Convergence theorems for mixed type asymptotically nonexpansive mappings in the intermediate sense

Transcription

1 Available online at J. Nonlinear Sci. Appl ), Research Article Convergence theorems for mixed type asymptotically nonexpansive mappings in the intermediate sense Gurucharan S. Saluja a, Mihai Postolache b,c,, Adrian Ghiura c a Department of Mathematics, Govt. N. P. G. College of Science, Raipur C.G.), India. b China Medical University, No.91, Hsueh-Shih Road, Taichung, Taiwan. c Department of Mathematics & Informatics, University Politehnica of Bucharest, Splaiul Independentei 313, Bucharest , Romania. Communicated by R. Saadati Abstract We provide a new two-step iteration scheme of mixed type for two asymptotically nonexpansive self mappings in the intermediate sense two asymptotically nonexpansive non-self mappings in the intermediate sense establish some strong weak convergence theorems for mentioned scheme mappings in uniformly convex Banach spaces. Our results extend generalize the corresponding results of Chidume et al. [C. E. Chidume, E. U. Ofoedu, H. Zegeye, J. Math. Anal. Appl., ), ] [C. E. Chidume, N. Shahzad, H. Zegeye, Numer. Funct. Anal. Optim., ), ], Guo et al. [W. Guo, W. Guo, Appl. Math. Lett., ), ] [W. Guo, Y. J. Cho, W. Guo, Fixed Point Theory Appl., ), 15 pages], Saluja [G. S. Saluja, J. Indian Math. Soc. N.S.), ), ], Schu [J. Schu, Bull. Austral. Math. Soc., ), ], Tan Xu [K. K. Tan, H. K. Xu, J. Math. Anal. Appl., ), ], Wang [L. Wang, J. Math. Anal. Appl., ), ], Wei Guo [S. I. Wei, W. Guo, Commun. Math. Res., ), ] [S. Wei, W. Guo, J. Math. Study, ), ]. c 2016 All rights reserved. Keywords: Asymptotically nonexpansive self non-self mapping in intermediate sense, new two-step iteration scheme of mixed type, common fixed point, uniformly convex Banach space, strong convergence, weak convergence MSC: 47H09, 47H10, 65J15, 47J Introduction preinaries Let K be a nonempty subset of a real Banach space E. Let T : K K be a nonlinear mapping, then Corresponding author addresses: saluja1963@gmail.com Gurucharan S. Saluja), mihai@mathem.pub.ro Mihai Postolache), adrianghiura25@gmail.com Adrian Ghiura) Received

2 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), we denote the set of all fixed points of T by F T ). The set of common fixed points of four mappings S 1, S 2, T 1 T 2 will be denoted by F = F S 1 ) F S 2 ) F T 1 ) F T 2 ). A mapping T : K K is said to be asymptotically nonexpansive in the intermediate sense [1] if it is continuous the following inequality holds: sup sup x,y K T n x) T n y) x y ) 0. From the above definition, it follows that an asymptotically nonexpansive mapping must be an asymptotically nonexpansive mapping in the intermediate sense. But the converse does not hold as the following example shows. Example 1.1. Let E = R be a normed linear space K = [0, 1]. For each x K, we define { kx, if x 0, T x) = 0, if x = 0, where 0 < k < 1. Then T n x T n y = k n x y x y for all x, y K n N. Thus T is an asymptotically nonexpansive mapping with constant sequence {1} sup{ T n x T n y x y } = sup{k n x y x y } 0, because k n = 0 as 0 < k < 1, for all x, y K, n N T is continuous. asymptotically nonexpansive mapping in the intermediate sense. Example 1.2. Let E = R, K = [ 1 π, 1 π ] λ < 1. For each x K, consider Hence T is an Clearly F T ) = {0}. Since T x = λ x sin1/x), T x) = { λ x sin1/x), if x 0, 0, if x = 0. T 2 x = λ 2 x sin1/x) sin we obtain {T n x} 0 uniformly on K as n. Thus sup { T n x T n y x y 0 1 λ x sin1/x) } = 0 ),..., for all x, y K. Hence T is an asymptotically nonexpansive mapping in the intermediate sense ANI in short), but it is not a Lipschitz mapping. In fact, suppose that there exists λ > 0 such that T x T y λ x y for all x, y K. If we take x = 2 5π y = 2 3π, then T x T y = λ 2 5π sin 5π 2 ) λ 2 3π sin 3π 2 whereas λ x y = λ 2 5π 2 = 4λ 3π 15π, hence it is not an asymptotically nonexpansive mapping. ) = 16λ 15π, Definition 1.3. A subset K of a Banach space E is said to be a retract of E if there exists a continuous mapping P : E K called a retraction) such that P x) = x for all x K. If, in addition P is nonexpansive, then P is said to be a nonexpansive retract of E.

