Báo cáo hóa học: " COMMON FIXED POINTS OF A FINITE FAMILY OF ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPS" potx

COMMON FIXED POINTS OF A FINITE FAMILY OF ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPS M. O. OSILIKE AND B. G. AKUCHU Received 3 December 2003 and in revised form 13 February 2004 Convergence theorems for approximation of common fixed points of a finite family of asymptotically pseudocontractive mappings are proved in Banach spaces using an aver- aging implicit iteration process. 1. Introduction Let E be a real Banach space and let J denote the normalized duality mapping from E into 2 E ∗ given by J(x) ={f ∈ E ∗ : x, f =x 2 =f  2 },whereE ∗ denotes the dual space of E and ·,· denotes the generalized duality pairing. If E ∗ is strictly convex, then J is single-valued. In the sequel, we will denote the single-valued duality mapping by j. Let K be a nonempty subset of E.AmappingT : K → K is said to be asymptoti- cally pseudocontractive (see, e.g., [3]) if there exists a sequence {a n } ∞ n=1 ⊆ [1,∞)suchthat lim n→∞ a n = 1and  T n x − T n y, j(x − y)  ≤ a n x − y 2 , ∀n ≥ 1, (1.1) for all x, y ∈ K, j(x − y) ∈ J(x − y). In Hilbert spaces H, a self-mapping T of a nonempty subset K of H is asymptotically pseudocontractive if it satisfies the simpler inequality   T n x − T n y   2 ≤ a n x − y 2 +   x − y −  T n x − T n y    2 , ∀n ≥ 1 (1.2) for all x, y ∈ K and for some sequence {a n } ∞ n=1 ⊆ [1,∞) such that lim n→∞ a n = 1. The class of asymptotically pseudocontractive mappings contains the important class of asymptot- ically nonexpansive mappings (i.e., mappings T : K → K such that   T n x − T n y   ≤ a n x − y, ∀n ≥ 1, ∀x, y ∈ K, (1.3) and for some sequence {a n } ∞ n=1 ⊆ [1,∞) such that lim n→∞ a n = 1). T is called asymp- totically quasi-nonexpansive if F(T) ={x ∈ K : Tx = x} =∅and (1.3) is satisfied for all x ∈ K and for all y ∈ F(T). If there exists L>0suchthatT n x − T n y≤Lx − y for Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:2 (2004) 81–88 2000 Mathematics Subject Classification: 47H09, 47H10, 47J05, 65J15 URL: http://dx.doi.org/10.1155/S1687182004312027 82 Asymptotically pseudocontractive maps all n ≥ 1andforallx, y ∈ K,thenT is said to be uniformly L-Lipschitzian.Amapping T : K → K is said to be semicompact (see, e.g ., [4]) if for any sequence {x n } ∞ n=1 in K such that lim n→∞ x n − Tx n =0, there exists a subsequence {x n j } ∞ j=1 of {x n } ∞ n=1 such that {x n j } ∞ j=1 converges strongly to some x ∗ ∈ K. In [5], Xu and Ori introduced an implicit iteration process and proved weak con- vergence theorem for approximation of common fixed points of a finite family of non- expansive mappings (i.e., a subclass of asymptotically nonexpansive mappings for which Tx− Ty≤x − y∀x, y ∈ K). In [4], Sun modified the implicit iteration process of Xu and Ori and applied the mod- ified averaging iteration process for the approximation of fixed points of asymptotically quasi-nonexpansive maps. If K is a nonempty closed convex subset of E,and{T i } N i=1 is N asymptotically quasi-nonexpansive self-maps of K,thenforx 0 ∈ K and {α n } ∞ n=1 ⊆ (0,1), the iteration process is generated as follows: x 1 = α 1 x 0 +  1 − α 1  T 1 x 1 , x 2 = α 2 x 1 +  1 − α 2  T 2 x 2 , . . . x N = α N x N−1 +  1 − α N  T N x N , x N+1 = α N+1 x N +  1 − α N+1  T 2 1 x N+1 , x N+2 = α N+2 x N+1 +  1 − α N+2  T 2 2 x N+2 , . . . x 