Convergence Theorems for a Maximal Monotone Operator and a V-Strongly Nonexpansive Mapping in a Banach Space

and Applied Analysis 3 Lemma 2.2 cf., 4 . Let D be a nonempty subset of a reflexive, strictly convex, and smooth Banach space E. Let R be a retraction from E onto D. Then R is sunny and generalized nonexpansive if and only if 〈 x − Rx, JRx − Jy ≥ 0, 2.2 for all x ∈ E and y ∈ D. A generalized resolvent Jr of a maximal monotone operator B ⊂ E∗ × E is defined by Jr I rBJ −1 for any real number r > 0. It is well known that Jr : E → E is single valued if E is reflexive, smooth, and strictly convex see 8 . It is also known that Jr satisfies 〈 x − Jrx − ( y − Jry ) , JJrx − JJry 〉 ≥ 0, ∀x, y ∈ E. 2.3 This implies that 〈 x − Jrx, JJrx − Jp 〉 ≥ 0, ∀x ∈ E, p ∈ F Jr . 2.4 Therefore, from Lemma 1.1 a , we obtain the following proposition. Proposition 2.3. a If a sunny retraction R is generalized nonexpansive, then R satisfies V x,Rx V ( Rx, y ) V ( x, y ) − 2x − Rx, JRx − Jy ≤ V x, y, ∀x, y ∈ D. 2.5 b For each r > 0, a generalized resolvent Jr satisfies V x, Jrx V ( Jrx, p ) ≤ V x, p, ∀x ∈ E, p ∈ F Jr . 2.6 Remark 2.4. The property in Proposition 2.3 b means that Jr is generalized nonexpansive for any r > 0. We recall some nonlinear mappings in Banach spaces see, e.g., 9–12 . Definition 2.5. Let D be a nonempty, closed, and convex subset of E. A mapping T : D → E is said to be firmly nonexpansive if ∥Tx − Ty∥2 ≤ x − y, jTx − Ty, 2.7 for all x, y ∈ D and some j Tx − Ty ∈ J Tx − Ty . 4 Abstract and Applied Analysis In 12 , Reich introduced a class of strongly nonexpansive mappings which is defined with respect to the Bregman distance D ·, · corresponding to a convex continuous function f in a reflexive Banach space E. Let S be a convex subset of E, and let T : S → S be a selfmapping of S. A point p in the closure of S is said to be an asymptotically fixed point of T if S contains a sequence {xn}which converges weakly to p and the sequence {xn−Txn} converges strongly to 0. F̂ T denotes the asymptotically fixed points set of T . Definition 2.6. The Bregman distance corresponding to a function f : E → R is defined by D ( x, y ) f x − fy − f ′yx − y, 2.8 where f is the Gâteaux differentiable and f ′ x stands for the derivative of f at the point x. We say that the mapping T is strongly nonexpansive if F̂ T / ∅ and D ( p, Tx ) ≤ Dp, x, ∀p ∈ F̂ T , x ∈ S, 2.9 and if it holds that limn→∞D Txn, xn 0 for a bounded sequence {xn} such that limn→∞ D p, xn −D p, Txn 0 for any p ∈ F̂ T . We remark that the symbols xn → u and xn ⇀ u mean that {xn} converges strongly and weakly to u, respectively. Taking the function ‖ · ‖ as the convex, continuous, and Gâteaux differentiable function f , we obtain the fact that the Bregman distance D ·, · coincides with V ·, · . Especially in a Hilbert space, D x, y V x, y ‖x − y‖. Bruck and Reich defined strongly nonexpansive mappings in a Hilbert space H as follows cf., 10 . Definition 2.7. A mapping T : D → H is said to be strongly nonexpansive if T is nonexpansive with F T / ∅ and if it holds that ( xn − yn ) − Txn − Tyn ) −→ 0 2.10 when {xn} and {yn} are sequences in D such that {xn − yn} is bounded and limn→∞ ‖xn − yn‖ − ‖Txn − Tyn‖ 0. The relation among firmly nonexpansive mappings, strongly nonexpansive mappings and V -strongly nonexpansive mappings is shown in the following proposition. Proposition 2.8. In a Hilbert space H, the following hold. a A firmly nonexpansive mapping is V -strongly nonexpansive with λ 1. b A V -strongly nonexpansive mapping T with F̂ T / ∅ is strongly nonexpansive. Abstract and Applied Analysis 5 Proof. a Suppose that T is firmly nonexpansive. Since J I in a Hilbert space, it holds thatand Applied Analysis 5 Proof. a Suppose that T is firmly nonexpansive. Since J I in a Hilbert space, it holds that 2 〈 x − y, Tx − Ty − ∥Tx − Ty∥2 ∥x − y∥2 − ∥ I − T x − I − T y∥2, 2.11 for all x, y ∈ D. Therefore, it is obvious that T is firmly nonexpansive if and only if T satisfies ∥ ∥x − y∥2 − ∥ T − I x − T − I y∥2 ≥ ∥Tx − Ty∥2, 2.12 for all x, y ∈ D. Hence we obtain a . b Suppose that T is V -strongly nonexpansive with λ. Then, it is trivial that T is nonexpansive and 2.9 holds. Suppose that the sequences {xn} and {yn} satisfy the conditions in Definition 2.7. Then {Txn − Tyn} is also bounded. Since T is V -strongly nonexpansive with λ, we have that 0 ≤ λ∥xn − yn − ( Txn − Tyn )∥2 λV ( I − T xn, I − T yn ) ≤ V xn, yn ) − V Txn, Tyn ) ∥xn − yn ∥2 − ∥Txn − Tyn ∥2 ∥xn − yn ∥ ∥Txn − Tyn ∥∥xn − yn ∥ − ∥Txn − Tyn ∥) −→ 0. 2.13 Hence, xn−yn − Txn−Tyn → 0 for λ > 0. This means that T is strongly nonexpansive. In a Banach space, V -strongly nonexpansive mappings have the following properties. Proposition 2.9. In a smooth Banach space E, the following hold. a For c ∈ −1, 1 , T cI is V -strongly nonexpansive. For c 1, T I is V -strongly nonexpansive for any λ > 0. For c ∈ −1, 1 , T cI is V -strongly nonexpansive for any λ ∈ 0, 1 c / 1 − c . b If T is V -strongly nonexpansive with λ, then, for any α ∈ −1, 1 with α/ 0, αT is also V -strongly nonexpansive with α2λ. c If T is V -strongly nonexpansive with λ ≥ 1, then A I − T is V -strongly nonexpansive with λ−1. d Suppose that T is V -strongly nonexpansive with λ and that α ∈ −1, 1 satisfies α2λ ≥ 1. Then I − αT is V -strongly nonexpansive with α2λ −1. Moreover, if Tα I − αT , then V ( Tαx, Tαy ) ≤ V x, y − λ−1V Tx, Ty. 2.14 6 Abstract and Applied Analysis Proof. a Let T cI for any c ∈ −1, 1 , and denote Il V Tx, Ty and Ir V x, y − λV I − T x, I − T y . Since J cx cJx, we have Il V ( Tx, Ty ) c2 { ‖x‖ ∥y∥2 − 2x, Jy } c2V ( x, y ) , Ir V ( x, y ) − λV ( I − T x, I − T y ‖x‖ − λ‖ 1 − c x‖ ∥y∥2 − λ∥ 1 − c y∥2 − 2x, Jy 2λ 1 − c x, J 1 − c y { 1 − λ 1 − c 2 }( ‖x‖ ∥y∥2 ) − 2 { 1 − λ 1 − c 2 }〈 x, Jy 〉 { 1 − λ 1 − c 2 }( ‖x‖ ∥y∥2 − 2x, Jy ) { 1 − λ 1 − c 2 } V ( x, y ) . 2.15 For c 1, it holds that Il ≤ Ir for all λ > 0. For c ∈ −1, 1 , we obtain Il ≤ Ir ⇐⇒ c2 ≤ 1 − λ 1 − c 2 ⇐⇒ 0 < λ 1 − c 2 ≤ 1 − c2 ⇐⇒ 0 < λ ≤ 1 − c 1 c 1 − c 2 1 c 1 − c . 2.16 Therefore, T cI is V -strongly nonexpansive for any λ ∈ 0, 1 c / 1 − c . b If T is V -strongly nonexpansive with λ > 0, then, for α ∈ −1, 1 with α/ 0, V ( αTx, αTy ) ‖αTx‖ ∥αTy∥2 − 2αTx, JαTy α2 { ‖Tx‖ ∥Ty∥2 − 2Tx, JTy }