3 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), If P : E K is a retraction, then P 2 = P. A retract of a Hausdorff space must be a closed subset. Every closed convex subset of a uniformly convex Banach space is a retract. In 2004, Chidume et al. [3] introduced the concept of non-self asymptotically nonexpansive mappings in the intermediate sense as follows. Definition 1.4. Let K be a nonempty subset of a real Banach space E let P : E K be a nonexpansive retraction of E onto K. A non-self mapping T : K E is said to be asymptotically nonexpansive in the intermediate sense if T is uniformly continuous sup sup x,y K T P T ) n 1 x) T P T ) n 1 y) x y ) 0. For the sake of convenience, we restate the following concepts results. 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 only if δ E ε) > 0 for all ε 0, 2]. Let S = {x E : x = 1} let E be the dual of E, that is, the space of all continuous linear functionals f on E. The space E has: i) Gâteaux differentiable norm if exists for each x y in S. x + ty x t 0 t ii) Fréchet differentiable norm [11] if for each x in S, the above it exists is attained uniformly for y in S in this case, it is also well-known that h, Jx) x x + h 2 1.1) h, Jx) x 2 + b x ) for all x, h E, where J is the Fréchet derivative of the functional at x E, is the pairing between E E, b is an increasing function defined on [0, ) such that t 0 bt) t = 0. iii) Opial condition [7] if for any sequence {x n } in E, x n converges to x weakly it follows that sup x n x < sup x n y for all y E with y x. Examples of Banach spaces satisfying Opial condition are Hilbert spaces all spaces l p 1 < p < ). On the other h, L p [0, 2π] with 1 < p 2 fails to satisfy Opial condition. A mapping T : K K is said to be demiclosed at zero, if for any sequence {x n } in K, the condition x n converges weakly to x K T x n converges strongly to 0 imply T x = 0. A mapping T : K K is said to be completely continuous if {T x n } has a convergent subsequence in K whenever {x n } is bounded in K. A Banach space E has the Kadec-Klee property [10] if for every sequence {x n } in E, x n x weakly x n x it follows that x n x 0. In 2003, Chidume et al. [2] studied the following iteration process for non-self asymptotically nonexpansive mappings: x 1 = x K, x n+1 = P 1 α n )x n + α n T P T ) n 1 x n ), n 1, 1.2) where {α n } is a sequence in 0, 1) proved some strong weak convergence theorems in the framework

4 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), of uniformly convex Banach spaces. In 2004, Chidume et al. [3] studied the following iteration scheme: x 1 = x K, x n+1 = P 1 α n )x n + α n T P T ) n 1 x n ), n 1, 1.3) where {α n } is a sequence in 0, 1), K is a nonempty closed convex subset of a real uniformly convex Banach space E, P is a nonexpansive retraction of E onto K, proved some strong weak convergence theorems for asymptotically nonexpansive non-self mappings in the intermediate sense in the framework of uniformly convex Banach spaces. In 2006, Wang [13] generalized the iteration process 1.2) as follows: x 1 = x K, x n+1 = P 1 α n )x n + α n T 1 P T 1 ) n 1 y n ), y n = P 1 β n )x n + β n T 2 P T 2 ) n 1 x n ), n 1, where T 1, T 2 : K E are two asymptotically nonexpansive non-self mappings {α n }, {β n } are real sequences in [0, 1), proved some strong weak convergence theorems for asymptotically nonexpansive non-self mappings. In 2012, Guo et al. [5] generalized the iteration process 1.3) as follows: x 1 = x K, x n+1 = P 1 α n )S n 1 x n + α n T 1 P T 1 ) n 1 y n ), y n = P 1 β n )S n 2 x n + β n T 2 P T 2 ) n 1 x n ), n 1, where S 1, S 2 : K K are two asymptotically nonexpansive self mappings T 1, T 2 : K E are two asymptotically nonexpansive non-self-mappings {α n }, {β n } are real sequences in [0, 1), proved some strong weak convergence theorems for mixed type asymptotically nonexpansive mappings. Recently, Wei Guo [15] studied the following. Let E be a real Banach space, K a nonempty closed convex subset of E P : E K a nonexpansive retraction of E onto K. Let S 1, S 2 : K K be two asymptotically nonexpansive self mappings T 1, T 2 : K E two asymptotically nonexpansive non-self-mappings. Then Wei Guo [15] defined the new iteration scheme of mixed type with mean errors as follows: x 1 = x K, 1.4) x n+1 = P α n S n 1 x n + β n T 1 P T 1 ) n 1 y n + γ n u n ), 1.5) y n = P α ns n 2 x n + β nt 2 P T 2 ) n 1 x n + γ nu n), n 1, where {u n }, {u n} are bounded sequences in E, {α n }, {β n }, {γ n }, {α n}, {β n}, {γ n} are real sequences in [0, 1) satisfying α n + β n + γ n = 1 = α n + β n + γ n for all n 1, proved some weak convergence theorems in the setting of real uniformly convex Banach spaces. If γ n = γ n = 0, for all n 1, then the iteration scheme 1.5) reduces to the scheme 1.4). The purpose of this paper is to study iteration scheme 1.5) for the mixed type asymptotically nonexpansive mappings in the intermediate sense which is more general than the class of asymptotically nonexpansive mappings in uniformly convex Banach spaces establish some strong weak convergence theorems for mentioned scheme mappings. Next we state the following useful lemmas to prove our main results. Lemma 1.5 [12]). Let {α n } n=1, {β n} n=1 {r n} n=1 be sequences of nonnegative numbers satisfying the inequality α n β n )α n + r n, n 1. If n=1 β n < n=1 r n <, then