2N = α 2N x 2N−1 +  1 − α 2N  T 2 N x 2N , x 2N+1 = α 2N+1 x 2N +  1 − α 2N+1  T 3 1 x 2N+1 , . . . (1.4) The iteration process can be expressed in a compact form as x n = α n x n−1 +  1 − α n  T k i x n , n ≥ 1, (1.5) where n = (k − 1)N + i, i ∈ I ={1,2, , N}. Assuming that the implicit iteration process is defined in K, Sun proved the following theorem. Theorem 1.1. Let E be a Banach space and let K be a nonempty closed convex subs et of E. Let {T i } N i=1 be N asymptotically quasi-nonexpansive self-maps of K (i.e., T n i x − p i ≤[1 + u in ]x − p i  for all n ≥ 1,forallx ∈ K,andforallp i ∈ F(T i ), i ∈ I). Let F =∩ N i=1 F(T i ) = ∅ and let  ∞ n=1 u in < ∞ for all i ∈ I.Letx 0 ∈ K, s ∈ (0,1),and{α n } ∞ n=1 ⊂ (s,1− s). Then the implicit iteration process (1.5) converges strongly to a common fixed point of the family {T i } N i=1 if and only if liminf n→∞ d(x n ,F) = 0,whered(x n ,F) = inf p∈F x n − p. Theorem 1.2. Let E be a real uniformly convex Banach space and K anonemptyclosed convex bounded subset of E.Let {T i } N i=1 be N uniformly Lipschitzian asymptotically quasi- nonexpansive self-maps of K such that  ∞ n=1 u in < ∞ for all i ∈ I.LetF =∩ N i=1 F(T i ) =∅ M. O. Osilike and B. G. Akuchu 83 and let one member of the family {T i } N i=1 be semicompact. Let x 0 ∈ K and let s and {α n } ∞ n=1 be as in Theorem 1.1. Then the iteration process (1.5) converges strongly to a common fixed point of the family {T i } N i=1 . Observe that if T : K → K is a uniformly L-Lipschitzian asymptotically pseudocon- tractive map with sequence {a n } ∞ n=1 ⊆ [1,∞)suchthatlim n→∞ a n = 1, then for e very fixed u ∈ K and t ∈ (L/(1 + L),1), the operator S t,n : K → K defined for all x ∈ K by S t,n x = tu+(1− t)T n x (1.6) satisfies   S t,n x − S t,n y   ≤ (1− t)Lx − y, ∀x, y ∈ K. (1.7) Since (1 − t)L ∈ (0,1), it follows that S t,n is a contraction map and hence has a unique fixed point x t,n in K. This implies that there exists a unique x t,n ∈ K such that x t,n = tu+(1− t)T n x t,n . (1.8) Thus the implicit iteration process (1.5)isdefinedinK for the family {T i } N i=1 of N uni- formly L i -Lipschitzian asymptotically pseudocontractive self-mappings of a nonempty convex subset K of a Banach space provided that α n ∈ (α,1) for all n ≥ 1, where α = L/(1 + L)andL = max 1≤i≤N {L i }. It is our purpose in this paper to first extend Theorem 1.1 to the class of uniformly L- Lipschitzian asymptotically pseudocontractive mappings. The condition  ∞ n=1 (a in − 1) < ∞ for all i ∈ I ={1,2, ,N} which is equivalent to the condition  ∞ n=1 u in < ∞ for all i ∈ I assumed in Theorems 1.1 and 1.2 is not imposed in our theorem. We do not want to make the general assumption that the iteration process is defined. If one assumes that the iteration process is always defined, our result will hold for even the more general class of asymptotically hemicontractive maps (i.e., mappings for which F(T) =∅and (1.1)holds for all x ∈ K and y ∈ F(T)). If E = H,aHilbertspace,weobtainastrongconvergence theorem similar to Theorem 1.2 fortheclassofuniformlyL-Lipschitzian asymptotically pseudocontractive maps. In the sequel we will need the following lemma. Lemma 1.3 [1, page 80]. Let {a n } ∞ n=1 , {b n } ∞ n=1 ,and{δ n } ∞ n=1 be sequences of nonnegative real numbers satisfying the inequality a n+1 ≤  1+δ n  a n + b n , n ≥ 1. (1.9) If  ∞ n=1 δ n < ∞ and  ∞ n=1 b n < ∞, then lim n→∞ a n