Strong and Weak Convergence of Modified Mann Iteration for New Resolvents of Maximal Monotone Operators in Banach Spaces

and Applied Analysis 3 2. Preliminaries Let E be a real Banach space with dual space E∗. When {xn} is a sequence in E, we denote strong convergence of {xn} to x ∈ E by xn → x and weak convergence by xn ⇀ x, respectively. As usual, we denote the duality pairing of E∗ by 〈x, x∗〉, when x∗ ∈ E∗ and x ∈ E, and the closed unit ball by UE, and denote by R and N the set of all real numbers and the set of all positive integers, respectively. The set R stands for 0, ∞ and R R ∪ { ∞}. An operator T ⊂ E×E∗ is said to be monotone if 〈x−y, x∗ −y∗〉 ≥ 0 whenever x, x∗ , y, y∗ ∈ T . We denote the set {x ∈ E : 0 ∈ Tx} by T−10. A monotone T is said to be maximal if its graph G T { x, y : y ∈ Tx} is not properly contained in the graph of any other monotone operator. If T is maximal monotone, then the solution set T−10 is closed and convex. If E is reflexive and strictly convex, then a monotone operator T is maximal if and only if R J λT E∗ for each λ > 0 see 8, 9 for more details . The normalized duality mapping J from E into E∗ is defined by J x { x∗ ∈ E∗ : 〈x, x∗〉 ‖x‖ ‖x∗‖2 } , ∀x ∈ E. 2.1 We recall 10 that E is reflexive if and only if J is surjective; E is smooth if and only if J is single-valued. Let E be a smooth Banach space. Consider the following function: see 11 φ ( x, y ) ‖x‖ − 2x, Jy ∥y∥2, ∀x, y ∈ E. 2.2 It is obvious from the definition of φ that ‖x‖ − ‖y‖ 2 ≤ φ x, y ≤ ‖x‖ ‖y‖ , for all x, y ∈ E. We also know that φ ( x, y ) φ ( y, x ) 2 〈 x − y, Jx − Jy, for each x, y ∈ E. 2.3 We recall 12 that the functional ‖ · ‖ is called totally convex at x if the function ν x, t : 0,∞ → 0,∞ defined by ν x, t inf { φ ( y, x ) : y ∈ E,∥y − x∥ t, 2.4 is positive whenever t > 0. The functional ‖ · ‖ is called totally convex on bounded sets if for each bounded nonempty subset A of E, the function ν A, t : 0,∞ → 0,∞ defined by ν A, t inf{ν x, t : x ∈ A} is positive on 0,∞ . It is well known that if a Banach space E is uniformly convex, then ‖ · ‖ is totally convex on any bounded nonempty set. It is known that see 12 if ‖ · ‖ is totally convex on a bounded set A, then ν A, ct ≥ cν A, t for c ≥ 1 and t ≥ 0, and ν A, · is strictly increasing on 0,∞ . Lemma 2.1 see 13 . Let E be a uniformly convex, smooth Banach space, and let {xn} and {yn} be sequences in E. If {xn} or {yn} is bounded and limn→∞φ xn, yn 0, then limn→∞‖xn − yn‖ 0. 