Common fixed points of two generalized asymptotically quasi-nonexpansive mappings

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1 An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) Tomul LXIII, 2017, f. 2 Common fixed points of two generalized asymptotically quasi-nonexpansive mappings Safeer Hussain Khan Isa Yildirim Received: 5.VIII.2013 / Accepted: 5.XI.2013 Abstract In this paper, we consider an iterative procedure for approximating common fixed points of two generalized asymptotically quasi-nonexpansive mappings and we prove some strong and weak convergence theorems for such mappings in uniformly convex Banach spaces.this will extend the results of Cholamjiak and Suantai, Khan and those generalized therein to the case of generalized asymptotically quasi-nonexpansive mappings and by a faster iterative procedure. Keywords generalized asymptotically quasi-nonexpansive mappings fixed point iterative procedure Mathematics Subject Classification (2010) 47H09 47H10 47J25 1 Introduction Throughout this paper, N denotes the set of all positive integers. Let C be a nonempty subset of a real Banach space E. Let T : C C be a mapping, then we denote the set of all fixed points of T by F (T ). The set of common fixed points of two mappings S and T will be denoted by F = F (S) F (T ). A mapping T : C C is said to be (i) nonexpansive if T x T y x y, for all x, y C; (ii) quasi-nonexpansive if T x p x p, for all x C and p F (T ); (iii) asymptotically nonexpansive if there exists a sequence k n 1, lim k n = 1 and T n x T n y k n x y, for all x, y C and n N. Note that {k n } is a nonincreasing bounded sequence (see [9]). Safeer Hussain Khan Department of Mathematics, Statistics and Physics, Qatar University, Doha 2713, Qatar safeer@qu.edu.qa; safeerhussain5@yahoo.com Isa Yildirim Department of Mathematics, Faculty of Science, Atatürk University, Erzurum, 25240, Turkey isayildirim@atauni.edu.tr

2 2 Safeer Hussain Khan, Isa Yildirim (iv) asymptotically quasi-nonexpansive if there exists a sequence k n 1, lim k n = 1 and T n x p k n x p, for all x C, p F (T ) and n N; (v) generalized asymptotically nonexpansive (see [18]) if there exist nonnegative real sequences {k n } and {c n } with k n 1, lim k n =1 and lim c n =0 such that T n x T n y k n x y +c n, for all x, y C and n N; (vi) generalized asymptotically quasi-nonexpansive (see [18]) if there exist nonnegative real sequences {k n } and {c n } with k n 1, lim k n = 1 and lim c n = 0 such that T n x p k n x p + c n, for all x C, p F (T ) and n N; (vii) uniformly L-Lipschitzian if there exists a constant L > 0 such that T n x T n y L x y, for all x, y C and n N. From the above definitions, it is clear that each of a nonexpansive, a quasi-nonexpansive, an asymptotically nonexpansive, an asymptotically quasi-nonexpansive, a generalized quasi-nonexpansive mappings is a generalized asymptotically quasinonexpansive mapping. However, the converse of each of above statements may be not true. For the fact that a generalized asymptotically quasi-nonexpansive mapping is not an asymptotically quasi-nonexpansive mapping (see, for example, [18]). In 1991, Schu [17] introduced the following Mann-type iterative process: { x1 = x C, x n+1 = (1 α n )x n + α n T n (1.1) x n, n N, where T : C C is an asymptotically nonexpansive mapping with a sequence {k n } such that n=1 (k n 1) < and {α n } is a sequence in (0, 1) satisfying the condition δ α n 1 δ for all n N and for some δ > 0. He concluded that the sequence {x n } converges weakly to a fixed point of T. Since 1972, weak and strong convergence problems of iterative procedures (with errors) for asymptotically nonexpansive type mappings in a Hilbert space or a Banach space have been studied by many authors (see, for example, [14, 15, 17]). In 2007, Agarwal et al. [1] introduced the following iteration procedure: x 1 = x C, x n+1 = (1 α n )T n x n + α n T n y n, (1.2) y n = (1 β n ) x n + β n T n x n, n N where {α n } and {β n } are in (0, 1). They showed that this iterative procedure converges at a rate same as that of Picard iterative procedure and faster than Mann for contractions. The above iterative procedure deals with one mapping only. The case of two mappings in iterative procedures has also remained under study since Das and Debata [7] gave and studied a two mappings procedure. Also see, for example, [13] and [21]. The problem of approximating common fixed points of finitely many mappings plays an important role in applied mathematics, especially in the theory of evolution equations and the minimization problems (see [2 4, 20], for example). Ishikawa-type iterative procedure x 1 = x C, x n+1 = (1 α n )x n + α n S n y n, (1.3) y n = (1 β n ) x n + β n T n x n, n N