exists. If in addition {a n } ∞ n=1 has a subse- quence which converges strongly to zero, then lim n→∞ a n = 0. Throughout the remaining part of this paper, {T i } N i=1 is a finite family of uniformly L i -Lipschitzian asymptotically pseudocontractive self-maps of a nonempty closed convex 84 Asymptotically pseudocontractive maps subset K of a Banach space so that  T n i x − T n i y, j(x − y)  ≤ a in x − y 2 , ∀n ≥ 1, ∀i ∈ I ={1,2, ,N}, ∀x, y ∈ K, (1.10) and for some sequences {a in } ∞ n=1 , i ∈ I, with lim n→∞ a in = 1, for all i ∈ I; T n i x − T n i y≤ L i x − y for all n ≥ 1, for all i ∈ I,forallx, y ∈ K,andforsomeL i > 0, i ∈ I. L = max 1≤i≤N {L i }. Theorem 1.4. Let E be a real Banach space and K a nonempty closed convex subset of E. Let {T i } N i=1 be N uniformly L i -Lipschitzian asymptotically pseudocontractive self-maps of K such that F =∩ N i=1 F(T i ) =∅.Letx 0 ∈ K and let {α n } ∞ n=1 be a real sequence in (α,1) satisfying the condition  ∞ n=1 (1 − α n ) < ∞,whereα = (1 + L)/(2 + L) (so that 2 α − 1 > 0). Then the implicit iteration sequence {x n } ∞ n=1 generated by (1.5)existsinK and converges strongly to a common fixed point of the family {T i } N i=1 if and only if liminf n→∞ d(x n ,F) = 0, where d(x n ,F) = inf p∈F x n − p. Proof. We will use the well-known inequality x + y 2 ≤x 2 +2  y, j(x + y)  (1.11) which holds for all x, y ∈ E and for all j(x − y) ∈ J(x − y) and which was first proved in [2]. Let p ∈ F, then using (1.1), (1.5), and (1.11), we obtain   x n − p   2 =   α n  x n−1 − p  +  1 − α n  T k i x n − p    2 ≤ α 2 n   x n−1 − p   2 +2  1 − α n  T k i x n − p, j  x n − p  ≤ α 2 n   x n−1 − p   2 +2  1 − α n  a ik   x n − p   2 . (1.12) Observe that since lim k→∞ a ik = 1foralli ∈ I, then there exists N 0 such that for all k>N 0 /N +1 (i.e., for all n ≥ N 0 ), we have a ik ≤ 1+(2α − 1)/4(1 − α)foralli ∈ I.Conse- quently, for all k>N 0 /N +1(foralln ≥ N 0 ), we have 1 − 2(1 − α n )a ik ≥ (1/2)(2α − 1) > 0. Let a = max 1≤i≤N {sup k≥1 {a 1k },sup k≥1 {a 2k }, ,sup k≥1 {a Nk }}.Thenforallk>N 0 / N +1(foralln ≥ N 0 ), it follows from the last inequality in (1.12)that   x n − p   2 ≤  α 2 n  1 − 2  1 − α n  a ik     x n−1 − p   2 =  1+ 2  1 − α n  a ik − 1   1 − 2  1 − α n  a ik  +  1 − α n  2  1 − 2  1 − α n  a ik     x n−1 − p   2 ≤  1+4a[2α − 1] −1  1 − α n  +2[2α − 1] −1  1 − α n  2    x n−1 − p   2 =  1+σ n    x n−1 − p   2 , (1.13) where σ n = 4a[2α − 1] −1 (1 − α n )+2[2α − 1] −1 (1 − α n ) 2 .Since  ∞ n=1 σ n < ∞,itfollows from the last equality in (1.13)andLemma 1.3 that lim n→∞ x n − p exists so that there M. O. Osilike and B. G. Akuchu 85 exists M>0suchthatx n − p≤M for all n ≥ 1. Consequently, we obtain from the last equality in (1.13)that   x n − p   ≤  1+σ n  1/2   x n−1 − p   ≤  1+σ n    x n−1 − p   ≤   x n−1 − p   + Mσ n . (1.14) It follows f rom (1.14)that d  x n ,F  ≤  1+σ n  d  x n−1 ,F  , (1.15) so that it again follows from Lemma 1.3 that lim n→∞ d(x n ,F) exists. If {x n } ∞ n=1 converges strongly to a common fixed point p of the family {T i } N i =1 ,then lim n→∞ x n − p=0. Since 0 ≤ d  x n ,F  ≤   x n − p   , (1.16) we have lim i nf d(x n ,F) = 0. Conversely suppose liminf n→∞ d(x n ,F) = 0, then we have lim n→∞ d(x n ,F) = 0. Thus for arbitrary  > 0, there exists a positive integer N 1 such that d  x n ,F  <  4 , ∀n ≥ N 1 . (1.17) Furthermore,  ∞ n=1 σ n < ∞ implies that there exists a positive integer