CONVERGENCE THEOREMS FOR STRICTLY ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN HILBERT SPACES. Gurucharan Singh Saluja

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1 Opuscula Mathematica Vol 30 No CONVERGENCE THEOREMS FOR STRICTLY ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN HILBERT SPACES Gurucharan Singh Saluja Abstract In this paper, we establish the weak and strong convergence theorems for a k-strictly asymptotically pseudo-contractive mapping in the framework of Hilbert spaces Our result improve and extend the corresponding result of Acedo and Xu, Liu, Marino and Xu, Osilike and Akuchu, and some others Keywords: strictly asymptotically pseudo-contractive mapping, implicit iteration scheme, common fixed point, strong convergence, weak convergence, Hilbert space Mathematics Subject Classification: 47H09, 47H10 1 INTRODUCTION AND PRELIMINARIES Let H be a real Hilbert space with the scalar product and norm denoted by the symbols, and respectively, and C be a closed convex subset of H Let T be a (possibly) nonlinear mapping from C into C We now consider the following classes: (1) T is contractive, ie, there exists a constant k < 1 such that for all x, y C (2) T is nonexpansive, ie, T x T y k x y, (11) T x T y x y, (12) for all x, y C (3) T is uniformly L-Lipschitzian, ie, there exists a constant L > 0 such that for all x, y C and n N T n x T n y L x y, (13) 485

2 486 Gurucharan Singh Saluja (4) T is pseudo-contractive, ie, T x T y, x y x y 2, (14) for all x, y C (5) T is k-strictly pseudo-contractive, ie, if there exists a constant k [0, 1) such that T x T y 2 x y 2 + k (x T x) (y T y) 2, (15) for all x, y C (6) T is asymptotically nonexpansive [3], ie, if there exists a sequence {r n } [0, ) with lim n r n = 0 such that T n x T n y (1 + r n ) x y, (16) for all x, y C and n N (7) T is k-strictly asymptotically pseudo-contractive [10], ie, if there exists a sequence {r n } [0, ) with lim n r n = 0 such that T n x T n y 2 (1 + r n ) 2 x y 2 + k (x T n x) (y T n y) 2 (17) for some k [0, 1) for all x, y C and n N Remark 11 ([10]) If T is a k-strictly asymptotically pseudo-contractive mapping, then it is uniformly L-Lipschitzian, but the converse does not hold The class of strictly pseudo-contractive mappings have been studied by several authors (see, for example [2, 4, 8, 12] and references therein) In the case of a contractive mapping, the Banach Contraction Principle guarantees not only the existence of a unique fixed point, but also obtain the fixed point by successive approximation (or Picard iteration) But outside the class of contractive mappings, the classical iteration scheme no longer applies So some other iteration scheme is required Two iteration processes are often used to approximate the fixed point of nonexpansive and pseudo-contractive mappings The first iteration process is known as Mann s iteration [9], where {x n } is defined as x n+1 = α n x n + (1 α n )T x n, n 0, (18) where the initial guess x 0 is taken in C arbitrary and the sequence {α n } is in the interval [0, 1] The second iteration process is known as Ishikawa iteration process [5] which is defined by x n+1 = α n x n + (1 α n )T y n, (19) y n = β n x n + (1 β n )T x n ; n 0, where the initial guess x 0 is taken in C arbitrary and {α n } and {β n } are sequences in the interval [0, 1]

