Convergence and stability results for a class of asymptotically quasi-nonexpansive mappings in the intermediate sense

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1 Functional Analysis, Approximation and Computation 7 1) 2015), Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: Convergence and stability results for a class of asymptotically quasi-nonexpansive mappings in the intermediate sense G. S. Saluja a a Department of Mathematics, Govt. Nagarjuna P.G. College of Science, Raipur C.G.), India Abstract. In this paper, we study the notion of a class of asymptotically quasi-nonexpansive mappings in the intermediate sense and define a k-step iterative sequence for approximating common fixed point in a class of above mentioned mappings. Also, we study its stability in real Banach spaces. The results presented in this paper are improvement and generalization of several known corresponding results in the existing literature see, e.g., [3], [4], [9], [12]-[14], [17], [20]-[26]). 1. Introduction Let E be an arbitrary real Banach space and K be a nonempty closed and convex subset E. Definition 1.1. Let T : K K be a mapping. Suppose that, for any x 0 E, x n+1 = f T, x n ) 1) yields a sequence of points x n } in K, where f denotes the iterative process involving T and x n. Suppose that FT) = x K : Tx = x} and x n } converges strongly to p FT). Let y n } be a sequence in K and ε n } be a sequence in [0, ) defined by ε n = y n+1 f T, y n ). 2) If lim ε n = 0 implies that lim y n = p, then the iterative process defined by 1) is said to be T-stable with respect to T see, for example, [6]-[8], [15], [16], [18], [19] and references therein). We say that the iterative process x n } defined by 1) is almost T-stable or almost stable with respect to T if n=0 ε n < implies that lim y n = p see [16]). It is easy to see from above that an iterative process which is T-stable is almost T-stable. The example in [16] showed that an iterative process which is almost T-stable need not be T-stable Mathematics Subject Classification. 47H05, 47H09, 47H10, 49M05. Keywords. A class of asymptotically quasi-nonexpansive mappings in the intermediate sense; k-step iterative sequence; common fixed point; Banach space; stability. Received: 20 September 2014; Accepted: 30 April 2015 Communicated by Dijana Mosić address: saluja1963@gmail.com G. S. Saluja)

2 G. S. Saluja / FAAC 7 1) 2015), Stability results for many iterative processes for some kinds of nonlinear mappings have been shown in recent papers by many authors see, for example, [1], [2], [6]-[8], [15], [16], [18], [19] and the references therein). The concept of quasi-nonexpansive mapping was initiated by Tricomi in 1941 for real function. The concept of an asymptotically nonexpansive mapping and an asymptotically nonexpansive type mapping were introduced by Goebel and Kirk [5] and Kirk [11], respectively, which are closely related to the theory of fixed points in Banach spaces. In 2003, Sahu and Jung [20] studied Ishikawa and Mann iteration process in Banach spaces and they proved some weak and strong convergence theorems for asymptotically quasi-nonexpansive type mapping. In 2006, Shahzad and Udomene [22] gave the necessary and sufficient condition for convergence of common fixed point of two-step modified Ishikawa iterative sequence for two asymptotically quasi-nonexpansive mappings in real Banach space. Recently, Yao and Liou [25] gave the notion of asymptotically quasi-nonexpansive mappings in the intermediate sense and gave necessary and sufficient condition for the iterative sequence to converge to the common fixed points for two asymptotically quasi-nonexpansive mappings in the intermediate sense in the framework of real Banach spaces. The iterative approximating problem of fixed points for asymptotically nonexpansive mappings or asymptotically quasi-nonexpansive mappings have been studied by many authors see, for example, [3], [4], [9], [12]-[14], [17], [20]-[26] and the references therein). In this paper, we study the notion of a class of asymptotically quasi-nonexpansive mappings in the intermediate sense and define a k-step iterative sequence for approximating common fixed point in a class of said mappings. Also, we study its stability in real Banach spaces. Our results extend, improve and unify the corresponding results of [3], [4], [9], [12]-[14], [17], [20]-[26]. 2. Preliminaries Definition 2.1. Let E be a real Banach space and K be a nonempty closed and convex subset of E. Let T : K K be a mapping. Denote FT) the set of fixed points of T, that is, FT) = x K : Tx = x}. 1) T is said to be nonexpansive if Tx Ty x y for all x, y K. 3) 2) T is said to be quasi-nonexpansive if FT) and Tx p x p for all x K and p FT). 4) 3) T is said to be asymptotically nonexpansive if there exists a sequence u n } [0, ) with u n 0 as n such that T n x T n y 1 + u n ) x y 5) for all x, y K and n 0. 4) T is said to be asymptotically quasi-nonexpansive if FT) and there exists a sequence u n } [0, ) with u n 0 as n such that T n x p 1 + u n ) x p for all x K, p FT) and n 0. 6)