3 A two-step iterative procedure for common fixed points 3 for two mappings has aslo been studied by many authors including [7,13,21]. Recently, Cholamjiak and Suantai [6] have used the above procedure for finitely many generalized asymptotically quasi-nonexpansive mappings. Recently, Khan et al. [11] modified the iterative procedure (1.2) to the case of two mappings as follows. x 1 = x C, x n+1 = (1 α n )T n x n + α n S n y n, (1.4) y n = (1 β n ) x n + β n T n x n, n N where {α n } and {β n } are in (0, 1). Remark 1.1 (i) Note that (1.4) reduces to (1.2) when S = T. Similarly, the procedure (1.4) reduces to (1.1) when T = I. (ii) The procedure (1.2) does not reduce to (1.1) but (1.4) does. Thus (1.4) not only covers the results proved by (1.2) but also by (1.1) which are not covered by (1.2). (iii) The procedure (1.4) is independent of (1.3): neither of them reduces to the other. Following the method of Agarwal et al. [1], it can be shown that (1.4) converges faster than (1.3) for contractions (for details, see [12]). Khan [10] has used the above iterative procedure for asymptotically quasi-nonexpansive mappings. In this paper, we prove some weak and strong convergence theorems for two generalized asymptotically quasi-nonexpansive mappings using (1.4). This will generalize the results of [6, 10] and those generalized therein. Moreover, our results are independent of [6] but our procedure converges at a better rate as remarked above. We believe that once the results for two mappings have been obtained, extension to finitely many mappings is a routine work. Thus we will use only two mappings. 2 Preliminaries For the sake of convenience, we restate the following concepts and results. Let E be Banach space with its dimension greater than or equal to 2. The modulus of E is the function δ E (ε) : (0, 2] [0, 1] defined by { } δ E (ε) = inf 1 1 (x + y) 2 : 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]. We recall the following. 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. The space E has : x+ty x (i) Gâteaux differentiable norm if lim t 0 t exists for each x and y in S; (ii) Fréchet differentiable norm (see e.g. [19]) if for each x in S, the above limit exists and is attained uniformly for y in S and in this case, it is also well-known that h, J(x) x x + h 2 h, J(x) x 2 + b( h ), (2.1)

