Báo cáo hóa học: "Research Article Approximate Fixed Points for Nonexpansive and Quasi-Nonexpansive Mappings in Hyperspaces" pot

Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 520976, 16 pages doi:10.1155/2009/520976 Research Article Approximate Fixed Points for Nonexpansive and Quasi-Nonexpansive Mappings in Hyperspaces Zeqing Liu, 1 Jeong Sheok Ume, 2 and Shin Min Kang 3 1 Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China 2 Department of Applied Mathematics, Changwon National University, Changwon 641-773, South Korea 3 Department of Mathematics, Research Institute of Natural Science, Gyeongsang National University, Chinju 660-701, South Korea Correspondence should be addressed to Jeong Sheok Ume, jsume@changwon.ac.kr Received 9 May 2009; Accepted 14 December 2009 Recommended by W. A. Kirk This paper provides a few convergence results of the Ishikawa iteration sequence with errors for nonexpansive and quasi-nonexpansive mappings in hyperspaces. The results presented in this paper improve and generalize some results in the literature. Copyright q 2009 Zeqing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and Preliminaries Browder 1 and Kirk 2 established that a nonexpansive mapping T which maps a closed bounded convex subset C of a uniformly convex Banach space into itself has a fixed point in C. Since then, many researchers have studied, under various conditions, the convergence of the Mann and Ishikawa iteration methods dealing with nonexpansive and quasi-nonexpansive mappings see 3–11 and the references therein. Rhoades 9 pointed out that the Picard iteration schemes for nonexpansive mappings need not converge. Senter and Dotson 10 obtained conditions under which the Mann iteration schemes generated by nonexpansive and quasi-nonexpansiv mappings in uniformly convex Banach spaces, converge to fixed points of these mappings, respectively. Ishikawa 7 established that the Mann iteration methods can be used to approximate fixed points of nonexpansive mappings in Banach spaces. Deng 3 obtained similar results for Ishikawa iteration processes in normed linear spaces and Banach spaces. Our aim is to prove several convergence theorems of the Ishikawa iteration sequence with errors for nonexpansive and quasi-nonexpansive mappings in hyperspaces. Our results presented in this paper extend substantially the results due to Deng 3, Ishikawa 7,and Senter and Dotson 10. 