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Volume 2010, Article ID 982352, 13 pagesdoi:10.1155/2010/982352 Research Article Iterative Algorithms with Variable Coefficients for Multivalued Generalized Φ-Hemicontractive Mappings wi

Volume 2010, Article ID 982352, 13 pages

doi:10.1155/2010/982352

Research Article

Iterative Algorithms with Variable

Coefficients for Multivalued Generalized

Φ-Hemicontractive Mappings without

Generalized Lipschitz Assumption

Ci-Shui Ge

Department of Mathematics and Physics, Anhui University of Architecture, Hefei, Anhui 230022, China

Correspondence should be addressed to Ci-Shui Ge,gecishui@sohu.com

Received 17 August 2010; Accepted 8 November 2010

Academic Editor: Tomonari Suzuki

Copyrightq 2010 Ci-Shui Ge 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

We introduce and study some new Ishikawa-type iterative algorithms with variable coefficients for multivalued generalizedΦ-hemicontractive mappings Several new fixed-point theorems for multivalued generalizedΦ-hemicontractive mappings without generalized Lipschitz assumption

are established in p-uniformly smooth real Banach spaces A result for multivalued generalized

Φ-hemicontractive mappings with bounded range is obtained in uniformly smooth real Banach spaces As applications, several theorems for multivalued generalizedΦ-hemiaccretive mapping equations are given

1 Introduction

Let X be a real Banach space and X∗ the dual space of X ∗, ∗ denotes the generalized duality pairing between X and X∗ J is the normalized duality mapping from X to 2 X∗ given

by Jx

J x :f ∈ X∗: 

x, f

f  · x, f  x, x ∈ X. 1.1

Let D be a nonempty convex subset of X and CBD the family of all nonempty bounded closed subsets of D H·, · denotes the Hausdorff metric on CBD defined by

H A, B : max

 sup

y ∈Binf

x ∈Ax − y, sup

y ∈Binf

x ∈Ax − y, A, B ∈ CBD. 1.2

We use FT to denote the fixed-point set of T, that is, FT : {x : x ∈ Tx} N denotes the

set of nonnegative integers

Recall that a mapping T : D → D is called to be a generalized Lipschitz mapping 1 ,

if there exists a constant L > 0 such that

Similarly, a multivalued mapping T : D → CBD is said to be a generalized Lipschitz mapping, if there exists a constant L > 0 such that

H

Tx, Ty ≤ L 1 , ∀x, y ∈ D. 1.4

A multivalued mapping T : D → 2Dis said to be a bounded mapping if for any bounded

subset A of D,

T A :x : x ∈ T y , ∃y ∈ A 1.5

is a bounded subset of D.

Clearly, every mapping with bounded range is a generalized Lipschitz mapping1, Example Furthermore, every generalized Lipschitz mapping is a bounded mapping The following example shows that the class of generalized Lipschitz mappings is a proper subset

of the class of bounded mappings

Example 1.1 Take D  0, ∞ and define T : D → D by

where sgn· denotes sign function Then, T is a bounded mapping but not a generalized Lipschitz mapping

Definition 1.2 see 2  Let D be a nonempty subset of X T : D → 2 D is said to be a multivalued Φ-hemicontractive mapping if the fixed point set FT of T is nonempty, and

there exists a strictly increasing functionΦ : 0, ∞ → 0, ∞ with Φ0  0 such that for each

x ∈ D and x∗∈ FT, there exists a jx − x∗ ∈ Jx − x∗ such that



u − x∗, j x − x∗≤ x − x∗2− Φx − x∗ · x − x∗, 1.7

for all u ∈ Tx.

T is said to be a multivalued hemiaccretive mapping if I − T is a multivalued

Φ-hemicontractive mapping

Definition 1.3 Let D be a nonempty subset of X T:D → 2D is said to be a multivalued generalized Φ-hemicontractive mapping if the fixed point set FT of T is nonempty,

and there exists a strictly increasing functionΦ : 0, ∞ → 0, ∞ with Φ0  0 such that for each x ∈ D and x∗∈ FT, there exists a jx − x∗ ∈ Jx − x∗ such that



u − x∗, j x − x∗≤ x − x∗2− Φx − x∗, 1.8

for all u ∈ Tx.

