Journal of Inequalities in Pure and Applied Mathematics STRONG CONVERGENCE THEOREMS FOR ITERATIVE SCHEMES WITH ERRORS FOR ASYMPTOTICALLY DEMICONTRACTIVE MAPPINGS IN ARBITRARY REAL NORMED LINEAR SPACES volume 3, issue 5, article 83, 2002. 1 2 YEOL JE CHO , HAIYUN ZHOU AND SHIN MIN KANG 1 Department of Mathematics, Gyeongsang National University, Chinju 660-701, Korea. EMail: yjcho@nongae.gsnu.ac.kr 2 Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, People’s Republic of China. EMail: luyao_846@163.com 1 Received 30 April, 2002; accepted 5 September, 2002. Communicated by: S.S. Dragomir Abstract Contents JJ J II I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 040-02 Quit Abstract In the present paper, by virtue of new analysis technique, we will establish several strong convergence theorems for the modified Ishikawa and Mann iteration schemes with errors for a class of asymptotically demicontractive mappings in arbitrary real normed linear spaces. Our results extend, generalize and improve the corresponding results obtained by Igbokwe [1], Liu [2], Osilike [3] and others. 2000 Mathematics Subject Classification: Primary 47H17; Secondary 47H05, 47H10 Key words: Asymptotically demicontractive mapping; Modified Mann and Ishikawa iteration schemes with errors; arbitrary linear space. Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces This work was supported by grant No. (2000-1-10100-003-3) from the Basic Research Program of the Korea Science & Engineering Foundation. Yeol Je Cho, Haiyun Zhou and Shin Min Kang Contents Title Page 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 References Contents JJ J II I Go Back Close Quit Page 2 of 16 1. Introduction Let X be a real normed linear space and let J denote the normalized duality ∗ mapping from X into 2X given by J(x) = {f ∈ X ∗ : hx, f i = kxk2 = kf k2 }, x ∈ X, where X ∗ denotes the dual space of X and h·, ·i denotes the generalized duality pairing of elements between X and X ∗ . Let F (T ) denote the set of all fixed points of a mapping T . Let C be a nonempty subset of X. A mapping T : C → C is said to be k-strictly asymptotically pseudocontractive with a sequence {kn } ⊂ [0, ∞), kn ≥ 1 and kn → 1 as n → ∞ if there exists k ∈ [0, 1) such that (1.1) n n 2 kT x − T yk ≤ kn2 kx 2 n n 2 − yk + k k(x − T x) − (y − T y)k for all n ≥ 1 and x, y ∈ C. The mapping T is said to be asymptotically demicontractive with a sequence {kn } ⊂ [1, ∞) and limn→∞ kn = 1 if F (T ) 6= ∅ and there exists k ∈ [0, 1) such that (1.2) kT n x − pk2 ≤ kn2 kx − pk2 + kkx − T n xk2 for all n ≥ 1, x ∈ C and p ∈ F (T ). The classes of k-strictly asymptotically pseudocontractive and asymptotically demicontractive mappings, as a natural extension to the class of asymptotically non-expansive mappings, were first introduced in Hilbert spaces by Liu Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang Title Page Contents JJ J II I Go Back Close Quit Page 3 of 16 [5] in 1996. By using the modified Mann iterates introduced by Schu [4, 5], he established several strong convergence results concerning an iterative approximation to fixed points of k-strictly asymptotically pseudocontractive and asymptotically demicontractive mappings in Hilbert spaces. In 1998, Osilike [3], by virtue of normalized duality mapping, first extended the concepts of k-strictly asymptotically pseudocontractive and asymptotically demicontractive maps from Hilbert spaces to the much more general Banach spaces, and then proved the corresponding convergence theorems which generalized the results of Liu [2]. A mapping T : C → C