@ Appl. Gen. Topol. 20, no. 1 (2019), 119-133doi:10.4995/agt.2019.10360 c© AGT, UPV, 2019 Existence of fixed points for pointwise eventually asymptotically nonexpansive mappings M. Radhakrishnan a and S. Rajesh b a Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, India. (radhariasm@gmail.com ) b Department of Mathematics, Indian Institute of Technology, Tirupati 517 506, India. (srajeshiitmdt@gmail.com) Communicated by E. A. Sánchez-Pérez Abstract Kirk introduced the notion of pointwise eventually asymptotically non- expansive mappings and proved that uniformly convex Banach spaces have the fixed point property for pointwise eventually asymptotically nonexpansive maps. Further, Kirk raised the following question: “Does a Banach space X have the fixed point property for pointwise eventually asymptotically nonexpansive mappings whenever X has the fixed point property for nonexpansive mappings?”. In this paper, we prove that a Banach space X has the fixed point property for pointwise eventually asymptotically nonexpansive maps if X has uniform normal sturcture or X is uniformly convex in every direction with the Maluta constant D(X) < 1. Also, we study the asymptotic behavior of the sequence {T nx} for a pointwise eventually asymptotically nonexpansive map T defined on a nonempty weakly compact convex subset K of a Banach space X whenever X satisfies the uniform Opial condition or X has a weakly continuous duality map. 2010 MSC: 47H10; 47H09. Keywords: fixed points; pointwise eventually asymptotically nonexpansive mappings; uniform normal structure; uniform Opial condition; duality mappings. Received 06 June 2018 – Accepted 12 December 2018 http://dx.doi.org/10.4995/agt.2019.10360 M. Radhakrishnan and S. Rajesh 1. Introduction Let K be a nonempty weakly compact convex subset of a Banach space X. A mapping T : K → K is said to be asymptotically nonexpansive if there exists a sequence {αn} ⊆ [1, ∞) with lim n→∞ αn = 1 such that for each integer n ≥ 1, (1.1) ‖T nx − T ny‖ ≤ αn‖x − y‖, for all x, y ∈ K. If αn = 1 in (1.1) for all n ∈ N, then T is said to be nonexpansive. If for each x ∈ K, the following inequality holds lim sup n→∞ ( sup y∈K {‖T nx − T ny‖ − ‖x − y‖} ) ≤ 0, then T is said to be asymptotically nonexpansive type. Kirk [13] proved that if K is a nonempty weakly compact convex set in a Banach space X with normal structure, then every nonexpansive map T on K has a fixed point. Goebel and Kirk [9] further proved that if X is a uniformly convex Banach space, then every asymptotically nonexpansive map T on K has a fixed point. Later, this result was extended to mappings of asymptotically nonexpansive type by Kirk [14]. However, it remains open that whether normal structure condition on a Banach space X guarantees the existence of fixed points of asymptotically nonexpansive mapping. Kim and Xu [12] proved that if X is a Banach space with uniform normal structure, then every asymptotically nonexpansive map T on K has a fixed point. Li and Sims [8] proved the existence of fixed points of asymptotically nonexpansive type mappings in the setting of Banach spaces having uniform normal structure. Gossez and Lami Dozo [11] studied the class of spaces which satisfies Opial’s condition and observed that all such spaces have normal struc- ture. Hence, if X is a Banach space satisfying Opial’s condition, then every nonexpansive map T on K has a fixed point. However, it is not clear whether Opial’s condition implies the existence of fixed points for