This article was published in an Elsevier journal. The attached copy is furnished to the author for non-commercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: http://www.elsevier.com/copyright

Author's personal copy

Nonlinear Analysis 67 (2007) 3058–3063 www.elsevier.com/locate/na

On a Mann type implicit iteration process for continuous pseudo-contractive mappings Yisheng Song ∗ College of Mathematics and Information Science, Henan Normal University, 453007, PR China Received 24 April 2006; accepted 29 September 2006

Abstract Let K be a nonempty closed convex subset of a Banach space E, T : K → K a continuous pseudo-contractive mapping. Suppose that {αn } is a real sequence in [0, 1] satisfying appropriate conditions; then for arbitrary x0 ∈ K , the Mann type implicit iteration process {xn } given by xn = αn xn−1 + (1 − αn )T xn , n ≥ 0, strongly and weakly converges to a fixed point of T , respectively. c 2006 Elsevier Ltd. All rights reserved.

MSC: 47H05; 47H10; 47H17 Keywords: Continuous pseudo-contractive mappings; Strong and weak convergence; Mann type implicit iteration process

1. Introduction and preliminaries ∗

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 ∗ , hx, f i = kxkk f k, kxk = k f k},

∀ x ∈ E,

where E ∗ denotes the dual space of E and h·, ·i denotes the generalized duality pairing. In the sequel, we shall denote the single-valued duality mapping by j, and define F(T ) = {x ∈ E; T x = x}. When {xn } is a sequence in E, then ∗ xn → x (respectively, xn * x, xn * x) will denote strong (respectively, weak, weak*) convergence of the sequence {xn } to x. Let E be a real Banach space and T be a mapping with domain D(T ) and range R(T ) in E. T is called nonexpansive if for any x, y ∈ D(T ), kT x − T yk ≤ kx − yk. T is called (respectively, strongly) pseudo-contractive if for any x, y ∈ D(T ), there exists j (x − y) ∈ J (x − y) such that hT x − T y, j (x − y)i ≤ kx − yk2 ,

(1.1)

∗ Corresponding address: College of Mathematics and Information Science, Henan Normal University, No. 67 Jianshedong Lu, 453007 Xinxiang, Henan Province, PR China. Tel.: +86 03733326148; fax: +86 03733326174. E-mail address: [email protected].

c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.09.059

Author's personal copy

3059

Y. Song / Nonlinear Analysis 67 (2007) 3058–3063

(respectively, hT x − T y, j (x − y)i ≤ βkx − yk2 for some 0 < β < 1.)

(1.2)

If E = H is a Hilbert space (J = I , the identity operator), it is easy to show that Eqs. (1.1) and (1.2) are respectively equivalent to the following inequalities (see [2,3]): kT x − T yk2 ≤ kx − yk2 + k(x − T x) − (y − T y)k2

(1.3)

kT x − T yk2 ≤ kx − yk2 + βk(x − T x) − (y − T y)k2 .

(1.4)

and T is called hemicontractive if for any x ∈ D(T ) and p ∈ F(T ), there exists j (x − p) ∈ J (x − p) such that hT x − p, j (x − p)i ≤ kx − pk2 . In Hilbert space, this is equivalent to the following inequality: kT x − pk2 ≤ kx − pk2 + kx − T xk2 ,

∀x ∈ D(T ), p ∈ F(T ).

