Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings

Mathematica Moravica Vol. 20:1 (2016), 125 144 Weak and strong convergence theorems of modified SP-iterations for generalized asymptotically quasi-nonexpansive mappings G.S. Saluja Abstract. The aim of this paper is to establish some strong and weak convergence theorems of modified SP-iterations for three generalized asymptotically quasi-nonexpansive mappings in the framework of Banach spaces. Our results extend and generalize several results from the current existing literature. 1. Introduction Let E be a Banach space and let C be a nonempty subset E. We denote the set of all fixed points of T by F (T ). The set of common fixed points of three mappings T 1, T 2 and T 3 will be denoted by F = 3 i=1 F (T i). Let T : C C be a nonlinear mapping. Definition 1.1. A mapping T : C C is said to be: (i) nonexpansive if (1) T x T y x y, x, y C; (ii) quasi-nonexpansive if (2) T x p x p, x C, p F (T ); (iii) asymptotically nonexpansive [4] if there exists a sequence {r n } in [0, ) with lim r n = 0 such that (3) T n x T n y (1 + r n ) x y, x, y C, n N; (iv) asymptotically quasi-nonexpansive [9] if F (T ) and there exists a sequence {r n } in [0, ) with lim r n = 0 such that (4) T n x p (1 + r n ) x p, x C, p F (T ), n N; 2000 Mathematics Subject Classification. 47H09, 47H10, 47J25. Key words and phrases. Generalized asymptotically quasi-nonexpansive mapping, modified SP-iteration process, common fixed point, Banach space, strong convergence, weak convergence. 125 c 2016 Mathematica Moravica

126 Weak and strong convergence theorems of modified SP-iterations... (v) generalized asymptotically quasi-nonexpansive [8, 11] if there exist two sequences {r n } and {s n } in [0, ) with lim r n = 0 and lim s n = 0 such that (5) T n x p (1 + r n ) x p + s n x T n x, (6) for all x C, p F (T ) and n N. From above definitions we have the following implications: let C 1 = {T : T : C C is a nonexpansive mapping}, C 2 = {T : T : C C is a quasi-nonexpansive mapping}, C 3 = {T : T : C C is an asymptotically nonexpansive mapping}, C 4 = {T : T : C C is an asymptotically quasi-nonexpansive mapping}, C 5 = {T : T : C C is generalized asymptotically then, we have the following (7) quasi-nonexpansive mapping}, C 1 C 2. C 1 C 3 C 4 C 5. Example 1.1. Let E = R and let T be defined by { x T (x) = 2 cos 1 x, if x 0, 0, if x = 0. If x 0 and T x = x, then x = x 2 cos 1 x. Thus 2 = cos 1 x. This is impossible. T is a quasi-nonexpansive mapping since if x E and z = 0, then T x z = T x 0 = x cos 1 x < x = x z = x z, 2 x 2 and hence T is an asymptotically quasi-nonexpansive mapping with constant sequence {k n } = {1}. Hence by implication relation (7), T is generalized asymptotically quasi-nonexpansive mapping. But it is not a nonexpansive mapping as T x T y x y is not satisfied for x = 2 3π and y = 1 π and hence is not asymptotically nonexpansive mapping. Thus the class of generalized asymptotically quasi-nonexpansive mappings is larger than the class of nonexpansive, quasi-nonexpansive, asymptotically nonexpansive and asymptotically quasi-nonexpansive mappings. (1) Modified S-iteration [1]: In 2007, Agarwal et al. [1] introduced the following iteration scheme: (8) x 1 = x C, x n+1 = (1 α n )T n x n + α n T n y n, y n = (1 β n )x n + β n T n x n, n 1

