Attractive points and convergence theorems of generalized hybrid ...
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 8 (2015), 354–362 Research Article
Attractive points and convergence theorems of generalized hybrid mapping Yuchun Zheng∗ School of Mathematics and Information Science, Henan Normal University, XinXiang, P.R. China, 453007
Abstract In this paper, by means of the concept of attractive points of a nonlinear mapping, we prove strong convergence theorem of the Ishikawa iteration for an (α, β)−generalized hybrid mapping in a uniformly convex Banach space, and obtain weak convergence theorem of the Ishikawa iteration for such a mapping in a Hilbert space. Keywords: Attractive points, generalized hybrid mapping, Ishikawa iteration, Mann iteration, Xu’s inequality. 2010 MSC: 47H10, 54H25, 49J40, 47H05, 47H04, 65J15, 47H10. 1. Introduction Let E be a Banach space with the norm k · k and let K be a nonempty subset of E. In 2010, Kocourek, Takahashi and Yao [5] firstly introduced the concept of the generalized hybrid mapping, which contains the classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings. A mapping T : K → K is called (α, β)−generalized hybrid if there exist α, β ∈ R such that αkT x − T yk2 + (1 − α)kx − T yk2 ≤ βkT x − yk2 + (1 − β)kx − yk2
(1.1)
for all x, y ∈ K, where R is the set of real numbers. T is said to be nonexpansive if T is (1, 0)−generalized hybrid; T is called hybrid (Takahashi [10]) if T is ( 23 , 12 )−generalized hybrid, i.e. 3kT x − T yk2 ≤ kx − T yk2 + kT x − yk2 + kx − yk2 ∀x, y ∈ K; T is called nonspreading (Kohsaka and Takahashi [6]) if T is (2, 1)−generalized hybrid, i.e. 2kT x − T yk2 ≤ kx − T yk2 + kT x − yk2 ∀x, y ∈ K. ∗
Corresponding author Email address: [email protected] (Yuchun Zheng)
Received 2015-1-16
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
355
Baillon [1] proved the first nonlinear ergodic theorem: suppose that K is a nonempty closed convex subset of Hilbert space E and T : K → K is nonexpansive mapping such that F (T ) 6= ∅, then ∀x ∈ K, the Ces`aro means n 1 X i Tn x = T x n+1 i=0
weakly converges to a fixed point of T . Bruck [2, 3] studied the property of Ces` aro means for nonexpansive mapping in uniformly convex Banach space. Takahashi and Yao [11] proved the nonlinear ergodic theorem for both hybrid and nonspreading mappings in a Hilbert space. Kocourek, Takahashi and Yao [5] showed that both the nonlinear ergodic theorem and the weak convergence theorem of the Mann iteration for (α, β)−generalized hybrid mapping. The Mann iteration is the original definition of Mann [7] for a nonexpansive mapping T , xn+1 = αn xn + (1 − αn )T xn {αn } ⊂ (0, 1), x1 ∈ K. Takahashi and Takeuchi [12] obtained the nonlinear ergodic theorem without convexity for (α, β)−generalized hybrid mappings. Hojo and Takahashi [4] showed the strong convergence of the Halpern iteration of Ces` aro means for (α, β)−generalized hybrid mapping T under some proper conditions, n
xn+1
1 X i = αn u + (1 − αn ) T xn {αn } ⊂ (0, 1), u, x1 ∈ K. n+1 i=0
In this paper, we will deal with strong and weak convergence of the Ishikawa iteration for finding attractive points of (α, β)−generalized hybrid mappings under some conditions on the sequences {αn } and {βn } in (0, 1), ( xn+1 = αn xn + (1 − αn )T yn (1.2) yn = βn xn + (1 − βn )T xn Our results obviously develop and complement the corresponding ones of Kocourek, Takahashi and Yao [5], Takahashi and Yao [11] , Takahashi and Takeuchi [12], Takahashi [10] and others. 2. Preliminaries and basic results Let N and R be the sets of positive integers and real numbers, respectively. Let K be a nonempty subset of a Banach space E with the norm k · k and let T be a mapping T from K to E. A point y ∈ E is called an attractive point of T if for all x ∈ K kT x − yk ≤ kx − yk. We denote by A(T ) the set of all attractive points of T , i.e., A(T ) = {y ∈ E; kT x − yk ≤ kx − yk ∀x ∈ K}. Takahashi and Takeuchi [12] used this concept and proved the closed and convex property of A(T ) in a Hilbert space H. For more details, see Takahashi and Takeuchi [12]. A Banach space E is said to be uniformly convex if for all ε ∈ [0, 2], ∃δε > 0 such that
kx + yk < 1 − δε whenever kx − yk ≥ ε. 2 The following lemmas are well-known which can be found in [13]. kxk = kyk = 1 implies
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
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Lemma 2.1. (Xu [13, Theorem 2]) Let q > 1 and r > 0 be two fixed real numbers. Then a Banach space is uniformly convex if and only if there exists a continuous strictly increasing convex function g : [0, +∞) → [0, +∞) with g(0) = 0 such that kλx + (1 − λ)ykq ≤ λkxkq + (1 − λ)kykq − ωq (λ)g(kx − yk),
(2.1)
for all x, y ∈ Br (0) = {x ∈ E; kxk ≤ r} and λ ∈ [0, 1], where ωq (λ) = λq (1 − λ) + λ(1 − λ)q . Note that the inequality in Lemma 2.1 is known as Xu’s inequality. Let H be a real Hilbert space with the norm k · k and the inner produce h·, ·i. Obviously, the Xu’s inequality is replaced by the following equality in a Hilbert space H, for x, y ∈ H and t ∈ R, ktx + (1 − t)yk2 = tkxk2 + (1 − t)kyk2 − t(1 − t)kx − yk2 .
