Journal of the Indian Math. Soc. Vol. 81, Nos. 1-2, (2014), 147–153.
IMPLICIT MANN ITERATION SCHEME FOR HEMICONTRACTIVE MAPPINGS ARIF RAFIQ AND MOHAMMAD IMDAD Abstract. The purpose of this paper is to characterize conditions for the convergence of the implicit Mann iterative scheme due to Xu and Ori (An implicit iterative process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001), 767-773) to the unique fixed point of φ-hemicontractive mappings defined on a nonempty convex subset of an arbitrary Banach space.
(Received: July 23, 2012, Accepted: October 2, 2012) 1. Introduction and Preliminaries Let K be a nonempty subset of an arbitrary Banach space E and E ∗ be its dual space. The symbols D(T ), R(T ) and F (T ) respectively stand for the domain, the range and the set of fixed points of T (for a single-valued map T : D(T ) → X) wherein x ∈ D(T ) is called a fixed point of T iff T (x) = x. We ∗ denote by J the normalized duality mapping from E to 2E defined by 2
2
J(x) = {f ∗ ∈ E ∗ : hx, f ∗ i = kxk = kf ∗ k }. Definition 1.1. (cf. [1, 4, 10, 19]) Let T : D(T ) ⊆ X → X be an operator. Then T is said to be (i) strongly pseudocontractive if there exists a t > 1 such that for each x, y ∈ D(T ), there exists j(x − y) ∈ J(x − y) satisfying 1 kx − yk2 , t (ii) strictly hemicontractive if F (T ) 6= ∅ and if there exists a t > 1 such that for each x ∈ D(T ) and q ∈ F (T ), there exists j(x−q) ∈ J(x−q) satisfying Re hT x − T y, j(x − y)i ≤
Re hT x − q, j(x − q)i ≤
1 2 kx − qk , t
2010 Mathematics Subject Classification. 54H25, 47H10. Key words and phrases: Implicit iterative scheme, φ-hemicontractive mappings, Banach spaces ISSN 0019–5839
c Indian Mathematical Society, 2014 .
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(iii) φ-strongly pseudocontractive if there exists a strictly increasing function φ : [0, ∞) → [0, ∞) with φ(0) = 0 such that for each x, y ∈ D(T ), there exists j(x − y) ∈ J(x − y) satisfying 2
Re hT x − T y, j(x − y)i ≤ kx − yk − φ(kx − yk) kx − yk , (iv) φ-hemicontractive if F (T ) 6= ∅ and also there exists a strictly increasing function φ : [0, ∞) → [0, ∞) with φ(0) = 0 such that for each x ∈ D(T ) and q ∈ F (T ), there exists j(x − q) ∈ J(x − q) satisfying 2
Re hT x − q, j(x − q)i ≤ kx − qk − φ(kx − qk) kx − qk . Clearly, each strictly hemicontractive operator is φ-hemicontractive. Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of T in case T is a Lipschitz strongly pseudo-contractive mapping from a bounded closed convex subset K of Lp (or lp ) into itself. Afterwards, several authors generalized this result of Chidume in various directions (e.g.[4, 6-9, 15-16, 18-19]). In 2001, Xu and Ori [20] introduced the following implicit iteration process for a finite family of nonexpansive mappings {Ti : i ∈ I} where I = {1, 2, . . . , N } with {αn } a real sequence in (0, 1), and x0 ∈ K an initial point: x1 = (1 − α1 )x0 + α1 T1 x1 , x2 = (1 − α2 )x1 + α2 T2 x2 , .. . xN = (1 − αN )xN −1 + αN TN xN , xN +1 = (1 − αN +1 )xN + αN +1 TN +1 xN +1 , .. . which can be written in the following compact form: xn = (1 − αn )xn−1 + αn Tn xn , for all n ≥ 1, where Tn = Tn (mod
N)
(XO)
(here the mod N function takes values in I ). Xu and
Ori [20] proved the weak convergence of this process to a common fixed point of the finite family defined in a Hilbert space. They further remarked that it is yet unclear what assumptions on the mappings and/or the parameters {αn } are sufficient to ensure the strong convergence of the sequence {xn }. In [11], Oslilike proved the following: Theorem 1.1. Let E be a real Banach space and K be a nonempty closed convex subset of E. Let {Ti : i ∈ I} be N strictly pseudocontractive self-mappings
IMPLICIT MANN ITERATION SCHEME
of K with F =
N \
149
∞
F (Ti ) 6= ∅. Let {αn }n=1 be a real sequence satisfying the
i=1
conditions:
(i) 0 < αn < 1, (ii)
∞ X
(1 − αn ) = ∞, (iii)
n=1
∞ X
(1 − αn )2 < ∞.