5 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), i) α n exists; ii) in particular, if {α n } n=1 has a subsequence which converges strongly to zero, then α n = 0. Lemma 1.6 [9]). Let E be a uniformly convex Banach space 0 < α t n β < 1 for all n N. Suppose further that {x n } {y n } are sequences of E such that sup x n a, sup y n a t n x n + 1 t n )y n = a hold for some a 0. Then x n y n = 0. Lemma 1.7 [10]). 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 p, q w w x n ) where w w x n ) denotes the set of all weak subsequential its of {x n }). Suppose tx n + 1 t)p q exists for all t [0, 1]. Then p = q. Lemma 1.8 [10]). 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 φ0) = 0 such that for each Lipschitzian mapping T : K K with the Lipschitz constant L, for all x, y K all t [0, 1]. tt x + 1 t)t y T tx + 1 t)y) Lφ 1 x y 1 L T x T y ) 2. Strong convergence theorems In this section, we prove some strong convergence theorems of iteration scheme 1.5) for the mixed type asymptotically nonexpansive mappings in the intermediate sense in real uniformly convex Banach spaces. First, we shall need the following lemma. Lemma 2.1. Let E be a real uniformly convex Banach space, K a nonempty closed convex subset of E. Let S 1, S 2 : K K be two asymptotically nonexpansive self-mappings in the intermediate sense T 1, T 2 : K E two asymptotically nonexpansive non-self-mappings in the intermediate sense. Put G n = max { 0, sup S1 n x S1 n y x y x, y K, n 1 ) )}, sup S2 n x S2 n y x y, 2.1) x, y K, n 1 { ) H n = max 0, sup x, y K, n 1 T 1 P T 1 ) n 1 x) T 1 P T 1 ) n 1 y) x y, 2.2) sup x, y K, n 1 )} T 2 P T 2 ) n 1 x) T 2 P T 2 ) n 1 y), such that n=1 G n < n=1 H n <. Let {x n } be the sequence defined by 1.5), where {u n }, {u n} are bounded sequences in E, {α n }, {β n }, {γ n }, {α n}, {β n}, {γ n} are real sequences in [0, 1) satisfying α n + β n + γ n = 1 = α n + β n + γ n for all n 1, n=1 γ n < n=1 γ n <. Assume that F = F S 1 ) F S 2 ) F T 1 ) F T 2 ). Then x n q dx n, F ) both exist for any q F. Proof. Let q F. From 1.5), 2.1) 2.2), we have y n q = P α ns n 2 x n + β nt 2 P T 2 ) n 1 x n + γ nu n) P q) α ns n 2 x n + β nt 2 P T 2 ) n 1 x n + γ nu n q α n S n 2 x n q + β n T 2 P T 2 ) n 1 x n q + γ n u n q 2.3) α n[ x n q + G n ] + β n[ x n q + H n ] + γ n u n q α n + β n) x n q + G n + H n + γ n u n q

6 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), = 1 γ n) x n q + G n + H n + γ n u n q x n q + A n, where A n = G n +H n +γ n u n q. Since by assumptions n=1 G n <, n=1 H n < n=1 γ n <, it follows that n=1 A n <. Again from 1.5), 2.1) 2.2), we have x n+1 q = P α n S n 1 x n + β n T 1 P T 1 ) n 1 y n + γ n u n ) P q) By using equation 2.3) in 2.4), we obtain α n S n 1 x n + β n T 1 P T 1 ) n 1 y n + γ n u n q α n S n 1 x n q + β n T 1 P T 1 ) n 1 y n q + γ n u n q 2.4) α n [ x n q + G n ] + β n [ y n q + H n ] + γ n u n q α n x n q + G n + H n + β n y n q + γ n u n q. x n+1 q α n x n q + β n [ x n q + A n ] + G n + H n + γ n u n q = α n + β n ) x n q + β n A n + G n + H n + γ n u n q 2.5) = 1 γ n ) x n q + β n A n + G n + H n + γ n u n q x n q + B n, where B n = β n A n +G n +H n +γ n u n q. Since by hypothesis n=1 G n <, n=1 H n <, n=1 γ n < n=1 A n <, it follows that n=1 B n <. For any q F, from 2.5), we obtain the following inequality dx n+1, F ) dx n, F ) + B n. 2.6) By applying Lemma 1.5 in 2.5) 2.6), we have x n q dx n, F ) both exist. This completes the proof. Lemma 2.2. Let E be a real uniformly convex Banach space, K a nonempty closed convex subset of E. Let S 1, S 2 : K K be two asymptotically nonexpansive self-mappings in the intermediate sense T 1, T 2 : K E two asymptotically nonexpansive non-self-mappings in the intermediate sense G n H n be taken as in Lemma 2.1. Assume that F = F S 1 ) F S 2 ) F T 1 ) F T 2 ). Let {x n } be the sequence defined by 1.5), where {u n }, {u n} are bounded sequences in E, {α n }, {β n }, {γ n }, {α n}, {β n}, {γ n} are real sequences in [0, 1) satisfying α n + β n + γ n = 1 = α n + β n + γ n for all n 1, n=1 γ n < n=1 γ n <. If the following conditions hold: i) {β n } {β n} are real sequences in [ρ, 1 ρ] for all n 1 for some ρ 0, 1), ii) x T i P T i ) n 1 y S n i x T ip T i ) n 1 y for all x, y K i = 1, 2, then x n S i x n = x n T i x n = 0 for i = 1, 2. Proof. By Lemma 2.1, x n q exists for all q F therefore {x n } is bounded. Thus, there exists a real number r > 0 such that {x n } K = B r 0) K, so that K is a closed convex subset of K. Let x n q = c. Then c > 0 otherwise there is nothing to prove. Now 2.3) implies that Also, we have sup y n q c. 2.7) S n 2 x n q x n q + G n, n 1,