G.S. Saluja 127 where {α n } and {β n } are sequences in (0, 1) and they established some weak convergence theorems under additional conditions for nearly asymptotically nonexpansive self mappings in the framework of uniformly convex Banach spaces. (9) (2) Noor iteration [12]: Chose x 1 C and define z n = (1 γ n )x n + γ n T x n y n = (1 β n )x n + β n T z n x n+1 = (1 α n )x n + α n T y n, n 1, where {α n }, {β n } and {γ n } are sequences in [0,1]. (10) (3) Modified Noor iteration [26]: Chose x 1 C and define z n = (1 γ n )x n + γ n T n x n y n = (1 β n )x n + β n T n z n x n+1 = (1 α n )x n + α n T n y n, n 1, where {α n }, {β n } and {γ n } are sequences in [0,1]. Recently, Phuengrattana and Suantai [15] introduced the following iteration scheme. (11) (4) SP-iteration [15]: Chose x 1 C and define z n = (1 γ n )x n + γ n T x n y n = (1 β n )z n + β n T z n x n+1 = (1 α n )y n + α n T y n, n 1, where {α n }, {β n } and {γ n } are sequences in [0,1]. Inspired and motivated by [15], we modify iteration scheme (11) for three generalized asymptotically quasi-nonexpansive self mappings of C as follows: (12) (5) Modified SP-iteration: Chose x 1 C and define z n = (1 γ n )x n + γ n T n 3 x n y n = (1 β n )z n + β n T2 n z n x n+1 = (1 α n )y n + α n T1 n y n, n 1, where {α n }, {β n } and {γ n } are sequences in [0,1]. Remark 1.1. If we take T1 n = T 2 n = T 3 n = T for all n 1, then (12) reduces to the SP -iteration scheme (11) for generalized asymptotically quasinonexpansive self mapping of C.

128 Weak and strong convergence theorems of modified SP-iterations... The three-step iterative approximation problems were studied extensively by Noor [12, 13], Glowinsky and Le Tallec [5], and Haubruge et al [6]. It has been shown [5] that three-step iterative scheme gives better numerical results than the two step and one step approximate iterations. Thus we conclude that three step scheme plays an important and significant role in solving various problems, which arise in pure and applied sciences. Recently, many papers have appeared on the iterative approximation of fixed point and common fixed points of asymptotically quasi-nonexpansive mappings and generalized asymptotically quasi-nonexpansive mappings throu gh various iteration schemes in Banach spaces (see, e.g., [2, 7, 9, 16, 17, 18, 19, 20, 23]). The purpose of this paper is to establish a necessary and sufficient condition for {x n } generated by (12) to converge to common fixed point for three generalized asymptotically quasi-nonexpansive mappings. Also we establish some strong convergence theorems of {x n } and the weak convergence theorems in uniformly convex Banach spaces, which satisfies the Opial property, or whose dual space has the Kadec-Klee property (KK-property). In fact, a dual space of a reflexive Banach space with a Fréchet differentiable norm or the Opial property also satisfies the Kadec-Klee property [24]. Our results extend and generalize the previous works from the current existing literature. 2. Preliminaries For our main results, we shall need the following definitions and lemmas. Let E be a Banach space with its dimension greater than or equal to 2. The modulus of convexity of E is the function δ E (ε): (0, 2] [0, 1] defined by δ E (ε) = inf {1 1 } 2 (x + y) : x = 1, y = 1, ε = x y. A Banach space E is uniformly convex if and only if δ E (ε) > 0 for all ε (0, 2]. Let S = {x E : x = 1} and let E be the dual of E, that is, the space of all continuous linear functionals f on E. Definition 2.1. (i) Opial condition: The space E has Opial condition [14] if for any sequence {x n } in E, x n converges to x weakly it follows that lim sup x n x < lim sup x n y for all y E with y x. Examples of Banach spaces satisfying Opial condition are Hilbert spaces and all spaces l p (1 < p < ). On the other hand, L p [0, 2π] with 1 < p 2 fail to satisfy Opial condition. (ii) A mapping T : C C is said to be demiclosed at zero, if for any sequence {x n } in K, the condition x n converges weakly to x C and T x n