4 Abstract and Applied Analysis Let E be a reflexive, strictly convex, smooth Banach space, and J the duality mapping from E into E∗. Then J−1 is also single-valued, one-to-one, surjective, and it is the duality mapping from E∗ into E. We make use of the following mapping V studied in Alber 11 : V x, x∗ ‖x‖ − 2〈x, x∗〉 ‖x∗‖2 2.5 for all x ∈ E and x∗ ∈ E∗. In other words, V x, x∗ φ x, J−1 x for all x ∈ E and x∗ ∈ E∗. Lemma 2.2 see 7 . Let E be a reflexive, strictly convex, smooth Banach space, and let V be as in 2.5 . Then V x, x∗ 2〈y, Jx − x∗〉 ≤ V x y, x∗ 2.6 for all x, y ∈ E and x∗ ∈ E∗. Let E be a smooth Banach space and let D be a nonempty closed convex subset of E. A mapping R : D → D is called generalized nonexpansive if F R / ∅ and φ Rx, y ≤ φ x, y for each x ∈ D and y ∈ F R , where F R is the set of fixed points of R. Let C be a nonempty closed subset of E. A mapping R : E → C is said to be sunny if R Rx t x − Rx Rx, ∀x ∈ E, ∀t ≥ 0. 2.7 A mapping R : E → C is said to be a retraction if Rx x, for all x ∈ C. If E is smooth and strictly convex, then a sunny generalized nonexpansive retraction of E onto C is uniquely decided if it exists see 14 . We also know that if E is reflexive, smooth, and strictly convex and C is a nonempty closed subset of E, then there exists a sunny generalized nonexpansive retraction RC of E onto C if and only if J C is closed and convex. In this case, RC is given by RC J−1ΠJ C J see 15 . Let C be a nonempty closed subset of a Banach space E. Then C is said to be a sunny generalized nonexpansive retract resp., a generalized nonexpansive retract of E if there exists a sunny generalized nonexpansive retraction resp, a generalized nonexpansive retraction of E onto C see 14 for more detials . The set of fixed points of such a generalized nonexpansive retraction is C. The following lemma was obtained in 14 . Lemma 2.3 see 14 . Let C be a nonempty closed subset of a smooth and strictly convex Banach space E. Let RC be a retraction of E onto C. Then RC is sunny and generalized nonexpansive if and only if 〈x − RCx, JRCx − Jy〉 ≥ 0, 2.8 for each x ∈ E and y ∈ C, where J is the duality mapping of E. Let E be a reflexive, strictly convex, and smooth Banach space with its dual E∗. If a monotone operator B ⊂ E∗ × E is maximal, then BJ −10 is closed and E R I rBJ for all r > 0 see 14 . So, for each r > 0 and x ∈ E, we can consider the set Jr x {z ∈ E : x ∈ z rBJz}. From 14 , Jrx consists of one point. We denote such a Jr by I rBJ −1. Abstract and Applied Analysis 5 However Jr is called a generalized resolvent of B.We also know that BJ −10 F Jr for each r > 0, where F Jr is the set of fixed points of Jr and Jr is generalized nonexpansive for each r > 0 see 14 . The Yosida approximtion of B is defined by Ar I − Jr /r. We know that J Jrx,Arx ∈ B; see 14 for more detials . The following result was obtained in 14 . Theorem 2.4 