3 Convergence theorems for strictly asymptotically pseudocontractive mappings 487 Process (19) is indeed more general than the process (18) But research has been concentrated on the later, probably due to the reason that process (18) is simpler and that a convergence theorem for process (18) may possibly lead to a convergence theorem for process (19), provided that the sequence {β n } satisfy certain appropriate conditions If T is a nonexpansive mapping with a fixed point and the control sequence {α n } is chosen so that n=0 α n(1 α n ) =, then the sequence {x n } generated by Mann s iteration process (18) converges weakly to a fixed point of T (this is also valid in a uniformly convex Banach space with the Fréchet differentiable norm [14]) Recently, Marino and Xu [8] extended the results of Reich [14] from nonexpansive mappings to strict pseudo-contractions and obtained a weak convergence theorem in Hilbert spaces More precisely, they gave the following results Theorem 12 ([8]) Let C be a closed convex subset of a Hilbert space H Let T : C C be a k-strict pseudo-contraction for some 0 k < 1 and assume that T admits a fixed point in C Let {x n } n=0 be the sequence generated by Mann s algorithm (18) Assume that the control sequence {α n } n=0 is chosen so that k < α n < 1 for all n and n=0 (α n k)(1 α n ) = Then {x n } converges weakly to a fixed point of T In 2001, Xu and Ori [15] have introduced an implicit iteration process for a finite family of nonexpansive mappings in a Hilbert space H Let C be a nonempty subset of H Let T 1, T 2,, T N be self-mappings of C and suppose that F = N i=1 F (T i), the set of common fixed points of T i, i = 1, 2,, N An implicit iteration process for a finite family of nonexpansive mappings is defined as follows, with {t n } a real sequence in (0, 1), x 0 C: x 1 = t 1 x 0 + (1 t 1 )T 1 x 1, x 2 = t 2 x 1 + (1 t 2 )T 2 x 2, x N = t N x N 1 + (1 t N )T N x N, x N+1 = t N+1 x N + (1 t N+1 )T 1 x N+1, which can be written in the following compact form: x n = t n x n 1 + (1 t n )T n x n, n 1, (110) where T k = T k mod N (Here the mod N function takes values in {1, 2,, N}) And they proved the weak convergence of the process (110) Very recently, Acedo and Xu [1] still in the framework of Hilbert spaces introduced the following cyclic algorithm Let C be a closed convex subset of a Hilbert space H and let {T i } N 1 i=0 be N k-strict pseudo-contractions on C such that F = N 1 i=0 F (T i) Let x 0 C and

4 488 Gurucharan Singh Saluja let {α n } be a sequence in (0, 1) The cyclic algorithm generates a sequence {x n } n=1 in the following way: In general, {x n+1 } is defined by x 1 = α 0 x 0 + (1 α 0 )T 0 x 0, x 2 = α 1 x 1 + (1 α 1 )T 1 x 1, x N = α N 1 x N 1 + (1 α N 1 )T N 1 x N 1, x N+1 = α N x N + (1 α N )T 0 x N, x n+1 = α n x n + (1 α n )T [n] x n, (111) where T [n] = T i with i = n (mod N), 0 i N 1 They also proved a weak convergence theorem for k-strict pseudo-contractions in Hilbert spaces by cyclic algorithm (111) More precisely, they obtained the following theorem: Theorem 13 ([1]) Let C be a closed convex subset of a Hilbert space H Let N 1 be an integer Let for each 0 i N 1, T i : C C be a k i -strict pseudo-contraction for some 0 k i < 1 Let k = max{k i : 1 i N} Assume the common fixed point set N 1 i=0 F (T i) of {T i } N 1 i=0 is nonempty Given x 0 C, let {x n } n=0 be the sequence generated by the cyclic algorithm (111) Assume that the control sequence {α n } is chosen so that k + ɛ < α n < 1 ɛ for all n and for some ɛ (0, 1) Then {x n } converges weakly to a common fixed point of the family {T i } N 1 i=0 Motivated by Xu and Ori [15], Acedo and Xu [1] and some others we introduce and study the following: Let C be a closed convex subset of a Hilbert space H and let {T i } N 1 i=0 be N k-strictly asymptotically pseudo-contractions on C such that F = N 1 i=0 F (T i) Let x 0 C and let {α n } be a sequence in (0, 1) The implicit iteration scheme generates a sequence {x n } n=0 in the following way: x 1 = α 0 x 0 + (1 α 0 )T 0 x 0, x 2 = α 1 x 1 + (1 α 1 )T 1 x 1, x N = α N 1 x N 1 + (1 α N 1 )T N 1 x N 1, x N+1 = α N x N + (1 α N )T 2 0 x 0, x 2N = α 2N 1 x 2N 1 + (1 α 2N 1 )T 2 N 1x 2N 1, x 2N+1 = α 2N x 2N + (1 α 2N )T 3 0 x 0,