G.S. Saluja 129 converges strongly to 0 imply T x = 0. (iii) A Banach space E has the Kadec-Klee property [24] if for every sequence {x n } in E, x n x weakly and x n x it follows that x n x 0. Definition 2.2. Condition (A): The mapping T : C C with F (T ) is said to satisfy condition (A) [22] if there is a nondecreasing function f : [0, ) [0, ) with f(0) = 0, f(t) > 0 for all t (0, ) such that x T x f(d(x, F (T ))) for all x C, where d(x, F (T )) = inf{ x p : p F (T )}. Now, we modify Condition (A) for three mappings. Definition 2.3. Condition (B): Three mappings T 1, T 2, T 3 : C C are said to satisfy condition (B) if there is a nondecreasing function f : [0, ) [0, ) with f(0) = 0, f(t) > 0 for all t (0, ) such that a 1 x T 1 x + a 2 x T 2 x +a 3 T 3 x f(d(x, F )) for all x C, where d(x, F ) = inf{ x p : p F = 3 i=1 F (T i)}, where a 1, a 2 and a 3 are nonnegative real numbers such that a 1 + a 2 + a 3 = 1. Note that condition (B) reduces to condition (A) when T 1 = T 2 = T 3 = T and hence is more general than the demicompactness of T 1, T 2 and T 3 [22]. A mapping T : C C is called: (1) demicompact if any bounded sequence {x n } in C such that {x n T x n } converges has a convergent subsequence; (2) semicompact (or hemicompact) if any bounded sequence {x n } in C such that {x n T x n } 0 as n has a convergent subsequence. Every demicompact mapping is semicompact but the converse is not true in general. Senter and Dotson [22] have approximated fixed points of a nonexpansive mapping T by Mann iterates whereas Maiti and Ghosh [10] and Tan and Xu [25] have approximated the fixed points using Ishikawa iterates under the condition (A) of [22]. Tan and Xu [25] pointed out that condition (A) is weaker than the compactness of C. We shall use condition (B) instead of compactness of C to study the strong convergence of {x n } defined by iteration scheme (13). Lemma 2.1. (See [25]) Let {α n } n=1, {β n} n=1 and {r n} n=1 be sequences of nonnegative numbers satisfying the inequality α n+1 (1 + β n )α n + r n, n 1. If n=1 β n < and n=1 r n <, then (i) lim α n exists; (ii) In particular, if {α n } n=1 has a subsequence which converges strongly to zero, then lim α n = 0. Lemma 2.2. (See [21]) Let E be a uniformly convex Banach space and 0 < α t n β < 1 for all n N. Suppose further that {x n } and {y n } are

130 Weak and strong convergence theorems of modified SP-iterations... sequences of E such that lim sup x n a, lim sup y n a and lim t n x n +(1 t n )y n = a hold for some a 0. Then lim x n y n = 0. Lemma 2.3. (See [24]) Let E be a real reflexive Banach space with its dual E has the Kadec-Klee property. Let {x n } be a bounded sequence in E and p, q w w (x n ) (where w w (x n ) denotes the set of all weak subsequential limits of {x n }). Suppose lim tx n + (1 t)p q exists for all t [0, 1]. Then p = q. Lemma 2.4. (See [24]) Let K be a nonempty convex subset of a uniformly convex Banach space E. Then there exists a strictly increasing continuous convex function φ: [0, ) [0, ) with φ(0) = 0 such that for each Lipschitzian mapping T : C C with the Lipschitz constant L, tt x + (1 t)t y T (tx + (1 t)y) Lφ 1( x y 1 L T x T y ) for all x, y K and all t [0, 1]. Proposition 2.1. Let C be a nonempty subset of a Banach space E and T 1, T 2, T 3 : C C be three generalized asymptotically quasi-nonexpansive mappings. Then there exist nonnegative real sequences {r n } and {s n } in [0, ) with r n 0 and s n 0 such that (13) T n 1 x p (1 + r n ) x p + s n x T n x, (14) (15) and T n 2 x p (1 + r n ) x p + s n x T n x, T n 3 x p (1 + r n ) x p + s n x T n x, for all x C, p F (T ) and n 1. Proof. Since T 1, T 2, T 3 : C C are three generalized asymptotically quasinonexpansive mappings, there exist nonnegative real sequences {r n1 }, {r n2 }, {r n3 }, {s n1 }, {s n2 } and {s n3 } in [0, ) with r ni 0 and s ni 0 as n for all i = 1, 2, 3 such that (16) T n 1 x p (1 + r n1 ) x p + s n1 x T n x, (17) and (18) T n 2 x p (1 + r n2 ) x p + s n2 x T n x, T n 3 x p (1 + r n3 ) x p + s n3 x T n x, for all x C, p F (T ) and n 1. Setting r n = max{r ni : i = 1, 2, 3}, s n = max{s ni : i = 1, 2, 3}