is said to be k-strictly asymptotically pseudocontractive with a sequence {kn } ⊂ [0, ∞), kn ≥ 1 and kn → 1 as n → ∞ if there exist k ∈ [0, 1) and j(x − y) ∈ J(x − y) such that (1.3) h(I − T n )x − (I − T n )y, j(x − y)i 1 1 ≥ (1 − k)k(I − T n )x − (I − T n )yk2 − (kn2 − 1)kx − yk2 2 2 for all n ≥ 1 and x, y ∈ C. The mapping T is called an asymptotically demicontractive mapping with a sequence {kn } ⊂ [0, ∞), limn→∞ kn = 1 if F (T ) 6= ∅ and there exist k ∈ [0, 1) and j(x − y) ∈ J(x − y) such that (1.4) 1 1 hx − T n x, j(x − p)i ≥ (1 − k)kx − T n xk2 − (kn2 − 1)kx − pk2 2 2 for all n ≥ 1, x ∈ C and p ∈ F (T ). Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang Title Page Contents JJ J II I Go Back Close Quit Page 4 of 16 Furthermore, T is said to be uniformly L-Lipschitzian if there is a constant L ≥ 1 such that (1.5) kT n x − T n yk ≤ Lkx − yk for all x, y ∈ C and n ≥ 1. Remark 1.1. The definitions above may be stated in the setting of a real normed linear space. In the case of X being a Hilbert space, (1.1) and (1.2) are equivalent to (1.3) and (1.4), respectively. Recall that there are two iterative schemes with errors which have been used extensively by various authors. Let X be a normed linear space, C be a nonempty convex subset of X and T : C → C be a given mapping. Then the modified Ishikawa iteration scheme {xn } with errors is defined by x1 ∈ C, yn = (1 − βn )xn + βn T n (xn ) + vn , n ≥ 1, xn+1 = (1 − αn )xn + αn T n (yn ) + un , n ≥ 1, where {αn }, {βn } are some suitable sequences in [0, 1] and {un }, {vn } are two summable sequences in X. With X, C, {αn } and x1 as above, the modified Mann iteration scheme {xn } with errors is defined by x1 ∈ C, (1.6) xn+1 = (1 − αn )xn + αn T n (xn ) + un , n ≥ 1. Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang Title Page Contents JJ J II I Go Back Close Quit Page 5 of 16 Let X, C and T be as in above. Let {an }, {bn }, {cn }, {a0n }, {b0n } and {c0n } be real sequences in [0, 1] satisfying an +bn +cn = 1 = a0n +b0n +c0n and let {un } and {vn } be bounded sequences in C. Define the modified Ishikawa iteration schemes {xn } with errors generated from an arbitrary x1 ∈ C as follows: yn = an xn + bn T n (xn ) + cn un , n ≥ 1, (1.7) xn+1 = a0n xn + b0n T n yn + c0n vn , n ≥ 1. In particular, if we set bn = cn = 0 in (1.7), we obtain the modified Mann iteration scheme {xn } with errors given by x1 ∈ C, (1.8) xn+1 = a0n xn + b0n T n xn + c0n vn , n ≥ 1. Osilike [3] proved the following convergence theorems for k-strictly asymptotically demicontractive mappings: Theorem 1.1. [3] Let q > 1 and let E be a real q-uniformly smooth Banach space. Let K be a nonempty closed convex and bounded subset of E and T : K → K be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with a sequence {kn } ⊂ [1, ∞) for all P∞ n ≥ 1, kn → 1 as n → ∞ and n=1 (kn 2 − 1) < ∞. Let {αn } and {βn } be real sequences in [0, 1] satisfying the conditions: (i) 0 < ≤ cq αn some > 0, q−1 ≤ 1 {q(1 2 −(q−2) − k)(1 + L) } − for all n ≥ 1 and for Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang Title Page Contents JJ J II I Go Back Close Quit Page 6 of 16 (ii) P∞ n=1 βn < ∞. Then the sequence {xn } defined by (1.6) with un ≡ 0 and vn ≡ 0 for all n ≥ 1 converges strongly to a fixed point of T . Very recently, Igbokwe [2] extended the above Theorem 1.1 to Banach spaces. More precisely, he proved the following results: Theorem 1.2. [2] Let E be a real Banach space and K be a nonempty closed convex subset of E. Let T : K → K be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive P mapping with a sequence 