(2.2)
Lemma 2.2. Let K be a nonempty closed and convex subset of a real uniformly convex Banach space E and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the Ishikawa iteration ( xn+1 = αn xn + (1 − αn )T yn (2.3) yn = βn xn + (1 − βn )T xn where the sequences {αn } and {βn } in (0, 1) such that lim inf βn (1 − αn )(1 − βn ) > 0. n→∞
(2.4)
Then (i) the sequence {xn } is bounded; (ii) the limit lim kxn − uk exists for each u ∈ A(T ); n→∞
(iii) lim kxn − T xn k = 0. n→∞
Proof. Take u ∈ A(T ). By the definition of the attractive point, we have kxn+1 − uk = kαn (xn − u) + (1 − αn )(T yn − u)k ≤ αn kxn − uk + (1 − αn )kT yn − uk ≤ αn kxn − uk + (1 − αn )kyn − uk ≤ αn kxn − uk + (1 − αn )(βn kxn − uk + (1 − βn )kT xn − uk) ≤ (αn + (1 − αn )(βn + (1 − βn )))kxn − uk ≤ kxn − uk .. . ≤ kx1 − uk. So the sequence {xn } is bounded and the sequence {kxn − uk} is monotone non-increasing, and hence the limit lim kxn − uk exists for each u ∈ A(T ). n→∞
Now we show (iii). Let r ≥ max kxn − uk. n∈N
Then kT yn − uk ≤ kyn − uk ≤ kxn − uk ≤ r and kT xn − uk ≤ kxn − uk ≤ r.
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357
It follows from Lemma 2.1(q = 2) that kxn+1 − uk2
= kαn (xn − u) + (1 − αn )(T yn − u)k2 ≤ αn kxn − uk2 + (1 − αn )kT yn − uk2 ≤ αn kxn − uk2 + (1 − αn )kyn − uk2 = αn kxn − uk2 + (1 − αn )kβn (xn − u) + (1 − βn )(T xn − u)k2 ≤ αn kxn − uk2 + (1 − αn )(βn kxn − uk2 + (1 − βn )kT xn − uk2 −βn (1 − βn )g(kT xn − xn k)) ≤ αn kxn −
uk2
+ (1 − αn )(βn kxn − uk2 + (1 − βn )kxn − uk2 −βn (1 − βn )g(kT xn − xn k))
≤ kxn −
uk2
− (1 − αn )βn (1 − βn )g(kT xn − xn k).
(2.5)
Then we have (1 − αn )βn (1 − βn )g(kT xn − xn k) ≤ kxn − uk2 − kxn+1 − uk2 , and so,
∞ X
(1 − αn )βn (1 − βn )g(kT xn − xn k) ≤ kx1 − uk2 < +∞.
n=1
From the condition ((2.4)), it follows that lim g(kT xn − xn k) = 0.
n→∞
By the property of the function g, we have lim kT xn − xn k = 0.
n→∞
This completes the proof. When αn = 0 for all n, the following conclusions hold obviously. Corollary 2.3. Let K be a nonempty closed and convex subset of a real uniformly convex Banach space E and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the following iteration xn+1 = T (βn xn + (1 − βn )T xn )
(2.6)
where the sequence {βn } in (0, 1) such that lim inf βn (1 − βn ) > 0. n→∞
(2.7)
Then (i) the sequence {xn } is bounded; (ii) the limit lim kxn − uk exists for each u ∈ A(T ); n→∞
(iii) lim kxn − T xn k = 0. n→∞
3. Strongly Convergent Theorems Let K be a nonempty subset of a Banach space E. A mapping T : K → K is said to satisfy Condition I if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0 for r ∈ (0, ∞) such that kx − T xk ≥ f (d(x, A(T ))) for all x ∈ K, where d(x, A(T )) = inf{kx − yk; y ∈ A(T )}. This concept was introduced by Senter and Dotson [9] and the examples of mappings that satisfy Condition I was given.
