Coefficients for Multivalued Generalized Φ -Hemicontractive Mappings without Generalized Lipschitz Assumption

Volume 2010, Article ID 982352,13pages 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,[email protected] Received 17 August 2010; Accepted 8 November 2010

Academic Editor: Tomonari Suzuki

Copyrightq2010 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 inp-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 andX^∗ the dual space of X.∗, ∗ denotes the generalized duality pairing betweenXandX^∗.Jis the normalized duality mapping fromXto 2^X∗ given byJx

Jx:

f∈X^∗: x, f

f· x, fx

, x∈X. 1.1

LetD be a nonempty convex subset ofX andCBDthe family of all nonempty bounded closed subsets ofD.H·,·denotes the Hausdorffmetric onCBDdefined by

HA, B:max

sup

y∈Binf

x∈Ax−y, sup

y∈Binf

x∈Ax−y

, A, B∈CBD. 1.2

We useFTto denote the fixed-point set ofT, that is,FT:{x: x∈Tx}.Ndenotes the set of nonnegative integers.

Recall that a mappingT :D → Dis called to be a generalized Lipschitz mapping1 , if there exists a constantL >0 such that

Tx−Ty≤L 1x−y, ∀x, y∈D. 1.3 Similarly, a multivalued mapping T : D → CBD is said to be a generalized Lipschitz mapping, if there exists a constantL >0 such that

H Tx, Ty

≤L 1x−y

, ∀x, y∈D. 1.4

A multivalued mappingT : D → 2^Dis said to be a bounded mapping if for any bounded subsetAofD,

TA:

x:x∈T y

, ∃y∈A

1.5 is a bounded subset ofD.

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. TakeD 0,∞and defineT :D → Dby

Txexpx xsgnsinx, 1.6

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 FTofT is nonempty, and there exists a strictly increasing functionΦ:0,∞ → 0,∞withΦ0 0 such that for each x∈Dandx^∗∈FT, there exists ajx−x^∗∈Jx−x^∗such that

u−x^∗, jx−x^∗

≤ x−x^∗2−Φx−x^∗· x−x^∗, 1.7

for allu∈Tx.

T is said to be a multivaluedΦ-hemiaccretive mapping if I−T is a multivaluedΦ- hemicontractive mapping.

Definition 1.3. Let D be a nonempty subset ofX. T:D → 2^D 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,∞withΦ0 0 such that for eachx∈Dandx^∗∈FT, there exists ajx−x^∗∈Jx−x^∗such that

u−x^∗, jx−x^∗

≤ x−x^∗2−Φx−x^∗, 1.8

for allu∈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. LetX R²with the Euclidean norm · , whereRdenotes the set of the real numbers. DefineT :X → Xby

Tx x²

1x²x. 1.9

Thus, FT {0,0}/∅. It is easy to verify that T is a generalized Φ-hemicontractive mapping withΦt t²/1t². However,Tis notΦ-hemicontractive. Indeed, if there exists a strictly increasing functionφ:0,∞ → 0,∞withφ0 0 such that for eachx∈Xand x^∗ 0,0∈FT,

Tx−x^∗, Jx−x^∗ ≤ x−x^∗2−φx−x^∗· x−x^∗, 1.10

then we getφt≤t/1t²for allt∈0,∞. Thus, limt→ ∞φt 0. This is in contradiction with the hypotheses thatφtis strictly increasing andφ0 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 multivaluedφ-hemicontractive mappings, Hirano and Huang 2 obtained the following result.

Theorem HHSee2, Theorem 1 . LetEbe a uniformly smooth Banach space andT :E → 2^E be a multivaluedφ- hemicontractive operator with bounded range. Suppose{an},{bn},{cn}and{a_n}, {b_n},{c_n}are real sequences in0,1satisfying the following conditions:

ia_nb_nc_n a_nb_n c_n1, for all n∈N, iilimn→ ∞bnlimn→ ∞b_nlimn→ ∞cn0, iii_∞

n1b_n∞, ivc_nobn.