see 14 . Let E be a uniformly convex Banach space with a Fréchet differentiable norm and let B ⊂ E∗ × E be a maximal monotone operator with B−10/ ∅. Then the following hold: 1 for each x ∈ E, limr→∞ Jrx exists and belongs to BJ −10, 2 if Rx : limr→∞ Jrx for each x ∈ E, then R is a sunny generalized nonexpansive retraction of E onto BJ −10. Lemma 2.5 see 7 . Let E be a reflexive, strictly convex, and smooth Banach space, let B ⊂ E∗ × E be a maximal monotone operator with B−10/ ∅, and Jr I rBJ −1 for all r > 0. Then φ x, Jrx φ Jrx, u ≤ φ x, u , 2.9and Applied Analysis 5 However Jr is called a generalized resolvent of B.We also know that BJ −10 F Jr for each r > 0, where F Jr is the set of fixed points of Jr and Jr is generalized nonexpansive for each r > 0 see 14 . The Yosida approximtion of B is defined by Ar I − Jr /r. We know that J Jrx,Arx ∈ B; see 14 for more detials . The following result was obtained in 14 . Theorem 2.4 see 14 . Let E be a uniformly convex Banach space with a Fréchet differentiable norm and let B ⊂ E∗ × E be a maximal monotone operator with B−10/ ∅. Then the following hold: 1 for each x ∈ E, limr→∞ Jrx exists and belongs to BJ −10, 2 if Rx : limr→∞ Jrx for each x ∈ E, then R is a sunny generalized nonexpansive retraction of E onto BJ −10. Lemma 2.5 see 7 . Let E be a reflexive, strictly convex, and smooth Banach space, let B ⊂ E∗ × E be a maximal monotone operator with B−10/ ∅, and Jr I rBJ −1 for all r > 0. Then φ x, Jrx φ Jrx, u ≤ φ x, u , 2.9 for all r > 0, u ∈ BJ −10, and x ∈ E. Lemma 2.6 see 16 . Let {sn} be a sequence of nonnegative real numbers satisfying sn 1 ≤ 1 − αn sn αntn rn, n ≥ 1, 2.10 where {αn}, {tn}, and {rn} satisfy the conditions: {αn} ⊂ 0, 1 , ∑∞ n 1 αn ∞, lim supn→∞tn ≤ 0, and rn ≥ 0, ∑∞ n 1 rn < ∞. Then, limn→∞sn 0. Lemma 2.7 see 17 . Let {αn} and {βn} be sequence of nonnegative real numbers satisfying αn 1 ≤ αn βn, 2.11 for all n ∈ N. If∞n 1 βn < ∞. Then {αn} has a limit in 0, ∞ . 3. Convergence Theorems In this section, we first prove a strong convergence theorem for the algorithm 1.7 which extends the previous result of Ibaraki and Takahashi 7 and we next prove a weak convergence theorem for algorithm 1.7 under different conditions on data, respectively. Theorem 3.1. Let E be a uniformly convex Banach space whose norm is uniformly Gâteaux differentiable. Let B ⊂ E∗ ×E be a maximal monotone operator with B−10/ ∅ and let Jr I rBJ −1 for all r > 0. Let {xn} be a sequence generated by x1 x ∈ E and yn αnxn 1 − αn Jrnxn, xn 1 βnx ( 1 − βn ) yn, 3.1 6 Abstract and Applied Analysis for every n 1, 2, . . . , where {αn}, {βn} ⊂ 0, 1 , {rn} ⊂ 0,∞ satisfy limn→∞αn 0, limn→∞βn 0, ∑∞ n 1 βn ∞ and