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358
Theorem 3.1. Let K be a nonempty closed and convex subset of a uniformly convex Banach space E and and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅ and satisfying Condition I. Suppose that the sequence {xn } is defined by the Ishikawa iteration ( xn+1 = αn xn + (1 − αn )T yn (3.1) yn = βn xn + (1 − βn )T xn , where the sequences {αn } and {βn } in (0, 1) such that lim inf βn (1 − αn )(1 − βn ) > 0. n→∞
(3.2)
Then the sequence {xn } converges strongly to an attractive point z of T . Proof. It follows from Lemma 2.2 that the sequence {xn } is bounded and kxn+1 − uk ≤ kxn − uk for each u ∈ A(T ) and lim kxn − T xn k = 0. n→∞
(3.3)
Then Condition I implies lim f (d(xn , A(T ))) = 0, and hence n→∞
lim d(xn , A(T )) = 0.
n→∞
(3.4)
Next we show that the sequence {xn } is a Cauchy sequence of E. In fact, for any n, m ∈ N, without loss of generality, we may set m > n, then kxm − uk ≤ kxn − uk for each u ∈ A(T ) by ((3.3)), and so kxn − xm k ≤ kxn − uk + ku − xm k ≤ 2kxn − uk.
(3.5)
Since u is arbitrary, then we may take the infimum for u in ((3.5)), kxn − xm k ≤ 2 inf{kxn − uk; u ∈ A(T )} = 2d(xn , A(T )). From ((3.4)), it follows that as lim kxn − xm k = 0, which means that {xn } is a Cauchy sequence. So there n→∞ exists z ∈ E such that lim kxn − zk = 0. n→∞
By ((3.3)), we have lim kT xn − zk = 0.
n→∞
Now we prove z ∈ A(T ). In fact, it follows from the definition of (α, β)−generalized hybrid mapping that for all x ∈ K, αkT xn − T xk2 + (1 − α)kxn − T xk2 ≤ βkT xn − xk2 + (1 − β)kxn − xk2 .
(3.6)
Let n → ∞ in ((3.6)). Then by the continuity of the norm k · k and the function g(t) = t2 , we have αkz − T xk2 + (1 − α)kz − T xk2 ≤ βkz − xk2 + (1 − β)kz − xk2 , and hence kz − T xk ≤ kz − xk for all x ∈ K. So z ∈ A(T ) and lim kxn − zk = 0. The proof is completed. n→∞
A mapping T : K → E is said to be demicompact (Petryshyn [8]) provided whenever a sequence {xn } ⊂ K is bounded and the sequence {xn − T xn } strongly converges, then there is a subsequence {xnk } which strongly converges.
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359
Theorem 3.2. Let K be a nonempty closed and convex subset of a uniformly convex Banach space E and and let T : K → K be (α, β)−generalized hybrid and demicompact with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the Ishikawa iteration ( (3.1)) and the sequences {αn } and {βn } satisfy ( (3.2)). Then the sequence {xn } converges strongly to an attractive point z of T . Proof. It follows from Lemma 2.2 that the sequence {xn } is bounded and lim kxn − T xn k = 0.
n→∞
(3.7)
Then the demicompactness of T implies there is a subsequence {xnk } ⊂ {xn } and z ∈ E such that lim kxnk − zk = 0.
k→∞
(3.8)
By ((3.7)), we also have lim kT xnk − zk = 0.
k→∞
From the definition of (α, β)−generalized hybrid mapping, it follows that for all x ∈ K, αkT xnk − T xk2 + (1 − α)kxnk − T xk2 ≤ βkT xnk − xk2 + (1 − β)kxnk − xk2 .
(3.9)
Let k → ∞ in ((3.9)). Then by the continuity of the norm k · k and the function g(t) = t2 , we have kz − T xk ≤ kz − xk for all x ∈ K. So z ∈ A(T ). Since lim kxn − uk exists for each u ∈ A(T ) by Lemma 2.2 (ii), then we have n→∞
lim kxn − zk = 0.
n→∞
The proof is completed. The condition (3.2) contains αn ≡ 0 and lim inf βn (1 − βn ) > 0 as special cases. So the following result n→∞ is obtained easily. Corollary 3.3. Let K be a nonempty closed and convex subset of a real uniformly convex Banach space E and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the following iteration xn+1 = T (βn xn + (1 − βn )T xn )
(3.10)
where the sequence {βn } in (0, 1) such that lim inf βn (1 − βn ) > 0. n→∞
(3.11)
Assume that T either satisfies Condition I or is demicompact. Then the sequence {xn } converges strongly to an attractive point z of T . 4. Weakly Convergent Theorems Let {xn } is a sequence in E, then xn * x will denote weak convergence of the sequence {xn } to x.
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
360
Theorem 4.1. Let K be a nonempty closed and convex subset of a Hilbert space H and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the Ishikawa iteration ( xn+1 = αn xn + (1 − αn )T yn (4.1) yn = βn xn + (1 − βn )T xn , where the sequences {αn } and {βn } in (0, 1) such that lim inf βn (1 − αn )(1 − βn ) > 0. n→∞
(4.2)
Then the sequence {xn } converges weakly to an attractive point z of T . Proof. It follows from Lemma 2.2 that the sequence {xn } is bounded and lim kxn − uk exists for each u ∈ A(T ) and lim kxn − T xn k = 0.
n→∞
n→∞
(4.3)
Then there exists a subsequence {xnk } of {xn } and z ∈ H such that xnk * z. We claim z ∈ A(T ). In fact, it follows from the definition of (α, β)−generalized hybrid mapping that for all x ∈ K, αkT xnk − T xk2 + (1 − α)kxnk − T xk2 ≤ βkT xnk − xk2 + (1 − β)kxnk − xk2 .
