Iterative Algorithm and Δ-Convergence Theorems for Total Asymptotically Nonexpansive Mappings in CAT(0) Spaces

and Applied Analysis 3 Let {xn} be a bounded sequence in a CAT 0 space X. For x ∈ X, one sets r x, {xn} lim sup n→∞ d x, xn . 2.5 The asymptotic radius r {xn} of {xn} is given by r {xn} inf x∈X {r x, {xn} }, 2.6 the asymptotic radius rC {xn} of {xn}with respect to C ⊂ X is given by rC {xn} inf x∈C {r x, {xn} }, 2.7 the asymptotic center A {xn} of {xn} is the set A {xn} {x ∈ X : r x, {xn} r {xn} }, 2.8 the asymptotic center AC {xn} of {xn}with respect to C ⊂ X is the set AC {xn} {x ∈ C : r x, {xn} rC {xn} }. 2.9 Recall that a bounded sequence {xn} in X is said to be regular if r {xn} r {un} for every subsequence {un} of {xn}. Proposition 2.2 see 15 . If {xn} is a bounded sequence in a complete CAT(0) space X and C is a closed convex subset of X, then 1 there exists a unique point u ∈ C such that r u, {xn} inf x∈C r x, {xn} ; 2.10 2 A {xn} and AC {xn} are both singleton. Lemma 2.3 see 16 . If C is a closed convex subset of a complete CAT(0) space X and if {xn} is a bounded sequence in C, then the asymptotic center of {xn} is in C. Definition 2.4 see 17 . A sequence {xn} in a CAT 0 space X is said to Δ-converge to x ∈ X if x is the unique asymptotic center of {un} for every subsequence {un} of {xn}. In this case one writes Δ − limn→∞xn x and call x the Δ-limit of {xn}. Lemma 2.5 see 17 . Every bounded sequence in a complete CAT(0) space always has a Δconvergent subsequence. 4 Abstract and Applied Analysis Let {xn} be a bounded sequence in a CAT 0 space X and let C be a closed convex subset of X which contains {xn}. We denote the notation {xn} ⇀ w iff Φ w inf x∈C Φ x , 2.11 where Φ x : lim supn→∞d xn, x . Now one gives a connection between the “⇀” convergence and Δ-convergence. Proposition 2.6 see 13 . Let {xn} be a bounded sequence in a CAT(0) space X and let C be a closed convex subset of X which contains {xn}. Then 1 Δ − limn→∞xn x implies that {xn} ⇀ x; 2 {xn} ⇀ x and {xn} is regular imply that Δ − limn→∞xn x; Let C be a closed subset of a metric space X, d . Recall that a mapping T : C → C is said to be nonexpansive if d ( Tx, Ty ) ≤ dx, y, ∀x, y ∈ X. 2.12 T is said to be asymptotically nonexpansive if there is a sequence {kn} ⊂ 1, ∞ with limn→∞ kn 1 such that d ( Tx, Ty ) ≤ knd ( x, y ) , ∀n ≥ 1, x, y ∈ X. 2.13 T is said to be closed if, for any sequence {xn} ⊂ C with d xn, x → 0 and d Txn, y → 0, then Tx y. T is called L-uniformly Lipschitzian, if there exists a constant L > 0 such that d ( Tx, Ty ) ≤ Ldx, y, ∀x, y ∈ C, n ≥ 1. 2.14 Definition 2.7. Let X, d be a metric space and let C be a closed subset of X. A mapping T : C → C is said to be {vn}, {μn}, ζ −total asymptotically nonexpansive if there exist nonnegative real sequences {vn}, {μn} with vn → 0, μn → 0 n → ∞ and a strictly increasing continuous function ζ : 0, ∞ → 0, ∞ with ζ 0 0 such that d ( Tx, Ty ) ≤ dx, y vnζ ( d ( x, y )) μn, ∀n ≥ 1, x, y ∈ C. 2.15 Remark 2.8. 1 It is obvious that If T is uniformly Lipschitzian, then T is closed. 