G.S. Saluja 131 then we get that there exist nonnegative real sequences {r n } and {s n } with r n 0 and s n 0 as n such that T n 1 x p (1 + r n1 ) x p + s n1 x T n x (1 + r n ) x p + s n x T n x, T2 n x p (1 + r n2 ) x p + s n2 x T n x (1 + r n ) x p + s n x T n x, and T3 n x p (1 + r n3 ) x p + s n3 x T n x (1 + r n ) x p + s n x T n x, for all x C, p F (T ) and n 1. 3. Strong Convergence Theorems In this section, we prove some strong convergence theorems of iteration scheme (12) for three generalized asymptotically quasi-nonexpansive mappings in a Banach space. First, we shall need the following lemmas. Lemma 3.1. Let E be a Banach space and let C be a nonempty closed convex subset of E. Let T 1, T 2, T 3 : C C be three generalized asymptotically quasi-nonexpansive mappings with sequences {r n } and {s n } as defined in proposition 2.1 such that n=1 rn+2sn < and F = 3 i=1 F (T i). Let {x n } be the iteration scheme defined by (12). Then lim x n q and lim d(x n, F ) both exist for all q F. Proof. Let q F, then from (12), we have (19) Now, we have z n q = (1 γ n )x n + γ n T n 3 x n q = (1 γ n )(x n q) + γ n (T n 3 x n q) (1 γ n ) x n q + γ n T n 3 x n q (1 γ n ) x n q + γ n [(1 + r n ) x n q +s n x n T n 3 x n ] (1 + r n )[(1 γ n ) x n q + γ n x n q ] +s n x n T n 3 x n (1 + r n ) x n q + s n x n T n 3 x n. x n T n 3 x n x n q + T n 3 x n q x n q + (1 + r n ) x n q + s n x n T n 3 x n = (2 + r n ) x n q + s n x n T n 3 x n

132 Weak and strong convergence theorems of modified SP-iterations... which implies that (20) x n T n 3 x n Now using (20) in (19), we have (21) ( 2 + rn ) x n q. ( 2 + rn ) z n q (1 + r n ) x n q + s n x n q [ ( 2 + rn )] = (1 + r n ) + s n x n q ( 1 + rn + s ) n = x n q. Again from (12), we have (22) Now, we have y n q = (1 β n )z n + β n T n 2 z n q = (1 β n )(z n q) + β n (T n 2 z n q) (1 β n ) z n q + β n T n 2 z n q (1 β n ) z n q + β n [(1 + r n ) z n q +s n z n T n 2 z n ] (1 + r n )[(1 β n ) z n q + β n z n q ] +s n z n T n 2 z n (1 + r n ) z n q + s n z n T n 2 z n. z n T n 2 z n z n q + T n 2 z n q which implies that (23) z n q + (1 + r n ) z n q + s n z n T n 2 z n = (2 + r n ) z n q + s n z n T n 2 z n z n T n 2 z n Using (23) in (22), we have (24) ( 2 + rn ) z n q. ( 2 + rn ) y n q (1 + r n ) z n q + s n z n q [ ( 2 + rn )] = (1 + r n ) + s n z n q ( 1 + rn + s ) n = z n q. Now substituting (21) in (24), we get ( 1 + rn + s )( n 1 + rn + s ) n y n q x n q

G.S. Saluja 133 (25) = Finally, from (13), we have (26) Now, we have ( 1 + rn + s n ) 2 xn q. x n+1 q = (1 α n )y n + α n T n 1 y n q = (1 α n )(y n q) + α n (T n 1 y n q) (1 α n ) y n q + α n T n 1 y n q (1 α n ) y n q + α n [(1 + r n ) y n q +s n y n T n 1 y n ] (1 + r n )[(1 α n ) y n q + α n y n q ] +s n y n T n 1 y n (1 + r n ) y n q + s n y n T n 1 y n. y n T n 1 y n y n q + T n 1 y n q which implies that (27) y n q + (1 + r n ) y n q + s n y n T n 1 y n = (2 + r n ) y n q + s n y n T n 1 y n y n T n 1 y n Substituting (27) in (26), we obtain (28) ( 2 + rn ) y n q. ( 2 + rn ) x n+1 q (1 + r n ) y n q + s n y n q [ ( 2 + rn )] = (1 + r n ) + s n y n q ( 1 + rn + s ) n = y n q. Now substituting (25) in (28), we get ( 1 + rn + s )( n 1 + rn + s ) n 2 xn x n+1 q q ( 1 + rn + s ) n 3 xn = q [ ( rn + 2s )] n 3 xn = 1 + q (29) = (1 + h n ) 3 x n q (1 + Qh n ) x n q where h n = rn+2sn and for some Q > 0. Since by hypothesis n=1 rn+2sn, it follows that n=1 h n <. <