7 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), T 2 P T 2 ) n 1 x n q x n q + H n, n 1, Hence Next, gives by virtue of 2.6) that Also, it follows from S n 1 x n q x n q + G n, n 1. sup S2 n x n q c, sup T 2 P T 2 ) n 1 x n q c, sup S1 n x n q c. 2.8) T 1 P T 1 ) n 1 y n q y n q + H n, sup T 1 P T 1 ) n 1 y n q c. c = x n+1 q = α ns n 1 x n + β n T 1 P T 1 ) n 1 y n + γ n u n q = α n[s n 1 x n q + γ n 2α n u n q)] + β n [T 1 P T 1 ) n 1 y n q + γ n 2β n u n q)] = α n[s n 1 x n q + γ n 2α n u n q)] + 1 α n )[T 1 P T 1 ) n 1 y n q + γ n 2β n u n q)], Lemma 1.6 that Sn 1 x n T 1 P T 1 ) n 1 γn y n + γ ) n u n q) = 0. 2α n 2β n Since γn 2α n γn 2β n )u n q) = 0, we obtain that By condition ii), it follows that Sn 1 x n T 1 P T 1 ) n 1 y n = ) x n T 1 P T 1 ) n 1 y n S n 1 x n T 1 P T 1 ) n 1 y n, so, from 2.9), we have Now x n T 1 P T 1 ) n 1 y n = ) x n+1 q = α n S n 1 x n + β n T 1 P T 1 ) n 1 y n + γ n u n q = α n S n 1 x n q) + β n T 1 P T 1 ) n 1 y n S n 1 x n ) + γ n u n S n 1 x n ) α n S n 1 x n q + β n T 1 P T 1 ) n 1 y n S n 1 x n + γ n u n S n 1 x n,

8 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), yields that so that 2.8) gives On the other h, c inf Sn 1 x n q, Sn 1 x n q = c. S n 1 x n q S n 1 x n T 1 P T 1 ) n 1 y n + T 1 P T 1 ) n 1 y n q S n 1 x n T 1 P T 1 ) n 1 y n + y n q + H n, so, we have By using 2.7) 2.11), we obtain c inf y n q. 2.11) y n q = c. Thus Lemma 1.6 implies that c = y n q = α ns n 2 x n + β nt 2 P T 2 ) n 1 x n + γ nu n q = α n[s2 n x n q + γ n 2α n u n q)] + β n[t 2 P T 2 ) n 1 x n q + γ n 2β n u n q)] = α n[s2 n x n q + γ n 2α n u n q)] + 1 α n)[t 2 P T 2 ) n 1 x n q + γ n 2β n u n q)], γ Sn 2 x n T 2 P T 2 ) n 1 x n + n 2α n ) Since γ n 2α γ n n 2β u n n q) = 0, we obtain that By condition ii), it follows that so, from 2.12), we have γ ) n 2β n u n q) = 0. Sn 2 x n T 2 P T 2 ) n 1 x n = ) x n T 2 P T 2 ) n 1 x n S n 2 x n T 2 P T 2 ) n 1 x n, x n T 2 P T 2 ) n 1 x n = ) Since S n 2 x n = P S n 2 x n) P : E K is a nonexpansive retraction of E onto K, we have y n S n 2 x n = α ns n 2 x n + β nt 2 P T 2 ) n 1 x n + γ nu n S n 2 x n = 1 β n γ n)s n 2 x n + β nt 2 P T 2 ) n 1 x n + γ nu n S n 2 x n

9 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), β n S n 2 x n T 2 P T 2 ) n 1 x n + γ n u n S n 2 x n, so Again, we have y n S n 2 x n = ) y n x n y n S n 2 x n + S n 2 x n T 2 P T 2 ) n 1 x n + T 2 P T 2 ) n 1 x n x n. Thus, it follows from 2.12), 2.13) 2.14) that y n x n = ) Since x n T 1 P T 1 ) n 1 x n S n 1 x n T 1 P T 1 ) n 1 y n by condition ii) S n 1 x n T 1 P T 1 ) n 1 x n S n 1 x n T 1 P T 1 ) n 1 y n By using 2.9), 2.15) H n 0 as n, we have + T 1 P T 1 ) n 1 y n T 1 P T 1 ) n 1 x n S n 1 x n T 1 P T 1 ) n 1 y n + y n x n + H n. Sn 1 x n T 1 P T 1 ) n 1 x n = 0, 2.16) It follows from x n T 1 P T 1 ) n 1 x n = ) x n+1 S n 1 x n = P α n S n 1 x n + β n T 1 P T 1 ) n 1 y n + γ n u n ) P S n 1 x n ) α n S n 1 x n + β n T 1 P T 1 ) n 1 y n + γ n u n S n 1 x n β n S n 1 x n T 1 P T 1 ) n 1 y n + γ n u n S n 1 x n, 2.9) that In addition, we have x n+1 S n 1 x n = ) x n+1 T 1 P T 1 ) n 1 y n x n+1 S n 1 x n + S n 1 x n T 1 P T 1 ) n 1 y n. By using 2.9) 2.18), we have Now, by using 2.16), 2.17) the inequality x n+1 T 1 P T 1 ) n 1 y n = ) S n 1 x n x n S n 1 x n T 1 P T 1 ) n 1 x n + T 1 P T 1 ) n 1 x n x n, we have S n 1 x n x n = 0. It follows from 2.13) the inequality S n 1 x n T 2 P T 2 ) n 1 x n S n 1 x n x n + x n T 2 P T 2 ) n 1 x n, that Sn 1 x n T 2 P T 2 ) n 1 x n = )