For arbitraryx1, u1, v1∈E, define the sequence{xn}^∞_n1by

x_n1a_n x_nb_n η_nc_n u_n, ∃ηn∈Ty_n, n∈N,

y_na_n x_nb_n ξ_nc_n v_n, ∃ξn∈Tx_n, n∈N, 1.11

where{un}^∞_n1,{vn}^∞_n1are arbitrary bounded sequences inE. Then,{xn}^∞_n1converges strongly to the unique fixed point ofT.

Further, for general multivalued generalized Φ-hemicontractive mappings, C. E.

Chidume and C. O. Chidume1 gave the following interesting result.

Theorem CCsee1, Theorem 3.8 . LetEbe a uniformly smooth real Banach space. LetFT: {x∈E :x∈Tx}/∅. SupposeT :E → 2^Eis a multivalued generalized Lipschitz and generalized Φ-hemicontractive mapping. Let{an},{bn}and{cn}be real sequences in0,1satisfying the following conditions: (i)anbncn1, (ii)bncn ∞, (iii)

cn <∞,and (iv) limbn0. Let{xn} be generated iteratively from arbitraryx₀∈Eby

x_n1a_nx_nb_nη_nc_nu_n, ∃ηn∈Tx_n n≥0, 1.12 where{un}is an arbitray bounded sequence inE. Then, there existsγ0∈Rsuch that ifbncn≤γ0

for alln≥0, the sequence{xn}converges strongly to the unique fixed point ofT.

Remark 1.5. 1Theorem CC1, Theorem 3.8 is a multivalued version of Theorem 3.2 of1 . 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

αj≤ⁿ

j0

xj−x^∗2− x_j1−x^∗2 M

n j0

cj<∞ ∗

was obtained by using implicitly the following conditions:

xj−x^∗≤2Φ⁻¹a0, x_j1−x^∗>2Φ⁻¹a0, j0,1, . . . , 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 .

2The real numberγ0in 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 ofp-uniformly smooth real Banach spaces and the corresponding results of2 in uniformly smooth real Banach spaces.

2. Preliminaries

LetXbe a real Banach space of dimension dimX≥2. The modulus of smoothness ofXis the functionρ_X :0,∞ → 0,∞defined by

ρ_Xτ:sup

2⁻¹ xyx−y−1 :x1,y≤τ

, τ >0. 2.1 The functionρ_Xτis convex, continuous, and increasing, andρ_X0 0.

The spaceXis called uniformly smooth if and only if

τlim→0

ρXτ

τ 0. 2.2

The spaceXis calledp-uniformly smooth if and only if there exist a constantCpand a real number 1< p≤2, such that

ρ_Xτ≤C_pτ^p. 2.3

Typical examples of uniformly smooth spaces are the LebesgueL_p, the sequence _p, and Sobolev W_p^m spaces for 1 < p < ∞. In particular, for 1 < p ≤ 2, these spaces arep- uniformly smooth and for 2≤p <∞, they are 2-uniformly smooth.

It is well known that ifXis uniformly smooth, then the normalized duality mapping Jis single-valued and uniformly continuous on any bounded subset ofX.

Lemma 2.1see3,9 . IfXis a uniformly smooth Banach space, then for allx, y∈Xwithx ≤ R,y ≤R,

x−y, Jx−Jy

≤2L_FR²ρ_X

4x−y R

,

Jx−Jy≤8Rh_X

16LFx−y R

,

2.4

wherehXτ:ρXτ/τ,LF is the Figiel s constant, 1< LF <1.7.