asymptotically non- expansive mappings. In this direction, Lin et. al [18] proved that every asymptotically nonexpan- sive map T on K has a fixed point whenever X is a Banach space that satisfies the uniform Opial condition. Also, Lim and Xu [17] proved the existence of fixed points of an asymptotically nonexpansive map T on K in a Banach space X whenever the Maluta constant D(X) < 1 and T is weakly asymptotically regular on K. In 2008, Kirk and Xu [16] introduced the notion of pointwise asymptotically nonexpansive mappings and studied the existence of fixed points in the setting of uniformly convex Banach spaces. Definition 1.1 ([16]). A mapping T : K → K is said to be pointwise asymptot- ically nonexpansive if for each x ∈ K there exists a sequence {αn(x)} ⊆ [1, ∞) with lim n→∞ αn(x) = 1 such that for each integer n ≥ 1, ‖T nx − T ny‖ ≤ αn(x)‖x − y‖, for all y ∈ K. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 120 Existence of fixed points for pointwise eventually asymptotically nonexpansive mappings Theorem 1.2 ([16]). Let K be a nonempty closed bounded convex subset of a uniformly convex Banach space X and T : K → K be a pointwise asymptoti- cally nonexpansive map. Then T has a fixed point in K. Recently, Kirk [15] introduced the notion of pointwise eventually asymptot- ically nonexpansive mappings as follows: Definition 1.3 ([15]). A mapping T : K → K is said to be pointwise even- tually asymptotically nonexpansive if for each x ∈ K there exists a sequence {αn(x)} ⊆ [1, ∞) with lim n→∞ αn(x) = 1 and an integer N(x) ∈ N such that for n ≥ N(x), ‖T nx − T ny‖ ≤ αn(x)‖x − y‖, for all y ∈ K. Though the definition of pointwise asymptotically nonexpansive mappings and pointwise eventually asymptotically nonexpansive mappings are quite sim- ple to understand, examples of such mappings are rare. In [15] Kirk proved that Theorem 1.2 holds for pointwise eventually asymp- totically nonexpansive mappings. Further, Kirk [15] raised the following ques- tion: Does a Banach space X have the fixed point property for pointwise even- tually asymptotically nonexpansive mappings whenever X has the fixed point property for nonexpansive mappings? In this paper we give a partial answer to the above question. In section 3, we prove that a Banach space X have the fixed point property for pointwise eventually asymptotically nonexpansive mappings if X has uniform normal structure or X is uniformly convex in every direction with the Maluta constant D(X) < 1. In section 4, we study the asymptotic behavior of the sequence {T nx} for a pointwise eventually asymptotically nonexpansive map T defined on a nonempty weakly compact convex set K in a Banach space X whenever X satisfies the uniform Opial condition or X has a weakly continuous duality mapping. For results about asymptotic behavior of nonexpansive mappings and asymptotically nonexpansive mappings, one may refer to [2, 7, 17, 18]. 2. Preliminaries Let X be a Banach space and C be the collection of all closed bounded convex sets in X. For K ∈ C, define (1) for x ∈ X, δ(x, K) = sup{‖x − y‖ : y ∈ K}; (2) r(K) = inf{δ(x, K) : x ∈ K} and (3) δ(K) = diam(K) = sup{δ(x, K) : x ∈ K}. Definition 2.1 ([3]). A Banach space X is said to have normal structure if every nonempty closed bounded convex subset K of X with diam(K) > 0 has a point x0 ∈ K such that δ(x0, K) < diam(K). Also, Brodskii and Milman [3] gave a characterization for normal structure in terms of sequences as follows: c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 