limn→∞rn ∞. Then the sequence {xn} converges strongly to R BJ −10 x , where R BJ −10 is a sunny generalized nonexpansive retraction of E onto BJ −10. Proof. Note that B−10/ ∅ implies BJ −10/ ∅. In fact, if u∗ ∈ B−10, we obtain 0 ∈ Bu∗ and hence 0 ∈ BJ J−1u∗. So, we have J−1u∗ ∈ BJ −10. We denote a sunny generalized nonexpansive retraction R BJ −10 of E onto BJ −10 by R. Let z ∈ BJ −10. We first prove that {xn} is bounded. From Lemma 2.5 and the convexity of ‖ · ‖, we have φ ( yn, z ) φ αnxn 1 − αn Jrnxn, z ≤ αnφ xn, z 1 − αn φ Jrnxn, z ≤ αnφ xn, z 1 − αn { φ xn, z − φ xn, Jrnxn } ≤ αnφ xn, z 1 − αn φ xn, z φ xn, z , 3.2 for all n ∈ N. By 3.2 , we have φ xn 1, z φ ( βnx ( 1 − βn ) yn, z ) ≤ βnφ x, z ( 1 − βn ) φ ( yn, z ) ≤ βnφ x, z ( 1 − βn ) φ xn, z , 3.3 for all n ∈ N. Hence, by induction, we have φ xn, z ≤ φ x, z for all n ∈ N and, therefore, {φ xn, z } is bounded. This implies that {xn} is bounded. Since φ yn, z ≤ φ xn, z and φ Jrnxn, z ≤ φ xn, z for all n ∈ N, it follows that {yn} and {Jrnxn} are also bounded. We next prove that lim n→∞ sup〈x − Rx, Jxn − JRx〉 ≤ 0. 3.4 Put un xn 1 for all n ∈ N. Since {Jun} is bounded, without loss of generality, we have a subsequence {Juni} of {Jun} such that lim i→∞ 〈x − Rx, Juni − JRx〉 lim n→∞ sup〈x − Rx, Jun − JRx〉, 3.5 and {Juni} converges weakly to some v∗ ∈ E∗. From the definition of {xn}, we have un − yn βn ( x − yn ) , yn − Jrnxn αn xn − Jrnxn 3.6 for all n ∈ N. Since {yn} is bounded and βn → 0 as n → ∞, it follows that lim n→∞ ‖un − yn‖ lim n→∞ βn‖x − yn‖ 0. 3.7 Abstract and Applied Analysis 7 Moreover, we note that lim n→∞ ‖yn − Jrnxn‖ lim n→∞αn‖xn − Jrnxn‖ 0. 3.8and Applied Analysis 7 Moreover, we note that lim n→∞ ‖yn − Jrnxn‖ lim n→∞αn‖xn − Jrnxn‖ 0. 3.8 By 3.7 and 3.8 , we have lim n→∞ ‖un − Jrnxn‖ 0. 3.9 Since E has a uniformly Gâteaux differentiable norm, the duality mapping J is norm toweak∗ uniformly continuous on each bounded subset of E. Therefore, we obtain from 3.9 that Juni − JJrni xni ⇀ 0, as i −→ ∞. 3.10 This implies that J Jrni xni ⇀ v ∗ as i → ∞. On the other hand, from rn → ∞ as n → ∞, we have lim n→∞ ‖Arnxn‖ lim n→∞ 1 rn ‖xn − Jrnxn‖ 0. 3.11 If y∗, y ∈ B, then it holds from the monotonicity of B that 〈y −Arni xni , y∗ − JJrni xni〉 ≥ 0, 3.12 for all i ∈ N. Letting i → ∞, we get 〈y, y∗ −v∗〉 ≥ 0. Then, the maximal of B implies v∗ ∈ B−10. Put v J−1v∗. Applying Lemma 2.3, we obtain lim n→∞ sup〈x − Rx, Jun − JRx〉 lim i→∞ 〈x − Rx, Juni − JRx〉 〈x − Rx, v∗ − JRx〉 〈x − Rx, Jv − JRx〉 ≤ 0. 