The Equivalence of Convergence Results of Modified Mann and Ishikawa Iterations with Errors without Bounded Range Assumption

and Applied Analysis 3 The aim of this paper is to prove the equivalence of convergent results of above Ishikawa andMann iterations with errors for generalized asymptoticallyΦ-strongly pseudocontractive mappings without bounded range assumptions in uniformly smooth real Banach spaces. For this, we need the following concepts and lemmas. Definition 1.4 see 4 . The mapping T is called generalized asymptotically Φ-strongly pseudocontractive if 〈 Tx − Ty, j(x − y)〉 ≤ kn ∥∥x − y∥∥2 −Φ(∥∥x − y∥∥), n ≥ 0, 1.6 where j x − y ∈ J x − y , {kn} ⊂ 1, ∞ is converging to one and Φ : 0, ∞ → 0, ∞ is strictly increasing continuous function with Φ 0 0. Definition 1.5 see 4 . For arbitrary given x0 ∈ D, modified Ishikawa iterative process with errors {xn}n 0 defined by yn 1 − bn − dn xn bnTxn dnwn, n ≥ 0, xn 1 1 − an − cn xn anTyn cnvn, n ≥ 0, 1.7 where {vn}, {wn} are any bounded sequences in D; {an}, {bn}, {cn}, {dn} are four real sequences in 0, 1 and satisfy an cn ≤ 1, bn dn ≤ 1, for all n ≥ 0. If bn dn 0, we define modified Mann iterative process with errors {zn} by zn 1 1 − an − cn zn anTzn cnun, n ≥ 0, 1.8 where {un} is any bounded sequence in D. Lemma 1.6 see 7 . Let E be a uniformly smooth real Banach space and let J : E → 2E be a normalized duality mapping. Then ∥∥x y∥∥2 ≤ ‖x‖ 2〈y, J(x y)〉, 1.9 for all x, y ∈ E. Lemma 1.7 see 8 . Let {ρn}n 0 be a nonnegative sequence which satisfies the following inequality: ρn 1 ≤ 1 − λn ρn σn, n ≥ 0, 1.10 where λn ∈ 0, 1 with ∑∞ n 0 λn ∞, σn o λn . Then ρn → 0 as n → ∞. 2. Main Results First of all, we give a new concept. 4 Abstract and Applied Analysis Definition 2.1. Amapping T : D → D is called uniformly generalized Lipschitz if there exists a constant L > 0 such that ∥∥Tnx − Tny∥∥ ≤ L(1 ∥∥x − y∥∥), ∀x, y ∈ D, ∀n ≥ 0. 2.1 It is mentioned to notice that if T has bounded range, then it is uniformly generalized Lipschitz. In fact, since R T ⊆ R T , then supx∈D{‖Tx‖} ≤ supx∈D{‖Tx‖} M1, thus ‖Tnx − Tny‖ ≤ 2M1 ≤ L 1 ‖x − y‖ , where L 2M1. On the contrary, it is not true in general See 6 . In the following, we prove the main theorems of this paper. Theorem 2.2. Let E be an arbitrary uniformly smooth real Banach space, let D be a nonempty closed convex subset of E, and let T : D → D be a uniformly generalized Lipschitz generalized asymptotically Φ-strongly pseudocontractive mapping with q ∈ F T / ∅. Let {an}, {bn}, {cn}, {dn} be four real sequences in 0, 1 and satisfy the following conditions: i an cn ≤ 1, bn dn ≤ 