3 G. S. Saluja / FAAC 7 1) 2015), ) T is said to be asymptotically nonexpansive type [11], if lim sup sup T n x T n y x y )} 0. 7) x,y K 6) T is said to be asymptotically quasi-nonexpansive type [20], if FT) and lim sup T n x p x p )} 0. 8) sup x K, p FT) 7) T is said to be asymptotically quasi-nonexpansive mapping in the intermediate sense [25] provided that T is uniformly continuous and lim sup T n x p x p )} 0. 9) sup x K, p FT) From the above definitions, it follows that asymptotically nonexpansive mapping must be asymptotically quasi-nonexpansive and asymptotically quasi-nonexpansive mapping in the intermediate sense. But the converse does not hold as the following example. Example 2.2. Let X = R be a normed linear space and K = [0, 1]. For each x K, we define k x, if x 0, Tx) = 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 and n N. Thus T is an asymptotically nonexpansive mapping with constant sequence 1} and lim sup T n x T n y x y } = lim sup k n x y x y } 0, because lim k n = 0 as 0 < k < 1 and for all x, y K, n N. Hence T is an asymptotically nonexpansive mapping in the intermediate sense. Since FT) = 0}, and so T is an asymptotically quasi-nonexpansive mapping in the intermediate sense. Example 2.3. see [10]) Let X = R, K = [ 1 π, 1 π ] and λ < 1. For each x K, define λ x sin 1 Tx) = x ), if x 0, 0, if x = 0. Then T is an asymptotically nonexpansive mapping in the intermediate sense but it is not asymptotically nonexpansive mapping. Throughout this paper, let E be a real Banach space and K be a nonempty closed convex subset of E. Let T 1, T 2,..., T k : K K be mappings, FT i ) the set of fixed points of T i, k is a positive integer. Let m and n denote the nonnegative integers. Definition 2.4. Let E be a real Banach space and K be a nonempty closed convex subset of E. T 1, T 2,..., T k : K K are said to be a class of asymptotically quasi-nonexpansive mappings in the intermediate sense provided that T i i = 1, 2,..., k) is uniformly continuous and, for each i 1, 2,..., k}, lim sup sup x K, p F= k FT i) T n i x p x p )} 0. 10)

4 G. S. Saluja / FAAC 7 1) 2015), Remark 2.5. It is easy to see that the notion of a class of asymptotically quasi-nonexpansive mappings in the intermediate sense defined by Definition 2.4 reduces to that of asymptotically quasi-nonexpansive mapping in the intermediate sense defined by Definition 2.17) when T 1 = T 2 = = T k = T. In the sequel, we need the following lemma. Lemma 2.6. see [23]) Let a n }, b n } be sequences of nonnegative real numbers satisfying the inequality a n+1 a n + b n, n 1. If n=1 b n <. Then a) lim a n exists. b) If lim inf a n = 0, then lim a n = Main Results Theorem 3.1. Let E be a real Banach space, K a nonempty closed convex subset of E. Let T 1, T 2,..., T k : K K be a class of asymptotically quasi-nonexpansive mappings in the intermediate sense defined by Definition 2.4. Assume that, for each i 1, 2,..., k}, there exist L i and γ i > 0 such that T i x q L i x q γ i, x K, q F = k FT i). 11) Put, for each i 1, 2,..., k} D n = max max 1ik sup x K, q F T n i x q x q ), 0 }, 12) such that n=0 D n <. For any given x 0 K, define the k-step iterative sequence x n } by z k 1,n = 1 α k,n )x n + α k,n T n k x n, z k 2,n = 1 α k 1,n )x n + α k 1,n T n k 1 z k 1,n,. z 1,n = 1 α 2,n )x n + α 2,n T n 2 z 2,n, x n+1 = 1 α 1,n )x n + α 1,n T n 1 z 1,n, n 0, 13) where α i,n } is a sequence in [0, 1] for each i 1, 2,..., k}. Suppose that y n } is a sequence in K and define a sequence ε n } of positive real numbers by w k 1,n = 1 α k,n )y n + α k,n T n k y n, w k 2,n = 1 α k 1,n )y n + α k 1,n T n k 1 w k 1,n,. w 1,n = 1 α 2,n )y n + α 2,n T n 2 w 2,n, ε n = y n+1 1 α 1,n )y n α 1,n T n 1 w 1,n, n 0. 14) If F = k FT i), then we have the following: i) x n } converges strongly to a common fixed point q of T 1, T 2,..., T k in K if and only if lim inf dx n, F) = 0, where dx n, F) = inf p F x n p. ii) n=0 ε n < and lim inf dy n, F) = 0 imply that y n } converges strongly to a common fixed point q of T 1, T 2,..., T k in K. iii) If y n } converges strongly to a common fixed point q of T 1, T 2,..., T k in K, then lim ε n = 0.