5 Convergence theorems for strictly asymptotically pseudocontractive mappings 489 In general, {x n } is defined by x n+1 = α n x n + (1 α n )T s [n] x n, (112) where T s [n] = T s n (mod N) = T s i with n = sn + i and i I = {0, 1,, N 1} The purpose of this paper is to establish weak and strong convergence theorems of the implicit iteration process (112) for finite family of k-strictly asymptotically pseudo-contraction mappings in Hilbert spaces Our results extend the corresponding results of Reich [14], Marino and Xu [8], Acedo and Xu [1] and many others In the sequel, we will need the following lemmas Lemma 14 Let H be a real Hilbert space There hold the following identities: (i) x y 2 = x 2 y 2 2 x y, y x, y H (ii) tx + (1 t)y 2 = t x 2 +(1 t) y 2 t(1 t) x y 2, t [0, 1], x, y H (iii) If {x n } is a sequence in H that weakly converges to z, then lim sup x n y 2 = lim sup x n z 2 + z y 2 y H n n We use following notation: 1 for weak convergence and for strong convergence 2 ω w (x n ) = {x : x nj x} denotes the weak ω-limit set of {x n } Lemma 15 ([13]) Let {a n } n=1, {β n } n=1 and {r n } n=1 be sequences of nonnegative real numbers satisfying the inequality a n+1 (1 + r n )a n + β n, n 1 If n=1 r n < and n=1 β n <, then lim n a n exists If in addition {a n } n=1 has a subsequence which converges strongly to zero, then lim n a n = 0 Lemma 16 Let H be a real Hilbert space, let C be a nonempty closed convex subset of H, and let T i : C C be a k i -strictly asymptotically pseudocontractive mapping for i = 0, 1,, N 1 with a sequence {r ni } [0, ) such that n=1 r n i < and for some 0 k i < 1, then there exist constants L > 0 and k [0, 1) and a sequence {r n } [0, ) with lim n r n = 0 such that for any x, y C and for each i = 0, 1,, N 1 and each n 1, the following hold: and T n i x T n i y (1 + r n ) 2 x y 2 + k (x T n i x) (y T n i y) 2, (113) T n i x T n i y L x y (114) Proof Since for each i = 0, 1,, N 1, T i is k i -strictly asymptotically pseudocontractive, where k i [0, 1) and {r ni } [0, ) with lim n r ni = 0 By Remark 11, T i is L i -Lipschitzian Taking r n = max{r ni, i = 0, 1,, N 1} and k = max{k i, i = 0, 1,, N 1}, hence, for each i = 0, 1,, N 1, we have T n i x T n i y (1 + r ni ) 2 x y 2 + k i (x T n i x) (y T n i y) 2 (1 + r n ) 2 x y 2 + k (x T n i x) (y T n i y) 2 (115)

6 490 Gurucharan Singh Saluja The conclusion (113) is proved Again taking L = max{l i : i = 0, 1,, N 1} for any x, y C, we have This completes the proof of Lemma 16 T n i x T n i y L i x y L x y (116) 2 MAIN RESULTS Theorem 21 Let C be a closed convex subset of a Hilbert space H Let N 1 be an integer Let for each 0 i N 1, T i : C C be N k i -strictly asymptotically pseudo-contraction mappings for some 0 k i < 1, n=1 r n < and I T [n] is demiclosed at zero Let k = max{k i : 0 i N 1} and r n = max{r ni : 0 i N 1} Assume that F = N 1 i=0 F (T i) Given x 0 C, let {x n } n=0 be the sequence generated by an implicit iteration scheme (112) Assume that the control sequence {α n } is chosen so that k + ɛ < α n < 1 ɛ for all n and for some ɛ (0, 1) Then {x n } converges weakly to a common fixed point of the family {T i } N 1 i=0 Proof Let p F = N 1 i=0 F (T i) It follows from (112) and Lemma 11 (ii) that x n+1 p 2 = α n x n + (1 α n )T[n] s x n p 2 = = α n (x n p) + (1 α n )(T[n] s x n p) 2 = = α n x n p 2 + (1 α n ) T[n] s x n p 2 α n (1 α n ) x n T[n] s x n 2 [ α n x n p 2 + (1 α n ) (1 + r n ) 2 x n p k x n T[n] s x n 2 ] α n (1 α n ) x n T[n] s x n 2 [ α n (1 + r n ) 2 + (1 α n )(1 + r n ) 2] x n p 2 (α n k)(1 α n ) x n T[n] s x n 2 = = (1 + r n ) 2 x n p 2 (α n k)(1 α n ) x n T[n] s x n 2 = = (1 + d n ) x n p 2 (α n k)(1 α n ) x n T s 2, [n] x n where d n = r 2 n + 2r n Since k + ɛ < α n < 1 ɛ for all n, from (21) we have x n+1 p 2 (1 + d n ) x n p 2 ɛ 2 xn T s [n] x n (21) 2 (22)