10 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), Since x n+1 T 2 P T 2 ) n 1 y n x n+1 S n 1 x n + S n 1 x n T 2 P T 2 ) n 1 x n + x n y n + H n, from 2.15), 2.18), 2.20) H n 0 as n, it follows that x n+1 T 2 P T 2 ) n 1 y n = ) Since T i for i = 1, 2 is uniformly continuous, P is nonexpansive retraction, it follows from 2.21) that T i P T i ) n 1 y n 1 T i x n = T i [P T i )P T ) n 2 )y n 1 ] T i P x n ) 0 as n 2.22) for i = 1, 2. Moreover, we have x n+1 y n x n+1 T 1 P T 1 ) n 1 y n + T 1 P T 1 ) n 1 y n x n + x n y n. By using 2.10), 2.15) 2.19), we have In addition, we have x n+1 y n = ) x n T 1 x n x n T 1 P T 1 ) n 1 x n + T 1 P T 1 ) n 1 x n T 1 P T 1 ) n 1 y n 1 + T 1 P T 1 ) n 1 y n 1 T 1 x n x n T 1 P T 1 ) n 1 x n + x n y n 1 + H n + T 1 P T 1 ) n 1 y n 1 T 1 x n. Thus, it follows from 2.17), 2.22), 2.23) H n 0 as n, that x n T 1 x n = 0. Similarly, we can prove that Finally, we have x n T 2 x n = 0. x n S 1 x n x n T 1 P T 1 ) n 1 x n + S 1 x n T 1 P T 1 ) n 1 x n x n T 1 P T 1 ) n 1 x n + S n 1 x n T 1 P T 1 ) n 1 x n by condition ii)). Thus, it follows from 2.16) 2.17) that x n S 1 x n = 0. Similarly, we can prove that This completes the proof. x n S 2 x n = 0. Theorem 2.3. Under the assumptions of Lemma 2.2, if one of S 1, S 2, T 1 T 2 is completely continuous, then the sequence {x n } defined by 1.5) converges strongly to a common fixed point of the mappings S 1, S 2, T 1 T 2.

11 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), Proof. Without loss of generality we can assume that S 1 is completely continuous. Since {x n } is bounded by Lemma 2.1, there exists a subsequence {S 1 x nk } of {S 1 x n } such that {S 1 x nk } converges strongly to some q 1. Moreover, by definition of complete continuity from Lemma 2.2, we have which implies that x n k S 1 x nk = x n k S 2 x nk = 0, k k x n k T 1 x nk = x n k T 2 x nk = 0, k k x nk q 1 x nk S 1 x nk + S 1 x nk q 1 0, as k so x nk q 1 K. Thus, by the continuity of S 1, S 2, T 1 T 2, we have q 1 S i q 1 = k x n k S i x nk = 0, q 1 T i q 1 = k x n k T i x nk = 0 for i = 1, 2. Thus it follows that q 1 F S 1 ) F S 2 ) F T 1 ) F T 2 ). Again, since x n q 1 exists by Lemma 2.1, we have x n q 1 = 0. This shows that the sequence {x n } converges strongly to a common fixed point of the mappings S 1, S 2, T 1 T 2. This completes the proof. For our next result, we need the following definition. A mapping T : K K is said to be semi-compact if for any bounded sequence {x n } in K such that x n T x n 0 as n, then there exists a subsequence {x nr } {x n } such that x nr x K strongly as r. Theorem 2.4. Under the assumptions of Lemma 2.2, if one of S 1, S 2, T 1 T 2 is semi-compact, then the sequence {x n } defined by 1.5) converges strongly to a common fixed point of the mappings S 1, S 2, T 1 T 2. Proof. Since we know that from Lemma 2.2, x n S i x n = x n T i x n = 0 for i = 1, 2 one of S 1, S 2, T 1 T 2 is semi-compact, there exists a subsequence {x nj } of {x n } such that {x nj } converges strongly to some q K. Moreover, by the continuity of S 1, S 2, T 1 T 2, we have q S i q = j x nj S i x nj = 0 q T i q = j x nj T i x nj = 0 for i = 1, 2. Thus it follows that q F S 1 ) F S 2 ) F T 1 ) F T 2 ). Since x n q exists by Lemma 2.1, we have x n q = 0. This shows that the sequence {x n } converges strongly to a common fixed point of the mappings S 1, S 2, T 1 T 2. This completes the proof. Theorem 2.5. Under the assumptions of Lemma 2.2, if there exists a continuous function f : [0, ) [0, ) with f0) = 0 ft) > 0 for all t 0, ) such that fdx, F )) a 1 x S 1 x + a 2 x S 2 x + a 3 x T 1 x + a 4 x T 2 x for all x K, where F = F S 1 ) F S 2 ) F T 1 ) F T 2 ) a 1, a 2, a 3, a 4 are nonnegative real numbers such that a 1 + a 2 + a 3 + a 4 = 1, then the sequence {x n } defined by 1.5) converges strongly to a common fixed point of the mappings S 1, S 2, T 1 T 2. Proof. From Lemma 2.2, we know that x n S i x n = x n T i x n = 0 for i = 1, 2, we have fdx n, F )) = 0. Since f : [0, ) [0, ) is a nondecreasing function satisfying f0) = 0 ft) > 0 for all t 0, ) dx n, F ) exists by Lemma 2.1, we have dx n, F ) = 0. Now, we show that {x n } is a Cauchy sequence in K. Indeed, from 2.5), we have x n+1 q x n q + B n