1; ii an, bn, dn → 0 as n → ∞ and cn o an ; iii Σn 0an ∞. For some x0, z0 ∈ D, let {un}, {vn}, {wn} be any bounded sequences in D, and let {xn} and {zn} be Ishikawa and Mann iterative sequences with errors defined by 1.7 and 1.8 , respectively. Then the following conclusions are equivalent: 1 {xn} converges strongly to the unique fixed point q of T ; 2 {zn} converges strongly to the unique fixed point q of T . Proof. 1 ⇒ 2 is obvious, that is, let bn dn 0, 1.7 turns into 1.8 . We only need to show that 2 ⇒ 1 . Since T : D → D is a uniformly generalized Lipschitz generalized asymptotically Φ-strongly pseudocontractive mapping, then there exists a strictly increasing continuous function Φ : 0, ∞ → 0, ∞ with Φ 0 0 such that 〈 Tx − Ty, J(x − y)〉 ≤ kn ∥∥x − y∥∥2 −Φ(∥∥x − y∥∥), 2.2 that is, 〈 knI − T x − knI − T y, J ( x − y)〉 ≥ Φ(∥∥x − y∥∥), 2.3 ∥∥Tnx − Tny∥∥ ≤ L(1 ∥∥x − y∥∥), 2.4 for any x, y ∈ D. For convenience, denote k supn{kn}. Step 1. There exists x0 ∈ D and x0 / Tx0 such that r0 k L ‖x0−q‖ L‖x0−q‖ ∈ R Φ range of Φ . Indeed, if Φ r → ∞ as r → ∞, then r0 ∈ R Φ ; if sup{Φ r : r ∈ 0, ∞ } r1 < ∞with r1 < r0, then, for q ∈ D, there exists a sequence {νn} inD such that νn → q as n → ∞ with νn / q. Furthermore, there exists a natural number n0 such that k L ‖νn−q‖ L‖νn−q‖ < Abstract and Applied Analysis 5 r1/2 for n ≥ n0, thenwe redefine x0, r0 such that x0 νn0 , r0 k L ‖x0−q‖ L‖x0−q‖ ∈ R Φ . Hence, it is to ensure that Φ−1 r0 is well defined. Step 2. For any n ≥ 0, {xn} is a bounded sequence. Set R Φ−1 r0 . From 2.3 , we have 〈 kn ( x0 − q ) − Tnx0 − q ) , J ( x0 − q )〉 ≥ Φ∥x0 − q ∥∥), 2.5and Applied Analysis 5 r1/2 for n ≥ n0, thenwe redefine x0, r0 such that x0 νn0 , r0 k L ‖x0−q‖ L‖x0−q‖ ∈ R Φ . Hence, it is to ensure that Φ−1 r0 is well defined. Step 2. For any n ≥ 0, {xn} is a bounded sequence. Set R Φ−1 r0 . From 2.3 , we have 〈 kn ( x0 − q ) − Tnx0 − q ) , J ( x0 − q )〉 ≥ Φ∥x0 − q ∥∥), 2.5 that is, k L ‖x0 − q‖2 L‖x0 − q‖ ≥ Φ ‖x0 − q‖ . Thus, we obtain that ‖x0 − q‖ ≤ R. Denote B1 { x ∈ D : ∥∥x − q∥∥ ≤ R}, B2 { x ∈ D : ∥∥x − q∥∥ ≤ 2R}, M supn ∥vn − q ∥∥} supn ∥wn − q ∥∥}. 2.6 Next, we want to prove that xn ∈ B1 for any n ≥ 0 by induction. If n 0, then x0 ∈ B1. Now we assume that it holds for some n, that is, xn ∈ B1. We prove that xn 1 ∈ B1. Suppose that it is not the case, then ‖xn 1 − q‖ > R. Since J is uniformly continuous on bounded subset of E, then, for 0 Φ R/4 /24L 1 2R , there exists δ > 0 such that ‖Jx − Jy‖ < 0 when ‖x − y‖ < δ, for allx, y ∈ B2. Now denote τ0 min { R 2 L 1 2R 2R M , R 4 L 1 R 2R M , δ 2 L 1 2R 2R M , Φ R/4 24R2 , Φ R/4 24L 1 2R , Φ R/4 48MR } . 