3.13 Finally, we prove that xn → Rx as n → ∞. From Lemma 2.2, the convexity of ‖ · ‖ and 3.2 , we have φ xn 1, Rx V ( βnx ( 1 − βn ) yn, JRx ) ≤ V βnx ( 1 − βn ) yn − βn x − Rx , JRx ) − 2〈−βn x − Rx , Jxn 1 − JRx〉 V ( βnRx ( 1 − βn ) yn, JRx ) 2βn〈x − Rx, Jxn 1 − JRx〉 φ ( βnRx ( 1 − βn ) yn, Rx ) 2βn〈x − Rx, Jxn 1 − JRx〉 ≤ βnφ Rx,Rx ( 1 − βn ) φ ( yn, Rx ) 2βn〈x − Rx, Jxn 1 − JRx〉 ≤ 1 − βn ) φ xn, Rx βnσn, 3.14 8 Abstract and Applied Analysis for all n ∈ N, where σn 2〈x − Rx, Jxn 1 − JRx〉. It easily verified from the assumption and 3.4 that ∑∞ n 1 βn ∞ and lim supn→∞σn ≤ 0. Hence, by Lemma 2.6, limn→∞φ xn, Rx 0. Applying Lemma 2.1, we obtain limn→∞‖xn −Rx‖ 0. Therefore, {xn} converges strongly to R BJ −10 x . Put αn ≡ 0 in Theorem 3.1, then we obtain the following result. Corollary 3.2 see Ibaraki and Takahashi 7 . Let E be a uniformly convex and uniformly smooth Banach space and let B ⊂ E∗ × E be a maximal monotone operator with B−10/ ∅, let Jr I rBJ −1 for all r > 0, and let {xn} be a sequence generated by x1 x ∈ E and xn 1 βnx ( 1 − βn ) Jrnxn, 3.15 for every n 1, 2, . . . , where {βn} ⊂ 0, 1 , {rn} ⊂ 0,∞ satisfy limn→∞βn 0, ∑∞ n 1 βn ∞ and limn→∞rn ∞. Then the sequence {xn} converges strongly to R BJ −10 x , where R BJ −10 is the generalized projection of E onto BJ −10. Theorem 3.3. Let E be a uniformly convex and smooth Banach space whose duality mapping J is weakly sequentially continuous. Let B ⊂ E∗ × E be a maximal monotone operator with B−10/ ∅ and let Jr I rBJ −1 for all r > 0. Let {xn} be a sequence generated by x1 x ∈ E and yn αnxn 1 − αn Jrnxn, xn 1 βnx ( 1 − βn ) yn, 3.16 for every n 1, 2, . . . , where {αn}, {βn} ⊂ 0, 1 , {rn} ⊂ 0,∞ satisfy ∑∞ n 1 βn < ∞, lim supn→∞αn < 1 and lim infn→∞rn > 0. Then the sequence {xn} converges weakly to an element of BJ −10. Proof. Let v ∈ BJ −10. Then, from 3.3 , we have φ xn 1, v ≤ ( 1 − βn ) φ xn, v βnφ x, v ≤ φ xn, v βnφ x, v , 3.17 for all n ∈ N. By Lemma 2.7, limn→∞φ xn, v exists. From ‖xn‖ − ‖v‖ 2 ≤ φ xn, v and φ Jrnxn, v ≤ φ xn, v , we note that {xn} and {Jrnxn} are bounded. From 3.3 and 3.2 , we have φ xn 1, v ≤ βnφ x, v ( 1 − βn ) φ ( yn, v ) ≤ βnφ x, v ( 1 − βn ){ αnφ xn, v 1 − αn ( φ xn, v − φ xn, Jrnxn )} βnφ x, v ( 1 − βn ){ φ xn, v − 1 − αn φ xn, Jrnxn } βnφ x, v ( 1 − βn ) φ xn, v − ( 1 − βn ) 1 − αn φ xn, Jrnxn , 3.18 Abstract and Applied Analysis 9 for all n ∈ N and hence, ( 1 − βn ) 1 − αn φ xn, Jrnxn ≤ βnφ x, v ( 1 − βn ) φ xn, v − φ xn 1, v βn ( φ x, v − φ xn, v ) φ xn, v − φ xn 1, v , 3.19and Applied Analysis 9 for all n ∈ N and hence, ( 1 − βn ) 1 − αn φ xn, Jrnxn ≤ βnφ x, v ( 1 − βn ) φ xn, v − φ xn 1, v βn ( φ x, v − φ xn, v ) φ xn, v − φ xn 1, v , 3.19 for all n ∈ N. Since limn→∞βn 0 and lim supn→∞αn < 1, limn→∞φ xn, Jrnxn 0. Applying Lemma 2.1, we obtain lim n→∞ ‖xn − Jrnxn‖ 0. 