121 M. Radhakrishnan and S. Rajesh Theorem 2.2 ([3]). A Banach space X does not have normal structure if and only if there exists a nonconstant bounded sequence {xn} in X such that lim n→∞ d (xn+1, co{x1, . . . , xn}) = diam({xn}). Using this characterization of normal structure, Maluta [19] defined the con- stant D(X) of a given Banach space as follows: Definition 2.3 ([19]). Let X be a Banach space. The Maluta constant D(X) of X is defined as D(X) = sup    lim sup n→∞ d (xn+1, co{x1, . . . , xn}) diam({xn})    where the supremum is taken over all non-constant bounded sequences in X. It is known from [19] that a Banach space X with D(X) < 1 has normal structure. In [5] Bynum defined the concept of uniform normal structure as follows: Definition 2.4 ([5]). A Banach space X is said to have uniform normal struc- ture if N(X) < 1, where N(X) = sup { r(K) δ(K) : K ∈ C with δ(K) > 0 } . Remark 2.5. The following facts are known from [19]. (1) For a Banach space X, 0 ≤ D(X) ≤ 1 and D(X) ≤ N(X). (2) If X is nonreflexive Banach space, then D(X) = 1. Thus if D(X) < 1 then X is reflexive. Definition 2.6 ([20]). A Banach space X is said to satisfy Opial’s condition if for each weakly convergent sequence {xn} in X with limit x0 ∈ X, lim sup n→∞ ‖xn − x0‖ < lim sup n→∞ ‖xn − x‖, for all x ∈ X with x 6= x0. It is known that every Hilbert space, finite dimensional Banach spaces and the Banach space lp(N) for 1 < p < ∞ satisfy Opial’s condition. In [21] Prus introduced the notion of the uniform Opial condtion: Definition 2.7 ([21]). A Banach space X is said to satisfy the uniform Opial condition if for each c > 0, there exists an r > 0 such that 1 + r ≤ lim sup n→∞ ‖xn + x‖ for each x ∈ X with ‖x‖ ≥ c and each sequence {xn} in X such that w − lim n→∞ xn = 0 and lim sup n→∞ ‖xn‖ ≥ 1. Further, Prus [21] defined the Opial modulus of X denoted by rX, as follows rX(c) = inf { lim sup n→∞ ‖xn + x‖ − 1 } , c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 122 Existence of fixed points for pointwise eventually asymptotically nonexpansive mappings where c ≥ 0 and the infimum is taken over all x ∈ X with ‖x‖ ≥ c and sequences {xn} in X such that w − lim n→∞ xn = 0 and lim sup n→∞ ‖xn‖ ≥ 1. It is easy to see that X satisfies the uniform Opial condition if and only if rX(c) > 0 for all c > 0. Definition 2.8 ([10]). A continuous strictly increasing function ϕ : [0, ∞) → [0, ∞) is said to be gauge if ϕ(0) = 0 and lim t→∞ ϕ(t) = ∞. Definition 2.9 ([10]). Let X be a Banach space and ϕ be a gauge function. Then we associate with X a generalized duality map Jϕ : X → 2 X ∗ defined by Jϕ(x) = {x ∗ ∈ X∗ : 〈x, x∗〉 = ‖x‖ϕ(‖x‖) and ‖x∗‖ = ϕ(‖x‖)} , where 〈., .〉 is the pairing between X and X∗. Define Φ(t) = t ∫ 0 ϕ(s)ds for t ≥ 0. Then it is easy to see that Φ is a convex and gauge function on [0, ∞). Also, it is known from [4] that Jϕ(x) is the subdifferential of the convex function Φ(‖.‖) at x ∈ X. Definition 2.10 ([10]). A Banach space X is said to have a weakly continuous duality map if there exists a gauge function ϕ such that the duality map Jϕ is single-valued and continuous from X with the weak topology to X∗ with the weak∗ topology. It is clear from [10] that the Banach space lp(N) for 1 < p < ∞ has a weakly continuous duality map with the gauge function ϕ(t) = tp−1. Also, Browder [4] proved that a Banach space X with a weakly continuous duality map satisfies Opial’s condition. 