T is said to be a multivalued generalized Φ-hemiaccretive mapping if I − T is a

multivalued generalizedΦ-hemicontractive mapping

The following example shows that the class ofΦ-hemicontractive mappings is a proper subset of the class of generalizedΦ-hemicontractive mappings

Example 1.4 Let X  R2with the Euclidean norm · , where R denotes the set of the real

numbers Define T : X → X by

Tx x2

Thus, F T  {0, 0} / ∅ It is easy to verify that T is a generalized Φ-hemicontractive

mapping withΦt  t2/ 2 However, T is not Φ-hemicontractive Indeed, if there exists

a strictly increasing function φ : 0, ∞ → 0, ∞ with φ0  0 such that for each x ∈ X and

x∗ 0, 0 ∈ FT,

Tx − x∗, J x − x∗ ≤ x − x∗2− φx − x∗ · x − x∗, 1.10

then we get φ 2 for all t ∈ 0, ∞ Thus, lim t→ ∞φ t  0 This is in contradiction with the hypotheses that φt is strictly increasing and φ0  0.

In the last twenty years or so, numerous papers have been written on the existence and convergence of fixed points for nonlinear mappings, and strong and weak convergence theorems have been obtained by using some well-known iterative algorithmssee, e.g., 1 9 and the references therein

For multivalued φ-hemicontractive mappings, Hirano and Huang 2 obtained the following result

Theorem HH See 2, Theorem 1  Let E be a uniformly smooth Banach space and T : E → 2E

be a multivalued φ- hemicontractive operator with bounded range Suppose {a n }, {b n }, {c n }and{a n }, {b n }, {c n }are real sequences in 0, 1 satisfying the following conditions:

i a n n n  a n n n  1, for all n ∈ N,

ii limn→ ∞b n limn→ ∞b n limn→ ∞c n  0,

iii∞n1b n  ∞,

iv c n  ob n .

For arbitrary x1, u1, v1∈ E, define the sequence {x n}∞n1by

x n  a n x n n η n n u n , ∃η n ∈ Ty n , n ∈ N,

y n  a n x n n ξ n n v n , ∃ξ n ∈ Tx n , n ∈ N, 1.11

where {u n}∞n1, {v n}∞n1are arbitrary bounded sequences in E Then, {x n}∞n1converges strongly to the unique fixed point of T.

Further, for general multivalued generalized Φ-hemicontractive mappings, C E Chidume and C O Chidume1 gave the following interesting result

Theorem CC see 1, Theorem 3.8  Let E be a uniformly smooth real Banach space Let FT :

{x ∈ E : x ∈ Tx} / ∅ Suppose T : E → 2 E is a multivalued generalized Lipschitz and generalized

Φ-hemicontractive mapping Let {a n }, {b n }and{c n }be real sequences in0, 1 satisfying the following

conditions: (i) a n n n  1, (ii)b n n   ∞, (iii)c n < ∞, and (iv) lim b n  0 Let {x n}

be generated iteratively from arbitrary x0∈ E by

x n  a n x n n η n n u n , ∃η n ∈ Tx n n ≥ 0, 1.12

where {u n } is an arbitray bounded sequence in E Then, there exists γ0 ∈ R such that if b n n ≤ γ0

for all n ≥ 0, the sequence {x n } converges strongly to the unique fixed point of T.

Remark 1.5. 1 Theorem CC 1, Theorem 3.8 is a multivalued version of Theorem 3.2 of 1 Theorem 3.2 of 1 was obtained directly from Theorem 3.1 of 1 However, it seems that there exists a gap in the proof of Theorem 3.1 in1 Indeed, the following inequality in the proof of Theorem 3.1 in1

a0

n

j0

α j≤n

j0

x j − x∗2− x j − x∗2 n

j0

c j <∞ ∗

was obtained by using implicitly the following conditions:

x j − x∗ ≤ 2Φ−1a0, x j − x∗> 2Φ−1a0, j  0, 1, , n. 1.13

Thus, ∗ is dubious in the remainder of 1, Theorem 3.1 Hence, Theorem 3.1 of 1 is dubious, as is Theorem CC1, Theorem 3.8

