International Journal of Analysis and Applications ISSN 2291-8639 Volume 9, Number 2 (2015), 96-113 http://www.etamaths.com CONVERGENCE THEOREM FOR FINITE FAMILY OF TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS E.U. OFOEDU AND AGATHA CHIZOBA NNUBIA∗ Abstract. In this paper we introduce an explicit iteration process and prove strong convergence of the scheme in a real Hilbert space H to the common fixed point of finite family of total asymptotically nonexpansive mappings which is nearest to the point u ∈ H. Our results improve previously known ones ob- tained for the class of asymptotically nonexpansive mappings. As application, iterative method for: approximation of solution of variational Inequality prob- lem, finite family of continuous pseudocontractive mappings, approximation of solutions of classical equilibrium problems and approximation of solution- s of convex minimization problems are proposed. Our theorems unify and complement many recently announced results. 1. Introduction Let K be a nonempty subset of a real Hilbert space H. A mapping T : K −→ K is called nonexpansive if and only if for all x,y ∈ K, we have that ‖Tx−Ty‖≤‖x−y‖.(1) The mapping T 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, (2) ‖Tnx−Tny‖≤ (1 + µn)‖x−y‖ ∀ n ∈ N. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [11] as a generalisation of nonexpansive mappings. As further generalisation of class of nonexpansive mappings, Alber, Chidume and Zegeye [2] introduced the class of total asymptotically nonexpansive mappings, where a mapping T : K −→ K is called total asymptotically nonexpansive if and only if there exist two sequences {µn}n≥1,{ηn}n≥1 ⊂ [0, +∞), with lim n→∞ µn = 0 = lim n→∞ ηn and nondecreasing continous function φ : [0, +∞) −→ [0, +∞) with φ(0) = 0 such that for all x,y ∈ K, (3) ‖Tnx−Tny‖≤‖x−y‖ + µnφ(‖x−y‖) + ηn n ≥ 1 Observe that if φ(t) = 0 ∀ t ∈ [0, +∞), then equation (3) becomes (4) ‖Tnx−Tny‖≤‖x−y‖ + ηn n ≥ 1, so that if K is bounded and TN is continuous for some integer N ≥ 1, then the mapping T is of asymptotically nonexpansive type. The class of asymptotically 2010 Mathematics Subject Classification. 47H10, 47J25. Key words and phrases. Hilbert space; total asymptotically nonexpansive; nearest point ap- proximation; variational inequality; Viscosity approximation method; strong convergence. c©2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 96 TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 97 nonexpansive type mappings includes the class of mappings which are asymptoti- cally nonexpansive in the intermediate sense and the class of nearly asymptotically nonexpansive mappings. These classes of mappings had been studied extensively by several authors (see e.g.[11], [15], [31]). If φ(t) = t ∀ t ∈ [0, +∞), then equation (3) becomes (5) ‖Tnx−Tny‖≤ (1 + µn)‖x−y‖ + ηn n ≥ 1 In addition, if ηn = 0 for all n ∈ N, then we easily see that every asymptoti- cally nonexpansive mapping is total asymptotically nonexpansive. If µn = 0 and ηn = 0 ∀ n ≥ 1 we obtain from equation (3) the class of mappings which includes the class of nonexpansive mappings. The class of total asymptotically nonexpansive mappings properly includes the class of asymptotically nonexpansive mappings (See Example 2 of [20]). A point x0 ∈ K is called a fixed point of a mapping T : K −→ K if and only if Tx0 = x0. We denote the set of fixed points of T by F(T), that is, F(T) = {x ∈ K : Tx = x}. A point x∗ ∈ K is called a minimum norm fixed point of T if and only if x∗ ∈ F(T) and ‖x∗‖ = min{‖x‖ : x ∈ F(T)}. Let D1 and D2 be nonempty closed convex subsets of real Hilbert spaces H1 and H2, respectively. The split feasibility problem is formulated as finding a point x satisfying (6) x ∈ D1 such that Ax ∈ D2, where A is bounded linear operator from H1 into H2. A split feasibility problem in finite dimensional Hilbert spaces was first studied by Censor and Elfving [8] for modeling inverse problems which arise in medical image reconstruction, image restoration and radiation therapy treatment planning (see e.g [6], [7], [8]). It is clear that x ∈ D1 is a solution of the split feasibility problem (6) if and only if Ax−PD2Ax = 0, where PD2 is the metric projection from H2 onto D2. Consider the minimization problem: (7) find x∗ ∈ D1 such that 1 2 ‖Ax∗ −PD2Ax‖ 2 = min x∈D1 1 2 ‖Ax−PD2Ax‖ 2, then x∗ is a solution of (6) if and only if x∗ solves the minimization problem (7) with the minimum equal to zero. Suppose that problem (6) has solution and let Ω denote the (closed convex) set of solutions of (6) (or equivalently, solution of (7), then Ω is a singleton if and only if it is a set of solutions of the following variational inequality problem: (8) find x ∈ D1 such that 〈A∗(I −PD2 )Ax,y −x〉≥ 0 ∀ y ∈ D2, where A∗ is the adjoint of the linear operator A . Moreover, problem (8) can be rewritten as (9) find x ∈ D1 such that 〈x−rA∗(I −PD2 )Ax−x,y −x〉≤ 0 ∀ y ∈ D2, where r > 0 is any positive scalar. Using the nature of projection, (9) is equivalent to the fixed point equation (10) x = PD1 (x−rA ∗(I −PD2 )Ax). Thus, finding a solution of split feasibility problem (7) is equivalent to finding the minimum-norm fixed point of the mapping x −→ PD1 (x−rA∗(I −PD2 )Ax). Approximation of solutions of equations involving nonexpansive mappings and their 98 OFOEDU AND NNUBIA generalization by iterative methods has been of increasing research interest for numerous mathematicians in recent years. One of the first results of this nature was obtained by Browder [5] for nonexpansive self mappings in Hilbert spaces. Suppose K is a closed convex nonempty subset of a real Hilbert space H. Browder [5] studied the path u ∈ K, xt = tu+ (1−t)Txt, t ∈ (0, 1), where T : K −→ K is a nonexpansive mapping. In [5], Browder proved that lim t→0 xt exists and lim t→0 xt ∈ F(T). The result was extended by Reich [24] to uniformly smooth real Banach spaces. Reich [24] proved, in fact, that lim t→0 xt is a sunny nonexpansive retraction of K onto F(T). In [12], Halpern studied the convergence of the explicit iteration method defined from x1 ∈ K by (11) xn+1 = αnu + (1 −αn)Txn; n ≥ 1 in the frame work of real Hilbert spaces. Under appropriate conditions on the it- erative parameter αn, it had been shown by Halpern [12], Lions [16], Wittmann [26] and Banschke [3] that the sequence {xn} generated by (11) converges strongly to a fixed point of T nearest to u, that is, PF(T)u, Browder and Halpern itera- tive methods had motivated different iterative methods for approximation of fixed points of asymptotically nonexpansive mappings. In this regard, Lim and Xu [15] introduced and studied the following implicit iterative method for asymptotically nonexpansive mapping T, (12) zn = αnu + (1 −αn)Tnzn; n ≥ 1. They showed that the sequence {zn}n≥1 generated by (12) converges strongly to a fixed point of T in the frame work of uniformly smooth real Banach spaces under suitable conditions on the iterative parameters. In [10], Chidume, Li and Udomene proved the strong convergence of the explicit iterative method generated from x1,u ∈ K by (13) xn+1 = αnu + (1 −αn)Tnxn; n ≥ 1, where lim n→∞ αn = 0, ∞∑ n=0 αn = +∞ and T is asymptotically nonexpansive. Yao, Zhou and Lion [28], studied a modified Mann iteration algorithm {xn} gener- ated from x1,∈ H by νn = (1 − tn)xn, xn+1 = (1 −αn)xn + αnTνn, n ≥ 1,(14) where {tn}n≥1,{αn}n≥1 are sequences in (0, 1) satisfying appropriate conditions. They proved the strong convergence of the modified algorithm to the fixed point of a nonexpansive mapping T : H −→ H when F(T) 6= ∅. Osilike etal [23] modified the algorithm (14) with {xn} generated from x1,∈ K by νn = PK[(1 − tn)xn], xn+1 = (1 −αn)xn + αnTnνn; n ≥ 1,(15) where {tn}n≥1,{αn}n≥1 are sequences in (0, 1) satisfying appropriate conditions. They proved the strong convergence of the modified algorithm to the fixed point of assymptotically nonexpansive mapping T : K −→ K when F(T) 6= ∅. Recently, Alber, Espinola and Lorenzo [2] obtained strong convergence of (13) for TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 99 a total asymptotically nonexpansive self map T on K in the setting of smooth re- flexive real Banach space with weakly sequentially continuous duality mapping. In connection with the iterative approximation of minimum norm fixed point of the mapping T , Yang, Lion and Yao [27] introduced an explicit iterative method generated from x1 ∈ K by (16) xn+1 = βnTxn + (1 −βn)PK[(1 −αn)xn]; n ≥ 1, They proved under appropriate conditions on {αn}n≥1 and{βn}n≥1 that the se- quence {xn}n≥1 converges strongly to the minimum norm fixed point of T in Hilbert