3.20 Since {xn} is bounded, we have a subsequence {xni} of {xn} such that xni ⇀ w ∈ E as i → ∞. Then it follows from 3.20 that Jrni xni ⇀ w as i → ∞. On the other hand, from 3.20 and lim infn→∞rn > 0, we have lim n→∞ ‖Arnxn‖ lim n→∞ 1 rn ‖xn − Jrnxn‖ 0. 3.21 Let z∗, z ∈ B. Then, it holds from monotonicity of B that 〈z −Arni xni , z∗ − JJrni xni〉 ≥ 0, 3.22 for all i ∈ N. Since J is weakly sequentially continuous, letting i → ∞, we get 〈z, z∗ −Jw〉 ≥ 0. Then, the maximality of B implies Jw ∈ B−10. Thus, w ∈ BJ −10. Let {xni} and {xnj} be two subsequences of {xn} such that xni ⇀ w1 and xnj ⇀ w2. By similar argument as above, we obtain w1, w2 ∈ BJ −10. Put a : limn→∞ φ xn,w1 − φ xn,w2 . Note that φ xn,w1 − φ xn,w2 2〈xn, Jw2 − Jw1〉 ‖w1‖ − ‖w2‖, n 1, 2, . . . . From xni ⇀ w1 and xnj ⇀ w2, we have a 2〈w1, Jw2 − Jw1〉 ‖w1‖ − ‖w2‖, 3.23 a 2〈w2, Jw2 − Jw1〉 ‖w1‖ − ‖w2‖, 3.24 respectively. Combining 3.23 and 3.24 , we have 〈w1 −w2, Jw1 − Jw2〉 0. 3.25 Since J is strictly monotone, it follows that w1 w2. Therefore, {xn} converges weakly to an element of BJ −10. Put βn ≡ 0 in Theorem 3.3, then we obtain the following result. Corollary 3.4 see Ibaraki and Takahashi 7 . Let E be a uniformly convex and smooth Banach space whose duality mapping J is weakly sequentially continuous. Let B ⊂ E∗ × E be a maximal 10 Abstract and Applied Analysis monotone operator with B−10/ ∅, let Jr I rBJ −1 for all r > 0 and let {xn} be a sequence generated by x1 x ∈ E and xn 1 βnxn ( 1 − βn ) Jrnxn, 3.26 for every n 1, 2, . . . , where {αn},{βn} ⊂ 0, 1 , {rn} ⊂ 0,∞ satisfy ∑∞ n 1 βn < ∞, lim supn→∞αn < 1 and lim infn→∞rn > 0. Then the sequence {xn} converges weakly to an element of BJ −10. 4. Rate of Convergence for the Algorithm In this section, we study the rate of the convergence of the algorithm 1.7 . We use the following notations in 6, 18 : N0 : { φ : R → R | t → φ t is nondecreasing for t ≥ 0, φ 0 0 } , Ω0 : { φ : R → R | φ 0 0, lim t→ 0 φ t 0 } , Γ0 : { φ : R → R | φ is lsc and convex and φ t 0 ⇐⇒ t 0 } , Σ1 : { φ : R → R | φ is lsc and convex, φ 0 0, lim t→ 0 t−1φ t 0 } . 4.1 We recall 18 that, for a function φ : R → R satisfying φ 0 0, its pseudoconjugate φ# : R → R , defined by φ# s : sup { st − φ t | t ≥ 0 ∈ R, 4.2 is lower semicontinuous, convex and satisfies φ# 0 0, φ# s ≥ 0 for all s ≥ 0. For a function φ ∈ N0, its greatest quasi-inverse φ : R → R , defined by φ s : sup { t ≥ 0 | φ t ≤ s, 4.3 is nondecreasing. It is known 18 that φ ∈ N0 ∩Ω0 if φ t 0 ⇔ t 0. For a function φ : R → R, its lower semicontinuous convex hull, denoted by coφ, is defined by epi coφ cl co epiφ . It is obvious that coφ is lower semicontinuous convex and coφ ≤ φ. Proposition 4.1. Let E be uniformly convex and uniformly smooth. Then, for every r > 0, there exists σr ∈ Σ1 such that, for all x, y ∈ rUE, 〈y − x, Jy − Jx〉 ≤ σr ‖Jy − Jx‖. 