2 From the definitions, it is to know that, each nonexpansive mapping is a asymptotically nonexpansive mapping with sequence {kn 1}, and each asymptotically nonexpansive mapping is a total asymptotically nonexpansive mapping with vn kn − 1, μn 0, for all n ≥ 1, and ζ t t, t ≥ 0. Abstract and Applied Analysis 5 Lemma 2.9 demiclosed principle for total asymptotically nonexpansive mappings . Let C be a closed and convex subset of a complete CAT(0) space X and let T : C → C be a L-uniformly Lipschitzian and {vn}, {μn}, ζ −total asymptotically nonexpansive mapping. Let {xn} be a bounded sequence in C such that limn→∞ d xn, Txn 0 and xn ⇀ w. Then Tw w. Proof. By the definition, xn ⇀ w if and only if AC {xn} {w}. By Lemma 2.3, we have A {xn} {w}. Since limn→∞ d xn, Txn 0, by induction we can prove thatand Applied Analysis 5 Lemma 2.9 demiclosed principle for total asymptotically nonexpansive mappings . Let C be a closed and convex subset of a complete CAT(0) space X and let T : C → C be a L-uniformly Lipschitzian and {vn}, {μn}, ζ −total asymptotically nonexpansive mapping. Let {xn} be a bounded sequence in C such that limn→∞ d xn, Txn 0 and xn ⇀ w. Then Tw w. Proof. By the definition, xn ⇀ w if and only if AC {xn} {w}. By Lemma 2.3, we have A {xn} {w}. Since limn→∞ d xn, Txn 0, by induction we can prove that lim n→∞ d xn, Txn 0, ∀m ≥ 1. 2.16 In fact, it is obvious that, the conclusion is true for m 1. Suppose the conclusion holds for m ≥ 1, now we prove that the conclusion is also true for m 1. In fact, since T is a L-uniformly Lipschitzian mapping, we have d ( xn, T m xn ) ≤ d ( xn, Txn ) d ( Txn, TT xn ) ≤ d ( xn, Txn ) Ld ( xn, T xn ) −→ 0 as n −→ ∞ . 2.17 Equation 2.16 is proved. Hence for each x ∈ C and m ≥ 1 from 2.16 we have Φ x : lim sup n→∞ d xn, x lim sup n→∞ d Txn, x . 2.18 In 2.18 taking x Tw,m ≥ 1, we have Φ Tw lim sup n→∞ d Txn, Tw ≤ lim sup n→∞ ( d xn,w vmζ d xn,w μm ) . 2.19 Let m → ∞ and taking superior limit on the both sides, it gets that lim sup m→∞ Φ Tw ≤ Φ w . 2.20 Furthermore, for any n,m ≥ 1 it follows from inequality 2.3 with t 1/2 that d2 ( xn, w ⊕ Tw 2 ) ≤ 1 2 d2 xn,w 1 2 d2 xn, Tw − 14d 2 w, Tw . 2.21 6 Abstract and Applied Analysis Let n → ∞ and taking superior limit on the both sides of the above inequality, for any m ≥ 1 we get Φ ( w ⊕ Tw 2 )2 ≤ 1 2 Φ w 2 1 2 Φ Tw 2 − 1 4 d w, Tw . 2.22 Since A {xn} {w}, we have Φ w 2 ≤ Φ ( w ⊕ Tw 2 )2 ≤ 1 2 Φ w 2 1 2 Φ Tw 2 − 1 4 d w, Tw , ∀m ≥ 1, 2.23 which implies that d2 w, Tw ≤ 2Φ Tw 2 − 2Φ w . 2.24 By 2.20 and 2.24 , we have limm→∞ d w, Tw 0. This implies that limm→∞d w, T 1w 0. Since T is uniformly Lipschitzian, T is uniformly continuous. Hence we have Tw w. This completes the proof of Lemma 2.9. The following proposition can be obtained from Lemma 2.9 immediately which is a generalization of Kirk and Panyanak 17 and Nanjaras and Panyanak 13 . Proposition 2.10. Let C be a closed and convex subset of a complete CAT(0) space X and let T : C → C be an asymptotically nonexpansive mapping. Let {xn} be a bounded sequence in C such that limn→∞d xn, Txn 0 and Δ − limn→∞ xn w. Then T w w. Definition 2.11 see 18 . LetX be a CAT 0 space thenX is uniformly convex, that