact. (5) K is boundedly compact. Proposition 2.1. Let K be a nonexpansive retract of a uniformly convex Banach space X with nonexpansive retraction P . Let T : K −→ X uniformly continuous generalised asymptotically nonexpansive map with associated sequences {µn}n≥1,{ηn}n≥1 ⊂ [0, +∞) with ∞∑ n=0 (µn − 1) < ∞ ∞∑ n=0 ηn < ∞ , suppose that F(T) 6= ∅. Then F(T) is closed and convex (where F(T) is the fixed point set of T ). Int. J. Anal. Appl. 18 (1) (2020) 76 Proof. Let {xn} be a sequence in F(T) converging to x∗ ∈ K, then xn = Txn∀ n ≥ 0. By continuity of T,x∗ = lim n→∞ xn = lim n→∞ Txn = T( lim n→∞ xn) = Tx ∗. Thus, x∗ ∈ F(T) and F(T) is closed. Next, we show that F(T) is convex. For t ∈ (0, 1) and x,y ∈ F(T), put z = (1 − t)x + ty, we show that z = Tz. ‖z −T(PKT)n−1z‖2 = ‖z‖2 − 2〈z,T(PKT)n−1z〉 + ‖T(PKT)n−1z‖2 = ‖z‖2 − 2(1 − t)〈x,T(PKT)n−1z〉− 2t〈y,T(PKT)n−1z〉 +‖T(PKT)n−1z‖2 = ‖z‖2 + (1 − t)‖x−T(PKT)n−1z‖2 + t‖y −T(PKT)n−1z‖2 −(1 − t)‖x‖2 − t‖y‖2 ≤ ‖z‖2 + (1 − t)(µn‖x−z‖ + ηn)2 + t(µn‖y −z‖ + ηn)2 −(1 − t)‖x‖2 − t‖y‖2 = ‖z‖2 + (1 − t) ( µ2n‖x−z‖ 2 + (2µn‖x−z‖ + ηn)ηn ) +t ( µ2n‖y −z‖ 2 + (2µn‖y −z‖ + ηn)ηn ) − (1 − t)‖x‖2 − t‖y‖2 = ‖z‖2 + (1 − t)µ2n(‖x‖ 2 −‖z‖2 − 2〈x,z〉) +tµ2n(‖y‖ 2 −‖z‖2 − 2〈y,z〉) +2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) + η2n − (1 − t)‖x‖ 2 − t‖y‖2 = (1 + µ2n)‖z‖ 2 + (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 ) − 2µn ( (1 − t)〈x,z〉 +t〈y,z〉 ) + 2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) + η2n ≤ (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 + ‖z‖2 ) + 2µ2n‖z‖ 2 −2µ2n ( (1 − t)〈x,z〉 + t〈y,z〉 ) +2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) + η2n = (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 + ‖z‖2 ) − 2µ2n ( (1 − t)〈x,z〉 +t〈y,z〉− (1 − t)‖z‖2 − t‖z‖2 ) +2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) + η2n = (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 + ‖z‖2 ) − 2µ2n ( (1 − t)〈x−z,z〉 +t〈y −z,z〉 ) + 2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) + η2n = (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 + ‖z‖2 ) − 2µ2n〈(1 − t)(x−z) + t(y −z),z〉 +2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) + η2n Int. J. Anal. Appl. 18 (1) (2020) 77 = (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 + ‖z‖2 ) − 2µ2nt(1 − t)(〈x−y + y −x,z〉 +η2n + 2µnηn ( (1 − t)‖x−z‖ + t‖y −z‖ ) = (µ2n − 1) ( (1 − t)‖x‖2 + t‖y‖2 + ‖z‖2 ) + 4t(1 − t)‖x−y‖µnηn + η2n. Thus, lim n→∞ ‖z − T(PKT)n−1z‖ = 0, which implies that lim n→∞ T(PKT) n−1z = z and hence z = lim n→∞ T(PKT) n−1z = TPK ( lim n→∞ T(PKT) n−2z) = TPKz = Tz. Thus, z ∈ F(T). This completes the proof. Proposition 