12 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), for each n 1 with n=1 B n < q F. For any m, n, m > n 1, we have x m q x m 1 q + B m 1 x m 2 q + B m 1 + B m 2. m 1 x n q + x n q + i=n B i B i. Since dx n, F ) = 0 i=n B i <, for any given ε > 0 there exists a positive integer n 0 such that Therefore, there exists q 1 F such that dx n, F ) < ε 4, i=n i=n 0 B i < ε 4. x n0 q 1 < ε 4, i=n 0 B i < ε 4. Thus, for all m, n n 0, we get from the above inequality that x m x n x m q 1 + x n q 1 x n0 q x n0 q 1 + = 2 x n0 q 1 + ε < 2 4 4) + ε = ε. i=n 0 B i i=n 0 B i i=n 0 B i Thus, it follows that {x n } is a Cauchy sequence. Since K is a closed subset of E, the sequence {x n } converges strongly to some q K. It is easy to prove that F S 1 ), F S 2 ), F T 1 ) F T 2 ) are all closed so F is a closed subset of K. Since dx n, F ) = 0, we have q F. Thus, the sequence {x n } converges strongly to a common fixed point of the mappings S 1, S 2, T 1 T 2. This completes the proof. ) 3. Weak convergence theorems In this section, we prove some weak convergence theorems of iteration scheme 1.5) for the mixed type asymptotically nonexpansive mappings in the intermediate sense in real uniformly convex Banach spaces. Lemma 3.1. Under the assumptions of Lemma 2.1, for all p 1, p 2 F = F S 1 ) F S 2 ) F T 1 ) F T 2 ), the it tx n + 1 t)p 1 p 2 exists for all t [0, 1], where {x n } is the sequence defined by 1.5).

13 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), Proof. By Lemma 2.1, x n z exists for all z F therefore {x n } is bounded. By letting a n t) = tx n + 1 t)p 1 p 2 for all t [0, 1], we conclude that a n 0) = p 1 p 2 a n 1) = x n p 2 exists by Lemma 2.1. It, therefore, remains to prove Lemma 3.1 for t 0, 1). For all x K, we define the mapping W n : K K by: U n x) = P α ns n 2 x + β nt 2 P T 2 ) n 1 x + γ nu n), W n x) = P α n S n 1 x + β n T 1 P T 1 ) n 1 U n x) + γ n u n ). Then it follows that x n+1 = W n x n, W n p = p for all p F. Now from 2.3) 2.5) of Lemma 2.1, we see that U n x) U n y) x y + A n, with n=1 B n <. By setting From 3.1) 3.2), we have W n x) W n y) x y + B n, 3.1) Q n, m = W n+m 1 W n+m 2... W n, m 1, 3.2) b n, m = Q n, m tx n + 1 t)p 1 ) tq n, m x n + 1 t)q n,m p 2 ). Q n, m x) Q n, m y) = W n+m 1 W n+m 2... W n x) W n+m 1 W n+m 2... W n y) W n+m 2... W n x) W n+m 2... W n y) + B n+m 1 W n+m 3... W n x) W n+m 3... W n y). + B n+m 1 + B n+m 2 n+m 1 x y + for all x, y K, Q n, m x n = x n+m Q n, m p = p for all p F. Thus By using [4, Theorem 2.3], we have j=n B j a n+m t) = tx n+m + 1 t)p 1 p 2 b n, m + Q n, m tx n + 1 t)p 1 ) p 2 n+m 1 b n, m + a n t) + B j. j=n b n,m ϕ 1 x n u Q n,m x n Q n,m u ) ϕ 1 x n u x n+m u + u Q n,m u ) ϕ 1 x n u x n+m u Q n,m u u )), so the sequence {b n,m } converges uniformly to 0, i.e., b n,m 0 as n. Since B n = 0, therefore from 3.3), we have sup a n t) b n,m + inf a nt) = inf a nt). n,m This shows that a n t) exists, that is, tx n + 1 t)p 1 p 2 exists for all t [0, 1]. This completes the proof. 3.3)