n=1
From arbitrary x0 ∈ K, define the sequence {xn } by the implicit iteration process (XO). Then {xn } converges strongly to a common fixed point of the mappings {Ti : i ∈ I} if and only if lim inf d(xn , F ) = 0. n→∞
Remark 1.1. One can easily see that for αn = 1 −
1 1
n2
,
Hence, the results of Osilike [11] can be further improved.
P
(1 − αn )2 = ∞.
The purpose of this paper is to characterize conditions for the convergence of the implicit Mann iterative scheme mainly due to Xu and Ori [20] to the unique fixed point of φ-hemicontractive mappings in a nonempty convex subset of an arbitrary Banach space. Our results extend and improve a host of results of the existing literature especially those contained in [1-4,6-9, 11-13, 15-16, 18-19]. The following lemmas are now well known in the literature. Lemma 1.1. For all x, y ∈ X and j(x + y) ∈ J(x + y), ||x + y||2 ≤ ||x||2 + 2Re hy, j(x + y)i . Lemma 1.2. Let {θn } be a sequence of nonnegative real numbers, {λn } be a real sequence satisfying 0 ≤ λn ≤ 1,
∞ X
λn = ∞.
n=0
Suppose there exists a strictly increasing function φ : [0, ∞) → [0, ∞) with φ(0) = 0. If there exists a positive integer n0 such that 2 θn+1 ≤ θn2 − λn φ(θn+1 )θn+1 + σn + γn ,
for all n ≥ n0 , with σn ≥ 0, ∀n ∈ N, σn = 0(λn ) and
∞ P
n=0
limn→∞ θn = 0. 2. Main Results Now, we prove our main result as follows:
γn < ∞, then
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Theorem 2.1. Let K be a nonempty convex subset of an arbitrary Banach space X and let T : K → K be a uniformly continuous and φ-hemicontractive mapping. Suppose that {αn }∞ n=1 be any sequence in [0, 1] satisfying
(i) lim (1 − αn ) = 0, (ii) n→∞
∞ X
(1 − αn ) = ∞.
n=1
If {xn }∞ n=1 is the sequence generated from an arbitrary x0 ∈ K described by xn = αn xn−1 + (1 − αn )T xn , n ≥ 1,
(2.1)
then the following conditions are equivalent: (a) {xn }∞ n=1 converges strongly to the unique fixed point q of T , (b) {T xn }∞ n=1 is bounded . Proof. Since T is φ-hemicontractive, it follows that F (T ) is a singleton. Let F (T ) = {q} for some q ∈ K. Suppose that lim xn = q, then the uniform continuity of T yields that n→∞
lim T xn = q,
n→∞
so that {T xn }∞ n=1 is bounded. Put M1 = kx0 − qk + sup kT xn − qk .
(2.2)
n≥1
It is clear that ||x0 − q|| ≤ M1 . Let ||xn−1 − q|| ≤ M1 . Now, we wish to prove that ||xn − q|| ≤ M1 . Consider kxn − qk = kαn xn−1 + (1 − αn )T xn − qk = kαn (xn−1 − q) + (1 − αn )(T xn − q)k ≤ αn kxn−1 − qk + (1 − αn ) kT xn − qk ≤ αn M1 + (1 − αn ) kT xn − qk
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= αn kx0 − qk + sup kT xn − qk n≥1
+ (1 − αn ) kT xn − qk ≤ kx0 − qk + αn sup kT xn − qk + (1 − αn ) kT xn − qk n≥1
≤ kx0 − qk + αn sup kT xn − qk + (1 − αn )sup kT xn − qk n≥1
n≥1
= kx0 − qk + sup kT xn − qk n≥1
= M1 . So, from the above discussion, we can conclude that the sequence {xn − q}n≥1 is bounded. Thus there is a constant M > 0 satisfying M = sup kxn − qk + sup kT xn − qk , n≥1
n≥1
so that M < ∞. Owing to Lemma 1.1 and (2.1), we infer that ||xn − q||2 = ||αn xn−1 + (1 − αn )T xn − q||2 = ||αn (xn−1 − q) + (1 − αn )(T xn − q)||2 ≤ α2n kxn−1 − qk2 + 2(1 − αn )Re hT xn − q, j(xn − q)i 2
2
≤ α2n kxn−1 − qk + 2(1 − αn ) kxn − qk − 2(1 − αn )φ(kxn − qk) kxn − qk .