2 Fixed Point Theory and Applications Assume that X is a nonempty subset of a normed linear space E, · and CCX denotes the family of all nonempty convex compact subsets of X,andH is the Hausdorff metric induced by the norm ·. For x ∈ E, X ⊂ E, A, B ∈ CCX, I ⊆ CCX, T : I,H → CCX,H,andt ∈ R −∞, ∞,let d  x,,A   inf {  x − a  : a ∈ A } ,D  A, I   inf { H  A, C  : C ∈ I } , I X  {{ x } : x ∈ X } ,A B  { a  b : a ∈ A, b ∈ B } ,tA { ta : a ∈ A } , co  I    n  i1 t i A i : t i ≥ 0, n  i1 t i  1,A i ∈ I,n≥ 1  ,F  T   { A ∈ I : TA  A } . 1.1 It is easy to see that tA 1 − tA  A and tA 1 − tB ∈ CCE for all t ∈ 0, 1 and A, B ∈ CCE. Hence CCE is convex. Hu and Huang 12 proved that if E, · is a Banach space, then CCX,H is a complete metric space. Now we introduce the f ollowing concepts in hyperspaces. Definition 1.1. Let I be a nonempty subset of CCE and let T : I,H  → CCE,H be a mapping. Assume that {t n } n≥0 , {t  n } n≥0 , {s n } n≥0 ,and{s  n } n≥0 are arbitrary real sequences in 0, 1 satisfying t n  t  n ≤ 1ands n  s  n ≤ 1forn ≥ 1and{P n } n≥0 and {Q n } n≥0 are any bounded sequences of the elements in CCE. i For A 0 ∈ I, the sequence {A n } n≥0 defined by B n   1 − s n − s  n  A n  s n TA n  s  n P n , A n1   1 − t n − t  n  A n  t n TB n  t  n Q n ,n≥ 0 1.2 is called the Ishikawa iteration sequence with errors provided that {A n ,B n : n ≥ 0}⊆I. ii If s  n  t  n  0 for all n ≥ 0in1.2, the sequence {A n } n≥0 defined by B n   1 − s n  A n  s n TA n ,A n1   1 − t n  A n  t n TB n ,n≥ 0, 1.3 is called the Ishikawa iteration sequence provided that {A n ,B n : n ≥ 0}⊆I. iii If s n  s  n  0 for all n ≥ 0in1.2, the sequence {A n } n≥0 defined by A n1   1 − t n − t  n  A n  t n TA n  t  n Q n ,n≥ 0, 1.4 is called the Mann iteration sequence with errors provided that {A n : n ≥ 0}⊆I. iv If s  n  t  n  s n  0 for all n ≥ 0in1.2, the sequence {A n } n≥0 defined by A n1   1 − t n  A n  t n TA n ,n≥ 0, 1.5 is called the Mann iteration sequence provided that {A n : n ≥ 0}⊆I. Fixed Point Theory and Applications 3 Definition 1.2. Let I be a nonempty subset of CCE. A mapping T : I,H → CCE,H is said to be i nonexpansive if HTA,TB ≤ HA, B for all A, B ∈ I; ii quasi-nonexpansive if FT /  ∅and HTA,P ≤ HA, P for all A ∈ I and P ∈ FT. Definition 1.3. Let I be a nonempty subset of CCE. A mapping T : I,H → CCE,H with FT /  ∅ is said to be satisfy the following. i Condition A if there is a continuous function f : 0, ∞ → 0, ∞ with f00and ft > 0fort ∈ 0, ∞, such that HA, TA ≥ fDA, FT for all A ∈ I. ii Condition B if there is a nondecreasing function f : 0, ∞ → 0, ∞ with f00 and ft > 0fort ∈ 0, ∞, such that HA, TA ≥ fDA, FT for all A ∈ I. Remark 1.4. In case I  I X , where X is a nonempty subset of E,andT : I X → I E ⊆ CCE is a mapping, then Definitions 1.1, 1.2,and1.3ii reduce to the corresponding concepts in 1– 11, 13. It is well known that every nonexpansive