N 2 such that  ∞ j=n σ j < /4M for all n ≥ N 2 .ChooseN = max{N 0 ,N 1 ,N 2 }. Then d(x N ,F) ≤  /4and  ∞ j=N σ j < /4M.Itfollowsfrom(1.14)thatforalln,m ≥ N and for all p ∈ F,wehave   x n − x m   ≤   x n − p   +   x m − p   ≤   x N − p   + M n  j=N+1 σ j +   x N − p   + M m  j=N+1 σ j ≤ 2   x N − p   +2M ∞  j=N σ j . (1.18) Taking infinimum over all p ∈ F,weobtain   x n − x m   ≤ 2d  x N ,F  +2M ∞  j=N σ j < . (1.19) Thus {x n } ∞ n=1 is Cauchy. Suppose lim n→∞ x n = u.Thenu ∈ K since K is closed. Further- more, since F(T i )isclosedforalli ∈ I,wehavethatF is closed. Since lim n→∞ d(x n ,F) = 0, we have that u ∈ F.  Remark 1.5. Prototype for the iteration parameter {α n } in Theorem 1.4 is α n = α+n 2 (1 − α)/(n 2 +1)n ≥ 1. 86 Asymptotically pseudocontractive maps Theorem 1.6. Let H be a real Hilbert space and let K be a nonempty closed convex subset of H.Let{T i } N i=1 be N uniformly L i -Lipschitzian asymptotically pseudocontractive self-maps of K such that F =∩ N i=1 F(T i ) =∅and  ∞ n=1 (a in − 1) < ∞ for all i ∈ I.Letonememberofthe family {T i } N i=1 be semicompact. Let x 0 ∈ K and let {α n } ∞ n=1 be a sequence in (0,1) such that 0 <α≤ α n ≤ β<1 for all n ≥ 1,whereα = L/(1 + L). Then the implicit iteration sequence {x n } ∞ n=1 generated by (1.5) exists in K and converges strongly to a common fixed point of the family {T i } N i=1 . Proof. We will use the well-known identity   tx +(1− t)y   2 = tx 2 +(1− t)y 2 − t(1 − t)x − y 2 (1.20) which holds in Hilbert spaces H for all x, y ∈ H and for all t ∈ [0,1]. Let p ∈ F, then using (1.2)and(1.20), we obtain   x n − p   2 =   α n  x n−1 − p  +  1 − α n  T k i x n − p    2 = α n   x n−1 − p   2 +  1 − α n    T k i x n − p   2 − α n  1 − α n    x n−1 − T k i x n   2 ≤ α n   x n−1 − p   2 +  1 − α n   a ik   x n − p   2 +   x n − T k i x n   2  − α n  1 − α n    x n−1 − T k i x n   2 = α n   x n−1 − p   2 +  1 − α n  a ik   x n − p   2 − α n  1 − α n  2   x n−1 − T k i x n   2 . (1.21) Observe that since lim k→∞ a ik = 1foralli ∈ I, then there exists N 0 such that for all k> N 0 /N + 1 (i.e., for all n ≥ N 0 ), we have a ik ≤ 1+α 2 /(1 − α)foralli ∈ I. Consequently, for all k>N 0 /N +1(foralln ≥ N 0 ), we have 1 − (1 − α n )a ik ≥ α(1 − α) > 0. Thus for all k>N 0 /N +1(foralln ≥ N 0 ), it follows from (1.21)that   x n − p   2 ≤  α n  1 −  1 − α n  a ik     x n−1 − p   2 − α n  1 − α n  2   x n−1 − T k i x n   2 =  1+  1 − α n  a ik − 1   1 −  1 − α n  a ik     x n−1 − p   2 − α n  1 − α n  2   x n−1 − T k i x n   2 ≤  1+  α(1 − α)  −1  1 − α n  a ik − 1    x n−1 − p   2 − α n  1 − α n  2   x n−1 − T k i x n   2 ≤  1+  α(1 − α)  −1  a ik − 1    x n−1 − p   2 − α(1 − β) 2   x n−1 − T k i x n   2 =  1+σ ik    x n−1 − p   2 − α(1 − β) 2   x n−1 − T k i x n   2 , (1.22) where σ ik = [α(1 − α)] −1 (a ik − 1). Since  ∞ k=1 σ ik < ∞, it follows from the last equality in (1.22)andLemma 1.3 that lim n→∞ x n − p exists. Furthermore, there exists D>0such that x n − p≤D for a ll n ≥ 1. Thus from the last equality in (1.22), we obtain   x n − p   2 ≤   x n−1 − p   2 − α(1 − β) 2   x n−1 − T k i x n   2 + D 2 σ ik , (1.23) M. O. Osilike and B. G. Akuchu 87 from which it follows that lim n→∞ x n−1 − T k i x n =0. Thus lim n→∞ x n−1 − T k n x n =0. Furthermore,   x n − T k n x n   = α n   x n−1 − T k n x n   −→ 0asn −→ ∞ ,   x n − x n−1   =  1 − α n    x n−1 − T k n x n   −→ 0asn −→ ∞ . (1.24) Thus lim n→∞ x