In the last ten years or so, numerous papers have been published on the iterative approximation of fixed points of Lipschitz strongly pseudo-contractive (and correspondingly Lipschitz strongly accretive) mappings using the Mann iteration process (see, e.g., [15]). Results which had been known only for Hilbert spaces and only for Lipschitz mappings have been extended to more general Banach spaces (see, e.g., [4–12,14,16,17,23] and the references cited therein). This success, however, has not carried over to arbitrary Lipschitz pseudo-contraction T even when the domain of the operator T is a compact convex subset of a Hilbert space. In fact, it was a long standing open question whether or not the Mann iteration process converges in this setting. In 1974, Ishikawa [13] introduced an iteration process which, in some sense, is more general than that of Mann and which converges, in this setting, to a fixed point of T . In 1999, Chidume and Moore gave the following in [4]. 1. In connection with the iterative approximation of fixed points of pseudo-contractions, the following question is still open. Does the Mann iteration process always converge for continuous pseudo-contractions, or for even Lipschitz pseudo-contractions? 2. Let K be a nonempty compact convex subset of Banach space E and T : K → K be a Lipschitz pseudocontractive map. In this setting, even for E = H , a Hilbert space, the answer to the above question is not known. These questions have recently been resolved in the negative by Chidume and Mutangadura [9] who produced an example of a Lipschitz pseudo-contractive map defined on a compact convex subset of real Hilbert space with a unique fixed point for which no Mann sequence converges. Very recently, Rafiq [1] studied the strong convergence of the Mann type implicit process (1.5) for hemicontractive mapping T with αn ∈ [δ, 1 − δ] for some δ ∈ (0, 1): xn = αn xn−1 + (1 − αn )T xn .

(1.5)

However, Arif Rafiq’s control conditions exclude the natural choice of αn = But in studying his work, we discovered that there is a gap in the iteration process (1.5) for hemicontractive mapping T . That is, there was firstly given an initial point x0 ∈ C, then x1 defined by the following equation 1 n.

x1 = α1 x0 + (1 − α1 )T x1 . But the existence of x1 in C is not established, because for hemicontractive mapping T , we do not know whether the mapping S1 := α1 x0 + (1 − α1 )T must have fixed point x1 ∈ C. Similarly, the existence of x2 , x3 , . . . is also not arrived at. Thus the sequence {xn } in [1] is not well defined. Let K be a nonempty convex subset of Banach space E, and T a continuous pseudo-contractive mapping from K to itself. For every fixed u ∈ K and t ∈ (0, 1), the operator St : K → K defined by St x = tu + (1 − t)T x, ∀x ∈ K satisfies that, ∀x, y ∈ K , there exists j (x − y) ∈ J (x − y) such that hSt x − St y, j (x − y)i = (1 − t)hT x − T y, j (x − y)i ≤ (1 − t)kx − yk2 . Thus St is strongly pseudo-contractive. Since St is also continuous, so that St has a unique fixed point xt ∈ K by [21, Corollary 2], i.e. xt = tu + (1 − t)T xt , which implies a continuous pseudo-contractive mapping T satisfies the Mann type implicit iteration process (1.5).

Author's personal copy

3060

Y. Song / Nonlinear Analysis 67 (2007) 3058–3063

In this paper, we consider the Mann type implicit iteration scheme (1.5) for continuous pseudo-contractive mappings, and prove several weak and strong convergence results for the iterative scheme (1.5) in Hilbert space and Banach space, respectively (it includes the natural choice of αn = n1 ). For continuous pseudo-contractive mappings, the results presented in this paper not only overcome the gaps in [1], but also complement and improve on the corresponding ones of [1] (the iterative coefficient is simpler). 2. Weakly convergent theorems Lemma 2.1. Suppose K is a nonempty closed convex subset of Banach space E and T : K → K is a continuous pseudo-contractive mapping such that F(T ) 6= ∅. Assume that {αn } is a real sequence satisfying αn ∈ (0, b] ⊂ (0, 1) for some constant b ∈ (0, 1). Let x0 ∈ K and let {xn } be defined by xn = αn xn−1 + (1 − αn )T xn ,

n ≥ 0.

(2.1)

Then (i) for any fixed p ∈ F(T ), limn→∞ kxn − pk exists; (ii) both {xn } and {T xn } are bounded. Proof. Take p ∈ F(T ), it follows from Eq. (1.1) and the Cauchy–Schwarz inequality [22, Proposition 1.4.7] that for some j (xn − p) ∈ J (xn − p), kxn − pk2 = hαn xn−1 + (1 − αn )Tn xn − p, j (xn − p)i = (1 − αn )hTn xn − p, j (xn − p)i + αn hxn−1 − p, j (xn − p)i ≤ (1 − αn )kxn − pk2 + αn kxn−1 − pkkxn − pk. So kxn − pk2 ≤ kxn−1 − pkkxn − pk.