4.4 Abstract and Applied Analysis 11 Proof. Since E is uniformly convex, f x 1/2 ‖x‖ is uniform convex on rUE for all r > 0. Since the norm of E is Fréchet differentiable, its Fréchet derivative ∇f x Jx. In 18, Proposition 3.6.5 for f and B rUE, where r is an arbitrary positive real number, we get the function θr · : 0, ∞ → 0, ∞ , defined by θr t : inf { 1 2 ∥ ∥y ∥ ∥2 − 1 2 ‖x‖ − y − x, Jx : x ∈ rUE, y ∈ E, ‖y − x‖ t } , 4.5and Applied Analysis 11 Proof. Since E is uniformly convex, f x 1/2 ‖x‖ is uniform convex on rUE for all r > 0. Since the norm of E is Fréchet differentiable, its Fréchet derivative ∇f x Jx. In 18, Proposition 3.6.5 for f and B rUE, where r is an arbitrary positive real number, we get the function θr · : 0, ∞ → 0, ∞ , defined by θr t : inf { 1 2 ∥ ∥y ∥ ∥2 − 1 2 ‖x‖ − y − x, Jx : x ∈ rUE, y ∈ E, ‖y − x‖ t } , 4.5 satisfies that θr t 0 if and only if t 0, and t−1θr t is nondecreasing. Thus, 1 2 ∥ ∥y ∥ ∥2 − 1 2 ‖x‖ − 〈y − x, Jx〉 ≥ θr ‖y − x‖ 4.6 for all x ∈ rUE, y ∈ E and hence 1 2 ∥y x ∥2 ≥ 1 2 ‖x‖ y, Jx co θr ‖y‖ 4.7 for all x ∈ rUE, y ∈ E. It follows that 〈x, Jx〉 − 1 2 ‖x‖ 〈x, Jy − Jx〉 〈y, Jy − Jx〉 − coφθr ‖y‖ ≥ 〈y x, Jy〉 − 1 2 ∥y x ∥2 4.8 for all x ∈ rUE, y ∈ E. Since 〈x, Jx〉 1/2 ‖x‖ 1/2 ‖x‖, we have 1 2 ‖x‖ 〈y x, Jy − Jx〉 − coθr ‖y‖ ≥ 〈y x, Jy〉 − 1 2 ∥y x ∥2 4.9 for all x ∈ rUE and y ∈ E. Taking the supremum on both sides of 4.9 over y ∈ E, by 18, Lemma 3.3.1 v if f x : φ ‖x‖ , where φ ∈ N0, then f∗ x∗ φ# ‖x∗‖ , we get that 1 2 ‖x‖ 〈x, Jy − Jx〉 coθr # ‖Jy − Jx‖ ≥ 1 2 ∥y ∥2 4.10 for all x ∈ rUE and y ∈ E. Since θr t is nondecreasing and limt→∞t−1θr t ≥ θr 1 > 0, we have coθr ∈ Γ0. It follows from 20, Lemma 3.3.1 iii that coθr # ∈ Σ1. Interchanging x and y in 4.10 for x, y ∈ rUE, it also holds that 1 2 ∥y ∥2 〈y, Jx − Jy〉 coθr # ‖Jy − Jx‖ ≥ 1 2 ‖x‖. 4.11 Thus, by taking σr : 2 coθr # ∈ Σ1, and adding side by side 4.10 and 4.11 , we obtained 〈 y − x, Jy − Jx ≤ σr ‖Jy − Jx‖, ∀x, y ∈ rUE. 4.12 12 Abstract and Applied Analysis Theorem 4.2. Let E be a uniformly convex and uniformly smooth Banach space. Suppose that B ⊂ E∗ × E is maximal monotone with B−10 {v∗} and B−1 is Lipschitz continuous at 0 with modulus l ≥ 0. Let {xn} be a sequence generated by x1 x ∈ E and yn αnxn 1 − αn Jrnxn, xn 1 βnx ( 1 − βn ) yn, 4.13 for every n 1, 2, . . . , where {αn}, {βn} ⊂ 0, 1 , {rn} ⊂ 0,∞ satisfying limn→∞rn ∞. If either ∑∞ n 1 βn < ∞ and lim supn→∞αn < 1 or limn→∞βn 0, ∑∞ n 1 βn ∞ and limn→∞αn 0, then {xn} converges strongly to v : J−1v∗and φ xn, v converges to 0. Moreover, there exists an integerN > 0 such that φ xn 1, v ≤ τnφ x, v θn