spaces. Yang et al [27] proved that the explicit iterative method generated from x1 ∈ K defined by (17) xn+1 = PK[(1 −αn)Txn]; n ≥ 1, converges strongly to the minimum norm fixed point of nonexpansive mapping T : K −→ K provided that {αn}n≥1 satisfies appropriate condition. Recently, Zegeye and Shahzad [31] proved that the iterative method generated from arbitrary x1 ∈ K by yn = PK[(1 −αn)xn], xn+1 = βnxn + (1 −βn)Tnyn; n ≥ 1,(18) converges strongly to minimum norm fixed point of asymptotically nonexpansive self map T on K. Motivated by the results of these authors, it is our aim in this paper to prove strong convergence theorem to the common fixed point of finite family of total asymptotically nonexpansive mappings which is nearest to the point u ∈ H. Our theorems generalize and unify the corresponding results of Osilike etal [23], Yao, Zhou and Lion [28], Yang, Lion and Yao [27], Zegeye and Shahzad [31]. Our method of proof is of independent interest. 2. Preliminaries We shall make use of the following lemmas and propositions. Lemma 2.1. 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. 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. [25] Let K be a closed convex nonempty subset of a real Hilbert space H. Let x ∈ H, then x0 = PKx if and only if 〈z −x0,x−x0〉≤ 0 ∀ z ∈ K Let T : K −→ K be a mapping and I be the identity mapping of K, we say that (I −T) is demiclose at zero if and only if for any sequence {xn}n≥1 in K such that xn converges weakly to x and xn −Txn → 0, as n →∞, we have that x = Tx. 100 OFOEDU AND NNUBIA Lemma 2.4. (see Corollary 2.6 of [1]) Let E be a reflexive Banach space with weakly continuous normalized duality mapping. Let K be a closed convex subset of E and let T be a uniformly continuous total asymptotically nonexpansive mapping from K into itself with bounded orbit, then (I −T) is demiclose at zero. Lemma 2.5. [1] Let {an} be a sequence of nonegative real numbers satisfying the following relation: an+1 ≤ (1 −αn)an + δn; n ≥ 1. Suppose that for n ≥ 1, δn αn ≤ c1 and αn ≤ α (for some α,c1 > 0), then an ≤ max{a1, (1 +α)c1}. Moreover, if ∞∑ n=0 αn = ∞ and δn, = o(αn), then lim n→∞ an = 0. Lemma 2.6. (see [17]) Let {Γn} be sequence of real numbers that does not decrease at infinity in the sense that there exists a subsequence {Γnj}of {Γn} which satisfies Γnj < Γnj+1 ∀ j ∈ N. Define the sequence {τ(n)}n≥n0 of integers as follows τ(n) = max{k ≤ n : Γk < Γk+1}, where n0 ∈ N and that the set {k ≤ n0 : Γk < Γk+1} is not empty, then the following hold (i) τ(n0) ≤ τ(n0 + 1) and τ(n) →∞ as n →∞ (ii) Γτ(n) ≤ Γτ(n)+1 and Γn ≤ Γτ(n)+1 ∀ n ∈ N. Proposition 2.1. (see Proposition 8 of [21]) Let H be a real Hilbert space, let K be a nonempty closed convex subset of H and let Ti : K −→ K , where i ∈ I = {1, 2, ...,m}, be m uniformly continuous total asymptotically nonexpansive map- pings from K into itself with sequences {µn,i}n≥1,{ηn,i}n≥1 ⊂ [0, +∞) such that lim n→∞ µn,i = 0 = lim n→∞ ηn,i and with function φi : [0, +∞) −→ [0, +∞) satisfying φi(t) ≤ M0t ∀ t > M1 for some constants M0,M1 > 0. Let µn = max i∈I {µn,i} and ηn = max i∈I {ηn,i} and, φ(t) = max i∈I {φi(t)}∀ t ∈ [0,∞). Suppose that F(T) =⋂m i=1 F(Ti), then F(T) is closed and convex. Proposition 2.2. [20] Let K be a nonempty subset of a real normed space E and Ti : K −→ K , where i ∈ I = {1, 2, ...,m}, be m total asymptotically nonexpansive mappings, then there exist sequences {µn}n≥1,{ηn}n≥1 ⊂ [0, +∞), with lim n→∞ µn = 0 = lim n→∞ ηn and nondecreasing continous function φ : [0, +∞) −→ [0, +∞) with φ(0) = 0 such that for all x,y ∈ K, (19) ‖Tni x−T n i y‖≤‖x−y‖ + µnφ(‖x−y‖) + ηn; n ≥ 1,∀ i ∈ I. 3. Main results Let K be a nonempty closed and convex subset of a real Hilbert space H. Let Ti : K −→ K, where i ∈ I = {1, 2, ...,m}, be m total asymptotically nonexpan- sive mappings and {αn}n≥1,{βn}n≥1 be sequences in (0, 1), we define the explicit TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 101 iteration process {xn}n≥1 from x1 ∈ K,u ∈ H by y1 = PK[α1u + (1 −α1)x1], x2 = (1 −β1)x1 + β1T1y1, y2 = PK[α2u + (1 −α2)x2], x3 = (1 −β2)x2 + β2T2y2, ... ym−1 = PK[αm−1u + (1 −αm−1)xm−1], xm = (1 −βm−1)xm + βm−1Tm−1ym−1, ym = PK[αmu + (1 −α)xm], xm+1 = (1 −βm)xm + βmT1mym, ym+1 = PK[αm+1u + (1 −αm+1)xm+1],(20) xm+2 = (1 −βm+1)xm+1 + βm+1T21 ym+1 ym+2 = PK[αm+2u + (1 −αm+2)xm+2], xm+3 = (1 −βm+2)xm+2 + βm+2T22 ym+2 ... y2m−1 = PK[α2m−1u + (1 −α2m−1)x2m−1], x2m = (1 −β2m−1)x + β2m−1T2m−1y2m−1 y2m = PK[α2mu + (1 −α2m)x2m], x2m+1 = (1 −β2m)x2m + β2mT2my2m y2m+1 = PK[α2m+1u + (1 −α2m+1)x2m+1], x2m+2 = (1 −β2m+1)x2m+1 + β2m+1T31 y2m+1 ... (21) Since ∀z ∈ Z (where Z is the set of integers), there exists j(z) ∈ I such that z − j(z) is divisible by m (that is j(z) = z mod(m)), then there exists q(z) ∈ Z with lim z→∞ q(z) = +∞ such that (22) z = ( q(z) − 1 ) m + j(z) 102 OFOEDU AND NNUBIA so we may write (20) in a more compact form as x1 ∈ K,u ∈ H,yn = PK[αnu + (1 −αn)xn], xn+1 = (1 −βn)xn + βnT q(n) j(n) yn. (23) Remark 3.1. Since n−m ∈ Z∀n ∈ N, we obtain from (22) for z = n−m that (24) n−m = ( q(n−m) − 1 ) m + j(n−m). Also, substituting n ∈ N for z in (22) and subtracting m from both sides of the resulting equation gives (25) n−m = ( (q(n) − 1) − 1 ) m + j(n) Comparing (24) and (25) we obtain (by unique representation theorem) that (26) q(n−m) = q(n) − 1 and j(n−m) = j(n) ∀ n ∈ N. Theorem 3.1. Let H be a real Hilbert space, let K be a closed convex nonemp- ty subset of H and let Ti : K −→ K, where i ∈ I = {1, 2, ...,m}, be m uni- formly continuous total asymptotically nonexpansive mapping from K into itself with sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) such that lim n→∞ µin = 0 = lim n→∞ ηin and with function φi : [0, +∞) −→ [0, +∞) satisfying φi(t) ≤ M0t ∀ t > M1 for some constants M0,M1 > 0, Let µn = max i∈I {µin} and ηn = max i∈I {ηin} and, φ(t) = max i∈I {φi(t)}∀ t ∈ [0,∞). Suppose that F = ⋂m i=1 F(Ti) 6= ∅ and let {xn}n≥1 be a sequence generated iteratively by (23), where {αn}n≥1,{βn}n≥1 are sequences in (0, 1) satisfying the following conditions: ∞∑ n=1 αn = ∞, lim n→∞ αn = 0, lim n→∞ α−1n µn = lim n→∞ α−1n ηn = 0 and 0 < ζ < βn < � < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to PF (u). Proof. Let x∗ ∈ F, then from (23) and hypothesis on Ti we have that ‖yn −x∗‖ = ‖PK[αnu + (1 −αn)xn] −PKx∗‖ ≤ ‖αnu + (1 −αn)xn −x∗‖ ≤ (1 −αn)‖xn −x∗‖ + αn‖u−x∗‖(27) and ‖xn+1 −x∗‖ = ‖(1 −βn)xn + βnT q(n) j(n) yn −x∗‖ ≤ (1 −βn)‖xn −x∗‖ + βn‖T q(n) j(n) yn −x∗‖ ≤ (1 −βn)‖xn −x∗‖ + βn [ ‖yn −x∗‖ + µq(n)φ(‖yn −x∗‖) + ηq(n) ] .(28) Since φ is continuous, it follows that φ attains its maximum (say M) on the interval [0,M1], moreover, φ(t) ≤ M0t whenever t > M1. Thus, (29) φ(t) ≤ M + M0t ∀ t ∈ [0, +∞). TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 103 Using (27) and (29) we obtain from (28) that ‖xn+1 −x∗‖ ≤ (1 −βn)‖xn −x∗‖ +βn [ ‖yn −x∗‖ + µq(n) ( M + M0‖yn −x∗‖ ) + ηq(n) ] ≤ (1 −βn)‖xn −x∗‖ + βn [ (1 + µq(n)M0)‖yn −x∗‖ + µq(n)M + ηq(n) ] ≤ (1 −βn)‖xn −x∗‖ + βn [ (1 + µq(n)M0)‖yn −x∗‖ ] +βnµq(n)M + βnηq(n) ≤ (1 −βn)‖xn −x∗‖ + βnµq(n)M + βnηq(n) +βn [ (1 + µq(n)M0)(1 −αn)]‖xn −x∗‖ + αn‖u−x∗‖ ] = [ 1 −αnβn + (1 −αn)βnµq(n)M0 ] ‖xn −x∗‖ +αnβn(1 + µq(n)M0)‖u−x∗‖ + βnµq(n)M + βnηq(n) = [ 1 −αnβn + (1 −αn)βnµq(n)M0 ] ‖xn −x∗‖ + δn,(30) where δn = αnβn(1+µq(n)M0)‖u−x∗‖+βnµq(n)M+βnηq(n). Since lim n→∞ α−1n µq(n) = 0 = lim n→∞ α−1n ηq(n), we may assume without loss of generality that there exists k0 ∈ (0, 1) and M2 > 0 such that α−1n µq(n) < (1−k0) (1−αn)M0 and δn αnβn < M2. Thus, we obtain from (30) that (31) ‖xn+1 −x∗‖≤‖xn −x∗‖−k0αnβn‖xn −x∗‖ + δn. So, Lemma 2.5 gives ‖xn−x∗‖≤ max{‖x1−x∗‖, (1+k1)M2}. Therefore, {xn}n≥1 is bounded and by (27) we obtain that {yn}n≥1 is bounded. Moreover, using Lemma 2.1, we obtain that ‖yn −x∗‖2 = ‖PK[αnu + (1 −αn)xn] −PKx∗‖2 ≤ ‖αnu + (1 −αn)xn −x∗‖2 ≤ (1 −αn)2‖xn −x∗‖2 +2αn(1 −αn)〈u−x∗,xn −x∗〉 + 2α2n‖u−x ∗‖2.(32) Furthermore, using Lemma 2.2, we obtain that ‖xn+1 −x∗‖2 = ‖(1 −βn)xn + βnT q(n) j(z) yn −x∗‖2 = (1 −βn)‖xn −x∗‖2 + βn‖T q(n) j(n) yn −x∗‖2 −βn(1 −βn)‖xn −T q(n) j(n) yn‖2 (33) But ‖Tq(n) j(n) yn −x∗‖2 ≤ [ (1 + µq(n)M0)‖yn −x∗‖ + µq(n)M + ηq(n) ]2 = (1 + µq(n)M0) 2‖yn −x∗‖2 +(µq(n)M + ηq(n)) [ 2(1 + µq(n)M0)‖yn −x∗‖ + µq(n)M + ηq(n) ]2 (34) 104 OFOEDU AND NNUBIA so that putting (32) in (34), we have ‖Tq(n) j(n) yn −x∗‖2 ≤ (1 + µq(n)M0)2 ( (1 −αn)2‖xn −x∗‖2 +2αn(1 −αn)〈u−x∗,xn −x∗〉 + 2α2n‖u−x ∗‖2 ) +(µq(n)M + ηq(n)) [ 2(1 + µq(n)M0)‖yn −x∗‖ + µq(n)M + ηq(n) ]2 (35) and βn‖T q(n) j(n) yn −x∗‖2 ≤ βn(1 + µq(n)M0)2(1 −αn)2‖xn −x∗‖2 +2αnβn(1 + µq(n)M0) 2(1 −αn)〈u−x∗,xn −x∗〉 +2α2nβn(1 + µq(n)M0) 2‖u−x∗‖2 + (µq(n)M + ηq(n))[ 2(1 + µq(n)M0)‖yn −x∗‖ + µq(n)M + ηq(n) ] .