(2.3)
Consider 2
2
kxn − qk = kαn xn−1 + (1 − αn )T xn − qk
= kαn (xn−1 − q) + (1 − αn )(T xn − q)k2 2
≤ αn kxn−1 − qk + (1 − αn ) kT xn − qk
2
2
≤ kxn−1 − qk + M 2 (1 − αn ),
(2.4) 2
wherein the first inequality holds due to convexity of k.k . Making use of (2.4) in (2.3), we get kxn − qk2 ≤ [α2n + 2(1 − αn )] kxn−1 − qk2 + 2M 2 (1 − αn )2 − 2(1 − αn )φ(kxn − qk) kxn − qk 2 = 1 + (1 − αn )2 kxn−1 − qk + 2M 2 (1 − αn )2 − 2(1 − αn )φ(kxn − qk) kxn − qk 2
≤ kxn−1 − qk + 3M 2 (1 − αn )2
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− 2(1 − αn )φ(kxn − qk) kxn − qk 2
= kxn−1 − qk + (1 − αn )ln − 2(1 − αn )φ(kxn − qk) kxn − qk ,
(2.5)
where ln = 3M 2 (1 − αn ) → 0,
(2.6)
as n → ∞. Denote θn = ||xn−1 − q||, λn = 2(1 − αn ), σn = (1 − αn )ln . Condition (i) assures the existence of a rank n0 ∈ N such that λn = 2(1−αn ) ≤ 1, for all n ≥ n0 . Now, owing to condition (ii), (2.5) and Lemma 1.2, we obtain lim ||xn − q|| = 0,
n→∞
which concludes the proof.
The following corollary follows immediately from preceding theorem. Corollary 2.1. Let K be a nonempty convex subset of an arbitrary Banach space X and let T : K → K be a uniformly Lipschitz and φ-hemicontractive mapping. Suppose that {αn }∞ n=1 be any sequence in [0, 1] satisfying (i) lim (1 − αn ) = 0, n→∞ ∞ X
(ii)
(1 − αn ) = ∞.
n=1
For any xo ∈ K, define the sequence {xn }∞ n=1 inductively as follows: xn = αn xn−1 + (1 − αn )T xn , n ≥ 1. Then the following conditions are equivalent: (a) {xn }∞ n=1 converges strongly to the unique fixed point q of T , (b) {T xn }∞ n=1 is bounded. Acknowledgment: The authors are grateful to the anonymous referee for his/her careful reading and valuable suggestions towards the improvement of this paper. References [1] C. E. Chidume, Iterative approximation of fixed point of Lipschitz strictly pseudocontractive mappings, Proc. Amer. Math. Soc., 99 (1987), 283–288. [2] Lj. B. Ciric, J. S. Ume, Ishikawa iterative process for strongly pseudocontractive operators in Banach spaces, Mathematical Communications, (8) (2003), 43-48. ´ c,A. Rafiq, N. Caki´ [3] Lj. B. Ciri´ c, J. S. Ume,Implicit Mann fixed point iterations for pseudo-contractive mappings, Appl. Math. Lett., 22 (4) (2009), 581-584.
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ARIF RAFIQ, Hajvery University, 43-52 Industrial Area, Gulberg-III, Lahore, PAKISTAN.
[email protected]
MOHAMMAD IMDAD, Department of Mathematics, Aligarh Muslim University, Aligarh - 202 002, INDIA.
[email protected]