mapping with nonempty fixed point set is quasi-nonexpansive, but the converse is not true; see 8. Examples 3.1 and 3.4 in this paper reveal that the class of nonexpansive mappings with nonempty fixed point set is a proper subclass of quasi-nonexpansive mappings with both Condition A and Condition B. The following lemmas play important roles in this paper. Lemma 1.5 see 12. Let E, · be a Banach space and I a compact subset of CCE,H.Then  coI,H is compact, where coI stands for the closure of coI. Lemma 1.6 see 4. Suppose t hat {a n } n≥0 , {b n } n≥0 , and {c n } n≥0 are three sequences of nonnegative numbers such that a n1 ≤ 1  b n a n  c n for all n ≥ 0.If  ∞ n0 b n and  ∞ n0 c n converge, then lim n →∞ a n exists. Lemma 1.7 see 14. Let X, d be a metric space. Let A and B be compact subsets of X. Then for any a ∈ A, there exists b ∈ B such that da, b ≤ HA, B,whereH is the Hausdorff metric induced by d. Lemma 1.8. Let E, · be a normed linear space. Then H  1 − t − s  AtBsC,  1 − t − s  L  tM  sN  ≤  1 − t − s  H  A, L   tH  B, M   sH  C, N   1.6 for all A, B, C, L,M, N ∈ CCE and t, s ∈ 0, 1 with s  t ≤ 1. Proof. Set r   1 − t − s  H  A, L   tH  B, M   sH  C, N  . 1.7 For any a ∈ A, b ∈ B, c ∈ C,byLemma 1.7 we infer that there exist l ∈ L, m ∈ M, n ∈ N such that  a − l  ≤ H  A, L  ,  b − m  ≤ H  B, M  ,  c − n  ≤ H  C, N  , 1.8 4 Fixed Point Theory and Applications which imply that   1 − t − s  a  tb  sc −  1 − t − s  l − tm − sn  ≤  1 − t − s   a − l   t  b − m   s  c − n  ≤ r. 1.9 That is, sup { d  1 − t − s  a  tb  sc,  1 − t − s  L  tM  sN  : a ∈ A, b ∈ B, c ∈ C } ≤ r. 1.10 Similarly we have sup { d  1 − t − s  l  tm  sn,  1 − t − s  A  tB  sC  : l ∈ L, m ∈ M, n ∈ N } ≤ r. 1.11 Thus 1.6 follows from 1.10 and 1.11. This completes the proof. Lemma 1.9. Let E, · be a normed linear space and I a nonempty closed subset of CCE,H. If T : I,H → CCE,H is quasi-nonexpansive, then FT is closed. Proof. Let {P n } n≥0 be in FT with lim n →∞ HP n ,P0. Since T is quasi-nonexpansive, it follows that H  P, TP  ≤ H  P n ,P   H  P n ,TP  ≤ 2H  P n ,P  −→ 0 1.12 as n →∞. Hence P ∈ FT.Thatis,FT is closed. This completes the proof. 2. Main Results Our results are as follows. Theorem 2.1. Let E, · be a normed linear space and let I be a nonempty subset of CCE. Assume that T : I,H → CCE,H is nonexpansive and A 0 ∈ I. Suppose t hat there exists a constant t satisfying 0 <t n  t  n ≤ t<1,n≥ 0, 2.1 ∞  n0 t n  ∞, ∞  n0 s n < ∞, ∞  n0 s  n < ∞, ∞  n0 t  n < ∞, ∞  n0 t  n  t n  t  n  −1 < ∞. 