(4.4)
Then we have α(kxnk − T xk2 + 2hxnk − T x, T xnk − xnk i + kT xnk − xnk k2 ) + (1 − α)kxnk − T xk2 = αkT xnk − T xk2 + (1 − α)kxnk − T xk2 ≤ βkT xnk − xk2 + (1 − β)kxnk − xk2 ≤ β(kT xnk − xnk k2 + 2hT xnk − xnk , xnk − xi + kxnk − xk2 ) + (1 − β)kxnk − xk2 . Let k → ∞. Then by ((4.3)) ( lim kxnk − T xnk k = 0), we have k→∞
lim sup kxnk − T xk2 ≤ lim sup kxnk − xk2 . k→∞
k→∞
Since xnk * z and kxnk − xk2 = kxnk − T xk2 + 2hxnk − T x, T x − xi + kT x − xk2 , then lim sup kxnk − xk2 = lim sup kxnk − T xk2 + 2hz − T x, T x − xi + kT x − xk2 . k→∞
Since kz −
xk2
k→∞
= kz −
T xk2
+ 2hz − T x, T x − xi + kT x − xk2 , we have
2hz − T x, T x − xi + kT x − xk2 = kz − xk2 − kz − T xk2 , and hence lim sup kxnk − xk2 = lim sup kxnk − T xk2 + kz − xk2 − kz − T xk2 . k→∞
k→∞
From ((4.5)), it follows that lim sup kxnk − T xk2 ≤ lim sup kxnk − xk2 k→∞
k→∞
= lim sup kxnk − T xk2 + kz − xk2 − kz − T xk2 , k→∞
(4.5)
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
361
and so kz − T xk ≤ kz − xk for all x ∈ K. That is, z ∈ A(T ). Now we prove {xn } converges weakly to z. Suppose not, then there exists another subsequence {xni } of {xn } which weakly converges to some y 6= z. Again in the same way, we have y ∈ A(T ). From ((4.3)), it follows that both lim kxn − zk and lim kxn − yk exist. By the elementary properties n→∞ n→∞ in Hilbert space, we easily obtain lim kxn − zk2 = lim kxnk − zk2
n→∞
k→∞
= lim (kxnk − yk2 + 2hxnk − y, y − zi + ky − zk2 ) k→∞
= lim kxnk − yk2 + 2hz − y, y − zi + ky − zk2 k→∞
= lim kxnk − yk2 − ky − zk2 k→∞
= lim kxn − yk2 − ky − zk2 n→∞
= lim kxni − yk2 − ky − zk2 i→∞
= lim (kxni − zk2 + 2hxni − z, z − yi + kz − yk2 ) − ky − zk2 i→∞
= lim kxni − zk2 − 2kz − yk2 i→∞
= lim kxn − zk2 − 2kz − yk2 , n→∞
which implies z = y, a contradiction. Thus, {xn } converges weakly to an attractive point z of T . Take αn ≡ 0. We also obtained easily the following. Corollary 4.2. Let K be a nonempty closed and convex subset of a Hilbert space H and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the following iteration xn+1 = T (βn xn + (1 − βn )T xn ) (4.6) where the sequence {βn } in (0, 1) such that lim inf βn (1 − βn ) > 0. n→∞
(4.7)
Then the sequence {xn } converges weakly to an attractive point z of T . References [1] J. B. Baillon, Un th´eor`eme de type ergodique pour les contractions non lin´ears dans un espaces de Hilbert, C.R. Acad. Sci. Paris Ser. A-B, 280 (1975), 1511–1541. 1 [2] R. E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math., 32 (1979), 107–116. 1 [3] R. E. Bruck,On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, Israel J. Math., 38 (1981), 304–314. 1 [4] M. Hojo, W. Takahashi, Weak and strong convergence theorems for generalized hybrid mappings in Hilbert spaces, Sci. Math. Jpn., 73 (2011), 31–40. 1 [5] P. Kocourek, W. Takahashi, J. C. Yao, Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese J. Math., 14 (2010), 2497–2511. 1, 1, 1 [6] F. Kohsaka, W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel), 91 (2008), 166–177. 1 [7] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953) 506–510. 1 [8] W. V. Petryshyn, Construction of fixed points of demicompact mappings in Hilbert space, J. Math. Anal. Appl., 14 (1966), 276–284. 3
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[9] H. F. Senter, W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44 (1974), 375–380 3 [10] W. Takahashi, Fixed point theorems for new nonlinear mappings in a Hilbert space, J. Nonlinear Convex Anal., 11 (2010), 79–88. 1, 1 [11] W. Takahashi, J. C. Yao, Fixed point theorems and ergodic theorems for nonlinear mappings in Hilbert spaces, Taiwanese J. Math., 15 (2011), 457–472. 1, 1 [12] W. Takahashi, Y. Takeuchi, Nonlinear ergodic theorem without convexity for generalized hybrid mappings in a Hilbert space, J. Nonlinear Convex Anal., 12 (2011), 399–406. 1, 1, 2 [13] H. K. Xu, Inequality in Banach spaces with applications, Nonlinear Anal., 16 (1991) 1127–1138.