14 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), Lemma 3.2. Under the assumptions of Lemma 2.1, if E has a Fréchet differentiable norm, then for all p 1, p 2 F = F S 1 ) F S 2 ) F T 1 ) F T 2 ), the following it exists x n, Jp 1 p 2 ), where {x n } is the sequence defined by 1.5), if W w {x n }) denotes the set of all weak subsequential its of {x n }, then l 1 l 2, Jp 1 p 2 ) = 0 for all p 1, p 2 F l 1, l 2 W w {x n }). Proof. Suppose that x = p 1 p 2 with p 1 p 2 h = tx n p 1 ) in inequality 1.1). Then, we get t x n p 1, Jp 1 p 2 ) p 1 p tx n + 1 t)p 1 p 2 2 Since sup n 1 x n p 1 M for some M > 0, we have t x n p 1, Jp 1 p 2 ) p 1 p bt x n p 1 ). t sup x n p 1, Jp 1 p 2 ) p 1 p tx n + 1 t)p 1 p 2 2 That is, t inf x n p 1, Jp 1 p 2 ) p 1 p btm ). sup x n p 1, Jp 1 p 2 ) inf x n p 1, Jp 1 p 2 ) + btm ) tm M. If t 0, then x n p 1, Jp 1 p 2 ) exists for all p 1, p 2 F ; in particular, l 1 l 2, Jp 1 p 2 ) = 0 for all l 1, l 2 W w {x n }). Theorem 3.3. Under the assumptions of Lemma 2.2, if E has Fréchet differentiable norm, then the sequence {x n } defined by 1.5) converges weakly to a common fixed point of the mappings S 1, S 2, T 1 T 2. Proof. By Lemma 3.2, we obtain l 1 l 2, Jp 1 p 2 ) = 0 for all l 1, l 2 W w {x n }). Therefore, q p 2 = q p, Jq p ) = 0 implies q = p. Consequently, {x n } converges weakly to a common fixed point in F = F S 1 ) F S 2 ) F T 1 ) F T 2 ). This completes the proof. Theorem 3.4. Under the assumptions of Lemma 2.2, if the dual space E of E has the Kadec-Klee KK) property the mappings I S i I T i for i = 1, 2, where I denotes the identity mapping, are demiclosed at zero, then the sequence {x n } defined by 1.5) converges weakly to a common fixed point of the mappings S 1, S 2, T 1 T 2. Proof. By Lemma 2.1, {x n } is bounded since E is reflexive, there exists a subsequence {x nj } of {x n } which converges weakly to some q 1 K. By Lemma 2.2, we have x n j S i x nj = 0, x nj T i x nj = 0 j j for i = 1, 2. Since by hypothesis the mappings I S i I T i for i = 1, 2 are demiclosed at zero, therefore S i q 1 = q 1 T i q 1 = q 1 for i = 1, 2, which means q 1 F = F S 1 ) F S 2 ) F T 1 ) F T 2 ). Now, we show that {x n } converges weakly to q 1. Suppose {x ni } is another subsequence of {x n } converges weakly to some q 2 K. By the same method as above, we have q 2 F q 1, q 2 W w {x n }). By Lemma 3.1, the it tx n + 1 t)q 1 q 2 exists for all t [0, 1] so q 1 = q 2 by Lemma 1.7. Thus, the sequence {x n } converges weakly to q 1 F. This completes the proof.

15 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), Theorem 3.5. Under the assumptions of Lemma 2.2, if E satisfies Opial condition the mappings I S i I T i for i = 1, 2, where I denotes the identity mapping, are demiclosed at zero, then the sequence {x n } defined by 1.5) converges weakly to a common fixed point of the mappings S 1, S 2, T 1 T 2. Proof. Let q F, from Lemma 2.1 the sequence { x n q } is convergent 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 x K. From Lemma 2.2, we have x n k S i x nk = 0 x n k T i x nk = 0 k k for i = 1, 2. Since the mappings I S i I T i for i = 1, 2 are demiclosed at zero, therefore S i x = x T i x = x for i = 1, 2, which means x F. Finally, let us prove that {x n } converges weakly to x. Suppose by the contrary that there is a subsequence {x nj } {x n } such that {x nj } converges weakly to y K x y. Then by the same method as given above, we can also prove that y F. From Lemma 2.1 the its x n x x n y exist. By virtue of the Opial condition of E, we obtain x n x = x n n k k x < n k x n k y = x n y = n j x n j y < n j x n j x = x n x, which is a contradiction, so x = y. Thus, {x n } converges weakly to a common fixed point of the mappings S 1, S 2, T 1 T 2. This completes the proof. Example 3.6. Let R be the real line with the usual norm let K = [ 1, 1]. Define two mappings S, T : K K, respectively, by the formulas { 2 sin x T x) = 2, if x [0, 1], 2 sin x 2, if x [ 1, 0), Sx) = { x, if x [0, 1], x, if x [ 1, 0). Then both S T are asymptotically nonexpansive mappings with constant sequence {k n } = {1} for all n 1 uniformly L-Lipschitzian mappings with L = sup n 1 {k n } both are uniformly continuous on [-1,1] hence they are asymptotically nonexpansive mappings in the intermediate sense. Also the unique common fixed point of S T, that is, F = F S) F T ) = {0}. Furthermore, S T satisfy the condition ii) in Lemma 2.2 see, [5, Example 3.1]). Now, we give some more examples by taking two mappings, T 1 = T 2 = T S 1 = S 2 = S as follows. Example 3.7. Let E = R, K = [ 1 π, 1 π ], λ < 1 P be the identity mapping. For each x K, define the mappings S, T : K K by the formulas { λ x sin1/x), if x 0, T x) = 0, if x = 0,