2.2 If the Ishikawa iteration sequence with errors {A n } n≥0 is bounded, then lim n →∞ HA n ,TA n 0. Proof. Since T is nonexpansive, {A n } n≥0 , {P n } n≥0 ,and{Q n } n≥0 are bounded, it follows that a : sup { H  A, B  : A ∈ { A n ,B n ,P n ,Q n : n ≥ 0 } ,B ∈ { A n ,B n ,TA n ,TB n : n ≥ 0 }} < ∞. 2.3 Fixed Point Theory and Applications 5 Let n and k be arbitrary nonnegative integers. In view of 1.2, 2.3, Lemma 1.8,andthe nonexpansiveness of T, we conclude that H  B n ,A n  ≤ s n H  A n ,TA n   as  n , 2.4 H  TB n ,A n  ≤ H  TB n ,TA n   H  TA n ,A n  ≤  1  s n  H  A n ,TA n   as  n , 2.5 H  A n1 ,A n  ≤ t n H  TB n ,A n   at  n ≤ t n  1  s n  H  A n ,TA n   a  t n s  n  t  n  , 2.6 H  A n1 ,TA k  ≤  1 − t n − t  n  H  A n ,TA k   t n H  TB n ,TA k   at  n ≤  1 − t n − t  n  H  A n ,TA k   t n H  B n ,A k   at  n , 2.7 which yields that H  A n ,TA k  ≥  1 − t n − t  n  −1  H  A n1 ,TA k  − t n H  B n ,A k  − at  n  . 2.8 Using 1.2, 2.3–2.6, Lemma 1.8, and the nonexpansiveness of T, we have H  B n ,A nk1  ≤ H  B n ,A n1   k  i1 H  A ni ,A ni1  ≤  1 − s n − s  n  H  A n ,A n1   s n H  TA n ,A n1   as  n  k  i1   1  s ni  t ni H  A ni ,TA ni   a  t ni s  ni  t  ni  ≤  1 − s n − s  n  t n  1  s n  H  TA n ,A n   a  t n s  n  t  n   s n  1 − t n − t  n  H  TA n ,A n   t n H  TB n ,TA n   at  n   as  n  k  i1  1  s ni  t ni H  A ni ,TA ni   a k  i1  t ni s  ni  t  ni  ≤  t n  s n − s n t n − s 2 n t n − s  n t n − s n t  n − s  n t n s n  H  A n ,TA n   a  1 − s n − s  n  t n s  n  t  n   s n t n  s n H  A n ,TA n   as  n   as n t  n  as  n  k  i1  1  s ni  t ni H  A ni ,TA ni   a k  i1  t ni s  ni  t  ni  ≤ k  i0  t ni  s ni  H  A ni ,TA ni   a  s  n  k  i0  t ni s  ni  t  ni   , 2.9 6 Fixed Point Theory and Applications H  TA n1 ,A n1  ≤  1 − t n − t  n  H  A n ,TA n1   t n H  TB n ,TA n1   t  n H  Q n ,TA n1  ≤  1 − t n − t  n   H  A n1 ,TA n1   H  A n ,A n1   t n H  B n ,A n1   at  n ≤  1 − t n − t  n  H  A n1 ,TA n1    1 − t n − t  n   1  s n  t n H  A n ,TA n   a  t n s  n  t  n   t n   t n  s n  H  A n ,TA n   a  s  n  t n s  n  t  n   at  n ≤  1 − t n − t  n  H  A n1 ,TA n1   t n  1  2s n  H  A n ,TA n   2a  t n s  n  t  n  2.10 which implies that H  A n1 ,TA n1  ≤  t n  t  n  −1  t n  1  2s n  H  A n ,TA n   2a  t n s  n  t  n  ≤  1  2s n  H  A n ,TA n   2a  s  n  t n  t  n  t  n  −1  . 2.11 Lemma 1.6, 2.2,and2.11 yield that there exists a nonnegative constant r satisfying lim n →∞ H  A n ,TA n   r, 2.12 which implies that for any ε>0 there exists a positive integer N such that r −ε ≤ H  A n ,TA n  ≤ r  ε for n ≥ N. 2.13 Now we prove by induction that the following inequality holds for all n ≥ 1: H  A p ,TA pn  ≥  r  ε   1  n−1  i0 t pi  − 2ε n−1  i0  1 − t pi − t  pi  −1 −  r  ε  n−1  i0 ⎡ ⎣ t pi ⎛ ⎝ n−1  ji s pj ⎞ ⎠ i  k0  1 − t pk − t  