Attractive points and convergence theorems of generalized hybrid mapping Yuchun Zheng∗ School of Mathematics and Information Science, Henan Normal University, XinXiang, P.R. China, 453007
Abstract In this paper, by means of the concept of attractive points of a nonlinear mapping, we prove strong convergence theorem of the Ishikawa iteration for an (α, β)−generalized hybrid mapping in a uniformly convex Banach space, and obtain weak convergence theorem of the Ishikawa iteration for such a mapping in a Hilbert space. Keywords: Attractive points, generalized hybrid mapping, Ishikawa iteration, Mann iteration, Xu’s inequality. 2010 MSC: 47H10, 54H25, 49J40, 47H05, 47H04, 65J15, 47H10. 1. Introduction Let E be a Banach space with the norm k · k and let K be a nonempty subset of E. In 2010, Kocourek, Takahashi and Yao [5] firstly introduced the concept of the generalized hybrid mapping, which contains the classes of nonexpansive mappings, nonspreading mappings, and hybrid mappings. A mapping T : K → K is called (α, β)−generalized hybrid if there exist α, β ∈ R such that αkT x − T yk2 + (1 − α)kx − T yk2 ≤ βkT x − yk2 + (1 − β)kx − yk2
(1.1)
for all x, y ∈ K, where R is the set of real numbers. T is said to be nonexpansive if T is (1, 0)−generalized hybrid; T is called hybrid (Takahashi [10]) if T is ( 23 , 12 )−generalized hybrid, i.e. 3kT x − T yk2 ≤ kx − T yk2 + kT x − yk2 + kx − yk2 ∀x, y ∈ K; T is called nonspreading (Kohsaka and Takahashi [6]) if T is (2, 1)−generalized hybrid, i.e. 2kT x − T yk2 ≤ kx − T yk2 + kT x − yk2 ∀x, y ∈ K. ∗
Corresponding author Email address: [email protected] (Yuchun Zheng)
Received 2015-1-16
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
355
Baillon [1] proved the first nonlinear ergodic theorem: suppose that K is a nonempty closed convex subset of Hilbert space E and T : K → K is nonexpansive mapping such that F (T ) 6= ∅, then ∀x ∈ K, the Ces`aro means n 1 X i Tn x = T x n+1 i=0
weakly converges to a fixed point of T . Bruck [2, 3] studied the property of Ces` aro means for nonexpansive mapping in uniformly convex Banach space. Takahashi and Yao [11] proved the nonlinear ergodic theorem for both hybrid and nonspreading mappings in a Hilbert space. Kocourek, Takahashi and Yao [5] showed that both the nonlinear ergodic theorem and the weak convergence theorem of the Mann iteration for (α, β)−generalized hybrid mapping. The Mann iteration is the original definition of Mann [7] for a nonexpansive mapping T , xn+1 = αn xn + (1 − αn )T xn {αn } ⊂ (0, 1), x1 ∈ K. Takahashi and Takeuchi [12] obtained the nonlinear ergodic theorem without convexity for (α, β)−generalized hybrid mappings. Hojo and Takahashi [4] showed the strong convergence of the Halpern iteration of Ces` aro means for (α, β)−generalized hybrid mapping T under some proper conditions, n
xn+1
1 X i = αn u + (1 − αn ) T xn {αn } ⊂ (0, 1), u, x1 ∈ K. n+1 i=0
In this paper, we will deal with strong and weak convergence of the Ishikawa iteration for finding attractive points of (α, β)−generalized hybrid mappings under some conditions on the sequences {αn } and {βn } in (0, 1), ( xn+1 = αn xn + (1 − αn )T yn (1.2) yn = βn xn + (1 − βn )T xn Our results obviously develop and complement the corresponding ones of Kocourek, Takahashi and Yao [5], Takahashi and Yao [11] , Takahashi and Takeuchi [12], Takahashi [10] and others. 2. Preliminaries and basic results Let N and R be the sets of positive integers and real numbers, respectively. Let K be a nonempty subset of a Banach space E with the norm k · k and let T be a mapping T from K to E. A point y ∈ E is called an attractive point of T if for all x ∈ K kT x − yk ≤ kx − yk. We denote by A(T ) the set of all attractive points of T , i.e., A(T ) = {y ∈ E; kT x − yk ≤ kx − yk ∀x ∈ K}. Takahashi and Takeuchi [12] used this concept and proved the closed and convex property of A(T ) in a Hilbert space H. For more details, see Takahashi and Takeuchi [12]. A Banach space E is said to be uniformly convex if for all ε ∈ [0, 2], ∃δε > 0 such that
kx + yk < 1 − δε whenever kx − yk ≥ ε. 2 The following lemmas are well-known which can be found in [13]. kxk = kyk = 1 implies
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
356
Lemma 2.1. (Xu [13, Theorem 2]) Let q > 1 and r > 0 be two fixed real numbers. Then a Banach space is uniformly convex if and only if there exists a continuous strictly increasing convex function g : [0, +∞) → [0, +∞) with g(0) = 0 such that kλx + (1 − λ)ykq ≤ λkxkq + (1 − λ)kykq − ωq (λ)g(kx − yk),
(2.1)
for all x, y ∈ Br (0) = {x ∈ E; kxk ≤ r} and λ ∈ [0, 1], where ωq (λ) = λq (1 − λ) + λ(1 − λ)q . Note that the inequality in Lemma 2.1 is known as Xu’s inequality. Let H be a real Hilbert space with the norm k · k and the inner produce h·, ·i. Obviously, the Xu’s inequality is replaced by the following equality in a Hilbert space H, for x, y ∈ H and t ∈ R, ktx + (1 − t)yk2 = tkxk2 + (1 − t)kyk2 − t(1 − t)kx − yk2 .