4 4 Safeer Hussain Khan, Isa Yildirim for all x, h in E, where J is the Fréchet derivative of the functional at x X,.,. is the pairing between E and E, and b is an increasing function defined on [0, ) b(t) such that lim t 0 t = 0; (iii) Opial condition [16] if for any sequence {x n } in E, x n x implies 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. A mapping T : C C is said to be demiclosed at zero, if for any sequence {x n } in C, the conditions x n converges weakly to x C and T x n converges strongly to 0 imply T x = 0. Two mappings S, T : C C, where C is a subset of a normed space E, are said to satisfy the Condition (A ) (see [8]) if there exists a nondecreasing function f : [0, ) [0, ) with f(0) = 0, f(r) > 0 for all r (0, ) such that either x Sx f(d(x, F )) or x T x f(d(x, F )) for all x C where d(x, F ) = inf{ x p : p F }. We need the following useful known lemmas for the development of our results. Lemma 2.1 ([22]) If {r n }, {t n } and {s n } are sequences of nonnegative real numbers such that r n+1 (1 + t n ) r n + s n, n=1 t n < and n=1 s n <, then lim r n exists. Lemma 2.2 ([17]) Suppose that E is a uniformly convex Banach space and 0<p t n q<1 for all n N. Let {x n } and {y n } be two sequences of E such that lim sup x n r, lim sup y n r and lim t n x n + (1 t n )y n = r hold for some r 0. Then lim x n y n = 0. 3 Main results In this section, we prove some strong and weak convergence theorems of the iterative procedure (1.4) under some suitable conditions. Before proving our main results, we would like to remark as follows. Let T, S : C C be two generalized asymptotically quasi-nonexpansive mappings such that T n x p k 1n x p +c 1n and S n x p k 2n x p + c 2n. Throughout this paper, we assume that k n = max n N {k 1n, k 2n }, c n = max n N {c 1n, c 2n } and F where F = F (T ) F (S) is the set of common fixed points of the mappings T and S. We need the following lemma in order to prove our main theorems. Lemma 3.1 Let C be a nonempty closed convex subset of a real Banach space E. Let T and S be two generalized asymptotically quasi-nonexpansive self mappings of C with n=1 (k n 1) < and n=1 c n <. Let {x n } be defined by (1.4) and F. Then lim x n q exists for all q F. Proof. Let q F. Then x n+1 q = (1 α n )T n x n + α n S n y n q (1 α n ) T n x n q + α n S n y n q (1 α n ) (k n x n q + c n ) + α n (k n y n q + c n )

5 [ ] (1 αn ) x k n q + α n (1 β n ) x n q n +α n β n T n x n q k n [ (1 αn ) x n q + α n (1 β n ) x n q +α n β n (k n x n q + c n ) A two-step iterative procedure for common fixed points 5 + c n ] + c n k n [(1 α n + α n (1 β n ) + k n α n β n + c n ) x n q ] + c n = k n [1 + (k n 1)α n β n + c n ] x n q + c n k n [1 + k n 1 + c n ] x n q + c n = k 2 n x n q + (k n + 1) c n = [ 1 + ( k 2 n 1 )] x n q + (k n + 1) c n Since {k n } (and hence {k n + 1} ) is a nonincreasing bounded sequence, n=1 (k n 1) < and n=1 c n < imply that n=1 (k2 n 1) < and n=1 (k n + 1) c n <. It now follows from Lemma 2.1 that lim x n q exists for all q F. Theorem 3.2 Let C be a nonempty closed convex subset of a uniformly convex Banach space E. Let T and S be two uniformly L 1 and L 2 -Lipschitzian and generalized asymptotically quasi-nonexpansive self mappings of C with n=1 (k n 1)<, n=1 c n< and L= max{l 1, L 2 }. Let {x n } be defined by (1.4) and F. Then lim x n T x n = lim x n Sx n =0. Proof. By Lemma 3.1, lim x n q exists. Assume that lim x n q = c. If c = 0, the conclusion is obvious. Suppose c > 0. Now y n q = (1 β n )x n + β n T n x n q (1 β n ) x n q +β n (k n x n q + c n ) (1 + β n (k n 1)) x n q + c n implies that lim sup y n q c. (3.1) Since T is a generalized asymptotically quasi-nonexpansive mappings, we have T n x n q k n x n q + c n, for all n = 1, 2,.... Taking lim sup on both sides, we obtain lim sup T n x n q c. (3.2) In a similar way, we have S n y n q k n y n q + c n. By using (3.1), we obtain lim sup S n y n q c. Also, it follows from c = lim x n+1 q = lim (1 α n ) (T n x n q) + α n (S n y n q) and Lemma 2.2 that lim T n x n S n y n = 0. (3.3) Now x n+1 q = (1 α n )T n x n +α n S n y n q = (T n x n q)+α n (S n y n T n x n ) T n x n q + α n T n x n S n y n yields that c lim inf T n x n q. so that (3.2) gives lim T n x n q = c. On the other hand, T n x n q T n x n S n y n + S n y n q T n x n S n y n + k n y n q + c n, so we have By using (3.1) and (3.4), we obtain c lim inf y n q. (3.4) lim y n q = c. (3.5)