2 The real number γ0in Theorem CC is not easy to get

It is our purpose in this paper to try to obtain some fixed-point theorems for multivalued generalized Φ-hemicontractive mappings without generalized Lipschitz assumption as in Theorem CC Motivated and inspired by 1, 2, 5, 7 , we introduce and study some new Ishikawa-type iterative algorithms with variable coefficients for multivalued generalizedΦ-hemicontractive mappings Our results improve essentially the corresponding results of1 in the framework of p-uniformly smooth real Banach spaces and

the corresponding results of2 in uniformly smooth real Banach spaces

2 Preliminaries

Let X be a real Banach space of dimension dim X ≥ 2 The modulus of smoothness of X is the function ρ X :0, ∞ → 0, ∞ defined by

ρ X τ : sup2−1 x 2.1

The function ρ X τ is convex, continuous, and increasing, and ρ X0  0

The space X is called uniformly smooth if and only if

lim

τ→ 0

ρ X τ

The space X is called p-uniformly smooth if and only if there exist a constant C pand

a real number 1 < p≤ 2, such that

ρ X τ ≤ C p τ p 2.3

Typical examples of uniformly smooth spaces are the Lebesgue L p , the sequence p,

and Sobolev W m

p spaces for 1 < p < ∞ In particular, for 1 < p ≤ 2, these spaces are

p-uniformly smooth and for 2≤ p < ∞, they are 2-uniformly smooth.

It is well known that if X is uniformly smooth, then the normalized duality mapping

J is single-valued and uniformly continuous on any bounded subset of X.

Lemma 2.1 see 3,9  If X is a uniformly smooth Banach space, then for all x, y ∈ X with x ≤

R, y ≤ R,



x − y, Jx − Jy≤ 2L F R2ρ X



4x − y

R



,

Jx − Jy ≤ 8Rh X



16L Fx − y

R



,

2.4

where h X τ : ρ X τ/τ, L F is the Figiel s constant, 1 < L F < 1.7.

Lemma 2.2 see 1  Let X be a real Banach space and J be the normalized duality mapping Then,

for any given x, y ∈ X, we have

x 2

≤ x2 

y, j

x , ∀j x ∈ J x 2.5

Lemma 2.3 see 8  Let {α n}n≥1, {β n}n≥1and {γ n}n≥1be nonnegative sequences satisfying

α n ≤ 1 n α n n , n ≥ 1, ∞

n1

β n < ∞, ∞

n1

γ n < ∞. 2.6

Then, lim n→ ∞α n exists Moreover, if lim inf n→ ∞ α n  0, then lim n→ ∞α n  0.

Lemma 2.4 see 4  Let f, g : N → 0, ∞ be sequences and suppose that

g n ≤ 1, ∀n ∈ N, gn −→ 0, as n −→ ∞, ∞

n1

g n  ∞. 2.7

Then,

∞

n1

f n < ∞ ⇒ f  o g , as n −→ ∞. 2.8

The converse is false.

3 Main Results and Their Proofs

Theorem 3.1 Let X be a p-uniformly smooth real Banach space and D a nonempty convex subset of

X Suppose T : D → 2D is a multivalued generalized Φ-hemicontractive and bounded mapping For

any given x0, u0, v0 ∈ D, let {x n } be the sequence generated by the following Ishikawa-type iterative

algorithm with variable coefficients:

y n  a n x n n ξ n n v n , ∃ξ n ∈ Tx n ,

x n  α n x n n η n n u n , ∃η n ∈ Ty n , n ∈ N, 3.1

where {u n } and {v n } are arbitrary bounded sequences in D,

a n  1 − b n − c n , b n b n

r n2, c n c n

r n2, r n n n n ,

α n  1 − β n − γ n , β n β n

R2

n

, γ n γ n

R2

n

, R n  r n η n n , 3.2

{β n }, {γ n }, {b n } and {c n } are four sequences in 0, 1 satisfying the following conditions:

∞

n0

β n  ∞, ∞

n0

β n p < ∞, ∞

n0

γ n < ∞, b n ≤ O β n , c n ≤ O β n 3.3

Then, {x n } converges strongly to the unique fixed point of T.