16 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), Sx) = Then {T n x} 0 uniformly on K as n. Thus sup { x 2, if x 0, 0, if x = 0. { T n x T n y x y 0 } = 0 for all x, y K. Hence T is an asymptotically nonexpansive mapping in the intermediate sense ANI in short), but it is not a Lipschitz mapping S is an asymptotically nonexpansive mapping with constant sequence {k n } = {1} for all n 1 uniformly L-Lipschitzian with L = sup n 1 {k n }. Also, F T ) = {0} is the unique fixed point of T F S) = {0} is the unique fixed point of S, that is, F = F S) F T ) = {0} is the unique common fixed point of S T. Example 3.8. Let E = R, K = [0, 1], P be the identity mapping. For each x K, define the mappings S, T : K K by the formulas k x, if 0 x 1/2, k T x) = 2k 1 k x), if 1/2 x k, 0, if k x 1, Sx) = { x, if x 0, 0, if x = 0, where 1/2 < k < 1. Then F T ) = {0} {T n x} converges uniformly to 0 on K. Obviously, T is uniformly continuous. It follows that T is an asymptotically nonexpansive mapping in the intermediate sense ANI in short). Further, T is uniformly Lipschitzian. Indeed, if 0 x 1/2 1/2 y k, then T n x = k n x T n y = kn 2k 1 k y). Therefore, we see that T n x T n y = k n x kn 2 + kn 2 kn 2k 1 k y) = k n x 1 ) + kn [ k 1 ) k y)] 2 2k 1 2 k n x 1 + kn y 1 2 2k 1 2 k kn x y 2k 1 k x y. 2k 1 Hence T is uniformly 2k 1-Lipschitzian S is an asymptotically nonexpansive mapping with constant sequence {k n } = {1} for all n 1 uniformly L-Lipschitzian with L = sup n 1 {k n }. Also, F S) = {0} is the unique fixed point of S. Thus, F = F S) F T ) = {0} is the unique common fixed point of S T. 4. Concluding remarks In this paper, we study the mixed type iteration scheme for two asymptotically nonexpansive selfmappings in the intermediate sense two asymptotically nonexpansive non-self-mappings in the intermediate sense establish some strong convergence theorems by using some completely continuous semi-compactness conditions some weak convergence theorems by using the following conditions: i) the space E has a Fréchet differentiable norm;

17 G. S. Saluja, M. Postolache, A. Ghiura, J. Nonlinear Sci. Appl ), ii) dual space E of E has the Kadec-Klee KK) property; iii) the space E satisfies Opial condition. Our results extend generalize the corresponding results of Chidume et al. [2, 3], Guo et al. [5, 6], Saluja [8], Schu [9], Tan Xu [12], Wang [13], Wei Guo [14, 15]. Acknowledgment The authors are grateful to the anonymous referee for his/her careful reading useful suggestions that helped to improve the manuscript. References [1] R. Bruck, T. Kuczumow, S. Reich, Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property, Colloq. Math., ), [2] C. E. Chidume, E. U. Ofoedu, H. Zegeye, Strong weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl., ), , 4 [3] C. E. Chidume, N. Shahzad, H. Zegeye, Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense, Numer. Funct. Anal. Optim., ), , 1, 4 [4] J. García Falset, W. Kaczor, T. Kuczumow, S. Reich, Weak convergence theorems for asymptotically nonexpansive mappings semigroups, Nonlinear Anal., ), [5] W. Guo, Y. J. Cho, W. Guo, Convergence theorems for mixed type asymptotically nonexpansive mappings, Fixed Point Theory Appl., ), 15 pages. 1, 3.6, 4 [6] W. Guo, W. Guo, Weak convergence theorems for asymptotically nonexpansive nonself-mappings, Appl. Math. Lett., ), [7] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc., ), [8] G. S. Saluja, Convergence theorems for two asymptotically nonexpansive non-self mappings in uniformly convex Banach spaces, J. Indian Math. Soc. N.S.), ), [9] J. Schu, Weak strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., ), , 4 [10] K. Sitthikul, S. Saejung, Convergence theorems for a finite family of nonexpansive asymptotically nonexpansive mappings, Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math., ), , 1.7, 1.8 [11] W. Takahashi, G. E. Kim, Approximating fixed points of nonexpansive mappings in Banach spaces, Math. Japon., ), [12] K. K. Tan, H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., ), , 4 [13] L. Wang, Strong weak convergence theorems for common fixed point of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl., ), , 4 [14] S. I. Wei, W. Guo, Strong convergence theorems for mixed type asymptotically nonexpansive mappings, Commun. Math. Res., ), [15] S. Wei, W. Guo, Weak convergence theorems for mixed type asymptotically nonexpansive mappings, J. Math. Study, ), , 4