(2.2)

If kxn − pk = 0, the result is apparent. Next let kxn − pk > 0; it follows from Eq. (2.2) that (2.3)

kxn − pk ≤ kxn−1 − pk,

which implies that the sequence {kxn − pk} is monotone and non-increasing. Hence, limn→∞ kxn − pk exists, (i) is proved. It follows from Eq. (2.3) (or (i)) that {xn } is bounded. From Eq. (2.1), we have

1

α n

kT xn k =

1 − α xn − 1 − α xn−1 n n αn 1 kxn k + kxn−1 k ≤ 1 − αn 1 − αn b 1 kxn k + kxn−1 k, ≤ 1−b 1−b which implies that {T xn } is also bounded.  We recall that a Banach space E is said to satisfy Opial’s condition if for any sequence {xn } in E, xn * x(n → ∞) implies lim sup kxn − xk < lim sup kxn − yk, n→∞

∀y ∈ E with x 6= y.

n→∞

It is well known that every Hilbert space satisfies Opial’s condition [20]. Theorem 2.2. Let E, K , T and {xn } be the same as Lemma 2.1. Assume that {αn } ⊂ (0, 1) is a real sequence satisfying the condition limn→∞ αn = 0. Then (i) limn→∞ kxn − T xn k = 0; and if, in addition, E is both reflexive and satisfies Opial’s condition, then (ii) {xn } weakly converges to a fixed point of T . Proof. It follows from the condition limn→∞ αn = 0 that for b ∈ (0, 1), ∃N , a positive integer, ∀n ≥ N , such that αn ∈ (0, b]. Then from Lemma 2.1 (ii), we get that both {xn } and {T xn } are bounded. Combining this with Eq. (2.1),

Author's personal copy

Y. Song / Nonlinear Analysis 67 (2007) 3058–3063

3061

we have lim kxn − T xn k = lim αn kxn−1 − T xn k = 0.

n→∞

n→∞

(2.4)

Set A = (2I − T )−1 ; then from [18, Theorem 6], we obtain that A is a non-expansive self-mapping on K and F(T ) = F(A) (also see, e.g., [19–21]). It follows from Eq. (2.4) and A = (2I − T )−1 that xn − T xn = (2I − T )xn − xn = A−1 xn − xn xn = A A−1 xn = A−1 Axn kxn − Axn k = kA A−1 xn − Axn k ≤ kA−1 xn − xn k = kxn − T xn k → 0 (n → ∞). Hence, lim kxn − Axn k = 0.

n→∞

Since E is a reflexive Banach space and {xn } is bounded by Lemma 2.1 (ii), then {xn } has a subsequence {xn k }∞ k=1 which converges weakly to some x ∗ ∈ K , and also lim kxn k − Axn k k = 0.

k→∞

Now we show that x ∗ ∈ F(T ). In fact, suppose x ∗ 6= Ax ∗ ; since E satisfies Opial’s condition and A is nonexpansive, we obtain that lim sup kxn k − x ∗ k < lim sup kxn k − Ax ∗ k k→∞

k→∞

≤ lim sup(kxn k − Axn k k + kAxn k − Ax ∗ k) k→∞

≤ lim sup kxn k − x ∗ k. k→∞

This is a contradiction; we must have Ax ∗ = x ∗ . Hence x ∗ ∈ F(A) = F(T ). Secondly, we prove {xn } converges weakly to x ∗ . Suppose that {xn } does not converge weakly to x ∗ ; then there ∗ exists another subsequence {xn j }∞ j=1 of {x n } which is weakly convergent to some y 6= x , y ∈ K . We also have y ∈ F(T ). Because limn→∞ kxn − pk exists for all p ∈ F(T ), by Lemma 2.1(i) and E satisfies Opial’s condition, thus lim kxn − x ∗ k = lim kxn k − x ∗ k < lim kxn k − yk

n→∞

k→∞

k→∞

= lim kxn j − yk < lim kxn j − x ∗ k j→∞

j→∞

∗

= lim kxn − x k. n→∞

This is a contradiction; we must have y = x ∗ . Thus {xn } converges weakly to x ∗ ∈ F. The proof is finished.