5 G. S. Saluja / FAAC 7 1) 2015), In order to prove Theorem 3.1, we first give the following lemma. Lemma 3.2. Assume that all the assumptions in Theorem 3.1 hold and n=0 ε n <. Then i) y n+1 q y n q + k δn) p Dn + ε n, q F; ii) y m q y n q + ε s + k ) m 1 δ p n D s, q F, m > n; iii) lim dy n, F) exists. Proof. Take any q F, and since w i,n } K, it follows from 12), 13) and 14) that y n+1 q ε n + 1 α 1,n )y n q) + α 1,n T n w 1,n q) 1 α 1,n ) y n q + α 1,n T n w 1,n q + ε n 1 α 1,n ) y n q + α 1,n w 1,n q + α 1,n D n 15) and w 1,n q = 1 α 2,n )y n q) + α 2,n T n 2 w 2,n q) 1 α 2,n ) y n q + α 2,n T n 2 w 2,n q 1 α 2,n ) y n q + α 2,n w 2,n q + D n ) 1 α 2,n ) y n q + α 2,n w 2,n q + α 2,n D n. 16) Continuing in this way, we can deduce that w i,n q = 1 α i+1,n )y n q) + α i+1,n T n i+1 w i+1,n q) 1 α i+1,n ) y n q + α i+1,n T n i+1 w i+1,n q 1 α i+1,n ) y n q + α i+1,n w i+1,n q + D n ) 1 α i+1,n ) y n q + α i+1,n w i+1,n q + α i+1,n D n. 17) Since y n } K, we have w k 1,n q = 1 α k,n )y n q) + α k,n T n k y n q) 1 α k,n ) y n q + α k,n T n k y n q 1 α k,n ) y n q + α k,n y n q + D n ) y n q + α k,n D n. 18)

6 Substituting 17) and 18) into 15), for any q F, we have y n+1 q y n q + ε n + α 1,n D n G. S. Saluja / FAAC 7 1) 2015), α 1,n α 2,n D n + + α 1,n α 2,n... α k,n D n y n q + ε n + 1 } α i,n Dn + 2 } α i,n Dn + + k } α i,n Dn y n q + ε n + δ 1 nd n + δ 2 nd n + + δ k nd n y n q + ε n + k δn) p Dn 19) for all n 0, where δ l n = l α i,n. Hence the conclusion i) holds. From the conclusion i), we have y m q y m 1 q + ε m 1 + k δn) p Dm 1... y m 2 q + ε m 2 + ε m 1 + k δn) p Dm 2 + k δn) p Dm 1... y n q + ε s + k ) m 1 δ p n D s 20) for all q F and m > n. Thus the conclusion ii) holds. Again, it follows from conclusion i) that dy n+1, F) dy n, F) + ε n + k δn) p Dn. 21) Since by hypothesis D n < n=0 we have and ε n <, n=0 ) k εn + QD n <, where Q = δn) p > 0. n=0 Therefore, from Lemma 2.1 we know that lim dy n, F) exists. Thus, the conclusion iii) holds. This completes the proof of Lemma 3.2. The Proof of Theorem 3.1

7 G. S. Saluja / FAAC 7 1) 2015), The necessity of the conclusion i) is obvious and the sufficiency follows from conclusion ii) by setting ε n = 0 for all n 0 in 14) and considering 13). Now, we prove the conclusion ii) holds. It follows from Lemma 3.2iii) that lim dy n, F) exists. Since by hypothesis lim inf dy n, F) = 0, so by Lemma 2.6, we have lim dy n, F) = 0. First, we have to prove that y n } is a Cauchy sequence in E. In fact, it follows from 22), the assumptions n=0 ε n < and n=0 D n <, that for any given ε > 0 there exists a positive integer n 1 such that and dy n, F) < ε, n n 1 23) 1 D s < ε Q, where Q = n=n 1 ε n < ε. k δ p n, 24) By the definition of infimum, it follows from 23) that for any given n n 1 there exists an q n) F such that 22) 25) y n q n) < 2ε. 26) On the other hand, for any m, n n 1, without loss of generality m > n 1, it follows from Lemma 3.2ii) that y m y n y m q n) + y n q n) y n q n) + Q D s + + y n q n), where Q = ε s k δ p n, = 2 y n q n) + Q D s + ε s. 27) Therefore by 24) - 27), for any m > n n 1, we have y m y n < 4ε + ε + ε = 6ε. 28) This shows that y n } is a Cauchy sequence in E. Since E is complete, there exists an z E such that y n z as n. Now, we prove that z is a common fixed point of T 1, T 2,..., T k in K. Since y n z and dy n, F) 0 as n, for any given ε > 0, there exists a positive integer n 2 n 1 such that y n z < ε, dy n, F) < ε, 29) for all n n 2. The second inequality in 29) implies that there exists z F such that 1 y n2 z 1 < 2ε. 30)