6 6 Safeer Hussain Khan, Isa Yildirim Thus c = lim y n q = lim (1 β n ) (x n q) + β n (T n x n q) gives by Lemma 2.2 that Now y n x n = β n T n x n x n. Hence by (3.6), Also note that so that lim T n x n x n = 0. (3.6) lim y n x n = 0. (3.7) x n+1 x n = (1 α n ) T n x n + α n S n y n x n T n x n x n + α n T n x n S n y n 0 as n, (3.8) x n+1 y n x n+1 x n + y n x n (3.9) 0 as n. Furthermore, from x n+1 S n y n x n+1 x n + x n T n x n + T n x n S n y n we find that lim x n+1 S n y n = 0. (3.10) Then x n+1 T x n+1 x n+1 T n+1 x n+1 + T n+1 x n+1 T n+1 x n + T n+1 x n T x n+1 x n+1 T n+1 x n+1 +L x n+1 x n +L T n x n x n+1 = x n+1 T n+1 x n+1 + L x n+1 x n + Lα n T n x n S n y n yields lim x n T x n = 0. (3.11) Now x n S n x n x n x n+1 + x n+1 S n y n + S n y n S n x n x n x n+1 + x n+1 S n y n + L y n x n 0 as n. Thus x n+1 Sx n+1 xn+1 S n+1 x n+1 + S n+1 x n+1 Sx n+1 xn+1 S n+1 x n+1 + L S n x n+1 x n+1 ( ) xn+1 S n+1 x n+1 S + L n x n+1 S n y n + S n y n x n+1 xn+1 S n+1 x n+1 + L 2 x n+1 y n +L S n y n x n+1 implies lim x n Sx n = 0. Lemma 3.3 Assume that the conditions of Theorem 3.2 are satisfied. Then, for any p 1, p 2 F, lim x n, J(p 1 p 2 ) exists; in particular, p q, J(p 1 p 2 ) = 0 for all p, q ω w (x n ).

7 A two-step iterative procedure for common fixed points 7 Proof. Suppose that x = p 1 p 2 with p 1 p 2 and h = t(x n p 1 ) in the inequality (2.1). Then, we get t x n p 1, J(p 1 p 2 ) p 1 p tx n + (1 t)p 1 p 2 2 t x n p 1, J(p 1 p 2 ) p 1 p b(t x n p 1 ). Since sup n 1 x n p 1 M for some M > 0, we have t lim sup x n p 1, J(p 1 p 2 ) p 1 p lim tx n + (1 t)p 1 p 2 2 t lim inf x n p 1, J(p 1 p 2 ) p 1 p b(tm ). That is, lim sup x n p 1, J(p 1 p 2 ) lim inf x n p 1, J(p 1 p 2 ) + b(tm ) tm M. If t 0, then lim x n p 1, J(p 1 p 2 ) exists for all p 1, p 2 F ; in particular, we have p q, J(p 1 p 2 ) = 0 for all p, q ω w (x n ). We now give our weak convergence theorem. In the sequel, I : X X denotes the identity mapping. Theorem 3.4 Let E be a uniformly convex Banach space satisfying Opial condition and let C, T, S and {x n } be taken as in Theorem 3.2. If the mappings I T and I S are demiclosed at zero, then {x n } converges weakly to a common fixed point of T and S. Proof. Let q F, then according to 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 1 C. From Theorem 3.2, we have lim x nk T x nk = 0, lim x nk Sx nk = 0. Since the mappings I T and I S are demiclosed at zero, therefore T q 1 = q 1 and Sq 1 = q 1, which means q 1 F. Finally, let us prove that {x n } converges weakly to q 1. Suppose on contrary that there exists a subsequence { xnj } {xn } such that { x nj } converges weakly to q2 C and q 1 q 2. Then by the same method as given above, we can also prove that q 2 F. From Lemma 3.1, the limits lim x n q 1 and lim x n q 2 exist. By virtue of the Opial condition of E, we obtain lim x n q 1 = lim x n n i q 1 < lim x n i n i q 2 = lim x n q 2 i = lim x n n j q 2 < lim x n j n j q 1 = lim x n q 1 j which is a contradiction so q 1 = q 2. Therefore, {x n } converges weakly to a common fixed point of T and S. Theorem 3.5 Let E be a uniformly convex Banach space which has a Fréchet differentiable norm and let C, T, S and {x n } be taken as in Theorem 3.2. If F, then {x n } converges weakly to a common fixed point of T and S. Proof. By Lemma 3.3, p q, J(p 1 p 2 ) = 0 for all p, q ω w (x n ). Therefore q 1 q 2 2 = q 1 q 2, J(q 1 q 2 ) = 0 implies q 1 = q 2. Consequently, {x n } converges weakly to a point of F and this completes the proof. We have the following corollaries. Corollary 3.6 Let E be a uniformly convex Banach space satisfying Opial condition and let C and T be taken as in Theorem 3.2. Suppose that F (T ). If the mapping I T is demiclosed at zero, then {x n } defined by (1.2) converges weakly to a fixed point of T.