pk  −1 ⎤ ⎦ − a n−1  i0 ⎧ ⎨ ⎩ ⎡ ⎣ t pi ⎛ ⎝ s  pi  n−1  ji  t pj s  pj  t  pj  ⎞ ⎠  t  pi ⎤ ⎦ × i  k0  1 − t pk − t  pk  −1  ,p≥ N. 2.14 Fixed Point Theory and Applications 7 According to 1.2, 2.8, 2.9,and2.13, we derive that H  A p ,TA p1  ≥  1 − t p − t  p  −1  H  A p1 ,TA p1  − t p H  B p ,A p1  − at  p  ≥  1 − t p − t  p  −1  r −ε −  r  ε  t p  t p  s p  − at p  s  p  t p s  p  t  p  − at  p    1−t p −t  p  −1  r −ε−  r  ε   1 − 2  1 − t p    1 − t p  2  t p s p  − a  t p  s  p  t p s  p  t  p   t  p  ≥  r  ε   1  t p  − 2ε  1 − t p − t  p  −1 −  r  ε  t p s p  1 − t p − t  p  −1 − a  t p  s  p  t p s  p  t  p   t  p  1 − t p − t  p  −1 ,p≥ N. 2.15 Hence 2.14 holds for n  1. Suppose that 2.14 holds for n  m ≥ 1. That is, H  A p ,TA pm  ≥  r  ε   1  m−1  i0 t pi  − 2ε m−1  i0  1 − t pi − t  pi  −1 −  r  ε  m−1  i0 ⎡ ⎣ t pi ⎛ ⎝ m−1  ji s pj ⎞ ⎠ i  k0  1 − t pk − t  pk  −1 ⎤ ⎦ − a m−1  i0 ⎧ ⎨ ⎩ ⎡ ⎣ t pi ⎛ ⎝ s  pi  m−1  ji  t pj s  pj  t  pj  ⎞ ⎠  t  pi ⎤ ⎦ × i  k0  1 − t pk − t  pk  −1  ,p≥ N. 2.16 In view of 1.2, 2.8, 2.9,and2.16, we infer that H  A p ,TA pm1  ≥  1 − t p − t  p  −1  H  A p1 ,TA pm1  − t p H  B p ,A pm1  − at  n  ≥  1 − t p − t  p  −1   r  ε   1  m−1  i0 t p1i  − 2ε m−1  i0  1 − t p1i − t  p1i  −1 −  r  ε  m−1  i0 ⎡ ⎣ t p1i ⎛ ⎝ m−1  ji s p1j ⎞ ⎠ i  k0  1 − t p1k − t  p1k  −1 ⎤ ⎦ − a m−1  i0 ⎡ ⎣ ⎛ ⎝ t p1i ⎛ ⎝ s  p1i  m−1  ji  t p1j s  p1j  t  p1j  ⎞ ⎠  t  p1i ⎞ ⎠ × i  k0  1 − t p1k − t  p1k  −1  − t p m−1  i0  t pi  s pi   r  ε  −at p  s  p  m  i0  t pi s  pi  t  pi   − at  p  8 Fixed Point Theory and Applications  −2ε m  i0  1 − t pi − t  pi  −1   1 − t p − t  p  −1  r  ε  ×  1  m  i0 t p1i − 1  2  1 − t p  −  1 − t p  2 − t p m  i1 t pi − t p m  i0 s pi  −  r  ε  m−1  i0 ⎡ ⎣ t p1i ⎛ ⎝ m−1  ji s p1j ⎞ ⎠ i1  k0  1 − t pk − t  pk  −1 ⎤ ⎦ − a   t p  s  p  m  i0  t pi s  pi  t  pi    t  p   1 − t p − t  p  −1  m−1  i0 ⎡ ⎣ ⎛ ⎝ t p1i ⎛ ⎝ s  p1i  m−1  ji  t p1j s  p1j  t  p1j  ⎞ ⎠  t  p1i ⎞ ⎠ × i1  k0  1 − t pk − t  pk  −1 ⎤ ⎦ ⎫ ⎬ ⎭  −2ε m  i0  1 − t pi − t  pi  −1   r  ε   1 − t p   1 − t p − t  p  −1  1  m  i0 t pi  −  r  ε  ⎧ ⎨ ⎩ t p  1 − t p − t  p  −1 m  i0 s pi  m−1  i0 ⎡ ⎣ t p1i ⎛ ⎝ m−1  ji s p1j ⎞ ⎠ i1  k0  1 − t pk − t  pk  −1 ⎤ ⎦ ⎫ ⎬ ⎭ − a m  i0 ⎧ ⎨ ⎩ ⎡ ⎣ t pi ⎛ ⎝ s  pi  m  ji  t pj s  pj  t  pj  ⎞ ⎠  t  pi ⎤ ⎦ i  k0  1 − t pk − t  pk  −1 ⎫ ⎬ ⎭ ≥  r  ε   1  m  i0 t pi  − 2ε m  i0  1 − t pi − t  pi  −1   r  ε  m  i0 ⎡ ⎣ t pi ⎛ ⎝ m  ji s pj ⎞ ⎠ i  k0  1 − t pk − t  pk  −1 ⎤ ⎦ − a m  i0 ⎧ ⎨ ⎩ ⎡ ⎣ t pi ⎛ ⎝ s  pi  m  ji  t pj s  pj  t  pj  ⎞ ⎠  t  pi ⎤ ⎦ i  k0  1 − t pk − t  pk  −1 ⎫ ⎬ ⎭ ,p≥ N. 2.17 That is, 2.14 holds for n  m  1. Hence 2.14 holds for all n ≥ 1. We now assert that r  0. If not, then r>0. Let m be an arbitrary positive integer and ε  min  r, 2 −1 rt  2r  a  −1  1 − t  m ,r  1 − t  m  2  at −1  −1  . 