n − x n+i =0foralli ∈ I.Foralln>N,wehaveT n = T n−N so that   x n−1 − T n x n   ≤   x n−1 − T k n x n   +   T k n x n − T n x n   ≤   x n−1 − T k n x n   + L   T k−1 n x n − x n   ≤   x n−1 − T k n x n   + L    T k−1 n x n − T k−1 n−N x n−N   +   T k−1 n−N x n−N − x (n−N)−1   +   x (n−N)−1 − x n    ≤   x n−1 − T k n x n   + L 2   x n − x n−N   + L   x (n−N)−1 − T k−1 n−N x n−N   + L   x n − x (n−N)−1   −→ 0asn −→ ∞ . (1.25) Hence,   x n − T n x n   ≤   x n − x n−1   +   x n−1 − T n x n   −→ 0asn −→ ∞ . (1.26) Consequently, for all i ∈ I,wehave   x n − T n+i x n   ≤   x n − x n+i   +   x n+i − T n+i x n+i   + L   x n+i − x n   = (1 + L)   x n+i − x n   +   x n+i − T n+i x n+i   −→ 0asn −→ ∞ . (1.27) It follows that lim n→∞ x n − T i x n =0foralli ∈ I.Sinceonememberof{T i } N i=1 is semi- compact, then there exists a subsequence {x n j } ∞ j=1 of the sequence {x n } ∞ n=1 such that {x n j } ∞ j=1 converges strongly to u.SinceK is closed, u ∈ K, and furthermore,   u − T i u   = lim j→∞   x n j − T i x n j   = 0 ∀i ∈ I. (1.28) Thus u ∈ F.Since{x n j } ∞ j=1 converges strongly to u and lim n→∞ x n − u exists, it follows from Lemma 1.3 that {x n } ∞ n=1 converges strongly to u.  Remark 1.7. Prototype for the iteration parameter {α n } in Theorem 1.6 is α n = α+n(1 − α)/2(n +1)n ≥ 1, for which 0 <α<α+(1− α)/4 ≤ α n <α+(1− α)/2 < 1. Acknowledg ments This work was completed when the first author was visiting the Abdus Salam Interna- tional Centre for Theoretical Physics, Trieste, Italy. He is grateful to the Committee on Development and Exchanges (CDE) of the International Mathematical Union, and the University of Nigeria, Nsukka, for generous travel support. His research is supported by a grant from TWAS (99-181 RG/MATHS/AF/AC). The authors thank the referees for their useful comments on the original manuscript and for drawing the authors attention to reference [2]. 88 Asymptotically pseudocontractive maps References [1] M.O.Osilike,S.C.Aniagbosor,andB.G.Akuchu,Fixed points of asymptotically demicontrac- tive mappings in arbitrary Banach spaces, Panamer. Math. J. 12 (2002), no. 2, 77–88. [2] W. V. Petryshyn, A characterization of strict convexity of Banach spaces and othe r uses of duality mappings, J. Funct. Anal. 6 (1970), 282–291. [3] J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings,J.Math. Anal. Appl. 158 (1991), no. 2, 407–413. [4] Z H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 286 (2003), no. 1, 351–358. [5] H K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001), no. 5-6, 767–773. M. O. Osilike: Department of Mathematics, University of Nigeria, Nsukka, Nigeria E-mail address: osilike@yahoo.com B. G. Akuchu: Department of Mathematics, University of Nigeria, Nsukka, Nigeria E-mail address: akuchubg@yahoo.com . 2004 Convergence theorems for approximation of common fixed points of a finite family of asymptotically pseudocontractive mappings are proved in Banach spaces using an aver- aging implicit iteration process. 1 M.O.Osilike,S.C.Aniagbosor,andB.G.Akuchu ,Fixed points of asymptotically demicontrac- tive mappings in arbitrary Banach spaces, Panamer. Math. J. 12 (2002), no. 2, 77–88. [2] W. V. Petryshyn, A characterization. iteration process and proved weak con- vergence theorem for approximation of common fixed points of a finite family of non- expansive mappings (i.e., a subclass of asymptotically nonexpansive mappings