134 Weak and strong convergence theorems of modified SP-iterations... (30) For any q F, from (29), we obtain the following inequality d(x n+1, F ) (1 + Qh n )d(x n, F ). Applying Lemma 2.1 in (29) and (30), we have lim x n q and d(x n, F ) both exist. This completes the proof. Lemma 3.2. Let E be a uniformly convex Banach space and let C be a nonempty closed convex subset of E. Let T 1, T 2, T 3 : C C be three uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive mappings with sequences {r n } and {s n } as defined in proposition 2.1 such that n=1 rn+2sn < and F = 3 i=1 F (T i). Let {x n } be the iteration scheme defined by (12), where {α n }, {β n } and {γ n } are sequences in [ρ, 1 ρ] for all n N and for some ρ (0, 1). Then lim x n T i x n = 0 for all i = 1, 2, 3. Proof. By Lemma 3.1, lim x n q exists for all q F, so we can assume that lim x n q = c. Then c > 0 otherwise there is nothing to prove. (31) and (32) Also Now (21) and (25) implies that and so (33) Since lim sup z n p c lim sup y n p c. T n 1 y n p (1 + r n ) y n p + s n y n T n 1 y n lim sup T1 n y n p c. c = x n+1 p = (1 α n )(y n p) + α n (T n 1 y n p). It follows from Lemma 2.2 that (34) Again note that lim T n 1 y n y n = 0. T n 3 x n p (1 + r n ) x n p + s n x n T n 3 x n T n 2 z n p (1 + r n ) z n p + s n z n T n 2 z n. Hence, from above inequalities, we obtain (35) lim sup T3 n x n p c

G.S. Saluja 135 and (36) Further, note that lim sup T2 n z n p c. y n p y n T n 1 y n + T n 1 y n p It follows from (32) and (34) that y n T n 1 y n + (1 + r n ) y n p + s n y n T n 1 y n. (37) c lim inf y n p. From (32) and (37), we get (38) Now, we have lim y n p = c. (39) c = lim y n p = (1 β n )(z n p) + β n (T n 2 z n p). It follows from (31), (36) and Lemma 2.2 that (40) Again note that lim T n 2 z n z n = 0. z n p z n T n 2 z n + T n 2 z n p It follows from (31) and (40) that z n T n 2 z n + (1 + r n ) z n p + s n z n T n 2 z n. (41) c lim inf z n p. From (31) and (41), we get (42) Now, we see that lim z n p = c. (43) c = lim z n p = (1 γ n )(x n p) + γ n (T n 3 x n p). It follows from Lemma 2.2 that (44) Again note that (45) Using (43) in (45), we get (46) Further, note that lim T n 3 x n x n = 0. x n z n = γ n x n T n 3 x n (1 ρ) x n T n 3 x n. lim x n z n = 0. x n y n = β n z n T n 2 z n

136 Weak and strong convergence theorems of modified SP-iterations... (47) Using (40) in (47), we get (48) Note that (49) (1 ρ) z n T n 2 z n. lim x n y n = 0. x n T n 2 z n x n z n + z n T n 2 z n. Using (40) and (46) in (49), we get (50) lim x n T n 2 z n = 0. Since T 2 is uniformly L-Lipschitzian, we have (51) x n T n 2 x n x n T n 2 z n + T n 2 z n T n 2 x n Using (46) and (50) in (51), we get (52) Again notice that (53) x n T n 2 z n + L z n x n. lim x n T n 2 x n = 0. x n T n 1 y n x n y n + y n T n 1 y n. Using (34) and (48) in (53), we get (54) lim x n T n 1 y n = 0. Since T 1 is uniformly L-Lipschitzian, we obtain (55) x n T n 1 x n x n T n 1 y n + T n 1 x n T n 1 y n Using (48) and (54) in (55), we get x n T n 1 y n + L x n y n. (56) lim x n T n 1 x n = 0. By the definitions of x n+1, we have (57) x n x n+1 x n y n + T n 1 y n y n. Using (34) and (48) in (57), we get (58) lim x n x n+1 = 0. By (56), (57) and since T 1 is uniformly L-Lipschitzian, we have (59) x n T 1 x n x n x n+1 + x n+1 T1 n+1 x n+1 + T1 n+1 x n+1 T1 n+1 x n + T1 n+1 x n T 1 x n x n x n+1 + x n+1 T1 n+1 x n+1 +L x n+1 x n + L T1 n x n x n = (1 + L) x n x n+1 + x n+1 T1 n+1 x n+1 +L T1 n x n x n 0 as n.