8 8 Safeer Hussain Khan, Isa Yildirim Corollary 3.7 Let E be a uniformly convex Banach space which has a Fréchet differentiable norm and let C and T be taken as in Theorem 3.2. If F (T ), then {x n } defined by (1.2) converges weakly to a fixed point of T. Corollary 3.8 Let E be a uniformly convex Banach space satisfying Opial condition and let C and T be taken as in Theorem 3.2. Suppose that F (T ). If the mapping I T is demiclosed at zero, then {x n } defined by (1.1) converges weakly to a fixed point of T. Corollary 3.9 Let E be a uniformly convex Banach space which has a Fréchet differentiable norm and let C and T be taken as in Theorem 3.2. If F (T ), then {x n } defined by (1.1) converges weakly to a fixed point of T. Theorem 3.10 Let E be a real Banach space and let C, T, S, F, {x n } be taken as in Theorem 3.2. Then {x n } converges to a point of F 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, lim x n w exists for all w F, therefore lim d(x n, F ) exists. But by hypothesis, lim inf d(x n, F ) = 0, therefore we have lim d(x n, F ) = 0. Next, we show that {x n } is a Cauchy sequence in C. Let ɛ > 0 be arbitrarily chosen. Since lim d(x n, F ) = 0, there exists a positive integer n 0 such that d(x n, F ) < ɛ 4, n n 0. In particular, inf{ x n0 p : p F } < ɛ 4. Thus there must exist p F such that x n0 p < ɛ 2. Now, for all m, n n 0, we have x n+m x n x n+m p + x n p 2 x n0 p < 2( ɛ 2 ) = ɛ. Hence {x n} is a Cauchy sequence in a closed subset C of a Banach space E and so it must converge to a point q in C. Now, lim d(x n, F ) = 0 gives that d(q, F ) = 0. Since F is closed, so we have q F. Applying Theorem 3.10, we obtain strong convergence of the procedure (1.4) under the Condition (A ) as follows. Theorem 3.11 Let E be a real uniformly convex Banach space and let C, T, S, F, {x n } be taken as in Theorem 3.2. Let T, S satisfy the condition (A ), then {x n } converges strongly to a common fixed point of T and S. Proof. We proved in Theorem 3.2 that lim x n T x n = lim x n Sx n = 0. (3.12) From the Condition (A ) and (3.12), either lim f(d(x n, F )) lim x n T x n = 0, or lim f(d(x n, F )) lim x n Sx n = 0. Hence lim f(d(x n, F )) = 0. Since f : [0, ) [0, ) is a nondecreasing function satisfying f(0) = 0, f(r) > 0 for all r (0, ), therefore we have lim d(x n, F ) = 0. Now all the conditions of Theorem 3.10 are satisfied, therefore by its conclusion {x n } converges strongly to a point of F. Remark 3.1 (i) Results using the iterative procedures (1.1) and (1.2) can now be obtained as corollaries from Theorems 3.10 and (ii) The case of nonexpansive mappings using the iterative procedures (1.1), (1.2) and (1.4) can now be deduced from our above results. (iii) Theorems of this paper can also be proved with error terms.

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