2.18 Fixed Point Theory and Applications 9 According to 2.1, 2.2,and2.12, we know that there exists a positive integer N  Nε satisfying 2.13 and max  np  kn s k ,s  i  np  kn  t k s  k  t  k   <ε for n, i ≥ N, p ≥ 1. 2.19 It follows from 2.1, 2.2, 2.13, 2.14,and2.19 that H  A N ,TA Nm  ≥  r  ε   1  m−1  i0 t Ni  − 2ε m−1  i0  1 − t Ni − t  Ni  −1 −  r  ε  m−1  i0 ⎡ ⎣ t Ni ⎛ ⎝ m−1  ji s Nj ⎞ ⎠ i  k0  1 − t Nk − t  Nk  −1 ⎤ ⎦ − a m−1  i0 ⎧ ⎨ ⎩ ⎡ ⎣ t Ni ⎛ ⎝ s  Ni  m−1  ji  t Nj s  Nj  t  Nj  ⎞ ⎠  t  Ni ⎤ ⎦ × i  k0  1 − t Nk − t  Nk  −1  ≥  r  ε   1  m−1  i0 t Ni  − 2ε  1 − t  −m −  r  ε  ε m−1  i0 t Ni  1 − t  −i−1 − a m−1  i0   t Ni ε  t  Ni   1 − t  −i−1  ≥  r  ε   1  m−1  i0 t Ni  − 2ε  1 − t  −m −  r  ε  ε m−1  i0 ⎡ ⎣ t Ni i  j0  1 − t  −j−1 ⎤ ⎦ − aε m−1  i0 t Ni i  j0  1 − t  −j−1 − a m−1  i0 ⎡ ⎣  1 − t  −i−1 i  j0 t  Nj ⎤ ⎦ ≥  r  ε   1  m−1  i0 t Ni  − 2ε  1 − t  −m −  r  ε  εt −1  1 − t  −m m−1  i0 t Ni − aεt −1  1 − t  −m m−1  i0 t Ni − aε m−1  i0  1 − t  −i−1 ≥  r  ε −  r  ε  a  εt −1  1 − t  −m  m−1  i0 t Ni   r  ε − 2ε  1 − t  −m − aεt −1  1 − t  −m  ≥  r −  2r  a  εt −1  1 − t  −m  m−1  i0 t Ni   r −  2  at −1  ε  1 − t  −m  ≥ 2 −1 r m−1  i0 t Ni −→ ∞ 2.20 10 Fixed Point Theory and Applications as m →∞.Thus2.3 and 2.20 yield that a  ∞, which is absurd. Hence r  0. This completes the proof. Theorem 2.2. Let E, · be a Banach space and I a nonempty closed subset of CCE. Assume that T : I,H → CCE,H is nonexpansive and there exists a compact subset Ω of CCE such that TI ∪{P n ,Q n : n ≥ 0}⊆Ω. If 2.1 and 2.2 hold, then T has a fixed point in I. Moreover, given A 0 ∈ I, the Ishikawa iteration sequence with errors {A n } n≥0 converges to a fixed point of T. Proof. Setting I 0  co{A 0 }∪Ω,byLemma 1.5 and the compactness Ω we conclude that I 0 is compact. It is evident that {A n } n≥0 ⊆ I 0 , which yields that {A n } n≥0 is bounded. Since I is closed and {A n } n≥0 ⊆ I, we conclude that there exist a subsequence {A n i } i≥0 of {A n } n≥0 and A ∈ I such that lim i →∞ H  A n i ,A   0. 2.21 It follows from 2.21, Theorem 2.1, and the nonexpansiveness of T that H  A, TA  ≤ H  A, A n i   H  A n i ,TA n i   H  TA n i ,TA  ≤ 2H  A, A n i   H  A n i ,TA n i  −→ 0 2.22 as i →∞.Thatis,A  TA.Put b  sup { H  P n ,A  ,H  Q n ,A  : n ≥ 0 } . 