Lemma 2.2see1 . LetXbe a real Banach space andJbe the normalized duality mapping. Then, for any givenx, y∈X, we have

xy²≤ x²2

y, j xy

, ∀j xy

∈J xy

. 2.5

Lemma 2.3see8 . Let{αn}_n≥1,{βn}_n≥1and{γn}_n≥1be nonnegative sequences satisfying

α_n1 ≤ 1γn

αnβn, n≥1, ∞ n1

βn<∞, ^∞

n1

γn <∞. 2.6

Then, lim_{n→ ∞}α_nexists. Moreover, if lim inf_{n→ ∞} α_n0, then lim_{n→ ∞}α_n0.

Lemma 2.4see4 . Letf, g:N → 0,∞be sequences and suppose that gn≤1, ∀n∈N, gn−→0, as n−→ ∞, ^∞

n1

gn ∞. 2.7

Then,

∞ n1

fn<∞ ⇒f o g

, as n−→ ∞. 2.8

The converse is false.

3. Main Results and Their Proofs

Theorem 3.1. LetXbe a p-uniformly smooth real Banach space andDa nonempty convex subset of X. SupposeT :D → 2^Dis a multivalued generalizedΦ-hemicontractive and bounded mapping. For any givenx₀, u₀, v₀ ∈D, let{xn}be the sequence generated by the following Ishikawa-type iterative algorithm with variable coefficients:

ynan xnbnξncn vn, ∃ξn∈Txn,

x_n1αn xnβn ηnγn un, ∃ηn∈Tyn, n∈N, 3.1

where{un}and{vn}are arbitrary bounded sequences inD,

an1−bn−cn, bn bn

r_n², cn cn

r_n², rn2xnξnvn,

αn 1−βn−γn, βn β_n

R²_n, γn γ_n

R²_n, Rnrnηnun,

3.2

{βn},{γn},{bn}and{cn}are four sequences in0,1 satisfying the following conditions:

∞ n0

βn∞, ^∞

n0

β_n^p<∞, ^∞

n0

γn<∞, bn≤O βn

, cn≤O βn

. 3.3

Then,{xn}converges strongly to the unique fixed point ofT.

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,∞withΦ0 0 such that for eachx∈Dandx^∗∈FT, the following inequality holds:

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

Ifz∈FT, that is,z∈Tz, then, by3.4, we have

z−x^∗2z−x^∗, Jz−x^∗ ≤ z−x^∗2−Φz−x^∗. 3.5

So,T has a unique fixed point, sayx^∗.

From3.1and3.2, we havexn−x^∗ ≤rnx^∗,yn−x^∗ ≤rnx^∗,ξn−x^∗ ≤ r_nx^∗,ηn−x^∗ ≤R_nx^∗andxn1−x^∗ ≤R_nx^∗.

ByLemma 2.4and 3.3, we know γn oβn. SinceD is a convex subset ofX and T : D → 2^D, it follows from3.1,3.2, and3.3that

x_n1−y_nx_n1−x^∗− y_n−x^∗

1−βn−γn

xn−x^∗ βn ηn−x^∗

γn un−x^∗

−

1−b_n−c_n

xn−x^∗ b_n ξn−x^∗ c_n vn−x^∗

≤ O β_n

r_n² rnx^∗ O β_n

R²_n Rnx^∗

≤ O βn

rn −→0 n−→ ∞.

3.6

From3.6andyn−x^∗ ≤rnx^∗, we havexn1−x^∗ ≤rnx^∗ Oβn/rn. Considering 1< p≤2 andr_n≥2, byLemma 2.1, we have

Jx_n1−x^∗−J y_n−x^∗≤8

r_nx^∗O βn

rn

C_p·

16LFx_n1−yn rnx^∗O βn

/rn

_p−1

≤

rnx^∗ O β_n rn

_2−pO β^p−1_n r_n^p−1

≤

r_n²r_nx^∗O β_n^2−pO β^p−1n

rn

≤rn·O β_n^p−1

.