Proof Since T is generalized Φ-hemicontractive, then the fixed-point set FT of T is

nonempty and there exists a strictly increasing functionΦ : 0, ∞ → 0, ∞ with Φ0  0 such that for each x ∈ D and x∗∈ FT, the following inequality holds:

ξ − x∗, J x − x∗ ≤ x − x∗2− Φx − x∗, ∀ξ ∈ Tx. 3.4

If z ∈ FT, that is, z ∈ Tz, then, by 3.4, we have

z − x∗2 z − x∗, J z − x∗ ≤ z − x∗2− Φz − x∗. 3.5

So, T has a unique fixed point, say x∗

From3.1 and 3.2, we have x n − x∗ ≤ r n ∗, y n − x∗ ≤ r n ∗, ξ n − x∗ ≤

r n ∗, η n − x∗ ≤ R n ∗ and x n − x∗ ≤ R n ∗

T : D → 2D, it follows from3.1, 3.2, and 3.3 that

x n − y n   x n − x∗ − y n − x∗

 1− β

n − γ n



x n − x∗

n

η n − x∗

n u n − x∗

− 1− b n − c n



x n − x∗

n ξ n − x∗

n v n − x∗

≤ O

β n

r2

n

r n ∗ O

β n

R2

n

R n ∗

≤ O

β n

r n −→ 0 n −→ ∞.

3.6

From3.6 and y n − x∗ ≤ r n ∗, we have x n − x∗ ≤ r n ∗

n /r n

Considering 1 < p ≤ 2 and r n≥ 2, byLemma 2.1, we have

J x n − x∗ − J y n − x∗



r n ∗ O

β n

r n



C p·



16L Fx n − y n

r n ∗

β n /r n

p−1

≤



r n ∗ O

β n

r n

2−pO β p−1

n



r n p−1

≤ r n2 n x∗

β n 2−pO β

p−1

n



r n

≤ r n · O β n p−1

.

3.7

By3.1, 3.2, 3.3 andLemma 2.2, we have

y n − x∗2 ≤a

n x n − x∗

n

η n − x∗

n v n − x∗2

≤ a2

n x n − x∗2 

b n ξ n − x∗

n v n − x∗, J y n − x∗

≤ x n − x∗2

β n

3.8

From3.1, 3.2, 3.7, and 3.8 andLemma 2.2, it can be concluded that

x n − x∗2α

n x n − x∗

n

η n − x∗

n u n − x∗2

≤ α2

n x n − x∗2

n



η n − x∗, J x n − x∗ − J y n − x∗

n



η n − x∗, J

y n − x∗

n u n − x∗, J x n − x∗

≤ α2

n x n − x∗2

nη n − x∗ · Jx n − x∗ − J y n − x∗

n y n − x∗2− Φ y n − x∗

n u n − x∗ · Jx n − x∗

≤ 1− β n − γ n

2

x n − x∗2

n R n ∗ · r n · O β p n−1

n x n − x∗2

β n − 2β nΦ y n − x∗

n · R n ∗2

≤ x n − x∗2 β n n

2

x n − x∗2 β p n

β2n

γ n

− 2β nΦ y n − x∗

≤ x n − x∗2 β n2

x n − x∗2 β p n

γ n − 2β nΦ y n − x∗ .

3.9

From3.3 and 3.9, we have

x n − x∗2≤ 1 β n2

x n − x∗2 β p n

γ n 3.10

Thus, by3.3, 3.10 andLemma 2.3, we have{x n − x∗} bounded It implies the sequences

{x n } and {y n } are bounded Since T is a bounded mapping, we have T{x n } and T{y n}

bounded Since η n ∈ Ty n and ξ n ∈ Tx n,{R n } is bounded Let its bound be R > 0 From

3.9, there exists a number M > 0 such that

x n − x∗2 ≤ 1 2n

x n − x∗2 β n p n

−2β n

R2 Φ y n − x∗ . 3.11 Next, we will show

lim inf

n→ ∞ Φ y n − x∗ 3.12

If it is not true, then there exist a n0 ∈ N and a positive constant m0 such that for any

positive integer n ≥ n0

Φ y n − x∗

In view of3.11 and 3.13, for any positive integer n ≥ n0, we have

x n − x∗2≤ 1 2

n



x n − x∗2 β p n n

− 2m0 β n

R2 . 3.14

Taking n  n0 , n0 3.14 above, we have

k

n n0 x n − x∗2≤ k

n n0 x n − x∗2 k

n n0

Mβ2n ∗2

k

n n0

M β p n n



−k

n n0

2m0 β n

R2 .