2.23 In view of 1.2, Lemma 1.8 and the nonexpansiveness of T, we derive that H  A n1 ,A  ≤  1 − t n − t  n  H  A n ,A   t n H  TB n ,A   bt  n ≤  1 − t n − t  n  H  A n ,A   t n  1 − s n − s  n  H  A n ,A   s n H  TA n ,A   bs  n   bt  n 2.24 for n ≥ 0. It follows from Lemma 1.6, 2.2, 2.23,and2.24 that lim i →∞ HA n ,A exists. Using 2.21 we get that lim i →∞ HA n ,A0. This completes the proof. Theorem 2.3. Let E, · be a Banach space and I a nonempty closed subset of CCE. Suppose that T : I,H → CCE,H is a qusi-nonexpansive mapping and satisfies Condition A. Assume that 2.1 and 2.2 hold and A 0 is in I.IfFT is bounded, then the Ishikawa iteration sequence with errors {A n } n≥0 converges to a fixed point of T in I. Proof. Let b  sup{HP n ,A,HQ n ,A : n ≥ 0andA ∈ FT}. Then b<∞.Asinthe proof of Theorem 2.2,wegetthat2.24 holds and lim i →∞ HA n ,A exists, where A ∈ FT. Consequently, {A n } n≥0 is bounded and D  A n1 ,F  T  ≤ D  A n ,F  T   b  s  n  t  n  ∀n ≥ 0. 2.25 [...]... point of T in X Remark 2.11 Corollary 2.10 extends, improves, and unifies Theorem 4 in 3 , Theorem 2 in 7 and 8 in the following ways: i the Mann iteration method in 7, 8 , and Ishikawa iteration method in 3 are replaced by the more general Ishikawa iteration method with errors; ii the nonexpansive mappings in 3, 7, 8 are replaced by the more general quasinonexpansive mappings Fixed Point Theory and. .. fixed point theorem for mappings which do not increase distances,” The American Mathematical Monthly, vol 72, pp 1004–1006, 1965 3 L Deng, “Convergence of the Ishikawa iteration process for nonexpansive mappings, ” Journal of Mathematical Analysis and Applications, vol 199, no 3, pp 769–775, 1996 4 D Lei and L Shenghong, “Ishikawa iteration process with errors for nonexpansive mappings in uniformly convex... Journal of Mathematics and Mathematical Sciences, vol 14, no 1, pp 1–16, 1991 10 H F Senter and W G Dotson Jr., “Approximating fixed points of nonexpansive mappings, ” Proceedings of the American Mathematical Society, vol 44, pp 375–380, 1974 11 K.-K Tan and H K Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol... a nonexpansive mapping in a Banach space,” Proceedings of the American Mathematical Society, vol 59, no 1, pp 65–71, 1976 8 W V Petryshyn and T E Williamson Jr., “Strong and weak convergence of the sequence of successive approximations for quasi -nonexpansive mappings, ” Journal of Mathematical Analysis and Applications, vol 43, pp 459–497, 1973 9 B E Rhoades, “Some fixed point iteration procedures,” International... space E, · and let T : X, · → E, · be quasi -nonexpansive Assume that 2.1 and 2.2 hold and T satisfies Condition A If F T is bounded, then for any A0 ∈ X, the Ishikawa iteration sequence with errors {An }n≥0 converges to a fixed point of T in X Corollary 2.10 Let X be a nonempty closed subset of a Banach space E, · and let T : X, · → E, · be quasi -nonexpansive