(2.2)
Lemma 2.2. Let K be a nonempty closed and convex subset of a real uniformly convex Banach space E and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the Ishikawa iteration ( xn+1 = αn xn + (1 − αn )T yn (2.3) yn = βn xn + (1 − βn )T xn where the sequences {αn } and {βn } in (0, 1) such that lim inf βn (1 − αn )(1 − βn ) > 0. n→∞
(2.4)
Then (i) the sequence {xn } is bounded; (ii) the limit lim kxn − uk exists for each u ∈ A(T ); n→∞
(iii) lim kxn − T xn k = 0. n→∞
Proof. Take u ∈ A(T ). By the definition of the attractive point, we have kxn+1 − uk = kαn (xn − u) + (1 − αn )(T yn − u)k ≤ αn kxn − uk + (1 − αn )kT yn − uk ≤ αn kxn − uk + (1 − αn )kyn − uk ≤ αn kxn − uk + (1 − αn )(βn kxn − uk + (1 − βn )kT xn − uk) ≤ (αn + (1 − αn )(βn + (1 − βn )))kxn − uk ≤ kxn − uk .. . ≤ kx1 − uk. So the sequence {xn } is bounded and the sequence {kxn − uk} is monotone non-increasing, and hence the limit lim kxn − uk exists for each u ∈ A(T ). n→∞
Now we show (iii). Let r ≥ max kxn − uk. n∈N
Then kT yn − uk ≤ kyn − uk ≤ kxn − uk ≤ r and kT xn − uk ≤ kxn − uk ≤ r.
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It follows from Lemma 2.1(q = 2) that kxn+1 − uk2
= kαn (xn − u) + (1 − αn )(T yn − u)k2 ≤ αn kxn − uk2 + (1 − αn )kT yn − uk2 ≤ αn kxn − uk2 + (1 − αn )kyn − uk2 = αn kxn − uk2 + (1 − αn )kβn (xn − u) + (1 − βn )(T xn − u)k2 ≤ αn kxn − uk2 + (1 − αn )(βn kxn − uk2 + (1 − βn )kT xn − uk2 −βn (1 − βn )g(kT xn − xn k)) ≤ αn kxn −
uk2
+ (1 − αn )(βn kxn − uk2 + (1 − βn )kxn − uk2 −βn (1 − βn )g(kT xn − xn k))
≤ kxn −
uk2
− (1 − αn )βn (1 − βn )g(kT xn − xn k).
(2.5)
Then we have (1 − αn )βn (1 − βn )g(kT xn − xn k) ≤ kxn − uk2 − kxn+1 − uk2 , and so,
∞ X
(1 − αn )βn (1 − βn )g(kT xn − xn k) ≤ kx1 − uk2 < +∞.
n=1
From the condition ((2.4)), it follows that lim g(kT xn − xn k) = 0.
n→∞
By the property of the function g, we have lim kT xn − xn k = 0.
n→∞
This completes the proof. When αn = 0 for all n, the following conclusions hold obviously. Corollary 2.3. Let K be a nonempty closed and convex subset of a real uniformly convex Banach space E and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the following iteration xn+1 = T (βn xn + (1 − βn )T xn )
(2.6)
where the sequence {βn } in (0, 1) such that lim inf βn (1 − βn ) > 0. n→∞
(2.7)
Then (i) the sequence {xn } is bounded; (ii) the limit lim kxn − uk exists for each u ∈ A(T ); n→∞
(iii) lim kxn − T xn k = 0. n→∞
3. Strongly Convergent Theorems Let K be a nonempty subset of a Banach space E. A mapping T : K → K is said to satisfy Condition I if there is a nondecreasing function f : [0, ∞) → [0, ∞) with f (0) = 0, f (r) > 0 for r ∈ (0, ∞) such that kx − T xk ≥ f (d(x, A(T ))) for all x ∈ K, where d(x, A(T )) = inf{kx − yk; y ∈ A(T )}. This concept was introduced by Senter and Dotson [9] and the examples of mappings that satisfy Condition I was given.