7 Convergence theorems for strictly asymptotically pseudocontractive mappings 491 Now (22) implies that x n+1 p 2 (1 + d n ) x n p 2 (23) Since n=1 r n < thus n=1 d n <, it follows by Lemma 12, we know that lim n x n p exists and so {x n } is bounded Consider (22) again yields that x n T[n] s x n 2 1 ɛ 2 [ x n p 2 x n+1 p 2 ] + d n ɛ 2 x n p 2 (24) Since {x n } is bounded and d n 0 as n So, we get x n T s 0 as n (25) [n] x n From the definition of {x n }, we have x n+1 x n = (1 α n ) x n T s [n] x n 0, as n (26) So, x n x n+l 0 as n and for all l < N Now for n N, and since T is uniformly Lipschitzian (by Remark 11) with Lipschitz constant L > 0, so we have xn T [n] x n xn T[n] s x n + T[n] s x n T [n] x n x n T[n] s x n + L T s 1 [n] x n x n [ (27) x n T[n] s x T s 1 n + L [n] x n T s 1 [n N] x n N + ] + T s 1 [n N] x n N x n N + xn N x n Since for each n N, n (n N) (mod N) Thus T [n] = T [n N], therefore from (27), we have xn T [n] x n xn T[n] s x n + L 2 x n x n N + (28) + L T s 1 [n N] x n N x n N + L xn N x n From (25) and (28), we obtain xn T [n] x n 0 as n (29) Consequently, for any l I = {0, 1,, N 1}, xn T [n+l] x n xn x n+l + xn+l T [n+l] x n+l + T[n+l] x n+l T [n+l] x n (1 + L) x n x n+l + xn+l T [n+l] x n+l 0 as n (210) This implies that lim xn T [l] x n = 0, l I = {0, 1,, N 1} (211) n

8 492 Gurucharan Singh Saluja Since I T [n] is demiclosed at zero, (210) implies that x n x, where x is a weak limit of {x n } and hence ω w (x n ) F = N 1 i=0 F (T i) Now we show that {x n } is weakly convergent Let p 1, p 2 ω w (x n ) and {x ni } and {x mj } be subsequences of {x n } which converges weakly to some p 1 and p 2 respectively Since lim n x n z exists for every z F and since p 1, p 2 F, we have lim x n p 1 2 = lim n j xmj p 1 2 = lim xmj p p2 p 1 2 = j = lim i x ni p p 2 p 1 2 = lim n x n p p 2 p 1 2 Hence p 1 = p 2 Thus {x n } converges weakly to a common fixed point of the family {T i } N 1 i=0 This completes the proof Theorem 22 Let C be a closed convex compact subset of a Hilbert space H Let N 1 be an integer Let for each 0 i N 1, T i : C C be N k i -strictly asymptotically pseudo-contraction mappings for some 0 k i < 1 and n=1 r n < Let k = max{k i : 0 i N 1} and r n = max{r ni : 0 i N 1} Assume that F = N 1 i=0 F (T i) Given x 0 C, let {x n } n=0 be the sequence generated by an implicit iteration scheme (112) Assume that the control sequence {α n } is chosen so that k + ɛ < α n < 1 ɛ for all n and for some ɛ (0, 1) Then {x n } converges strongly to a common fixed point of the family {T i } N 1 i=0 Proof We only conclude the difference By compactness of C this immediately implies that there is a subsequence {x nj } of {x n } which converges to a common fixed point of {T i } N 1 i=0, say, p Combining (23) with Lemma 15, we have lim n x n p = 0 This completes the proof For our next result, we shall need the following definition: Definition 23 A mapping T : C C is said to be semi-compact, if for any bounded sequence {x n } in C such that lim n x n T x n = 0 there exists a subsequence {x ni } {x n } such that lim i x ni = x C Theorem 24 Let C be a closed convex subset of a Hilbert space H Let N 1 be an integer For each 0 i N 1, let T i : C C be N k i -strictly asymptotically pseudo-contraction mappings for some 0 k i < 1 and n=1 r n < Let k = max{k i : 0 i N 1} and r n = max{r ni : 0 i N 1} Assume that F = N 1 i=0 F (T i) Given x 0 C, let {x n } n=0 be the sequence generated by an implicit iteration scheme (112) Assume that the control sequence {α n } is chosen so that k + ɛ < α n < 1 ɛ for all n and for some ɛ (0, 1) Assume that one member of the family {T i } N 1 i=0 be semi-compact Then {x n } converges strongly to a common fixed point of the family {T i } N 1 i=0 Proof Without loss of generality, we can assume that T 1 is semi-compact It follows from (211) that lim xn T [1] x n = 0 (212) n