2.7 Since an, bn, cn, dn → 0 as n → ∞, and cn o an , without loss of generality, we assume that 0 ≤ an, bn, cn, dn ≤ τ0, cn < anτ0 for any n ≥ 0. Then we obtain the following estimates: ∥Tnxn − q ∥∥ ≤ L(1 ∥xn − q ∥∥) ≤ L 1 R , ∥yn − q ∥∥ ≤ 1 − bn − dn ∥xn − q ∥∥ bn ∥Tnxn − q ∥∥ dn ∥wn − q ∥∥ ≤ R bnL ( 1 ∥xn − q ∥∥) dnM ≤ R bnL 1 R dnM ≤ R τ0 L 1 R M ≤ 2R, ∥Tnyn − q ∥∥ ≤ L(1 ∥yn − q ∥∥) ≤ L 1 2R , 6 Abstract and Applied Analysis ‖xn − Txn‖ ≤ ∥xn − q ∥∥ ∥Tnxn − q ∥∥ ≤ L 1 L ∥xn − q ∥∥ ≤ L 1 L R, ∥xn − q ) − yn − q )∥∥ ≤ bn‖xn − Txn‖ dn ∥wn − q ∥∥ ∥xn − q ∥∥] ≤ bn L 1 L R dn M R ≤ τ0 L 1 R 2R M ≤ τ0 L 1 2R 2R M ≤ δ 2 < δ, ∥xn − q ∥∥ ≥ ∥xn 1 − q ∥∥ − an ∥Tnyn − xn ∥∥ − cn‖vn − xn‖ ≥ ∥xn 1 − q ∥∥ − an ∥Tnyn − q ∥∥ ∥xn − q ∥∥] − cn ∥xn − q ∥∥ ∥vn − q ∥∥] > R − an L 1 2R R − cn R M ≥ R − τ0 L 1 2R M 2R ≥ R − R 2 R 2 , ∥yn − q ∥∥ ≥ ∥xn − q ∥∥ − bn‖Txn − xn‖ − dn‖xn −wn‖ ≥ ∥xn − q ∥∥ − bn L 1 L R − dn ∥xn − q ∥∥ ∥wn − q ∥∥] ≥ ∥xn − q ∥∥ − bn L 1 L R − dn R M ≥ ∥xn − q ∥∥ − τ0 L 1 R 2R M > R 2 − R 4 R 4 , ∥xn 1 − q ∥∥ ≤ 1 − an − cn ∥xn − q ∥∥ an ∥Tnyn − q ∥∥ cn ∥vn − q ∥∥ ≤ R τ0 L 1 2R M ≤ 2R, ∥xn 1 − q ) − xn − q )∥∥ ≤ an ∥Tnyn − xn ∥∥ cn‖un − xn‖ ≤ an ∥Tnyn − q ∥∥ ∥xn − q ∥∥] cn ∥vn − q ∥∥ ∥xn − q ∥∥] ≤ an L 1 2R R cn M R ≤ τ0 L 1 2R 2R M ≤ δ 2 < δ. 2.8 Hence, ‖J xn − q − J yn − q ‖ < 0; ‖J xn 1 − q − J xn − q ‖ < 0. Abstract and Applied Analysis 7and Applied Analysis 7 Using Lemma 1.6 and formulas above, we obtain ∥xn 1 − q ∥∥2 ≤ 1 − an − cn 2 ∥xn − q ∥∥2 2an 〈 Tyn − q, J ( xn 1 − q )〉 2cn 〈 un − q, J ( xn 1 − q )〉 ≤ 1 − an 2 ∥xn − q ∥∥2 2an 〈 Tyn − q, J ( xn 1 − q ) − Jxn − q )〉 2an 〈 Tyn − q, J ( xn − q ) − Jyn − q )〉 2an 〈 Tyn − q, J ( yn − q )〉 2cn 〈 un − q, J ( xn 1 − q )〉 ≤ 1 − an 2 ∥xn − q ∥∥2 2an ∥Tnyn − q ∥∥ · ∥Jxn 1 − q ) − Jxn − q )∥∥ 2an ∥Tnyn − q ∥∥ · ∥Jxn − q ) − Jyn − q )∥∥ 2an ∥∥yn − q ∥∥2 −Φ∥yn − q ∥∥)] 2cn ∥un − q ∥∥ · ∥xn 1 − q ∥∥ ≤ 1 − an R2 4anL 1 2R 0 2an ∥∥yn − q ∥∥2 −Φ∥yn − q ∥∥)] 4cnMR, 2.9 ∥yn − q ∥∥2 ≤ 1 − bn − dn 2 ∥xn − q ∥∥2 2bn 〈 Txn − q, J ( yn − q )〉 2dn 〈 wn − q, J ( yn − q )〉 ≤ ∥xn − q ∥∥2 2bn 〈 Txn − q, J ( yn − q ) − Jxn − q )〉 2bn 〈 Txn − q, J ( xn − q )〉 2dn ∥wn − q ∥∥ · ∥yn − q ∥∥ ≤ ∥xn − q ∥∥2 2bn ∥∥Tx n − q ∥∥ · ∥Jyn − q ) − Jxn − q )∥∥ 2bn ∥∥xn − q ∥∥2 −Φ∥xn − q ∥∥)] 2dn ∥wn − q ∥∥ · ∥yn − q ∥∥ ≤ R2 2bnL 1 R 0 2bnR 4dnMR. 2.10 Substitute 2.10 into 2.9 ∥xn 1 − q ∥∥2 ≤ 1 − an R2 4anL 1 2R 0 2an [ R2 2bnL 1 R 0 2bnR 4dnMR ] − 2anΦ ∥yn − q ∥∥) 4cnMR ≤ R2 anR 4anL 1 2R 0 2an [ 2bnL 1 R 0 2bnR 4dnMR ] − 2anΦ ( R 4 ) 4cnMR 8 Abstract and Applied Analysis R2 2an [ an 2 R2 2L 1 2R 0 2bnL 1 R 0 2bnR 4dnMR 2cnMR an ]