8 Thus, from 12), 29) and 30) and for any n n 2, we have T n i z z T n i z z 1 + z z 1 D n + 2 z z 1 D n + 2 z y n2 + z 1 y n 2 ) G. S. Saluja / FAAC 7 1) 2015), < D n + 2ε + 2ε) = D n + 6ε = ε 1, 31) where ε 1 = D n + 6ε, since D n 0 as n and ε > 0, it follows that ε 1 > 0. The inequality 31) implies that T n i z z as n. Again since T n i z T i z T n i z z 1 + T iz z 1 D n + z z 1 + T iz z 1, 32) for all n n 2, by assumption 11) and using 29) and 30), we have T n i z T i z D n + z z 1 + L i z z 1 γ i D n + z y n2 + z 1 y n 2 +L i z y n2 + z 1 y n 2 ) γ i < D n + 3ε + L i 3ε) γ i = ε 1 33) where ε 1 = D n + 3ε + L i 3ε) γ i, since D n 0 as n, ε > 0 and L i > 0 for all i 1, 2,..., k}, it follows that ε 1 > 0, the inequality 33) shows that Tn i z T i z as n. By the uniqueness and the continuity of limit, we have T i z = z for all i 1, 2,..., k}, that is, z is a common fixed point of T 1, T 2,..., T k in K. Therefore, the conclusion ii) holds. From 12) and 14)-18), we have, for any given ε > 0, ε n y n+1 z + 1 α 1,n )y n z ) + α 1,n T n 1 w 1,n z ) y n+1 z + 1 α 1,n ) y n z + α 1,n T n 1 w 1,n z ) y n+1 z + 1 α 1,n ) y n z + α 1,n w1,n z + D n y n+1 z + 1 α 1,n ) y n z + α 1,n w 1,n z + α 1,n D n y n+1 z + y n z + α 1,n D n + α 1,n α 2,n D n α 1,n α 2,n... α k,n D n y n+1 z + y n z + 1 } α i,n Dn + 2 } α i,n Dn + + k } α i,n Dn y n+1 z + y n z + δ 1 nd n + δ 2 nd n + + δ k nd n y n+1 z + y n z + k δn) p Dn = y n+1 z + y n z + Q D n 34) for all n 0, where δ l n = l α i,n and Q = ) k δp n > 0. Since yn z, and n=0 D n <, it follows that lim ε n = 0. Thus the conclusion iii) holds. This completes the proof of Theorem 3.1. If we take T 1 = T 2 = = T k = T in Theorem 3.1, then we obtain the following conclusion.