(36) Now, substituting (36) in (33), we have ‖xn+1 −x∗‖2 = ‖(1 −βn)xn + βnT q(n) j(z) yn −x∗‖2 ≤ (1 −βn)‖xn −x∗‖2 + βn(1 + µq(n)M0)2(1 −αn)2‖xn −x∗‖2 +2αnβn(1 + µq(n)M0) 2(1 −αn)〈u−x∗,xn −x∗〉 +2α2nβn(1 + µq(n)M0) 2‖u−x∗‖2 −βn(1 −βn)‖xn −T q(n) j(n) yn‖2 +(µq(n)M + ηq(n)) [ 2(1 + µq(n)M0)‖yn −x∗‖ + µq(n)M + ηq(n) ] ≤ (1 −γn)‖xn −x∗‖2 + 2γn(1 −αn)〈u−x∗,xn −x∗〉 +θn −βn(1 −βn)‖xn −T q(n) j(n) yn‖2,(37) where γn = βnαn(1+µnM0) 2 and θn = 2α 2 nβn(1+µnM0) 2‖u2−x∗‖2+βnµq(n)M0(2+ µq(n)M0 sup n≥1 ‖xn−x∗‖2+βn(µq(n)M+ηq(n)) [ 2(1+µq(n)M0) sup n≥1 ‖yn−x∗‖+µq(n)M+ ηq(n) ] . Two cases arise Case 1: Suppose{‖xn − x∗‖}n≥1 is nonincreasing for n ≥ n0, for some n0 ∈ N, this implies that ‖xn+1 − x∗‖ ≤ ‖xn − x∗‖ ∀ n ≥ n0. Thus, lim n→∞ ‖xn−x∗‖ exist and lim n→∞ ( ‖xn+1−x∗‖−‖xn−x∗‖ ) = 0. Moreover, using the fact that 0 < ξ0 < βn < ζ0 < 1, we obtain that (38) lim n→∞ ‖xn −T q(n) j(n) yn‖ = 0. Next, we observe that ‖xn+1 −xn‖ = ‖(1 −βn)xn + βnT q(n) j(n) yn −xn‖ = βn‖T q(n) j(n) yn −xn‖.(39) Thus, by (38) (40) lim n→∞ ‖xn+1 −xn‖ = 0. TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 105 Observe that by(40) (41) lim n→∞ ‖xn −xn−i‖ = 0 = lim n→∞ ‖xn −xn+i‖, i ∈ I. Moreso, ‖yn −xn‖ = ‖PK[αnu + (1 −αn)xn] −PKxn‖ ≤ ‖αnu + (1 −αn)xn] −xn‖ = αn‖u−xn‖(42) so that by our hypothesis (43) lim n→∞ ‖yn −xn‖ = 0. Furthermore, (44) ‖yn+1 −yn‖≤‖yn+1 −xn+1‖ + ‖xn+1 −xn‖ + ‖xn −yn‖ which by (40) and (43) gives (45) lim n→∞ ‖yn+1 −yn‖ = 0. Observe that by (45) we have (46) lim n→∞ ‖yn+i −yn‖ = 0 = lim n→∞ ‖yn−i −yn‖ ∀ i ∈ I Now, (47) ‖yn −T q(n) j(n) yn‖≤‖yn −xn‖ + ‖xn −T q(n) j(n) yn‖. Using (38) and(43) in (47) gives (48) lim n→∞ ‖yn −T q(n) j(n) yn‖ = 0. By uniform continuity of Ti; i ∈ I there exists a continuous increasing function Πi : R −→ R with Πi(0) = 0 such that (49) ‖Tix−Tiy‖≤ Πi(‖x−y‖)∀x,y ∈ K. Thus, defining Π0 : R −→ R by Π0(t) = max i∈I {Πi(t)} ∀ t ∈ R, we have that Π0 is a continuous increasing function with Π0(0) = 0 and ‖yn −Tj(n)yn‖ ≤ ‖yn −T q(n) j(n) yn‖ + ‖T q(n) j(n) yn −Tj(n)yn‖ ≤ ‖yn −T q(n) j(n) yn‖ + Π0(‖T q(n)−1 j(n) yn −yn‖).(50) Consider the argument of Π0 in (50), ‖Tq(n)−1 j(n) yn −yn‖ ≤ ‖T q(n)−1 j(n) yn −T q(n)−1 j(n−m)yn−m‖ + ‖T q(n)−1 j(n−m)yn−m −yn−m‖ +‖yn−m −yn‖ .(51) By (26) we have that ‖Tq(n)−1 j(n) yn −T q(n)−1 j(n−m)yn−m‖ ≤ ‖T q(n)−1 j(n−m)yn −T q(n)−1 j(n−m)yn−m‖ ≤ ‖yn−m −yn‖ + µq(n)−1 + φ(‖yn−m −yn‖) +ηq(n)−1 .(52) Using (46) in (52) and by hypothesis we have that (53) lim n→∞ ‖Tq(n)−1 j(n) yn −T q(n)−1 j(n−m)yn−m‖ = 0. 106 OFOEDU AND NNUBIA Moreso, by (26) we have that (54) ‖Tq(n)−1 j(n−m)yn−m −yn−m‖ = ‖T q(n−m) j(n−m) yn−m −yn−m‖. Thus, (55) lim n→∞ ‖Tq(n)−1 j(n−m)yn−m −yn−m‖ = 0. Now, using (53) and (54) in (50) we obtain that lim n→∞ ‖Tq(n)−1 j(n) yn −yn‖ = 0.(56) Consequently, we obtain from (48) and (50) that lim n→∞ ‖yn −Tj(n)yn‖ = 0.(57) Furthermore, we obtain for i ∈ I that ‖yn −Tj(n)+iyn‖ ≤ ‖yn −yn+i‖ + ‖yn+i −Tj(n)+iyn+i‖ + ‖Tj(n)+iyn+i −Tj(n)+iyn‖ ≤ ‖yn −yn+i‖ + ‖yn+i −Tj(n)+iyn+i‖ + Π0(‖yn+i −yn‖).(58) So,using (46), (57) and (58) we have lim n→∞ ‖yn −Tj(n)+iyn‖ = 0 ∀i ∈ I.(59) But ∀i ∈ I there exists ϑi ∈ I such that j(n) + ϑi = i mod(m) so that from (59), we have that lim n→∞ ‖yn −Tiyn‖ = lim n→∞ ‖yn −Tj(n)+iyn‖ = 0 ∀i ∈ I.(60) But (61) ‖xn −Tixn‖ = ‖xn −yn‖ + ‖yn −Tiyn‖ + ‖Tiyn −Tixn‖ ∀ n ∈ N. Hence, using (43), uniform continuity of the mapping T and (60) we obtain from (61) that (62) lim n→∞ ‖xn −Tixn‖ = 0, ∀i ∈ I. Now, let {xnk}k≥1 be a subsequence of {xn}n≥1 such that (63) lim sup n→∞ 〈u−x∗,xn −x∗〉 = lim k→∞ 〈u−x∗,xnk −x ∗〉, then, there exist a subsequence{xnkj}j≥1 of {xnk}k≥1 that converges weakly to some z ∈ H. Thus, (63) gives (64) lim sup n→∞ 〈u−x∗,xn−x∗〉 = lim k→∞ 〈u−x∗,xnk −x ∗〉 = lim j→∞ 〈u−x∗,xnkj −x ∗〉. Furthermore, by (62) limj→∞‖xnkj − Tixnkj‖ = 0 and by Lemma 2.4, I − Ti is demiclose at 0, we obtain that z ∈ F. So using(64) and the fact that x∗ = PKu, we obtain from Lemma 2.3 that lim sup n→∞ 〈u−x∗,xn −x∗〉 = lim k→∞ 〈u−x∗,xnk −x ∗〉 = lim j→∞ 〈u−x∗,xnkj −x ∗〉 = 〈u−x∗,z −x∗〉≤ 0. .