G.S. Saluja 137 Similarly, we can prove that (60) This completes the proof. x n T 2 x n = 0 and x n T 3 x n = 0. Theorem 3.1. Let E be a real Banach space and C be a nonempty closed convex subset of E. Let T 1, T 2, T 3 : C C be three generalized asymptotically quasi-nonexpansive mappings with sequences {r n } and {s n } as defined in proposition 2.1 such that n=1 rn+2sn < and F = 3 i=1 F (T i) is closed. Let {x n } be the iteration scheme defined by (12), where {α n }, {β n } and {γ n } are sequences in [ρ, 1 ρ] for all n N and for some ρ (0, 1). Then {x n } converges strongly to a common fixed point of the mappings T 1, T 2 and T 3 if and only if lim inf d(x n, F ) = 0, where d(x, F ) = inf{ x p : p F } Proof. Necessity is obvious. Conversely, suppose that lim inf d(x n, F ) = 0. As proved in Lemma 3.1, for all q F, lim d(x n, F ) exists. Thus by the hypothesis lim d(x n, F ) = 0. Now, we show that {x n } is a Cauchy sequence in C. From (29), we know that (61) for some Q > 0 with n=1 h n <. x n+1 q (1 + Qh n ) x n q = x n q + Q h n, Now, given ε > 0, since lim d(x n, F ) = 0 and n=1 h n <, there exists a natural number n 1 > 0 such that for all n n 1, d(x n, F ) < ε 5 and j=1 h j < ε 4Q. So, we get d(x n1, F ) < ε 4 and j=n 1 h j < ε 4Q. This means that there exists a q 1 F such that x n1 q 1 ε 4. Hence, for all integers n n 1 and m 1, we obtain from (61) that x n+m x n x n+m q 1 + x n q 1 n+m 1 x n1 q 1 + Q x n1 q 1 + Q j=n 1 h j + x n1 q 1 + Q h j + x n1 q 1 + Q j=n 1 = 2 ( x ) n1 q 1 + Q ε < 2( 4 + ε ) Q. 4Q = ε. j=n 1 h j n+m 1 j=n 1 j=n 1 h j This proves that {x n } is a Cauchy sequence in C. Thus, the completeness of E implies that {x n } must be convergent. Assume that lim x n = p. We will prove that p is a common fixed point of T 1, T 2 and T 3, that is, we will h j

138 Weak and strong convergence theorems of modified SP-iterations... show that p F = 3 i=1 F (T i). Since C is closed, therefore p C. Next, we show that p F. Now lim d(x n, F ) = 0 gives that d(p, F ) = 0. Since F is closed, p F. Thus, p is a common fixed point of the mappings T 1, T 2 and T 3. This completes the proof. We deduce the following result as corollary from Theorem 3.1 as follows. Corollary 3.1. Let E be a real Banach space and C be a nonempty closed convex subset of E. Let T 1, T 2, T 3 : C C be three generalized asymptotically quasi-nonexpansive mappings with sequences {r n } and {s n } as defined in proposition 2.1 such that n=1 rn+2sn < and F = 3 i=1 F (T i) is closed. Let {x n } be the iteration scheme defined by (12), where {α n }, {β n } and {γ n } are sequences in [ρ, 1 ρ] for all n N and for some ρ (0, 1). Then {x n } converges strongly to a point p F if and only if there exists some subsequence {x nj } of {x n } which converges to a point p F. Theorem 3.2. Let E be a real Banach space and C be a nonempty closed convex subset of E. Let T 1, T 2, T 3 : C C be three generalized asymptotically quasi-nonexpansive mappings with sequences {r n } and {s n } as defined in proposition 2.1 such that n=1 rn+2sn < and F = 3 i=1 F (T i). Let {x n } be the iteration scheme defined by (12), where {α n }, {β n } and {γ n } are sequences in [ρ, 1 ρ] for all n N and for some ρ (0, 1). Then lim inf d(x n, F ) = lim sup d(x n, F ) = 0 if {x n } converges to a unique point in F. Proof. Let p F. Since {x n } converges to p, lim d(x n, p) = 0. So, for a given ε > 0, there exists n 1 N such that d(x n, p) < ε for all n n 1. Taking the infimum over p F, we obtain that d(x n, F ) < ε for all n n 1. This means that lim d(x n, F ) = 0. Thus we obtain that This completes the proof. lim inf d(x n, F ) = lim sup d(x n, F ) = 0. As an application of Theorem 3.1, we establish some strong convergence results as follows. Theorem 3.3. Let E be a real Banach space and C be a nonempty closed convex subset of E. Let T 1, T 2, T 3 : C C be three uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive mappings with sequences {r n } and {s n } as defined in proposition 2.1 such that n=1 rn+2sn < and F = 3 i=1 F (T i). Let {x n } be the iteration scheme defined by (12), where {α n }, {β n } and {γ n } are sequences in [ρ, 1 ρ] for all n N and for some ρ (0, 1). If one of the mappings in {T i : i = 1, 2, 3} is demicompact, then