9 Convergence theorems for strictly asymptotically pseudocontractive mappings 493 By the semi-compactness of T 1, there exists a subsequence {x nk } of {x n } such that x nk u C strongly Since C is closed, u C, and furthermore, lim x nk T [l] x nk = u T [l] u = 0, (213) n k for all l I = {0, 1,, N 1} Thus u F Since {x nk } converges strongly to u and lim n x n u exists, it follows from Lemma 15 that {x n } converges strongly to u This completes the proof Remark 25 Theorem 21 extends and improves the corresponding result of Reich [14] and Marino and Xu [8] from nonexpansive and strict pseudo-contraction mappings to the more general class of a finite family of k-strictly asymptotically pseudo-contraction mappings and implicit iteration schemes considered in this paper Remark 26 Theorem 21 also extends and improves the corresponding result of Acedo and Xu [1] from k-strictly pseudo-contraction mappings to the more general class of k-strictly asymptotically pseudo-contraction mappings Remark 27 Theorem 21 also extends and improves the corresponding result of Xu and Ori [15] from nonexpansive mappings to more the general class of k-strictly asymptotically pseudo-contraction mappings Remark 28 Theorem 22 extends and improves the corresponding result of Liu [7] in the following respects: (i) We removed the uniformly L-Lipschitzian condition (ii) The modified Mann iteration process is replaced by implicit iteration process for a finite family of mappings Remark 29 Theorem 24 extends and improves the corresponding result of Kim and Xu [6] Remark 210 Theorem 24 also extends and improves Theorem 16 of Osilike and Akuchu [11] from asymptotically pseudocontractive mappings to strictly asymptotically pseudocontractive mappings Acknowledgments The author sincerely thanks the referee for his valuable suggestions and comments on the manuscript REFERENCES [1] GL Acedo, HK Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal 67 (2007), [2] FE Browder, WV Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert spaces, J Math Anal Appl 20 (1967), [3] K Goebel, WA Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc Amer Math Soc 35 (1972),

10 494 Gurucharan Singh Saluja [4] TL Hicks, JR Kubicek, On the Mann iterative process in Hilbert space, J Math Anal Appl 59 (1977), [5] S Ishikawa, Fixed points by a new iteration method, Proc Amer Math Soc 44 (1974), [6] TH Kim, HK Xu, Convergence of the modified Mann s iteration method for asymptotically strictly pseudocontractive mapping, Nonlinear Anal (2007), doi:101016/jna [7] Q Liu, Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear Anal 26 (1996), [8] G Marino, HK Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J Math Anal Appl 329 (2007), [9] WR Mann, Mean value methods in iteration, Proc Amer Math Soc 4 (1953), [10] MO Osilike, Iterative approximation of fixed points of asymptotically demicontractive mappings, Indian J Pure Appl Math 29 (12), December 1998, [11] MO Osilike, BG Akuchu, Common fixed points of a finite family of asymptotically pseudocontractive maps, Fixed Point Theory Appl 2 (2004), [12] MO Osilike, A Udomene, Demiclosedness principle and convergence results for strictly pseudocontractive mappings of Browder-Petryshyn type, J Math Anal Appl 256 (2001), [13] MO Osilike, SC Aniagbosor, BG Akuchu, Fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces, Panamer Math J 12 (2002), [14] S Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J Math Anal Appl 67 (1979), [15] HK Xu, RG Ori, An implicit iteration process for nonexpansive mappings, Numer Funct Anal Optim 22 (2001), Gurucharan Singh Saluja saluja_1963@rediffmailcom Govt Nagarjun PG College of Science Department of Mathematics & Information Technology Raipur (Chhattisgarh), India Received: March 21, 2010 Revised: April 25, 2010 Accepted: April 29, 2010