2 {kn } ⊂ [1, ∞) for all n ≥ 1, kn → 1 as n → ∞ and ∞ n=1 (kn − 1) < ∞. Let the sequence {xn } be defined by (1.7) with the restrictions that (i) an + bn + cn = 1 = a0n + b0n + c0n , P 0 (ii) ∞ n=0 bn = ∞, P P∞ 0 P∞ P∞ 0 2 (iii) ∞ n=0 (b ) < ∞, n=0 cn < ∞, n=0 bn < ∞ and n=0 cn < ∞. Then the modified Ishikawa iteration {xn } defined by (1.6) and (1.7) converges strongly to a fixed point p of T . It is our purpose in this paper to extend and improve the above Theorem 1.2 from Banach spaces to real normed linear spaces. In the case of Banach spaces, we use the Condition (A) to replace the assumption that T is completely continuous. In the sequel, we will need the following lemmas: Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang Title Page Contents JJ J II I Go Back Close Quit Page 7 of 16 Lemma 1.3. Let X be a normed linear space and C be a nonempty convex subset of X. Let T : C → C be a uniformly L-Lipschitzian mapping and the sequence {xn } be defined by (1.7). Then we have (1.9) kT xn − xn k ≤ kT n xn − xn k + L(1 + L2 )kT n−1 xn−1 − xn−1 k + L(1 + L)c0n−1 kvn−1 − xn−1 k + L2 (1 + L)cn−1 kun−1 − xn k + Lc0n−1 kxn−1 − T n−1 xn−1 k, n ≥ 1. Proof. See Igbokwe [1, Lemma 1]. Lemma 1.4. Let X be a normed linear space and C be a nonempty convex subset of X. Let T : C → C be a uniformly L-Lipschitzian and asymptotically P∞ demicontractive mapping with a sequence {kn } such that kn ≥ 1 and n=1 (kn − 1) < ∞. Let the sequence {xn } be defined by (1.7) with the restrictions ∞ X n=1 b0n = ∞, ∞ X 2 (b0n ) < ∞, n=1 ∞ X c0n < ∞, n=1 Then we have the following conclusions: (i) limn→∞ kxn − pk exists for any p ∈ F (T ). ∞ X n=1 Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang Title Page cn < ∞. Contents JJ J II I Go Back (ii) limn→∞ d(xn , F (T )) exists. Close (iii) lim inf n→∞ kxn − T xn k = 0. Quit Page 8 of 16 Proof. It is very clear that (1.7) is equivalent to the following: yn = (1 − bn )xn + bn T n (xn ) + cn (un − xn ), n ≥ 1, (1.10) xn+1 = (1 − b0n )xn + b0n T n yn + c0n (vn − xn ), n ≥ 1. For any p ∈ F (T ), let M > 0 be such that M = max{sup{kun − pk}, sup{kvn − pk}}. Observe first that (1.11) kyn − pk ≤ (1 − bn )kxn − pk + bn Lkxn − pk + cn (M + kxn − pk) ≤ (1 + L)kxn − pk + M and (1.12) Yeol Je Cho, Haiyun Zhou and Shin Min Kang kT n yn − xn k ≤ Lkyn − pk + kxn − pk ≤ L[(1 + L)kxn − pk + M ] + kxn − pk ≤ [1 + L(1 + L)]kxn − pk + M L. Observe also that (1.13) Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces b0n [kT n yn kxn+1 − yn k ≤ kxn − yn k + − pk + kyn − pk] 0 + cn [kvn − pk + kyn − pk] ≤ [bn (1 + L) + cn ]kxn − pk + (cn + c0n )M + [b0n (1 + L) + c0n ][(1 + L)kxn − pk + M ] ≤ σn kxn − pk + ςn , Title Page Contents JJ J II I Go Back Close Quit Page 9 of 16 where σn = [bn (1 + L) + cn ] + [b0n (1 + L) + c0n ](1 + L) and ςn = M [b0n (1 + L) + 2c0n + cn ]. Thus we have (1.14) kxn+1 − xn k ≤ b0n kT n yn − xn k + c0n (M + kxn − pk) ≤ b0n σn kxn − pk + ςn + c0n (M + kxn − pk). Using iterates (1.10), we have (1.15) kxn+1 − pk2 ≤ kxn − pkkxn+1 − pk − b0n hxn − T n yn , j(xn+1 − p)i + c0n hvn − xn , j(xn+1 − p)ik 1 1 ≤ kxn − pk2 + kxn+1 − pk2 2 2 − b0n hxn+1 − T n xn+1 , j(xn+1 − p)i + b0n hxn+1 − xn + T n yn − T n xn+1 , j(xn+1 − p)i + c0n (M + kxn − pk)kxn+1 − pk, which implies that (1.16) kxn+1 − pk2 ≤ kxn − pk2 − (1 − k)b0n kxn+1 − T n xn+1 k2 + b0n (kn2 − 1)kxn+1 − pk2 + 2b0n (kxn+1 − xn k + kT n xn+1 − T n yn k)kxn+1 − pk + 2c0n (M + kxn − pk)kxn+1 − pk. Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang Title Page Contents JJ J II I Go Back Close Quit Page 10 of 16 Substituting (1.12) – (1.14) in (1.16) and, after some calculations, we obtain kxn+1 − pk2 ≤ (1 + γn )kxn − pk2 − (1 − k)b0n kxn+1 − T n xn+1 k2 P for all very large n, where the sequence {γn } satisfies that ∞ n=1 γn < ∞. A direct induction of (1.17) leads to (1.17) kxn+1 − pk2 ≤ kxn − pk2 + M γn − b0n kxn+1 − T n xn+1 k2 , (1.18) which implies that limn→∞ kxn − pk exists by Tan and Xu [7, Lemma 1] and so this proves the claim (i). The claim (ii) follows from (1.18). Now, we prove the claim (iii). It follows from (1.18) that ∞ X b0n kxn+1 − T n xn+1 k2 < ∞ n=1 and hence Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang lim inf kxn+1 − T n xn+1 k = 0 n→∞ 0 n bn since = ∞. Therefore, we have lim inf n→∞ kxn − T n xn k = 0 by (1.7) and so lim inf n→∞ kxn − T xn k = 0 by Lemma 1.3. This completes the proof. P A mapping T : C → C with a nonempty fixed point set F (T ) in C will be said to satisfy the Condition (A) on C if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0 and f (r) > 0 for all r ∈ (0, ∞) such that kx − T xk ≥ f (d(x, F (T ))) Title Page Contents JJ J II I Go Back Close Quit for all x ∈ C. Page 11 of 16 2. The Main Results Now we prove the main results of this paper. Theorem 2.1. Let X be a real normed linear space, C be a nonempty closed convex subset of X and T : C → C be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with a sequence P (k − 1) < ∞. Let the sequence {kn } ⊂ [1, ∞) such that kn ≥ 1 and ∞ n=1 n {xn } be defined by (1.7) with the restrictions ∞ X n=1 b0n = ∞, ∞ X 2 b0n < ∞, n=1 ∞ X c0n < ∞, n=1 ∞ X cn < ∞. n=1 Then {xn } converges strongly to a fixed point p of T . Proof. It follows from Lemma 1.4 that lim inf kxn − T xn k = 0. n→∞ Since T is completely continuous, we see that there exists an infinite subsequence {xnk } such that {xnk } converges strongly for some p ∈ C and T p = p. This shows that p ∈ F (T ). However, limn→∞ kxn − pk exists for any p ∈ F (T ) and so we must have that the sequence {xn } converges strongly to p. This completes the proof. Theorem 2.2. Let X be a real Banach space, C be a nonempty closed convex subset of X and T : C → C be a uniformly L-Lipschitzian and asymptotically demicontractive mapping with a sequence {kn } ⊂ [1, ∞) such that kn ≥ 1 Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang Title Page Contents JJ J II I Go Back Close Quit Page 12 of 16 P and ∞ n=1 (kn − 1) < ∞. Let the sequence {xn } be defined by (1.7) with the restrictions ∞ ∞ ∞ ∞ X X X X 2 b0n = ∞, b0n < ∞, c0n < ∞, cn < ∞. n=1 n=1 n=1 n=1 Suppose in addition that T satisfies the Condition (A), then the sequence {xn } converges strongly to a fixed point p of T . Proof. By Lemma 1.4, we see that lim inf kxn − T xn k = 0. n→∞ Since T satisfies the Condition (A), we have lim inf f (d(xn , F (T ))) = 0 n→∞ and hence Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang lim inf d(xn , F (T )) = 0. n→∞ By Lemma 1.4 (ii), we conclude that d(xn , F (T )) → 0 as n → ∞. Now we can take an infinite subsequence {xnj } of {xP n } and a sequence −j {pj } ⊂ F (T ) such that kxnj − pj k ≤ 2 . Set M = exp { ∞ n=1 γn } and write nj+1 = nj + l for some l ≥ 1. Then we have (2.1) kxnj+1 − pj k = kxnj +l − pj k ≤ [1 + γnj +l−1 ]kxnj +l−1 − pj k ( l−1 ) X M ≤ exp γnj +m kxnj − pj k ≤ j . 2 m=0 Title Page Contents JJ J II I Go Back Close Quit Page 13 of 16 It follows from (2.1) that 2M + 1 . 