2.2. Let K be a nonexpansive retract of a uniformly convex Banach space X with nonex- pansive retraction P . Let Ti : K −→ X (i = 1,...,m) be a finite family of uniformly continuous gen- eralised asymptotically nonexpansive map with associated sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) with ∞∑ n=0 (µin − 1) < ∞, ∞∑ n=0 ηin < ∞. Suppose that F = ⋂m i=1 F(Ti) 6= ∅. Then F is closed and convex. Proof. By Lemma 2.1, we have that F(Ti) is closed for each i. Now, F = ⋂m i=1 F(Ti) is a finite intersection of closed sets, hence closed. Also, by Lemma 2.1, we have that F(Ti) is convex for each i. Since F = ⋂m i=1 F(Ti) is a finite intersection of convex set, we have that F is convex. Proposition 2.3. Suppose that there exist c > 0,k > 0 constants such that φ(t) ≤ ct for all ≥ k, then T is total asymptotically nonexpansive if and only if T is generalised asymptotically nonexpansive. Proof It is known that every generalised asymptotically nonexpansive map is total asymptotically non- expansive, so it suffices to show that that every total asymptotically nonexpansive with the condition of our hypothesis is generalised asymptotically nonexpansive. Now, let T be such that ‖Tnx−Tny‖≤‖x−y‖ + µnφ(‖x−y‖) + ηn n ≥ 1 (2.2) Since φ is continuous, it follows that φ reaches its maximum (say c0) on the interval [0,k]; moreover, φ(t) ≤ ct whenever t > k. Thus, φ(t) ≤ c0 + ct ∀ t ∈ [0, +∞). (2.3) So, we have, ‖Tnx−Tny‖ ≤ ‖x−y‖ + µn(c0 + c‖x−y‖) + ηn n ≥ 1 = (1 + µnc)‖x−y‖ + µnc0 + ηn = (1 + νn)‖x−y‖ + γn where νn = µnc and γn = µnc0 + ηn. This completes the proof. Int. J. Anal. Appl. 18 (1) (2020) 78 Corollary 2.1. Let K be a nonexpansive retract of a uniformly convex Banach space X with nonexpansive retraction P . Let T : K −→ X be a uniformly continuous total asymptotically nonexpansive map with associated sequences {µn}n≥1,{ηn}n≥1 ⊂ [0, +∞) with ∞∑ n=0 (µn − 1) < ∞ ∞∑ n=0 ηn < ∞ . Suppose that there exist c > 0,k > 0 constants such that φ(t) ≤ ct∀ t ≥ k, and that F(T) 6= ∅ then F(T) is closed and convex. Corollary 2.2. Let K be a nonexpansive retract of a uniformly convex Banach space X with nonexpansive retraction P . Let Ti : K −→ X (i = 1,...,m) be a finite family of uniformly continuous total asymptotically nonexpansive maps with associated sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) with ∞∑ n=0 (µin−1) < ∞ ∞∑ n=0 ηin < ∞ . Suppose that there exist c > 0,k > 0 constants such that φ(t) ≤ ct for all t ≥ k, and that F =⋂m i=1 F(Ti) 6= ∅ then F is closed and convex. 