G.S. Saluja 139 {x n } converges strongly to a common fixed point of the mappings T 1, T 2 and T 3. Proof. Without loss of generality, we can assume that T 1 is demicompact. It follows from equation (59) in Lemma 3.2 that lim x n T 1 x n = 0 and {x n } is bounded, by demicompactness of T 1, there exists a subsequence {x nk } of {x n } that converges strongly to some q C as k. From equation (59) in Lemma 3.2 we have lim k x n k T 1 x nk = q T 1 q = 0. This implies that q F (T 1 ). Similarly, we can prove that q F (T 2 ) and q F (T 3 ). Thus, we obtain that q F = 3 i=1 F (T i). It follows from Lemma 3.1 and Theorem 3.1 that {x n } must converges strongly to a common fixed point of the mappings T 1, T 2 and T 3. This completes the proof. Theorem 3.4. Let E be a real Banach space and C be a nonempty closed convex subset of E. Let T 1, T 2, T 3 : C C be three uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive mappings with sequences < and F = 3 i=1 F (T i). Let {x n } be the iteration scheme defined by (12), where {α n }, {β n } and {γ n } are sequences in [ρ, 1 ρ] for all n N and for some ρ (0, 1). If T 1, T 2 and T 3 satisfy condition (B), then {x n } converges strongly to a common fixed point of the mappings T 1, T 2 and T 3. {r n } and {s n } as defined in proposition 2.1 such that n=1 rn+2sn Proof. By Lemma 3.2, we know that (62) lim x n T i x n = 0, for i = 1, 2, 3. From condition (B) and (62), we get f(d(x n, F ) a 1. x n T 1 x n + a 2. x n T 2 x n + a 3. x n T 3 x n = 0, that is, f(d(x n, F ) = 0. Since f : [0, ) [0, ) is a nondecreasing function satisfying f(0) = 0, f(t) > 0 for all t (0, ), therefore we obtain lim d(x n, F ) = 0. Therefore, Theorem 3.1 implies that {x n } converges strongly to a common fixed point of the mappings T 1, T 2 and T 3. This completes the proof. 4. Weak Convergence Theorems In this section, we prove some weak convergence theorems of iteration scheme (12) for three generalized asymptotically quasi-nonexpansive mappings in a uniformly convex Banach space such that either it satisfies the Opial property or its dual space has the Kadec-Klee property (KK-property).

140 Weak and strong convergence theorems of modified SP-iterations... Theorem 4.1. Let E be a uniformly convex Banach space satisfying Opial s condition and C be a nonempty closed convex subset of E. Let T 1, T 2, T 3 : C C be three uniformly L-Lipschitzian and generalized asymptotically quasinonexpansive mappings with sequences {r n } and {s n } as defined in proposition 2.1 such that n=1 rn+2sn < and F = 3 i=1 F (T i). Let {x n } be the iteration scheme defined by (12), where {α n }, {β n } and {γ n } are sequences in [ρ, 1 ρ] for all n N and for some ρ (0, 1). If the mappings I T i for all i = 1, 2, 3, where I denotes the identity mapping, are demiclosed at zero, then {x n } converges weakly to a common fixed point of the mappings T 1, T 2 and T 3. Proof. Let q F, from Lemma 3.1 the sequence { x n q } is convergent and hence bounded. Since E is uniformly convex, every bounded subset of E is weakly compact. Thus there exists a subsequence {x nk } {x n } such that {x nk } converges weakly to q C. From Lemma 3.2, we have lim x n k T i x nk = 0, for all i = 1, 2, 3. k Since the mappings I T i for all i = 1, 2, 3 are demiclosed at zero, therefore T i q = q for all i = 1, 2, 3, which means q F. Finally, let us prove that {x n } converges weakly to q. Suppose on contrary that there is a subsequence {x nj } {x n } such that {x nj } converges weakly to p C and q p. Then by the same method as given above, we can also prove that p F. From Lemma 3.1 the limits lim x n q and lim x n p exist. By virtue of the Opial condition of E, we obtain lim x n q = lim x n n k k q < lim n k x n k p = lim x n p = lim n j x n j p < lim n j x n j q = lim x n q which is a contradiction, so q = p. Thus {x n } converges weakly to a common fixed point of the mappings T 1, T 2 and T 3. This completes the proof. Lemma 4.1. Under the conditions of Lemma 3.2 and for any p, q F, lim tx n + (1 t)p q exists for all t [0, 1]. Proof. By Lemma 3.1, lim x n q exists for all q F and therefore {x n } is bounded. Letting a n (t) = tx n + (1 t)p q