Theorem 2.3. Let E = H be a Hilbert space. Suppose K , T , αn and {xn } are the same as Lemma 2.1. Then (i) limn→∞ kxn − T xn k = 0; (ii) {xn } weakly converges to a fixed point of T . Proof. We shall use the well known identity in Hilbert space H , for all x, y ∈ H and t ∈ [0, 1]: kt x + (1 − t)yk2 = tkxk2 + (1 − t)kyk2 − t (1 − t)kx − yk2 . For fixed p ∈ F(T ), we have kxn − pk2 = kαn (xn−1 − p) + (1 − αn )(T xn − p)k2 = αn kxn−1 − pk2 + (1 − αn )kT xn − pk2 − αn (1 − αn )kxn−1 − T xn k2 .

(2.5)

Author's personal copy

3062

Y. Song / Nonlinear Analysis 67 (2007) 3058–3063

From Eqs. (1.3) and (2.1), we get that kT xn − pk2 = kTn xn − T pk2 ≤ kxn − pk2 + kxn − T xn k2 = kxn − pk2 + αn2 kxn−1 − T xn k2 .

(2.6)

Substituting (2.6) in (2.5), we obtain kxn − pk2 ≤ αn kxn−1 − pk2 + (1 − αn )kxn − pk2 + (1 − αn )αn2 kxn−1 − T xn k2 − αn (1 − αn )kxn−1 − T xn k2 = αn kxn−1 − pk2 + (1 − αn )kxn − pk2 − (1 − αn )2 αn kxn−1 − T xn k2 , which implies kxn − pk2 ≤ kxn−1 − pk2 − (1 − αn )2 kxn−1 − T xn k2 . It follows from the condition αn ∈ (0, b] ⊂ (0, 1) for some b ∈ (0, 1) that (1 − b)2 kxn−1 − T xn k2 ≤ (1 − αn )2 kxn−1 − Tn xn k2 ≤ kxn−1 − pk2 − kxn − pk2 .

(2.7)

Taking the upper limit on both sides of Eq. (2.7), along with the existence of the limit limn→∞ kxn − pk by Lemma 2.1 (i), we get (1 − b)2 lim sup kxn−1 − T xn k2 ≤ lim sup(kxn−1 − pk2 − kxn − pk2 ) = 0. n→∞

n→∞

Thus lim kxn−1 − T xn k = 0.

n→∞

Combining Eq. (2.1), we have that lim kxn − T xn k = lim αn kxn−1 − T xn k = 0.

n→∞

n→∞

(2.8)

Set A = (2I − T )−1 ; since every Hilbert space H is a reflexive Banach space and satisfies Opial’s condition (see, e.g., [20]), the remainder of the proof is the same as the proof of Theorem 2.2; we omit it. This completes the proof.  3. Strong convergence theorems Theorem 3.1. Let T and {xn } be the same as Lemma 2.1. Assume that K is compact convex subset of real Banach space E, and {αn } ⊂ (0, 1) is a real sequence satisfying the condition limn→∞ αn = 0. Then {xn } strongly converges to a fixed point of T . Proof. Since K is compact convex subset of E and {xn } is bounded by Lemma 2.1(ii), then there exists a subsequence {xn k } ⊂ {xn } such that xn k → x ∗ ∈ K . By the continuity of the mapping T and the norm k · k, together with limk→∞ kxn k − T xn k k = 0, by Theorem 2.2 (i), we have that kx ∗ − T x ∗ k = lim kxn k − T xn k k = 0. k→∞

Therefore ∈ F(T ). Because limn→∞ kxn − pk exists for all p ∈ F(T ) by Lemma 2.1 (i), thus we obtain that x∗

lim kxn − x ∗ k = lim kxn k − x ∗ k = 0.

n→∞

The proof is completed.