3. Main results Proposition 3.1. Let H be a normed linear space, let K be a closed convex nonempty subset of H and let TiK −→ H (i ∈ I = {1, ...,m}) be a finite family of continuous generalised asymptotically nonexpansive map with sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) such that lim n→∞ µin = 1 and lim n→∞ ηn = 0 with ∞∑ n=0 (µin − 1) < ∞ ∞∑ n=0 ηin < ∞ . Suppose that F(T) 6= ∅ and let {xn}n≥1 be a sequence generated iteratively by starting with an arbitrary x0 ∈ K, define by yn,i = PK [(1 −αn)xn + αnTi(PTi)n−1yn,i−1], yn,0 = xn; yn,m = xn+1 = yn+1 n ≥ 0 , (3.1) where {αn}n≥1, is a sequence in (0, 1) such that 0 < ζ < βn < � < 1 ∀ n ≥ 1. Let x∗ ∈ F, then lim n→∞ ‖xn−x∗‖ exist Proof. Let x∗ ∈ F, j ∈ I then from (3.1) we have that ‖yn,j −x∗‖ = ‖PK [(1 −αn)xn + αnTj(PTj)n−1yn,j−1] −Px∗‖ (3.2) ≤ (1 −αn)‖xn −x∗‖ + αn‖Tj(PTj)n−1yn,j−1 −x∗‖ ≤ (1 −αn)‖xn −x∗‖ + αn(µn,j)‖yn,j−1 −x∗‖ + ηin). = (1 −αn)‖xn −x∗‖ + αnµn,j(‖yn,j−1 −x∗‖ + αnηn,j) (3.3) Int. J. Anal. Appl. 18 (1) (2020) 79 ‖yn,1 −x∗‖ ≤ (1 −αn)‖xn −x∗‖ + αnµn,1(‖yn,0 −x∗‖ + αnηn,1) ≤ ( 1 + αn(µn,1−1) ) ‖xn −x∗‖ + αnηn,1 (3.4) ‖yn,2 −x∗‖ ≤ (1 −αn)‖xn −x∗‖ + αnµn,2 ( (1 + α(µn, 1 − 1))‖xn −x∗‖ + αnηn,1 ) + αnηn,2. ≤ ( 1 + αn(µn,2 − 1) + α2nµ(n, 2)(µn,1 − 1) ) ‖xn −x∗‖ + α2nµn,2ηn,1 + αnηn,2. (3.5) ‖yn,3 −x∗‖ ≤ ( 1 + αn(µn,3 − 1) + α2nµ(n, 3)(µn,2 − 1) + α 3 nµ(n, 3)µn,2(µn,1 − 1) ) ‖xn −x∗‖ +αnηn,3 + α 2 nµn,3ηn,2 + α 3 nµn,3µn,2ηn,1. Hence, ‖yn,j −x∗‖ ≤ ( 1 + j∑ t=1 αtnΠ t−1 s=1µn,j−s+1(µn,j−t+1 − 1) ) ‖xn −x∗‖ + j∑ t=1 αtnΠ t−1 s=1µn,j−s+1ηn,j−t+1. ‖xn+1 −x∗‖ ≤ ( 1 + m∑ t=1 αtnΠ t−1 s=1µn,m−s+1(µn,m−t+1 − 1) ) ‖xn −x∗‖ + m∑ t=1 αtnΠ t−1 s=1µn,m−s+1ηn,m−t+1 ≤ ( 1 + qm−1b m∑ j=1 (µn,j − 1) ) ‖xn −x∗‖ + qm−1b m∑ j=1 ηn,j. (since there exists n0 such that µn,i ≤ q for all n ≥ n0,∀ j ∈ I) So, lim n→∞ ‖xn − x∗‖ exist; and hence {xn},{yn,j} are bounded. Theorem 3.1. Let K be a nonexpansive retract of a uniformly convex Banach space X with nonexpan- sive retraction P . Let Ti : K −→ E be a finite family of uniformly continuous generalised asymptotically nonexpansive maps with sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) such that lim n→∞ µin = 1, lim n→∞ ηin = 0, ∞∑ n=0 (µin − 1) < ∞ ∞∑ n=0 ηin < ∞ Int. J. Anal. Appl. 18 (1) (2020) 80 Suppose that F = ⋂N i=1 F(Ti) is not empty and let {xn}n≥1 be a sequence generated iteratively by (3.1 ) where {αn}n≥1 is a sequence in (0, 1) satisfying the following conditions: ∞∑ n=1 αn < ∞, 0 < ζ < αn < � < 1 ∀ n ≥ 1 , then ∀j ∈{1, 2, ...,m}, lim n→∞ ‖xn −Tjxn‖ = 0 and {xn}n≥1 converges weakly to a point of F . Proof. Let x∗ ∈ F ‖yn,j −x∗‖2 ≤ (1 −αn)‖xn −x∗‖2 + αn‖Tj(PTj)n−1yn,j−1 −x∗‖ (3.6) −αn(1 −αn)g(‖xn −Tj(PTj)n−1yn,j−1‖) ≤ (1 −αn)‖xn −x∗‖2 + αn(µn,j‖yn,j−1 −x∗‖ + ηn,j)2 −αn(1 −αn)g(‖xn −Tj(PTj)n−1yn,j−1‖) ≤ (1 −αn)‖xn −x∗‖2 + αnµ2n,j‖yn,j−1 −x ∗‖2 +αn(2µn,j‖yn,j−1 −x∗‖ + ηn,j)ηn,j −αn(1 −αn)g(‖xn −Tj(PTj)n−1yn,j−1‖) So ‖yn,1 −x∗‖2 ≤ ( 1 + αn(µ 2 n,1 − 1) ) ‖xn −x∗‖2 + αn(2µn,1‖xn −x∗‖ + ηn,1)ηn,1 −αn(1 −αn)g(‖xn −T1(PT1)n−1xn‖) ‖yn,2 −x∗‖2 ≤ ( 1 + αn(µ 2 n,2 − 1) + α 2 nµ 2 n,2(µ 2 n,1 − 1) ) ‖xn −x∗‖2 +αn(2µn,2‖yn,1 −x∗‖ + ηn,2)ηn,2 + α2nµ 2 n,2(2µn,1‖xn −x ∗‖ + ηn,1)ηn,1 −αn(1 −αn)g(‖xn −T2(PT2)n−1yn,1‖) −α2nµ 2 n,2(1 −αn)g(‖xn −T1(PT1) n−1xn‖) So, ‖yn,j −x∗‖2 ≤ ( 1 + j∑ t=1 αtnΠ t−1 s=1µ 2 n,j−s+1(µ 2 n,j−t+1 − 1) ) ‖xn −x∗‖2 + j∑ t=1 αtnΠ t−1 s=1µ 2 n,j−s+1(2µn,j−t+1‖yn,j−t−1 −x ∗‖ + ηn,j−t+1)ηn,j−t+1Πt−1s=0µ 2 n,j−s −(1 −αn) j∑ t=1 αtng(‖xn −Tj−t+1(PTj−t+1) n−1yn,j−t‖)Πt−1s=1µ 2 n,j−s+1 Int. J. Anal. Appl. 18 (1) (2020) 81 Hence, ‖xn+1 −x∗‖2 ≤ ( 1 + m∑ t=1 α t nΠ t−1 s=1µ 2 n,m−s+1(µ 2 n,m−t+1 − 1) ) ‖xn −x∗‖2 + m∑ t=1 α t nΠ t−1 s=1µ 2 n,m−s+1(2µn,m−t+1‖yn,m−t −x ∗‖ + ηn,m−t+1)ηn,m−t+1 −(1 −αn) m∑ t=1 α t nΠ t−1 s=1µ 2 n,m−s+1g ( ‖xn −Tm−t+1(PTm−t+1)n−1yn,m−t‖ ) ≤ ( 1 + q 2(m−1) b m∑ j=1 (µ 2 n,j − 1) ) ‖xn −x∗‖ + q2(m−1)b m∑ j=1 (2µn,j‖yn,j−1 −x∗‖ +ηn,j )ηn,j‖−am(1 −αn) m∑ j=1 g(‖xn −Tm−j+1(PTm−j+1)n−1yn,m−j‖) ≤ ( 1 + q 2(m−1) b m∑ j=1 (µ 2 n,j − 1) ) ‖xn −x∗‖ +q 2(m−1) b m∑ j=1 ηn,j −am(1 −αn) m∑ j=1 g(‖xn −Tm−j+1(PTm−j+1)n−1yn,m−j‖) So, ‖xn+1 −x∗‖2 ≤ ( 1 + d0 m∑ j=1 (µ2n,j − 1) ) ‖xn −x∗‖ +d1 m∑ j=1 ηn,j −d2 m∑ j=1 g(‖xn −Tj(PTj)n−1yn,j−1‖) So, lim n→∞ g(‖xn −Tj(PTj)n−1yn,j−1‖) = 0, thus lim n→∞ ‖xn −Tj(PTj)n−1yn,j−1‖ = 0 ∀ j = 1, ...,m. Now, ‖xn −Tj(PTj)n−1xn‖ ≤ ‖xn −Tj(PTj)n−1yn,j−1‖ + ‖Tj(PTj)n−1yn,j−1 −Tj(PTj)n−1xn‖ ≤ ‖xn −Tj(PTj)n−1yn,j−1‖ + µn,j‖yn,j−1 −xn‖ + ηn,j ≤ ‖xn −Tj(PTj)n−1yn,j−1‖ + µn,jαn‖xn −Tj−1(PTj−1)n−1yn,j−2‖ + ηn,j Hence, lim n→∞ ‖xn −Tj(PTj)n−1xn‖ = 0 ∀ j = 1, ...,m. Further, ‖xn −Tjxn‖≤‖xn −Tj(PTj)n−1yn,j−1‖ + ‖Tj(PTj)n−1yn,j−1 −Tjxn‖ ‖(PTj)n−1yn,j−1 −xn‖ ≤ ‖Tj(PTj)n−2yn,j−1 −xn‖ ≤ ‖Tj(PTj)n−2yn,j−1 −Tj(PTj)n−2yn−1,j−1‖ +‖Tj(PTj)n−2yn−1,j−1 −xn−1‖ + ‖xn−1 −xn‖ ≤ µn−1,j‖yn,j−1 −yn−1,j−1‖ + ηn−1,i +‖xn−1 −Tj(PTj)n−2yn−1,j−1‖ + ‖xn −xn−1‖ Int. J. Anal. Appl. 18 (1) (2020) 82 ‖yn,j −yn−1,j‖ ≤ ‖yn,j −xn‖ + ‖xn −xn−1‖ + ‖xn−1 −yn−1,j ≤ αn‖xn −Tj(PTj)n−1yn,j−1‖ + αn−1‖xn−1 −Tm(PTm)n−2yn−1,m−1‖ +αn−1‖xn−1 −Tj(PTj)n−2yn−1,j−1‖ So, lim n→∞ ‖yn,j −yn−1,j‖ = 0. Also, lim n→∞ ‖xn −xn−1‖ = 0 so that lim n→∞ ‖(PTj)n−1yn,j−1 −xn‖ = 0. Hence lim n→∞ ‖xn −Tjxn‖ = 0 ∀ j = 1, ...,m. By reflexivity ∃ z ∈ K and {xnj}⊂{xn} such that, {xnj}→w z as j →∞. Since, xnj−Tixnj → 0 as j →∞ ∀i then z ∈ F(Ti) ∀i and so z ∈ F = m⋂ i=1 F(Ti). Let ωw(xn) be subsequential limit set of the sequence {xn}. Let q ∈ ωw(xn) arbitrary. Then ∃ {xnr} ⊂ {xn} 3 {xnr} converges weakly q and xnr − Tixnr → 0 as r → ∞ ∀i. Thus, ωw(xn) ⊆ F. Thus {xn}n≥1 converges weakly to a point of F . Theorem 3.2. Let K,X,P, T ′is,F,{xn} be as in Theorem 3.1 Then, {xn} converges strongly to a fixed point of T if and only if lim infn→∞d(xn,F) = 0 (where F= F(T)). The Proof follows from Lemma 2.6, since from Theorem 3.1 and it’s proof, the conditions of the lemma are satisfied. Theorem 3.3. Let K,X,P, T ′is,F,{xn} be as in Theorem 3.1 Then, {xn} converges strongly to a common fixed point of Ti’s if one of the Ti’s satify condition B. The Proof follows from Lemma 2.7, since from the proof of Theorem 3.1, the conditions of the lemma are satisfied. Theorem 3.4. Let K,X,P,T ′is,F,{xn} be as in Theorem 3.1 Then, {xn} converges strongly to a common fixed point of Ti’s if {xn}n≥1 has a convergent subsequence {xnj}n≥1 The Proof follows from Lemma 2.8, since from the proof of Theorem 3.1, the conditions of the lemma are satisfied. As a result of the proposition 2.3 we have the following results. Theorem 3.5. Let K be a nonexpansive retract of a uniformly convex Banach space X with nonexpansive retraction P . Let Ti : K −→ E be a finite family of uniformly continuous total asymptotically nonexpansive maps with sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) such that lim n→∞ µin = 0 = lim n→∞ ηin, ∞∑ n=0 (µin − 1) < ∞ ∞∑ n=0 ηin < ∞ and with function φ : [0, +∞) −→ [0, +∞) satisfying φ(t) ≤ M0t for all t > M1, for some constants M0,M1 > 0. Suppose that F = ⋂m i=1 F(Ti) is not empty and let {xn}n≥1 be a sequence generated iteratively Int. J. Anal. Appl. 18 (1) (2020) 83 by (3.1 ) where {αn}n≥1 is a sequence in (0, 1) satisfying the following conditions: ∞∑ n=1 αn < ∞, 0 < ζ < αn < � < 1 ∀ n ≥ 1, then for all j ∈{1, 2, ...,m}, lim n→∞ ‖xn −Tjxn‖ = 0 and {xn}n≥1 converges weakly to a point of F . Theorem 3.6. Let K,X,P,T ′is,F,{xn} be as in Theorem 3.5 Then, {xn} converges strongly to a fixed point of T if and only if lim infn→∞d(xn,F) = 0. Theorem 3.7. Let K,X,P, T ′is,F,{xn} be as in Theorem 3.5. Then, {xn} converges strongly to a common fixed point of Ti’s if one of the Ti’s satify condition B. Theorem 3.8. Let K,X,P,T ′is,F,{xn} be as in Theorem 3.5 Then, {xn} converges strongly to a common fixed point of Ti’s if {xn}n≥1 has a convergent subsequence {xnj}n≥1. Our iterative process generalise some of the existing ones, our theorems improves, generalise and extend several known results and our method of proof is of independent interest. 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