G.S. Saluja 141 for all t [0, 1]. Then lim a n (0) = p q and lim a n (1) = x n q exists by Lemma 3.1. It, therefore, remains to prove the Lemma 4.1 for t (0, 1). For all x C, we define the mapping G n : C C by: and A n (x) = (1 γ n )x + γ n T n 3 x B n (x) = (1 β n )A n (x) + β n T n 2 A n (x) G n (x) = (1 α n )B n (x) + α n T n 1 B n (x). Then it follows that z n = A n x n, y n = B n x n, x n+1 = G n x n and G n p = p for all p F. Now from (21), (25) and (29) of Lemma 3.1, we see that ( 1 + rn + s ) n A n (x) A n (y) x y and (63) B n (x) B n (y) ( 1 + rn + s n ) 2 x y G n (x) G n (y) (1 + Qh n ) x n q = l n x y, for some Q > 0 and for all x, y C, where l n = 1 + Qh n with n=1 h n < and l n 1 as n. Setting (64) and U n, m = G n+m 1 G n+m 2... G n, m 1 b n, m = U n, m (tx n + (1 t)p) (tu n, m x n + (1 t)u n,m q). From (63) and (64), we have (65) (66) U n, m (x) U n, m (y) = G n+m 1 G n+m 2... G n (x) G n+m 1 G n+m 2... G n (y). l n+m 1 G n+m 2... G n (x) G n+m 2... G n (y) l n+m 1 l n+m 2 G n+m 3... G n (x) G n+m 3... G n (y) ( n+m 1 j=n = J n x y l j ) x y for all x, y C, where J n = n+m 1 j=n l j and U n, m x n = x n+m, U n, m p = p for all p F. Thus a n+m (t) = tx n+m + (1 t)p q b n, m + U n, m (tx n + (1 t)p) q

142 Weak and strong convergence theorems of modified SP-iterations... (67) b n, m + J n a n (t). By using [ [3], Theorem 2.3], we have b n,m ϕ 1 ( x n u U n,m x n U n,m u ) ϕ 1 ( x n u x n+m u + u U n,m u ) ϕ 1 ( x n u ( x n+m u U n,m u u )) and so the sequence {b n,m } converges uniformly to 0, i.e., b n,m 0 as n. Since lim J n = 1, therefore from (67), we have lim sup a n (t) lim b n,m + lim inf a n(t) = lim inf a n(t). n,m This shows that lim a n (t) exists, that is, lim tx n + (1 t)p q exists for all t [0, 1]. This completes the proof. Theorem 4.2. Let E be a real uniformly convex Banach space such that its dual E has the Kadec-Klee property and C be a nonempty closed convex subset of E. Let T 1, T 2, T 3 : C C be three uniformly L-Lipschitzian and generalized asymptotically quasi-nonexpansive mappings with sequences {r n } < and F = 3 i=1 F (T i). Let {x n } be the iteration scheme defined by (12), where {α n }, {β n } and {γ n } are sequences in [ρ, 1 ρ] for all n N and for some ρ (0, 1). If the mappings I T i for all i = 1, 2, 3, where I denotes the identity mapping, are demiclosed at zero, then {x n } converges weakly to a common fixed point of the mappings S and T. and {s n } as defined in proposition 2.1 such that n=1 rn+2sn Proof. By Lemma 3.1, {x n } is bounded and since E is reflexive, there exists a subsequence {x nj } of {x n } which converges weakly to some p C. By Lemma 3.2, we have lim j x n j T i x nj = 0 for all 1, 2, 3. Since by hypothesis the mappings I T i for all i = 1, 2, 3 are demiclosed at zero, therefore T i p = p for all i = 1, 2, 3, which means p F. Now, we show that {x n } converges weakly to p. Suppose {x ni } is another subsequence of {x n } converges weakly to some q C. By the same method as above, we have q F and p, q w w (x n ). By Lemma 4.1, the limit lim tx n + (1 t)p q exists for all t [0, 1] and so p = q by Lemma 2.3. Thus, the sequence {x n } converges weakly to p F. This completes the proof. Example 4.1. Let E = R with the usual norm. and C = [0, ). Let T 1, T 2, T 3 : E E be defined by T 1 (x) = x 3, T 2(x) = x cosx and T 3 (x) = x 2 sinx. Then T 1, T 2 and T 3 are asymptotically quasi-nonexpansive mappings with constant sequence {k n } = {1}. Hence by implication relation (7), they

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