Assume that 2.32 holds and A0 is in X If T... spaces,” International Journal of Mathematics and Mathematical Sciences, vol 24, no 1, pp 49–53, 2000 5 W G Dotson Jr., “On the Mann iterative process,” Transactions of the American Mathematical Society, vol 149, pp 65–73, 1970 6 W G Dotson Jr., Fixed points of quasi -nonexpansive mappings, ” Journal of the Australian Mathematical Society, vol 13, pp 167–170, 1972 7 S Ishikawa, Fixed points and iteration... respectively Example 3.2 Let E R2 with the usual norm | · | and let X 0, 1 2 For any a, b ∈ X, Δ 0, 0 a, 0 0, b stands for the triangle with vertices 0, 0 , a, 0 , and 0, b Let I {Δ 0, 0 a, 0 0, b : a, b ∈ X} and {Pn }n≥0 and {Qn }n≥0 be in I Define T : I → CC E by T Δ 0, 0 a, 0 0, b Δ 0, 0 2−1 a b ,0 0, 4−1 b2 − a2 for a, b ∈ X 3.4 14 Fixed Point Theory and Applications √ −1 −1 Put tn 2 n , tn 2 n7/4 , sn... 0 0, b , F T for a, b ∈ X Therefore the conditions of Theorem 2.3 are fulfilled b2 −1 Fixed Point Theory and Applications 15 Example 3.4 Let E, X, I, and A0 be as in Example 3.2 Define T : I → CC E , f : 0, ∞ → 0, ∞ and h : 0, 1 → 1, 2 by Δ 0, 0 2−1 ah a , 0 T Δ 0, 0 a, 0 0, b 8−1 t f t for a, b ∈ X, for t ≥ 0 , ⎧7 ⎪ , for x ∈ 0, 1 , ⎪ ⎪4 ⎨ 2 h x It follows that F T 0, 2−1 b2 ⎪ ⎪ ⎪1, ⎩ for x ∈ 3.8 1... Applications 13 3 Examples and Problems Now we construct a few nontrivial examples to illustrate the results in Section 2 The following example reveals that Corollary 2.10 extends properly Theorem 4 in 3 , Theorem 2 in 7 and 8 Example 3.1 Let E R with the usual norm | · | and let X f : 0, ∞ → 0, ∞ by ⎧3 ⎪ x, for x ∈ 0, 1 , ⎪ ⎪4 ⎨ 2 Tx and f t 1/4 t for t ≥ 0 Set tn F T {0} and |T x − 0| ≤ 3 |x − 0|,... 0| ≤ 3 |x − 0|, 4 0, 1 Define T : X → E and ⎪1 ⎪ ⎪ x, for x ∈ ⎩ 2 2 √ n −1 |x − T x| ≥ and sn 1 |x| 4 3.1 1 ,1 , 2 1 n2 −1 f d x, F T for n ≥ 0 and A0 ∈ X Then for x ∈ X 3.2 Thus the assumptions of Corollary 2.10 are satisfied However, Theorem 4 in 3 , Theorem 2 in 7 and 8 are not applicable since T 17 1 −T 2 32 7 1 > 64 32 1 17 , − 2 32 3.3 that is, T is not nonexpansive The examples below show that . Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 520976, 16 pages doi:10.1155/2009/520976 Research Article Approximate Fixed Points for Nonexpansive and Quasi -Nonexpansive. Fixed points of quasi -nonexpansive mappings, ” Journal of the Australian Mathematical Society, vol. 13, pp. 167–170, 1972. 7 S. Ishikawa, Fixed points and iteration of a nonexpansive mapping. errors for nonexpansive and quasi -nonexpansive mappings in hyperspaces. The results presented in this paper improve and generalize some results in the literature. Copyright q 2009 Zeqing Liu