9 G. S. Saluja / FAAC 7 1) 2015), Theorem 3.3. Let E be a real Banach space, K a nonempty closed convex subset of E. Let T : K K be an asymptotically quasi-nonexpansive mapping in the intermediate sense defined by Definition 2.17). Assume that there exist constants L and γ > 0 such that Put Tx q L x q γ, x K, q FT). G n = max 0, sup T n x q x q )}, x K, q FT) such that n=0 G n <. For any given x 0 K, define the k-step iterative sequence x n } by z k 1,n = 1 α k,n )x n + α k,n T n x n, z k 2,n = 1 α k 1,n )x n + α k 1,n T n z k 1,n,. z 1,n = 1 α 2,n )x n + α 2,n T n z 2,n, x n+1 = 1 α 1,n )x n + α 1,n T n z 1,n, n 0, where α i,n } is a sequence in [0, 1]. Suppose that y n } is a sequence in K and define a sequence ε n } of positive real numbers by w k 1,n = 1 α k,n )y n + α k,n T n y n, w k 2,n = 1 α k 1,n )y n + α k 1,n T n w k 1,n,. w 1,n = 1 α 2,n )y n + α 2,n T n w 2,n, ε n = y n+1 1 α 1,n )y n α 1,n T n w 1,n, n 0. If FT), then we have the following: i) x n } converges strongly to a fixed point q of T in K if and only if lim inf dx n, FT)) = 0. ii) n=0 ε n < and lim inf dy n, FT)) = 0 imply that y n } converges strongly to a fixed point q of T in K. iii) If y n } converges strongly to a fixed point q of T in K, then lim ε n = 0. Remark 3.4. Theorem 3.1 and 3.3 extend, improve and unify the corresponding results of [3], [4], [9], [12]-[14], [17], [20]-[26] to the case of more general class of quasi-nonexpansive, asymptotically nonexpansive, asymptotically quasi-nonexpansive, asymptotically quasi-nonexpansive type mappings and k-step iterative sequence considered in this paper. 4. Conclusion The class of asymptotically quasi-nonexpansive mappings in the intermediate sense is more general than the class of quasi nonexpansive, asymptotically nonexpansive, asymptotically quasi-nonexpansive and asymptotically quasi-nonexpansive type mappings. Therefore, the results presented in this paper are improvement and generalization of several known results in the current existing literature.

10 G. S. Saluja / FAAC 7 1) 2015), References [1] R.P. Agarwal, N.J. Huang and Y.J. Cho, Stability of iterate procedures with errors for nonlinear equations of ϕ-strongly accretive type operators, Numer. Funct. Anal. Optimiz ), [2] R.P. Agarwal, Y.J. Cho, J. Li and N.J. Huang, Stability of iterate procedures with errors approximating common fixed point for a couple of quasi-contractive mappings in a q-uniformly smooth Banach spaces, J. Math. Anal. Appl ), [3] S.S. Chang, On the approximating problem of fixed points for asymptotically nonexpansive mappings, Indian J. Pure and Appl. 329) 2001), [4] M.K. Ghosh, L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings, J. Math. Anal. Appl ), [5] K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc ), [6] A.M. Harder, Fixed point theory and stability results for fixed point iteration procedures, Ph.D. Thesis, University of Missouru- Rolla, [7] A.M. Harder and T.L. Hicks, A stable iteration procedure for nonexpansive mappings, Math. Japon ), [8] A.M. Harder and T.L. Hicks, Stability results for fixed point iteration procedures, Math. Japon ), [9] Z.Y. Huang, Mann and Ishikawa iterations with errors for asymptotically nonexpansive mappings, Comput. Math. Appl ), 1 7. [10] G.E. Kim and T.H. Kim, Mann and Ishikawa iterations with errors for non-lipschitzian mappings in Banach spaces, Comput. Math. Appl ), [11] W.A. Kirk, Fixed point theorems for non-lipschitzian mappings of asymptotically nonexpansive type, Israel J. Math ), [12] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl ), 1 7. [13] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math. Anal. Appl ), [14] Q.H. Liu, Iteration sequences for asymptotically quasi-nonexpansive mappings with error member of uniformly convex Banach spaces, J. Math. Anal. Appl ), [15] M.O. Osilike, Stable iteration procedures for strong pseudocontractions and nonlinear equations of the accretive type, J. Math. Anal. Appl ), [16] M.O. Osilike, Stability of the Mann and Ishikawa iteration procedures for ϕ-strong pseudocontractions and nonlinear equations of the ϕ-strongly accretive type, J. Math. Anal. Appl ), [17] W.V. Petryshyn, T.E. Williamson, Strong and weak convergence of the sequence of successive approximations for quasinonexpansive mappings, J. Math. Anal. Appl ), [18] B.E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures, Indian J. Pure and Appl ), 1 9. [19] B.E. Rhoades, Fixed point theorems and stability results for fixed point iteration procedures II, Indian J. Pure and Appl ), [20] D.R. Sahu and J.S. Jung, Fixed point iteration processes for non-lipschitzian mappings of asymptotically quasi-nonexpansive type, Int. J. Math. Math. Sci ), [21] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl ), [22] N. Shahzad and A. Udomene, Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2006), Article ID 18909, Pages [23] K.K. Tan and H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl ), [24] K.K. Tan and H.K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc ), [25] Y. Yao and Y.-C. Liou, New iterative scheme for asymptotically quasi-nonexpansive mappings, J. Inequal. Appl. 2010), Article ID , 9 pages, doi: /2010/ [26] L.C. Zeng, A note on approximating fixed points of nonexpansive mapping by the Ishikawa iterative process, J. Math. Anal. Appl ),