(65) Therefore, defining TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 107 (66) νn = max{0,〈u−x∗,xn −x∗〉}, Then it is easy to see that lim n→∞ νn = 0. Moreover we obtain from (37)(using (66)) that ‖xn+1 −x∗‖2 ≤ ‖xn −x∗‖2 −γn‖xn −x∗‖2 + 2γn(1 −αn)〈u−x∗,xn −x∗〉 + θn ≤ (1 −γn)‖xn −x∗‖2 + 2γn(1 −αn)νn + θn = (1 −γn)‖xn −x∗‖2 + σn(67) where σn = 2γn(1 −αn)νn + θn. Conditions on our iterative parameter easily give that σn = o(γn). Hence, we obtain from (67) using Lemma 2.5 that {xn}n≥0 con- verges strongly to x∗ = PKu CASE 2: Suppose there exists a subsequence {xni} of {xn} such that ‖xni −x∗‖≤ ‖xni+1 − x∗‖ ∀ i ∈ N, then by lemma 2.6 there exist a nondecreasing sequence {τ(n)}⊂ N such that (i) lim n→∞ τ(n) = ∞ (ii) ‖xτ(n)−x∗‖≤‖xτ(n)+1−x∗‖ ∀ n ∈ N. So, from (37), we have that γn‖xτ(n) −x∗‖2 ≤ ‖xτ(n) −x∗‖2 −‖xτ(n)+1 −x∗‖2 +2γτ(n)〈u−x∗,xτ(n) −x∗〉 + θτ(n) ∀ n ∈ N(68) Thus, using the fact that γτ(n) > 0, we have that (69) ‖xτ(n) −x∗‖2 ≤ 2〈u−x∗,xτ(n) −x∗〉 + θτ(n) γτ(n) ∀ n ∈ N. Observe that following the argument of case 1 we have that lim n→∞ ‖xτ(n+1)−xτ(n)‖ = lim n→∞ ‖yτ(n) −Tiyτ(n)‖ = 0 ∀ i ∈ I and lim sup n→∞ 〈u−x∗,xn −x∗〉 ≤ 0. Thus, setting ντ(n) = max{0,〈u−x∗,xτ(n) −x∗〉}, we obtain that ντ(n) → 0 as n →∞. Furthermore, from conditions on our iterative parameters, we obtain that θτ(n) γτ(n) → 0. So we obtain from (69) that ‖xτ(n) − x∗‖2 ≤ 2ντ(n) + θτ(n) γτ(n) ∀ n ∈ N. Thus, lim n→∞ ‖xτ(n) −x∗‖ = 0. Also from Lemma 2.6 we have that ‖xn −x∗‖≤‖xτ(n+1) − x∗‖2 ∀ n ∈ N. Thus we obtain using sandwich theorem that lim n→∞ ‖xn − x∗‖ = 0. Hence, xn converges strongly to x ∗ = PKu. Remark 3.2. Observe that if Ti, i ∈ I in Theorem 3.1 were asymptotically non- expansive mappings, the condition there exist M0 > 0 and M1 > 0 such that φ(t) ≤ M0t ∀ t > M1 is not needed. Moreso, every asymptotically nonexpansive mapping Ti : K −→ K is uniformly L-Lipschitzian thus uniformly continuous. Hence we have the folowing theorems as an easy corollaries of Theorem 3.1 above: Theorem 3.2. Let K be a closed convex nonempty subset of a real Hilbert space H and let Ti : K −→ K, i ∈ I, be asymptotically nonexpansive mappings such that F = ⋂m i=1 F(Ti) 6= ∅ and let {xn}n≥1 be a sequence generated iteratively by (23), where {αn}n≥1,{βn}n≥1 are sequences in (0, 1) satisfying the following conditions: 108 OFOEDU AND NNUBIA ∞∑ n=1 αn = ∞, lim n→∞ αn = 0, lim n→∞ α−1n µn = 0 and 0 < ζ < βn < � < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to PF (u). Theorem 3.3. Let K be a closed convex nonempty subset of a real Hilbert space H and let Ti : K −→ K, i ∈ I, be finite family of nonexpansive mappings from K into itself. Suppose that F = ⋂m i=1 F(Ti) 6= ∅ and let {xn}n≥1 be a sequence generated iteratively by (23), where {αn}n≥1,{βn}n≥1 are sequences in (0, 1) satisfying the following conditions: ∞∑ n=1 αn = ∞, lim n→∞ αn = 0 and 0 < ζ < βn < � < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to PFu. . Corollary 3.1. Suppose in our Theorems the finite family is a singleton (that is if m = 1), our results hold. Remark 3.3. If u = 0 in the recursion formulas of our theorems, we obtain what authors now call the Minimum norm iteration process. We observe that all our theorems in this paper carry over trivially to the so called minimum norm iteration process. Remark 3.4. If f : K −→ K is a contraction map and we replace u by f(xn) in the recursion formulas of our theorems, we obtain what some authors now call viscosity iteration process. We observe that all our theorems in this paper carry over trivially to the so-called viscosity process. 4. Application to approximation of fixed points of continuous pseudocontractive mappings The most important generalization of the class of nonexpansive mappings is, perhaps, the class of pseudocontractive mappings. These mappings are intimate- ly connected with the important class of nonlinear monotone operators. For the importance of monotone operators and their connections with evolution equations, the reader may consult [9], [19]. Due to the above connection, fixed point theory of pseudocontractive mappings became a flourishing area of intensive research for several authors. Recently, H. Zegeye [29] established the following Lemmas: Lemma 4.1. [29] Let K be a nonempty closed convex subset of a real Hilbert space H. Let T : K −→ H be a continuous pseudocontractive mapping, then for all r > 0 and x ∈ H, there exists z ∈ K such that (70) 〈y −z,Tz〉− 1 r 〈y −z, (1 + r)z −x〉≤ 0; ∀ y ∈ K Lemma 4.2. [29] Let K be a nonempty closed convex subset of a real Hilbert space H. Let T : K −→ K be a continuous pseudocontractive mapping, then for all