3. Existence Result In this section, we prove that a pointwise eventually asymptotically nonex- pansive mappings defined on a weakly compact convex subset K of a Banach space X always has a fixed point if X has uniform normal structure or X is uniformly convex in every direction with the Maluta constant D(X) < 1. We use the following two Lemmas in the sequel. Lemma 3.1 ([6]). Let X be a Banach space with uniform normal structure. Then for every bounded sequence {xn} in X there exists a point z ∈ ∞ ⋂ k=1 co{xn : n ≥ k} such that (1) lim sup n→∞ ‖xn − z‖ ≤ N(X)δ(co({xn})) (2) ‖z − y‖ ≤ lim sup n→∞ ‖xn − y‖, for all y ∈ X. Lemma 3.2 ([22]). Let K be a nonempty weakly compact convex subset of a Banach space X and T : K → K be an asymptotically nonexpansive type mapping. Let K0 be minimal with respect to being a nonempty closed convex subset of K such that for every x ∈ K0 we have ωw(x) ⊆ K0, where ωw(x) is c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 123 M. Radhakrishnan and S. Rajesh the set of all weak subsequential limit points of the sequence {T nx : n ∈ N}. Then there exists a constant ρ0 ≥ 0 such that lim sup n→∞ ‖T nx − y‖ = ρ0 for all x, y ∈ K0. Remark 3.3. We infer that if K is bounded, then a pointwise eventually asymp- totically nonexpansive mapping on K is of asymptotically nonexpansive type. Theorem 3.4. Let K be a nonempty weakly compact convex subset of a Ba- nach space X with uniform normal structure and T : K → K be a pointwise eventually asymptotically nonexpansive map. Then T has a fixed point in K. Proof. Assume that K0 is minimal with respect to being a nonempty closed convex subset of K with the property that for every x ∈ K0, ωw(x) ⊆ K0. By Lemma 3.2, there is a constant ρ0 ≥ 0 such that lim sup n→∞ ‖T nx − y‖ = ρ0, for all x, y ∈ K0. First note that if ρ0 = 0, then lim n→∞ ‖T nx − x‖ = 0 and lim n→∞ ‖T nx − T x‖ = 0 for x ∈ K0, and it follows that K0 = {x} with T x = x. To see that ρ0 = 0 we break the proof into two assertions. Assertion I : If {T nx} has a convergent subsequence for some x ∈ K0, then ρ0 = 0. Proof. Assume ρ0 > 0, and suppose that there exists a x ∈ K0 such that lim i→∞ T nix = y for some y ∈ K0, and choose r1 > 0 so that (1 + r1)N(X) < 1. Since αn(y) → 1, there exists a natural number N1 ≥ N(y) such that αn(y) < 1 + r1, for all n ≥ N1. Define F = co{T ny : n ≥ N1}. For l, m ∈ N with l > m ≥ N1, ‖T l(y) − T m(y)‖ = lim i→∞ ‖T l+ni(x) − T my‖ ≤ lim sup n→∞ ‖T nx − T my‖ ≤ lim sup n→∞ αm(y)‖T n−mx − y‖ = αm(y)ρ0 < (1 + r1)ρ0. This gives that δ(F) ≤ (1 + r1)ρ0. Now by Lemma 3.1, there exists a z ∈ F ∩ K0 such that ρ0 = lim sup n→∞ ‖T ny − z‖ ≤ N(X)δ(F) ≤ N(X)(1 + r1)ρ0 < ρ0. Hence assertion I is proved. Assertion II : There exists a x ∈ K0 such that {T nx} has a convergent subsequence. Proof. Let x0 ∈ K0 and define D1 = co{T nx0 : n = 0, 1, 2, . . .}. By Lemma 3.1, there exists a x1 ∈ D1 ∩ K0 such that 0 ≤ β1 = lim sup n→∞ ‖x1 − T n x0‖ ≤ N(X)δ(D1). Choose r0 > 0 so that r = (1 + r0) 2N(X) < 1. Since αn(x1) → 1, there exists a natural number n1 ≥ N(x1) such that αn(x1) < 1 + r0, for all n ≥ n1. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 124 Existence of fixed points for pointwise eventually asymptotically nonexpansive mappings Define D2 = co{T kn1(x1) : k = 0, 1, 2, . . .