δn, ∀n ≥ N, 4.14 where τn βn ∑n−1 i N βi ∏n j i 1 1 − βj αj , θn ∏n i N 1 − βi αi, δn l 1 − αn /rn ∑n−1 i N 1 − αi /ri ∏n j i 1 1 − βj−1 αj and limn→∞τn limn→∞θn limn→∞δn 0. Also, one obtains φ xn 1, v ≤ βnφ x, v ( αn ◦ ( l rn )) k ( φ xn, v ) , 4.15 for all n ≥ N, where k t max{t, ν r t } ∈ N0 ∩ Ω0, and νr t : ν rUE, t , ν r is the greatest quasi-inverse of νr t , and r is a positive number such that {v} ∪ {xn} ∪ {Jrnxn} ⊂ rUE. Proof. Put v J−1v∗. Since B−10 {v∗}, we have BJ −10 {v}. We separate the proof into two cases. Case 1. ∑∞ n 1 βn < ∞, and lim supn→∞ αn < 1. According to Theorem 3.3, we have limn→∞φ xn, v exists, {xn} and {Jrn} are bounded, and hence {v} ∪ {xn} ∪ {Jrnxn} ⊂ rUE for some r > 0. Since B−1 is Lipschitz continuous at 0 with modulus l ≥ 0, for some τ > 0, we have ‖z∗ − v∗‖ ≤ l‖w‖ whenever z∗ ∈ B−1 w and ‖w‖ ≤ τ . Since rn → ∞, we may assume l/rn < 1 for all n ≥ 1. From Theorem 3.3, we have ‖xn − Jrnxn‖ → 0 and ‖Arnxn‖ → 0 as n → ∞. Hence, there exists an integer N > 0 such that ‖Arnxn‖ ≤ τ for all n ≥ N. Since J Jrnxn ∈ BArnxn, we have ‖JJrnxn − v∗‖ ≤ l‖Arnxn‖, ∀n ≥ N. 4.16 By ‖Jxn −v∗‖ ≤ ‖Jxn −JJrnxn‖ ‖JJrnxn −v∗‖ for all n ∈ N. Since J is uniformly continuous on each bounded set, 3.20 , 3.21 and 4.16 , we obtain limn→∞‖Jxn − v∗‖ 0. By the uniform smoothness of E∗, we have limn→∞‖xn − v‖ limn→∞‖J−1Jxn − J−1v∗‖ 0. Since φ xn, v φ v, xn 2〈xn − v, Jxn − Jv〉 2〈xn − v, Jxn − v∗〉 for all n ∈ N, we get φ xn, v ≤ 2〈xn − v, Jxn − v∗〉 ≤ 2‖xn − v‖‖Jxn − v∗‖ −→ 0, as n −→ ∞. 4.17 Abstract and Applied Analysis 13 Hence, φ xn, v → 0 as n → ∞. It follows from Proposition 4.1 and 4.16 that there exists σ t ∈ Σ1, which implies σ lt ≤ lσ t for all t ≥ 0 and l ∈ 0, 1 , such that φ Jrnxn, v ≤ φ Jrnxn, v φ v, Jrnxn 2〈Jrnxn − v, J Jrnxn − Jv〉 ≤ σ ‖J Jrnxn − Jv‖ 2σ ‖J Jrnxn − v∗‖ ≤ 2σ l‖Arnxn‖ 2σ ( l ∥ ∥ ∥ ∥ xn − Jrnxn rn ∥ ∥ ∥ ∥ ) ≤ 2l rn σ ‖xn − Jrnxn‖ 4.18and Applied Analysis 13 Hence, φ xn, v → 0 as n → ∞. It follows from Proposition 4.1 and 4.16 that there exists σ t ∈ Σ1, which implies σ lt ≤ lσ t for all t ≥ 0 and l ∈ 0, 1 , such that φ Jrnxn, v ≤ φ Jrnxn, v φ v, Jrnxn 2〈Jrnxn − v, J Jrnxn − Jv〉 ≤ σ ‖J Jrnxn − Jv‖ 2σ ‖J Jrnxn − v∗‖ ≤ 2σ l‖Arnxn‖ 2σ ( l ∥ ∥ ∥ ∥ xn − Jrnxn rn ∥ ∥ ∥ ∥ ) ≤ 2l rn σ ‖xn − Jrnxn‖ 4.18 for all n ≥ N. It follows from σ t ∈ Σ1 and 3.20 that lim n→∞ σ ‖xn − Jrnxn‖ 0. 4.19 From 3.3 , 3.2 , and 4.18 , we have φ xn 1, v ≤ βnφ x, v ( 1 − βn ){ αnφ xn, v 1 − αn φ Jrnxn, v } ≤ βnφ x, v ( 1 − βn ) αnφ xn, v ( 1 − βn ) 1 − αn 2l rn σ ‖xn − Jrnxn‖ 4.20 for all n ≥ N. Since φ xn, v → 0 and 4.19 , we may assume φ xn, v ≤ 1 and 2σ ‖xn − Jrnxn‖ ≤ 1 for all n ≥ N. By 4.20 and induction, we obtain φ xn 1, v ≤ ⎛ ⎝βn n−1 ∑