r > 0 and x ∈ H, there exists z ∈ K, define a mapping Fr : H −→ K by (71) Fr(x) = {z ∈ K : 〈y −z,Tz〉− 1 r 〈y −z, (1 + r)z −x〉≤ 0 ∀ y ∈ K} then the following hold: TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 109 (1) Fr is single-valued (2) Fr is firmly nonexpansive type mapping i.e for all x,y,z ∈ H (72) ‖Fr(x) −Fr(y)‖2 ≤〈Fr(x) −Fr(y),x−y〉 (3) Fix(Fr) is closed and convex; and Fix(Fr) = Fix(T); for all r > 0. Remark 4.1. We observe that Lemmas 4.1 and 4.2 hold in particular for r = 1. Thus, if Ti, i ∈ I = {1, 2, ...,m} is finite family of continuous pseudocontractive mapping and we define F1(i) : H −→ K by (73) F1(i)(x) = {z ∈ K : 〈y −z,Tiz〉−〈y −z, 2z −x〉≤ 0 ∀ y ∈ K} then F1(i) satisfies the conditions of Lemma 4.2 ∀ i ∈ I. Hence, we easily see that F1(i) is nonexpansive and Fix(F1(i)) = Fix(Ti)∀ i ∈ I. Thus, we have the following theorem. Theorem 4.1. Let K be a closed convex nonempty subset of a real Hilbert space H and let Ti : K −→ K i ∈ I be finite family of continous pseudocontractive mappings from K into itself. Suppose that F ′ = ⋂N i=1 F(Ti) 6= ∅ and let {xn}n≥1 be a sequence generated iteratively by x1 ∈ K,u ∈ H,yn = PK[αnu + (1 −αn)xn], xn+1 = (1 −βn)xn + βnF q(n) 1j(n) yn, (74) where {αn}n≥1,{βn}n≥1 are sequences in (0, 1) satisfying the following conditions: ∞∑ n=1 αn = ∞, lim n→∞ αn = 0 and 0 < ζ < βn < � < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to PF′u. Furthermore, if u = 0,{xn}n≥1 converges strongly to a minimum norm fixed point of the finite family. 5. Application to approximation of solutions of classical equilibrium problems Let K be a closed convex nonempty subset of a real Hilbert space H. Let f : KXK −→ R be a bifunction. The classical equilibrium problem (abbreviated EP) for f is to find u∗ ∈ K such that (75) f(u∗,y) ≥ 0 ∀ y ∈ K The set of solutions of classical equilibrium problem is denoted by EP(f), where EP(f) = {u ∈ K : f(u,y) ≥ 0 ∀ y ∈ K}. The classical equilibrium problem (EP) includes as special cases the monotone inclusion problems, saddle point problems, variational inequality problems, mini- mization problems, optimization problems, vector equilibrium problems, Nash equilibria in noncooperative games. Further- more, there are several other problems, for example, the complementarity problems and fixed point problems, which can also be written in the form of the classical e- quilibrium problem. In other words, the classical equilibrium problem is a unifying model for several problems arising from engineering, physics, statistics, computer 110 OFOEDU AND NNUBIA science, optimization theory, operations research, economics and countless other fields. For the past 20 years or so, many existence results have been published for various equilibrium problems (see e.g.[4], [14],[30]). In the sequel, we shall require that the bifunction f : KxK −→ R satisfies the fol- lowing conditions: (A1) f(x,x) = 0 ∀ x ∈ K; (A2)f is monotone, in the sense that f(x,y)+f(y,x) ≤ 0 ∀ x,y ∈ K; (A3)lim sup t→0+ f(tz+(1−t)x,y) ≤ f(x,y) ∀ x,y,z ∈ K;. (A4) the function y 7→ f(x,y) is convex and lower semicontinuous for all x ∈ K Lemma 5.1. [(compare with lemma 2.4 of [14])] Let K be a nonempty closed convex subset of a real Hilbert space H. Letfi : KxK −→ R be finite family of bifunction satisfying conditions (A1) - (A4) for each i ∈ I = {1, 2, ...,m} then for all r > 0 and x ∈ H, there exists u ∈ K such that (76) fi(u,y) + 1 r 〈y −u,u−x〉≥ 0 ∀ y ∈ K i ∈ I. moreover’ if for all x ∈ H we define Gir : H −→ 2K by (77) Gir(x) = {u ∈ K : fi(u,y) + 1 r 〈y −u,u−x〉≥ 0 ∀ y ∈ K.} then the following hold: (1) Gir is single-valued for all r ≥ 0 i ∈ I (2) Fix(Gir) = EP(fi) for all r > 0 (3) EP(fi) is closed and convex Remark 5.1. We observe that Lemmas 5.1 holds in particular for r = 1. Thus, if we define Gi1 : H −→ 2K by (78) Gi1(x) = {u ∈ K : fi(u,y) + 〈y −u,u−x〉≥ 0 ∀ y ∈ K.