}. For k ≥ 1, we have ‖T kn1(x1) − T n(x0)‖ ≤ αkn1(x1)‖x1 − T n−kn1(x0)‖. Letting lim sup n→∞ on both sides, we get lim sup n→∞ ‖T kn1(x1) − T n(x0)‖ ≤ αkn1 (x1) lim sup n→∞ ‖x1 − T n−kn1(x0)‖ = αkn1 (x1) lim sup n→∞ ‖x1 − T n(x0)‖ = αkn1 (x1)β1 and for l, m ∈ N with l > m, ‖T mn1(x1) − T ln1(x1)‖ ≤ αmn1(x1)‖x1 − T (l−m)n1(x1)‖ ≤ αmn1(x1) lim sup n→∞ ‖T (l−m)n1(x1) − T nx0‖ = αmn1(x1)α(l−m)n1(x1)β1 < (1 + r0) 2 β1 ≤ (1 + r0) 2N(X)δ(D1) = rδ(D1). This gives that δ(D2) ≤ rδ(D1). Now by Lemma 3.1, there exists a x2 ∈ D2 ∩ K0 such that 0 ≤ β2 = lim sup k→∞ ‖x2 − T kn1(x1)‖ ≤ N(X)δ(D2). Since αkn1(x2) → 1 as k → ∞, we can choose k1 ∈ N such that n2 = k1n1 ≥ N(x2) and αkn1(x2) < 1 + r0, for all k ≥ k1. Define D3 = co{T kn2(x2) : k = 0, 1, 2, . . .}. For l ≥ 1, we have ‖T ln2(x2) − T kn1(x1)‖ ≤ αln2(x2)‖x2 − T kn1−ln2(x1)‖ = αln2(x2)‖x2 − T (k−lk1)n1(x1)‖. This implies that lim sup k→∞ ‖T ln2(x2) − T kn1(x1)‖ ≤ αln2(x2)β2. So for l, m ∈ N with l > m, ‖T mn2(x2) − T ln2(x2)‖ ≤ αmn2(x2)‖x2 − T (l−m)n2(x2)‖ ≤ αmn2(x2) lim sup k→∞ ‖T (l−m)n2(x2) − T kn1(x1)‖ ≤ αmn2(x2)α(l−m)n2(x2)β2 < (1 + r0) 2β2 ≤ (1 + r0) 2 N(X)δ(D2) ≤ r2δ(D1). This gives that δ(D3) ≤ r 2δ(D1). By continuing the above process, we obtain a sequence {xm} and a sequence of sets {Dm} with the following properties: c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 125 M. Radhakrishnan and S. Rajesh (1) There exists a xm ∈ Dm ∩ K0 such that 0 ≤ βm = lim sup k→∞ ‖xm − T knm−1(xm−1)‖ ≤ N(X)δ(Dm) where Dm = co{T knm−1(xm−1) : k = 0, 1, 2, . . .} and nm−1 = km−2nm−2 ≥ N(xm−1) for some km−2 ∈ N. (2) δ(Dm) ≤ rδ(Dm−1) and hence δ(Dm) ≤ r m−1δ(D1). Note that xm−1, xm ∈ Dm and ‖xm − xm−1‖ ≤ δ(Dm) ≤ r m−1δ(D1). This implies that {xm} is a Cauchy sequence in K0. Thus, there exists a x ∈ K0 such that x = lim m→∞ xm. Now, for k ≥ N(x), ‖T kx − x‖ ≤ ‖T kx − T kxm‖ + ‖T kxm − x‖ ≤ αk(x)‖x − xm‖ + ‖T kxm − xm+1‖ + ‖xm+1 − x‖ Letting lim inf k→∞ on both sides, we get lim inf k→∞ ‖T kx − x‖ ≤ ‖x − xm‖ + lim inf k→∞ ‖T kxm − xm+1‖ + ‖xm+1 − x‖ ≤ ‖x − xm‖ + lim inf k→∞ ‖T knmxm − xm+1‖ + ‖xm+1 − x‖ ≤ ‖x − xm‖ + βm+1 + ‖xm+1 − x‖. As βm → 0 and xm → x, we have lim inf k→∞ ‖T kx − x‖ = 0. Thus, there exists a subsequence {T nix} of {T nx} such that lim i→∞ T nix = x. Hence assertion II is proved. Therefore ρ0 = 0 and K0 is singleton. Hence T has a fixed point in K. � Next, we use the ultrafilter techniques to prove the existence of fixed points for a pointwise eventually asymptotically nonexpansive mapping T on K in a Banach space X with the Maluta constant D(X) < 1. This result (Theorem 3.6) is motivated by the following result of Lim and Xu [17]: If T is an asymp- totically nonexpansive map defined on a nonempty weakly compact convex set K in a Banach space with the Maluta constant D(X) < 1 and T is weakly asymptotically regular on K, then T has a fixed point in K. In the following remark, we recall some facts about ultrafilter which are used in the proof of Theorem 3.6. For more information about ultrafilters, one may refer to [1, 10]. Remark 3.5 ([1]). A filter F on N is a nonempty collection of subsets of N satisfying (1) if A, B ∈ F then A ∩ B ∈ F; (2) if A ∈ F and A ⊆ B then B ∈ F. If F is a filter on N and ∅ ∈ F, then F = 2N and is called an improper filter. By usual set inclusion, the family P of all proper filters on N is a partially ordered set and every chain in P has an upper bound. By Zorn’s lemma, we get a maximal proper filter in P. A maximal filter in P is called an ultrafilter on N. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 126 Existence of fixed points for pointwise eventually asymptotically nonexpansive mappings A sequence {xn} in a Banach space X converges to x ∈ X over the filter F if for every neighbourhood V of x, the set {n ∈ N : xn ∈ V } belongs to F and it is denoted by lim F xn = x. A trivial ultrafilter Fn0 on N is the collection of subsets of N which contains an element n0 ∈ N, where n0 ∈ N is fixed and all other ultrafilters on N are said to be non-trivial. It is known that nontrivial ultrafilters always exist (Zorn’s lemma), and a sequence {xn} in a compact set always converges relative to any nontrivial ultrafilter over N. Theorem 3.6. Let K be a nonempty weakly compact convex subset of a Banach space X with the Maluta constant D(X) < 1 and T : K → K be a pointwise eventually asymptotically nonexpansive map. Further, assume that T is weakly asymptotically regular on K ( i.e., w − lim n→∞ (T nx − T n+1x) = 0 for all x ∈ K ) . Then T has a fixed point in K. Proof. Let U be a non-trivial ultrafilter on N. Define a mapping S on K by S(x) = w−lim U T nx, for x ∈ K. Since K is weakly compact, S(x) is well defined and there exists a subsequence {T nix} of {T nx} such that {T nix} converges weakly to S(x). For x, y ∈ K, we have ‖Sx − Sy‖ ≤ lim inf k→∞ ‖T nkx − T nky‖ ≤ lim sup n→∞ ‖T nx − T ny‖ ≤ lim sup n→∞ αn(x)‖x − y‖ = ‖x − y‖. Hence S is a nonexpansive map on K. Since D(X) < 1, X has normal structure. Therefore, the nonexpansive map S : K → K has a fixed point in K. That is, there exists a x ∈ K such that w − lim U T nx = x. This implies that there exists a subsequence {T n ′ k(x)} of {T nx} converges weakly to x. Without loss of generality, we may assume that n′ k ≥ N(x) for all k ∈ N. Choose r > 0 so that (1+r)2D(X) < 1. Since αn(x) → 1 as n → ∞, we can find a subsequence {ni} of {n ′ k} such that αni(x) < 1 + r and αni+1−ni(x) < 1 + r for all i ∈ N. From the definition of D(X), lim sup i→∞ ‖T nix − x‖ ≤ D(X)δ({T nix}). Since T is weakly asymptotically regular at x ∈ K, for fixed i > j it must be the case that T nt+(ni−nj)(x) converges weakly to x as t → ∞. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 127 M. Radhakrishnan and S. Rajesh Now observe that ‖T nix − T nj x‖ ≤ αnj (x)‖T ni−nj (x) − x‖ ≤ (1 + r) lim sup t→∞ ‖T ni−nj (x) − T nt+(ni−nj)(x)‖ ≤ (1 + r)αni−nj (x) lim sup t→∞ ‖x − T nt(x)‖ ≤ (1 + r)2 lim sup t→∞ ‖x − T nt(x)‖ Therefore lim sup i→∞ ‖x − T ni(x)‖ ≤ (1 + r)2D(X) lim sup t→∞ ‖x − T nt(x)‖ and since (1 + r)2D(X) < 1, we conclude lim sup i→∞ ‖x − T ni(x)‖ = 0. As T m is continuous at x for all m ≥ N(x), {T ni+m(x)} converges weakly to T m(x) and since T is weakly asymptotically regular at x this in turn implies x = T m(x) for all m ≥ N(x). Therefore x = T m+1x = T (T mx) = T x. � Theorem 3.7. Let K be a nonempty weakly compact convex subset of a Banach space X, which is uniformly convex in every direction and T : K → K be a pointwise eventually asymptotically nonexpansive map. Further, assume that the Maluta constant D(X) < 1. Then T has a fixed point in K. Proof. Assume that K0 is minimal with respect to being a nonempty closed convex subset of K with the property that for every x ∈ K0, ωw(x) ⊆ K0. By Lemma 3.2, there is a constant ρ0 ≥ 0 such that lim sup n→∞ ‖T nx − y‖ = ρ0, for all x, y ∈ K0. Since X is uniformly convex in every direction, K0 is singleton, say K0 = {x} and so the sequence {T nx} converges weakly to x. Now choose r > 0 so that (1+r)2D(X) < 1. Since αn(x) → 1 as n → ∞, we can find a subsequence {ni} of {N(x), N(x) + 1, . . . } such that αni(x) < 1 + r and αni+1−ni(x) < 1 + r for all i ∈ N. From the definition of D(X), lim sup i→∞ ‖x − T nix‖ ≤ D(X)δ({T nix}). Now for i > j, we have ‖T nix − T nj x‖ ≤ αnj (x)‖T ni−nj (x) − x‖ ≤ (1 + r) lim sup t→∞ ‖T ni−nj (x) − T nt+(ni−nj)(x)‖ ≤ (1 + r)αni−nj (x) lim sup t→∞ ‖x − T nt(x)‖ ≤ (1 + r)2 lim sup t→∞ ‖x − T nt(x)‖ Thus, lim sup i→∞ ‖x − T ni(x)‖ ≤ (1 + r)2D(X) lim sup t→∞ ‖x − T nt(x)‖. Since (1 + r)2D(X) < 1, we have lim sup i→∞ ‖x − T ni(x)‖ = 0. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 128 Existence of fixed points for pointwise eventually asymptotically nonexpansive mappings Finally, by the definition of T, we have {T ni+m(x)} converging weakly to T m(x) for all m ≥ N(x). But, we know that the sequence {T nx} converges weakly to x. This implies that x = T m(x) for all m ≥ N(x). Hence x is a fixed point of T. � 4. Asymptotic Behavior In this section, we investigate the asymptotic behavior of the sequence {T nx} for a pointwise eventually asymptotically nonexpansive map T defined on a nonempty weakly compact convex subset K of a Banach space X whenever X satisfies the uniform Opial condition or X has a weakly continuous duality map. Lemma 4.1. Let K be a nonempty weakly compact convex subset of a Banach space X satisfying Opial’s condition and T : K → K be a pointwise eventually asymptotically nonexpansive map. Further, assume that T is weakly asymptot- ically regular at some x ∈ K. For m ∈ N, define bm = lim sup i→∞ ‖T ni+m(x) − y‖ where y = w − lim i→∞ T nix. Then the sequence {bm} converges. Proof. Let y = w − lim i→∞ T nix. Since T is weakly asymptotically regular at x ∈ K, we have y = w − lim i→∞ T ni+m(x) for m ∈ N. For any j ≥ N(y), we have bm+j = lim sup i→∞ ‖T ni+m+j(x) − y‖ ≤ lim sup i→∞ ‖T ni+m+j(x) − T jy‖ ≤ αj(y) lim sup i→∞ ‖T ni+m(x) − y‖ = αj(y)bm. We claim that lim m→∞ bm exists. Note that there exists two subsequences {mi} and {m′i} of N such that lim i→∞ bmi = lim sup m→∞ bm and lim i→∞ bm′ i = lim inf m→∞ bm. This gives that there exists k0 ∈ N such that mj > m ′ 1 +N(y), for all j ≥ k0. i.e., mj = m ′ 1 + N(y) + nj, for some nj ∈ N. For any j ≥ k0, we have bmj = bm′1+N(y)+nj ≤ αN(y)+nj (y)bm′1 = αmj−m′1(y)bm ′ 1 So lim sup m→∞ bm = lim j→∞ bmj ≤ bm′1. Similarly, for each i ∈ N, we have lim sup m→∞ bm ≤ bm′ i and we get lim sup m→∞ bm ≤ lim inf m→∞ bm. Hence the sequence {bm} converges. � Theorem 4.2. Let K be a nonempty weakly compact convex subset of a Banach space X satisfying the uniform Opial condition and T : K → K be a pointwise c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 129 M. Radhakrishnan and S. Rajesh eventually asymptotically nonexpansive map. Then given an x ∈ K, {T nx} converges weakly to a fixed point of T if and only if T is weakly asymptotically regular at x. Proof. If {T nx} converges weakly to a fixed point of T, then {T nx} is obviously weakly asymptotically regular at x. Conversely, assume that T is weakly asymptotically regular at x ∈ K. Then we claim that ωw(x) ⊆ F(T ), where F(T ) is the set of all fixed point of T and ωw(x) is singleton. To see that ωw(x) ⊆ F(T ), let y ∈ ωw(x). Then there exists a subsequence {T nix} of {T nx} such that y = w− lim i→∞ T nix. Since T is weakly asymptotically regular at x, we have y = w − lim i→∞ T ni+m(x) for m ∈ N. Now, let bm = lim sup i→∞ ‖T ni+m(x) − y‖. By Lemma 4.1, the sequence {bm} converges to b ≥ 0. Assume b = 0. For m ≥ N(y), we have ‖T my − y‖ ≤ ‖T my − T ni+2m(x)‖ + ‖T ni+2m(x) − y‖ ≤ lim sup i→∞ ‖T my − T ni+2m(x)‖ + lim sup i→∞ ‖T ni+2m(x) − y‖ ≤ αm(y)bm + b2m. Letting m → ∞ on both sides, we get T my → y. Now, for m ≥ n ≥ N(y), we have ‖T my − T ny‖ ≤ αn(y)‖y − T m−ny‖. It follows that if n ≥ N(y), then T my → T ny as m → ∞. This implies that y = T ny, for n ≥ N(y) and hence T y = y. Now, suppose b > 0 and let z (m) i = T ni+m(x) − y bm . Then for each m ≥ 0, w − lim i→∞ z (m) i = 0 and lim sup i→∞ ‖z (m) i ‖ = 1. By the uniform Opial condition of X, we have 1 + rX(c) ≤ lim sup i→∞ ‖z (m) i + z‖, for all z ∈ X with ‖z‖ ≥ c. Take z = y − T my b2m . Then for m ≥ N(y), 1 + rX ( ‖y − T my‖ b2m ) ≤ lim sup i→∞ ∥ ∥ ∥ ∥ T ni+2m(x) − T my b2m ∥ ∥ ∥ ∥ and hence b2m ( 1 + rX ( ‖y − T my‖ b2m )) ≤ lim sup i→∞ ‖T ni+2m(x) − T my‖ ≤ αm(y)bm. Letting m → ∞, we get b  1 + rX   lim sup m→∞ ‖y − T my‖ b     ≤ b; rX   lim sup m→∞ ‖y − T my‖ b   ≤ 0. c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 130 Existence of fixed points for pointwise eventually asymptotically nonexpansive mappings Since X satisfies the uniform Opial condition, we have lim m→∞ ‖T my − y‖ = 0. This implies that T y = y and thus ωw(x) ⊆ F(T ). Now, it remains to prove that ωw(x) is a singleton. First to observe that lim n→∞ ‖T nx−p‖ exists for every p ∈ F(T ). For this, let p ∈ F(T ). For m ≥ N(p) and n ≥ 1, we have ‖T n+mx − p‖ = ‖T n+mx − T mp‖ ≤ αm(p)‖T nx − p‖ Letting lim sup m→∞ on both sides, we get lim sup m→∞ ‖T mx − p‖ ≤ ‖T nx − p‖ for all n ≥ 1. Thus, we have lim n→∞ ‖T nx − p‖ exists for all p ∈ F(T ). Suppose that there exists two subsequences {ni} and {mi} of N such that w − lim i→∞ T ni(x) = p1 and w − lim i→∞ T mi(x) = p2 for p1 6= p2. lim n→∞ ‖T nx − p1‖ = lim i→∞ ‖T ni(x) − p1‖ < lim i→∞ ‖T ni(x) − p2‖ = lim i→∞ ‖T mi(x) − p2‖ < lim i→∞ ‖T mi(x) − p1‖ = lim n→∞ ‖T nx − p1‖ which is a contradiction. Hence ωw(x) is singleton and the sequence {T nx} converges weakly to a fixed point of T. � Theorem 4.3. Let K be a nonempty weakly compact convex subset of a Banach space X with a weakly continuous duality map Jϕ and T : K → K be a pointwise eventually asymptotically nonexpansive map. Then given an x ∈ K, {T nx} converges weakly to a fixed point of T if and only if T is weakly asymptotically regular at x. Proof. If {T nx} converges weakly to a fixed point of T, then {T nx} is obviously weakly asymptotically regular at x. Conversely, assume that T is weakly asymptotically regular at x ∈ K. Then we claim that ωw(x) ⊆ F(T ) and ωw(x) is singleton. Since X has weakly continuous duality map Jϕ, so it satisfies Opial’s condition. By Theorem 4.2, it is enough to show that wω(x) ⊆ F(T ). Let y ∈ ωw(x). Then there exists a subsequence {T nix} of {T nx} such that y = w − lim i→∞ T nix. Since T is weakly asymptotically regular at x, we have y = w − lim i→∞ T ni+m(x) for m ∈ N. Let bm = lim sup i→∞ ‖T ni+m(x) − y‖. By Lemma 4.1, the sequence {bm} con- verges to b. For m ≥ N(y), we have c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 131 M. Radhakrishnan and S. Rajesh Φ ( ‖T ni+2m(x) − y‖ ) = Φ ( ‖T ni+2m(x) − T my + T my − y‖ ) = Φ ( ‖T ni+2m(x) − T my‖ ) + 1 ∫ 0 〈 T my − y, Jϕ(T ni+2m(x) − T my + t(T my − y)) 〉 dt ≤ Φ ( αm(y)‖T ni+m(x) − y‖ ) + 1 ∫ 0 〈 T my − y, Jϕ(T ni+2m(x) − T my + t(T my − y)) 〉 dt Letting lim sup i→∞ on both sides, we get Φ (b2m) ≤ Φ (αm(y)bm) + 1 ∫ 0 〈T my − y, Jϕ(y − T my + t(T my − y))〉 dt = Φ (αm(y)bm) − 1 ∫ 0 ‖y − T my‖ϕ ((1 − t)‖y − T my‖) dt = Φ (αm(y)bm) − Φ (‖y − T m y‖) Letting lim sup m→∞ on both sides, we get Φ ( lim sup m→∞ ‖y − T my‖ ) ≤ Φ(b) − Φ(b). Thus, T my → y as m → ∞ and T y = y. Hence ωw(x) ⊆ F(T ) and the sequence {T nx} converges weakly to a fixed point of T. � Acknowledgements. The authors would like to thank the anonymous ref- eree for the comments and suggestions. 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