k→∞



Author's personal copy

3063

Y. Song / Nonlinear Analysis 67 (2007) 3058–3063

Like in the argument of Theorem 3.1, we easily get the following theorem. Since the proof is the same as that for Theorem 3.1, it is omitted. Theorem 3.2. Let E = H , T , αn and {xn } be the same as Theorem 2.3. Assume that K is a compact convex subset of H . Then {xn } strongly converges to a fixed point of T . Theorems 3.1 and 3.2 actually proved the following results. Corollary 3.3. Let E, T , αn and {xn } be the same as for Theorem 3.1 or Theorem 3.2. Suppose that K is a nonempty closed convex subset of E. Then {xn } strongly converges to a fixed point of T if and only if {xn } exists as a strongly convergent subsequence. Remark 1. There are many hypotheses about mapping T (or set K ) to guarantee that the sequence {xn } has a strongly convergent subsequence. For example, (i) the mapping T is completely continuous, that is, T maps any bounded subset of E to a relatively compact subset of E; (ii) the mapping T is semi-compact, i.e., any bounded sequence {xn } in K satisfying kxn − T xn k → 0 as n → ∞ has a strongly convergent subsequence. Remark 2. It is clear that the condition αn ∈ (0, b] (or limn→∞ αn = 0) includes the natural choice of αn = overcomes the restriction in Rafiq [1] (αn ∈ [δ, 1 − δ] excludes αn = n1 ).

1 n

which

Acknowledgment The author is grateful to the anonymous referee for valuable suggestions which helped to improve the manuscript. References [1] A. Rafiq, On Mann iteration in Hilbert spaces, Nonlinear Anal. (2006). doi:10.1016/j.na.2006.03.012. [2] M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudo-contractive maps, J. Math. Anal. Appl. 294 (2004) 73–81. [3] M.O. Osilike, A. Udomene, Demiclosedness principle results for strictly pseudo-contractive mappings of Browder–Petryshyn type, J. Math. Anal. Appl. 256 (2001) 431–445. [4] C.E. Chidume, C. Moore, Fixed point iteration for pseudo-contractive maps, Proc. Amer. Math. Soc. 127 (4) (1999) 1163–1170. [5] C.E. Chidume, Iterative approximation of Lipschitz strictly pseudo-contractive mappings, Proc. Amer. Math. Soc. 99 (2) (1987) 283–288. [6] C.E. Chidume, Approximation of fixed points of strongly pseudo-contractive mappings, Proc. Amer. Math. Soc. 120 (2) (1994) 545–551. [7] C.E. Chidume, Global iteration schemes for strongly pseudo-contractive maps, Proc. Amer. Math. Soc. 126 (9) (1998) 2641–2649. [8] C.E. Chidume, M.O. Osilike, Ishikawa iteration process for nonlinear Lipschitz strongly accretive mappings, J. Math. Anal. Appl. 192 (1995) 727–741. [9] C.E. Chidume, S.A. Mutangadura, An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc. 129 (8) (2001) 2359–2363. [10] L. Deng, On Chidume’s open problems, J. Math. Anal. Appl. 174 (2) (1993) 441–449. [11] L. Deng, X.P. Ding, Iterative approximation of Lipschitz strictly pseudo-contractive mappings in uniformly smooth Banach spaces, Nonlinear Anal. 24 (7) (1995) 981–987. [12] T.L. Hicks, J.R. Kubicek, On the Mann iteration process in Hilbert space, J. Math. Anal. Appl. 59 (1977) 498–504. [13] S. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 4 (1) (1974) 147–150. [14] L.S. Liu, Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995) 114–125. [15] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953) 506–610. [16] L. Qihou, On Naimpally and Singh’s open questions, J. Math. Anal. Appl. 124 (1987) 157–164. [17] L. Qihou, The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings, J. Math. Anal. Appl. 148 (1990) 55–62. [18] R.H. Martin, Differential equations on closed subsets of a Banach space, Trans. Amer. Math. Soc. 179 (1973) 399–414. [19] S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal Appl. 75 (1980) 287–292. [20] W. Takahashi, Nonlinear Functional Analysis—Fixed Point Theory and its Applications, Yokohama Publishers Inc., Yokohama, 2000 (in Japanese). [21] K. Deimling, Zero of accretive operators, Manuscripta Math. 13 (1974) 365–374. [22] R.E. Megginson, An Introduction to Banach Space Theory, Springer-Verlag, New York, Inc., 1998. [23] R. Chen, Y. Song, H. Zhou, Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings, J. Math. Anal. Appl. 314 (2006) 701–709.