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358
Theorem 3.1. Let K be a nonempty closed and convex subset of a uniformly convex Banach space E and and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅ and satisfying Condition I. Suppose that the sequence {xn } is defined by the Ishikawa iteration ( xn+1 = αn xn + (1 − αn )T yn (3.1) yn = βn xn + (1 − βn )T xn , where the sequences {αn } and {βn } in (0, 1) such that lim inf βn (1 − αn )(1 − βn ) > 0. n→∞
(3.2)
Then the sequence {xn } converges strongly to an attractive point z of T . Proof. It follows from Lemma 2.2 that the sequence {xn } is bounded and kxn+1 − uk ≤ kxn − uk for each u ∈ A(T ) and lim kxn − T xn k = 0. n→∞
(3.3)
Then Condition I implies lim f (d(xn , A(T ))) = 0, and hence n→∞
lim d(xn , A(T )) = 0.
n→∞
(3.4)
Next we show that the sequence {xn } is a Cauchy sequence of E. In fact, for any n, m ∈ N, without loss of generality, we may set m > n, then kxm − uk ≤ kxn − uk for each u ∈ A(T ) by ((3.3)), and so kxn − xm k ≤ kxn − uk + ku − xm k ≤ 2kxn − uk.
(3.5)
Since u is arbitrary, then we may take the infimum for u in ((3.5)), kxn − xm k ≤ 2 inf{kxn − uk; u ∈ A(T )} = 2d(xn , A(T )). From ((3.4)), it follows that as lim kxn − xm k = 0, which means that {xn } is a Cauchy sequence. So there n→∞ exists z ∈ E such that lim kxn − zk = 0. n→∞
By ((3.3)), we have lim kT xn − zk = 0.
n→∞
Now we prove z ∈ A(T ). In fact, it follows from the definition of (α, β)−generalized hybrid mapping that for all x ∈ K, αkT xn − T xk2 + (1 − α)kxn − T xk2 ≤ βkT xn − xk2 + (1 − β)kxn − xk2 .
(3.6)
Let n → ∞ in ((3.6)). Then by the continuity of the norm k · k and the function g(t) = t2 , we have αkz − T xk2 + (1 − α)kz − T xk2 ≤ βkz − xk2 + (1 − β)kz − xk2 , and hence kz − T xk ≤ kz − xk for all x ∈ K. So z ∈ A(T ) and lim kxn − zk = 0. The proof is completed. n→∞
A mapping T : K → E is said to be demicompact (Petryshyn [8]) provided whenever a sequence {xn } ⊂ K is bounded and the sequence {xn − T xn } strongly converges, then there is a subsequence {xnk } which strongly converges.
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359
Theorem 3.2. Let K be a nonempty closed and convex subset of a uniformly convex Banach space E and and let T : K → K be (α, β)−generalized hybrid and demicompact with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the Ishikawa iteration ( (3.1)) and the sequences {αn } and {βn } satisfy ( (3.2)). Then the sequence {xn } converges strongly to an attractive point z of T . Proof. It follows from Lemma 2.2 that the sequence {xn } is bounded and lim kxn − T xn k = 0.
n→∞
(3.7)
Then the demicompactness of T implies there is a subsequence {xnk } ⊂ {xn } and z ∈ E such that lim kxnk − zk = 0.
k→∞
(3.8)
By ((3.7)), we also have lim kT xnk − zk = 0.
k→∞
From the definition of (α, β)−generalized hybrid mapping, it follows that for all x ∈ K, αkT xnk − T xk2 + (1 − α)kxnk − T xk2 ≤ βkT xnk − xk2 + (1 − β)kxnk − xk2 .
(3.9)
Let k → ∞ in ((3.9)). Then by the continuity of the norm k · k and the function g(t) = t2 , we have kz − T xk ≤ kz − xk for all x ∈ K. So z ∈ A(T ). Since lim kxn − uk exists for each u ∈ A(T ) by Lemma 2.2 (ii), then we have n→∞
lim kxn − zk = 0.
n→∞
The proof is completed. The condition (3.2) contains αn ≡ 0 and lim inf βn (1 − βn ) > 0 as special cases. So the following result n→∞ is obtained easily. Corollary 3.3. Let K be a nonempty closed and convex subset of a real uniformly convex Banach space E and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the following iteration xn+1 = T (βn xn + (1 − βn )T xn )
(3.10)
where the sequence {βn } in (0, 1) such that lim inf βn (1 − βn ) > 0. n→∞
(3.11)
Assume that T either satisfies Condition I or is demicompact. Then the sequence {xn } converges strongly to an attractive point z of T . 4. Weakly Convergent Theorems Let {xn } is a sequence in E, then xn * x will denote weak convergence of the sequence {xn } to x.
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360
Theorem 4.1. Let K be a nonempty closed and convex subset of a Hilbert space H and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the Ishikawa iteration ( xn+1 = αn xn + (1 − αn )T yn (4.1) yn = βn xn + (1 − βn )T xn , where the sequences {αn } and {βn } in (0, 1) such that lim inf βn (1 − αn )(1 − βn ) > 0. n→∞
(4.2)
Then the sequence {xn } converges weakly to an attractive point z of T . Proof. It follows from Lemma 2.2 that the sequence {xn } is bounded and lim kxn − uk exists for each u ∈ A(T ) and lim kxn − T xn k = 0.
n→∞
n→∞
(4.3)
Then there exists a subsequence {xnk } of {xn } and z ∈ H such that xnk * z. We claim z ∈ A(T ). In fact, it follows from the definition of (α, β)−generalized hybrid mapping that for all x ∈ K, αkT xnk − T xk2 + (1 − α)kxnk − T xk2 ≤ βkT xnk − xk2 + (1 − β)kxnk − xk2 .