is, for any given r > 0, ∈ 0, 2 and λ ∈ 0, 1 , there exists a η r, 2/8 such that, for all x, y, z ∈ X, d x, z ≤ r d ( y, z ) ≤ r d ( x, y ) ≥ r ⎫ ⎬ ⎭ ⇒ d 1 − λ x ⊕ λy, z ≤ ( 1 − 2λ 1 − λ 2 8 ) r , 2.25 where the function η : 0,∞ × 0, 2 → 0, 1 is called the modulus of uniform convexity of CAT 0 . Lemma 2.12 see 14 . If {xn} is a bounded sequence in a complete CAT(0) space with A {xn} {x}, {un} is a subsequence of {xn} withA {un} {u}, and the sequence {d xn, u } converges, then x u. Lemma 2.13. Let {an}, {bn}, and {δn} be sequences of nonnegative real numbers satisfying the inequality an 1 ≤ 1 δn an bn. 2.26 If Σn 1δn < ∞ and Σn 1bn < ∞, then {an} is bounded and limn→∞an exists. Abstract and Applied Analysis 7 Lemma 2.14 see 13 . Let X be a CAT(0) space, x ∈ X be a given point and {tn} be a sequence in b, c with b, c ∈ 0, 1 and 0 < b 1 − c ≤ 1/2. Let {xn} and {yn} be any sequences in X such thatand Applied Analysis 7 Lemma 2.14 see 13 . Let X be a CAT(0) space, x ∈ X be a given point and {tn} be a sequence in b, c with b, c ∈ 0, 1 and 0 < b 1 − c ≤ 1/2. Let {xn} and {yn} be any sequences in X such that lim sup n→∞ d xn, x ≤ r, lim sup n→∞ d ( yn, x ) ≤ r, lim n→∞ d ( 1 − tn xn ⊕ tnyn, x ) r, 2.27 for some r ≥ 0. Then lim n→∞ d ( xn, yn ) 0. 2.28 3. Main Results In this section, we will prove our main theorem. Theorem 3.1. Let C be a nonempty bounded closed and convex subset of a complete CAT(0) space X. Let S : C → C be a asymptotically nonexpansive mapping with sequence {kn} ⊂ 1,∞ , kn → 1 and T : C → C be a uniformly L-Lipschitzian and {vn}, {μn}, ζ −total asymptotically nonexpansive mapping such that F F S ∩F T / ∅. From arbitrary x1 ∈ C, defined the sequence {xn} as follows: yn αnSxn ⊕ 1 − αn xn, xn 1 βnTyn ⊕ ( 1 − βn ) xn 3.1 for all n ≥ 1, where {βn} is a sequence in (0, 1). If the following conditions are satisfied: i Σn 1vn < ∞;Σn 1μn < ∞;Σn 1 kn − 1 < ∞; ii there exists a constant M∗ > 0 such that ζ r ≤ M∗r, r ≥ 0; iii there exist constants b, c ∈ 0, 1 with 0 < b 1 − c ≤ 1/2 such that {αn} ⊂ b, c ; iv Σn 1 sup{d z, Sz : z ∈ B} < ∞ for each bounded subset B of C. Then the sequence {xn}Δ-converges to a fixed point of F. Proof. We divide the proof of Theorem 3.1 into four steps. I First we prove that for each p ∈ F the following limit exists lim n→∞ d ( xn, p ) . 3.2 8 Abstract and Applied Analysis In fact, for each p ∈ F, we have d ( yn, p ) d ( αnS xn ⊕ 1 − αn xn, p ) ≤ αnd ( Sxn, p ) 1 − αn d ( xn, p ) αnd ( Sxn, S p ) 1 − αn d ( xn, p ) ≤ αnknd ( xn, p ) 1 − αn d ( xn, p ) 1 αn kn − 1 d ( xn, p ) , d ( xn 1, p ) d ( βnT yn ⊕ ( 1 − βn ) xn, p ) ≤ βnd ( Tyn, p ) ( 1 − βn ) d ( xn, p ) βnd ( Tyn, T p ) ( 1 − βn ) d ( xn, p ) ≤ βn ( d ( yn, p ) vnζ ( d ( yn, p )) μn ) ( 1 − βn ) d ( xn, p ) ≤ βn ( d ( yn, p ) vnM∗d ( yn, p ) μn ) ( 1 − βn ) d ( xn, p ) ≤ dxn, p ) βnαn kn − 1 d ( xn, p ) βnvnM∗ 1 αn kn − 1 d ( xn, p ) μn ≤ 1 kn − 1 vnM∗ 1 αn kn − 1 d ( xn, p ) μn. 3.3 It follows from Lemma 2.13 that {d xn, p } is bounded and limn→∞d xn, p exists. Without loss of generality, we can assume limn→∞d xn, p c ≥ 0. II Next we prove that lim n→∞ d xn, Txn 0. 3.4