2j+1 Hence {pj } is a Cauchy sequence. Assume that pj → p as j → ∞. Then p ∈ F (T ) since F (T ) is closed and this in turn implies that xj → p as j → ∞. This completes the proof. kpj+1 − pj k ≤ Remark 2.1. We remark that, if T : C → C is completely continuous, then it must be demicompact (cf. [6]) and, if T is continuous and demicompact, it must satisfy the Condition (A) (cf. [6]). In view of this observation, our Theorem 2.1 improves Theorem 1.2 in the following aspects: (i) X may be not a Banach space. (ii) T may be not completely continuous. Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang (iii) Our proof methods are simpler than those of Igbokwe [1, Theorem 2]. As corollaries of Theorems 2.1 and 2.2, we have the following: Corollary 2.3. Let X be a real normed linear space, C be a nonempty closed convex subset of X and T : C → C be a completely continuous and uniformly L-Lipschitzian asymptotically demicontractive mapping with a sequence P {kn } ⊂ [0, ∞) such that kn ≥ 1 and ∞ (k − 1) < ∞. Let the sequence n=1 n {xn } be defined by (1.8) with the restrictions ∞ X n=1 b0n = ∞, ∞ X n=1 2 b0n < ∞, ∞ X n=1 Then {xn } converges strongly to a fixed point p of T . c0n < ∞. Title Page Contents JJ J II I Go Back Close Quit Page 14 of 16 Proof. By taking bn , cn ≡ 0 for n ≥ 1 in Theorem 2.1, we can obtain the desired conclusion. Corollary 2.4. Let X be a real Banch space, C be a nonempty closed convex subset of X and T : C → C be a uniformly L-Lipschitzian asymptotically demicontractive mapping with a sequence {kn } ⊂ [0, ∞) such that kn ≥ 1 P∞ and n=1 (kn − 1) < ∞. Let the sequence {xn } be defined by (1.8) with the restrictions ∞ ∞ ∞ X X X 0 2 bn = ∞, bn < ∞, c0n < ∞. n=1 n=1 n=1 If T satisfies the Condition (A) on the sequence {xn }, then {xn } converges strongly to a fixed point p of T . Proof. It follows from Theorem 2.2 by taking bn , cn ≡ 0 for all n ≥ 1. Remark 2.2. Using the same methods as in Lemma 1.4, Theorems 2.1 and 2.2, we can prove several convergence results similar to Theorems 2.1 and 2.2 concerning on the modified Ishikawa iteration schemes with errors defined by (1.6). Remark 2.3. Igbokwe [1, Corollary 1] has shown that, if T : C → C is asymptotically pseudocontractive, then it must be uniformly L-Lipschitzian and hence our Theorems 2.1 and 2.2 hold for asymptotically pseudocontractive mappings with a nonempty fixed point set F (T ). Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang Title Page Contents JJ J II I Go Back Close Quit Page 15 of 16 References [1] D.I. IGBOKWE, Approximation of fixed points of asymptotically demicontractive mappings in arbitrary Banach spaces, J. Ineq. Pure and Appl. Math., 3(1) (2002), Article 3. [ONLINE: http://jipam.vu.edu.au/v3n1/043_01.html ] [2] Q.H. LIU, Convergence theorems of sequence of iterates for asymptotically demicontractive and hemicontractive mappings, Nonlinear Anal. Appl., 26 (1996), 1835–1842. [3] M.O. OSILIKE, Iterative approximations of fixed points asymptotically demicontractive mappings, Indian J. Pure Appl. Math., 29 (1998), 1291– 1300. [4] J. SCHU, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal.Appl., 158 (1991), 407–413. [5] J. SCHU, Weak and strong convergence of fixed points of asymptotically non-expansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153–159. [6] H.F. SENTER AND W.G. DOTSON, JR., Approximating fixed points of non-expansive mappings, Proc. Amer. Math. Soc., 44 (1974), 375–380. [7] K.K. TAN AND H.K. XU, Fixed point iteration processes for asymptotically non-expansive mappings, Proc. Amer. Math. Soc., 122 (1994), 733–739. Strong Convergence Theorems for Iterative Schemes with Errors for Asymptotically Demicontractive Mappings in Arbitrary Real Normed Linear Spaces Yeol Je Cho, Haiyun Zhou and Shin Min Kang Title Page Contents JJ J II I Go Back Close Quit Page 16 of 16