} then Gi1 satisfies the conditions of Lemma 5.1 ∀ i ∈ I. Hence, we easily see that Gi1 is nonexpansive and Fix(Gi1) = EP(fi) ∀ i ∈ I. Thus, we have the following theorem: Theorem 5.1. Let K be a closed convex nonempty subset of a real Hilbert space H and let Letfi : KxK −→ R be finite family of bifunction satisfying conditions (A1) - (A4) for each i ∈ I = {1, 2, ...,m}. Suppose that F ′′ = ⋂m i=1 EP(fi) 6= ∅ and let {xn}n≥1 be a sequence generated iteratively by x1 ∈ K,u ∈ H,yn = PK[αnu + (1 −αn)xn], xn+1 = (1 −βn)xn + βnG q(n) 1j(n) yn n ≥ 0 (79) where {αn}n≥1,{βn}n≥1 are sequences in (0, 1) satisfying the following conditions: ∞∑ n=1 αn = ∞, lim n→∞ αn = 0 and 0 < ζ < βn < � < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to PF′′u. . Furthermore, if u = 0,{xn}n≥1 converges strongly to a minimum norm fixed point of the finite family. TOTAL ASYMPTOTICALLY NONEXPANSIVE MAPPINGS 111 Remark 5.2. Several authors (see e.g.[14], [18] and references therein) have studied the following problem:Let K be a closed convex nonempty subset of a real Hilbert space H. Let f : KxK −→ R be a bifunction and Φ : K −→ R be a proper extended real valued function, where R denotes the real numbers. let Θ : K −→ H be a nonlinear monotone mapping. The generalised mixed equilibrium problem (abbreviated GMEP) for f, ΦandΘ is to find u∗ ∈ K such that (80) f(u∗,y) + Φ(y) − Φ(u∗) + 〈Θu∗,y −u∗〉≥ 0 ∀ y ∈ K Observe that if we defineΓ : KxK −→ R (81) Γ(x,y) = f(x,y) + Φ(y) − Φ(x) + 〈Θx,y −x〉 then it could be easily checked that Γ is a bi-function and satisfies properties (A1)to(A4). Thus, the so called generalized mixed equilibrium problem reduces to the classical equilibrium problem for the bifunction Γ. Thus, consideration of the so called gener- alized mixed equilibrium problem in place of the classical equilibrium problem studied in this section leads to no further generalization. 6. Applications to Convex optimization) Let us look at the problem of minimizing a continuously Frechet-differentiable convex functional with minimum norm in Hilbert spaces. Let K be a closed convex subset of a real Hilbert space H, Consider the minimiza- tion problem given by (82) min x∈K φ(x) where φ is a Frechet-differentiable convex functional. Let Ω the solution set of (82) be nonempty. It is known that a point z ∈ K is a solution of (82) if and only if the following optimality condition holds: (83) z ∈ K, 〈∇φ(z),x−z〉≥ 0,x ∈ K, where ∇ is the gradient of φ at x ∈ K. It is also known that the optimality condition (83) is equivalent to the following fixed point problem: (84) z = Tγ(z), whereTγ := PK(I −γ∇φ), for all γ > 0. So, we have the following corollary deduced from theorem 3.1 Theorem 6.1. Let H be a real Hilbert space, let K be a closed convex nonemp- ty subset of H. Let ψ be a continuously Frechet-differentiable convex functional on K such that Tγ(i) := PK(I − γ(i)∇ψ) be finite family of uniformly continu- ous total asymptotically nonexpansive mapping from K into itself with sequences {µin}n≥1,{ηin}n≥1 ⊂ [0, +∞) such that lim n→∞ µin = 0 = lim n→∞ ηin and with func- tion φi : [0, +∞) −→ [0, +∞) satisfying φi(t) ≤ M0t ∀ t > M1 for some constants M0,M1 > 0, Let µn = max i∈I {µin} and ηn = max i∈I {ηin} and φ(t) = max i∈I {φi(t)}∀ t ∈ [0,∞). Suppose that F = ⋂N i=1 F(Tγ(i) ) 6= ∅ and {xn}n≥1 is a sequence generated iteratively by x1 ∈ K, yn = PK[(1 −αn)xn], xn+1 = (1 −βn)xn + βn[PK(I −γ(i)∇ψ)] q(n) j(n) yn; n ≥ 1(85) 112 OFOEDU AND NNUBIA where {αn}n≥1,{βn}n≥1 are sequences in (0, 1) satisfying the following conditions: ∞∑ n=1 αn = ∞, lim n→∞ αn = 0, lim n→∞ α−1n µn = 0 and 0 < ζ < βn < � < 1 ∀ n ≥ 1, then {xn}n≥1 converges strongly to the minimum norm solution of the minimiza- tion problem (82). A prototype of φi : [0, +∞) −→ [0, +∞) in Theorem 3.1 is Φ(λ) = λs, where 0 < s ≤ 1. Moreso, prototype of the sequences used in the same Theorem 3.1 are: Take αn = 1 n , µn = 1 n1+� , for � > 0,ηn = 1 n log n . Remark 6.1. Our results extends and unify most of the results that have been proved for the class of assymptotically nonexpansive mappings of which the results obtained in [10], [15],[27], [31] are examples. References [1] Y. Alber, R. Espinola and P. Lorenzo, Strongly Convergent Approximations to fixed points of total asymptotically nonexpansive mappings, Acta Mathematica Sinica, English Series, vol. 24 no. 6 (2008) 1005-1022. [2] Ya. Alber, C. E. Chidume and H. Zegeye, Approximating fixed points of total asymptotically nonexpansive mappings.Fixed Point Theory and Appl. 2006 (2006), article ID 10673. [3] H. H. Bauschke, The Approximation of fixed points of compositions of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 202 (1996) 150-159. [4] E. Blum and W. 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