(4.4)
Then we have α(kxnk − T xk2 + 2hxnk − T x, T xnk − xnk i + kT xnk − xnk k2 ) + (1 − α)kxnk − T xk2 = αkT xnk − T xk2 + (1 − α)kxnk − T xk2 ≤ βkT xnk − xk2 + (1 − β)kxnk − xk2 ≤ β(kT xnk − xnk k2 + 2hT xnk − xnk , xnk − xi + kxnk − xk2 ) + (1 − β)kxnk − xk2 . Let k → ∞. Then by ((4.3)) ( lim kxnk − T xnk k = 0), we have k→∞
lim sup kxnk − T xk2 ≤ lim sup kxnk − xk2 . k→∞
k→∞
Since xnk * z and kxnk − xk2 = kxnk − T xk2 + 2hxnk − T x, T x − xi + kT x − xk2 , then lim sup kxnk − xk2 = lim sup kxnk − T xk2 + 2hz − T x, T x − xi + kT x − xk2 . k→∞
Since kz −
xk2
k→∞
= kz −
T xk2
+ 2hz − T x, T x − xi + kT x − xk2 , we have
2hz − T x, T x − xi + kT x − xk2 = kz − xk2 − kz − T xk2 , and hence lim sup kxnk − xk2 = lim sup kxnk − T xk2 + kz − xk2 − kz − T xk2 . k→∞
k→∞
From ((4.5)), it follows that lim sup kxnk − T xk2 ≤ lim sup kxnk − xk2 k→∞
k→∞
= lim sup kxnk − T xk2 + kz − xk2 − kz − T xk2 , k→∞
(4.5)
Yuchun Zheng, J. Nonlinear Sci. Appl. 8 (2015), 354–362
361
and so kz − T xk ≤ kz − xk for all x ∈ K. That is, z ∈ A(T ). Now we prove {xn } converges weakly to z. Suppose not, then there exists another subsequence {xni } of {xn } which weakly converges to some y 6= z. Again in the same way, we have y ∈ A(T ). From ((4.3)), it follows that both lim kxn − zk and lim kxn − yk exist. By the elementary properties n→∞ n→∞ in Hilbert space, we easily obtain lim kxn − zk2 = lim kxnk − zk2
n→∞
k→∞
= lim (kxnk − yk2 + 2hxnk − y, y − zi + ky − zk2 ) k→∞
= lim kxnk − yk2 + 2hz − y, y − zi + ky − zk2 k→∞
= lim kxnk − yk2 − ky − zk2 k→∞
= lim kxn − yk2 − ky − zk2 n→∞
= lim kxni − yk2 − ky − zk2 i→∞
= lim (kxni − zk2 + 2hxni − z, z − yi + kz − yk2 ) − ky − zk2 i→∞
= lim kxni − zk2 − 2kz − yk2 i→∞
= lim kxn − zk2 − 2kz − yk2 , n→∞
which implies z = y, a contradiction. Thus, {xn } converges weakly to an attractive point z of T . Take αn ≡ 0. We also obtained easily the following. Corollary 4.2. Let K be a nonempty closed and convex subset of a Hilbert space H and let T : K → K be a (α, β)−generalized hybrid mapping with A(T ) 6= ∅. Suppose that the sequence {xn } is defined by the following iteration xn+1 = T (βn xn + (1 − βn )T xn ) (4.6) where the sequence {βn } in (0, 1) such that lim inf βn (1 − βn ) > 0. n→∞
(4.7)
Then the sequence {xn } converges weakly to an attractive point z of T . References [1] J. B. Baillon, Un th´eor`eme de type ergodique pour les contractions non lin´ears dans un espaces de Hilbert, C.R. Acad. Sci. Paris Ser. A-B, 280 (1975), 1511–1541. 1 [2] R. E. Bruck, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israel J. Math., 32 (1979), 107–116. 1 [3] R. E. Bruck,On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, Israel J. Math., 38 (1981), 304–314. 1 [4] M. Hojo, W. Takahashi, Weak and strong convergence theorems for generalized hybrid mappings in Hilbert spaces, Sci. Math. Jpn., 73 (2011), 31–40. 1 [5] P. Kocourek, W. Takahashi, J. C. Yao, Fixed point theorems and weak convergence theorems for generalized hybrid mappings in Hilbert spaces, Taiwanese J. Math., 14 (2010), 2497–2511. 1, 1, 1 [6] F. Kohsaka, W. Takahashi, Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces, Arch. Math. (Basel), 91 (2008), 166–177. 1 [7] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953) 506–510. 1 [8] W. V. Petryshyn, Construction of fixed points of demicompact mappings in Hilbert space, J. Math. Anal. Appl., 14 (1966), 276–284. 3
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