3.15

So,

2m0

R2

k

n n0

β n ∗2k

n n0

β2n

 k

n n0

β p n k

n n0

γ n



. 3.16

This leads to a contradiction as k → ∞ Hence, lim infn→ ∞ Φy n − x∗  0

By the definition ofΦ and 3.12, there exists a subsequence {y n i } of {y n} such that

{y n i } → x∗ as i → ∞ Thus, by 3.6, we have lim infn→ ∞ x n − x∗  0 Further, Using

Lemma 2.3and3.11, we obtain limn→ ∞xn − x∗  0 It means that {x n} converges strongly

to the unique fixed point of T The proof is finished.

Theorem 3.2 Let X be a p-uniformly smooth Banach space, D be a nonempty convex subset of X,

and T : D → 2D a multivalued generalized Φ-hemicontractive and bounded mapping For any given

x0, u0 ∈ D, let {x n } be the sequence generated by the following Mann-type iterative algorithm with

variable coefficients:

x n  α n x n n η n n u n , ∃ η n ∈ Tx n , n ∈ N, 3.17

where {u n } is an arbitrary bounded sequence in D,

α n  1 − β n − γ n , β n β n

R2

n

, γ n γ n

R2

n

, R n n η n n , 3.18

{β n } and {γ n } are sequences in 0, 1 satisfying the following conditions:

∞

n0

β n  ∞, ∞

n0

β p n < ∞, ∞

n0

γ n < ∞. 3.19

Then, {x n } converges strongly to the unique fixed point of T.

Remark 3.3 Theorems 3.1 and 3.2 improve Theorem CC 1, Theorem 3.8 in p-uniformly smooth real Banach spaces since the class of multivalued generalized Lipschitz mappings

is a proper subset of the class of bounded mappings and the number γ0 in Theorem CC1, Theorem 3.8 is dropped off

In uniformly smooth real Banach spaces, we have the following theorem

Theorem 3.4 Let X be a uniformly smooth real Banach space and D a nonempty convex subset of

X Suppose T : D → 2D is a multivalued generalized Φ-hemicontractive mapping with bounded

range For any given x0, u0, v0∈ D, let {x n } be the sequence generated by the following Ishikawa-type

iterative algorithm with variable coefficients:

y n  a n x n n ξ n n v n , ∃ξ n ∈ Tx n ,

x n  α n x n n η n n u n , ∃η n ∈ Ty n ,

n ∈ N, 3.20

where {u n } and {v n } are arbitrary bounded sequences in D,

a n  1 − b n − c n , b n b n

r2

n

, c n c n

r2

n

, r n n n n ,

α n  1 − β n − γ n , β n β n

R2

n

, γ n γ n

R2

n

, R n  r n η n n , 3.21

{β n }, {γ n }, {b n } and {c n } are four sequences in 0, 1 satisfying the following conditions:

∞

n0

β n  ∞, ∞

n0

β2

n < ∞, ∞

n0

γ n < ∞, b n ≤ O β n , c n ≤ O β n 3.22

Then, {x n } converges strongly to the unique fixed point of T.

Proof FromTheorem 3.1, T has a unique fixed point, say x∗ Let{x n }, {y n} be the sequences generated by the algorithm3.20 Since T has a bounded range, we set

d : supξ − η: x, y ∈ D, ξ ∈ Tx, η ∈ Ty

n − x∗, n ∈ N}

n − x∗ , n ∈ N}. 3.23 Obviously, d < ∞ Next, we will prove that for n ≥ 0, x n − x∗

0− x∗ In fact, for

n

there exists a η k ∈ Ty ksuch that

x k − x∗ ≤ α k x n − x∗

kη k − x∗

k u k − x∗

≤ α k 0− x∗

k d k d

0− x∗.

3.24

Remark 3.3 Theorems 3.1 and 3.2 improve Theorem CC 1, Theorem 3.8 in p-uniformly smooth... unique fixed point, say x∗

From3.1 and 3.2, we have x n −