3.7

By3.1,3.2,3.3andLemma 2.2, we have y_n−x^∗2 ≤a_n xn−x^∗ b_n η_n−x^∗

c_n vn−x^∗2

≤a²_n xn−x^∗22

b_nξn−x^∗ c_nvn−x^∗, J y_n−x^∗

≤ xn−x^∗2O βn

.

3.8

From3.1,3.2,3.7, and3.8andLemma 2.2, it can be concluded that x_n1−x^∗2α_nxn−x^∗ β_n η_n−x^∗

γ_nun−x^∗2

≤ α²_nxn−x^∗22βn

ηn−x^∗, Jxn1−x^∗−J yn−x^∗

2βn

ηn−x^∗, J yn−x^∗

2γnun−x^∗, Jxn1−x^∗

≤α²_nxn−x^∗22βnηn−x^∗·Jxn1−x^∗−J yn−x^∗

2β_ny_n−x^∗2−Φ y_n−x^∗ 2γ_nun−x^∗ · Jx_n1−x^∗

≤

1−β_n−γ_n2

xn−x^∗22β_nRnx^∗·r_n·O β^p−1_n 2βn

xn−x^∗2O βn

−2βnΦ yn−x^∗2γn·Rnx^∗2

≤ xn−x^∗2 βnγn

₂

xn−x^∗2O β^p_n

O β²_n

O γn

−2β_nΦ y_n−x^∗

≤ xn−x^∗2O β_n²

xn−x^∗2O β^p_n

O γn

−2βnΦ yn−x^∗.

3.9

From3.3and3.9, we have x_n1−x^∗2≤

1O β_n²

xn−x^∗2O β^p_n

O γ_n

. 3.10

Thus, by3.3,3.10andLemma 2.3, we have{xn−x^∗}bounded. It implies the sequences {xn} and {yn} are bounded. Since T is a bounded mapping, we have T{xn} and T{yn} bounded. Sinceηn ∈ Tyn and ξn ∈ Txn,{Rn} is bounded. Let its bound be R > 0. From 3.9, there exists a numberM >0 such that

xn1−x^∗2 ≤

1Mβ²_n

xn−x^∗2M

β_n^pγ_n

−2β_n

R² Φ y_n−x^∗. 3.11

Next, we will show

lim inf

n→ ∞ Φ y_n−x^∗0. 3.12

If it is not true, then there exist an₀ ∈Nand a positive constantm₀ such that for any positive integern≥n0

Φ y_n−x^∗≥m₀. 3.13

In view of3.11and3.13, for any positive integern≥n0, we have x_n1−x^∗2≤

1Mβ²_n

xn−x^∗2M

β^p_nγ_n

− 2m0βn

R² . 3.14

Takingnn₀, n₀1, . . . , kin3.14above, we have k

nn0

xn1−x^∗2≤ ^k

nn0

xn−x^{∗2 k}

nn0

Mβ²_nRx^∗2

^k

nn0

M β^p_nγn

−^k

nn0

2m0βn

R² .

3.15

So,

2m₀ R²

k nn0

βn≤MRx^∗2k

nn0

β²_nM _k

nn0

β^pn ^k

nn0

γn

. 3.16

This leads to a contradiction ask → ∞. Hence, lim infn→ ∞ Φyn−x^∗ 0.

By the definition ofΦand 3.12, there exists a subsequence{yni}of{yn}such that {yni} → x^∗ asi → ∞. Thus, by3.6, we have lim infn→ ∞ xn−x^∗ 0. Further, Using Lemma 2.3and3.11, we obtain limn→ ∞xn−x^∗0. It means that{xn}converges strongly to the unique fixed point ofT. The proof is finished.

FromTheorem 3.1, we can obtain the following theorems.

Theorem 3.2. LetX be a p-uniformly smooth Banach space,Dbe a nonempty convex subset ofX, andT :D → 2^Da multivalued generalizedΦ-hemicontractive and bounded mapping. For any given x0, u0 ∈D, let{xn}be the sequence generated by the following Mann-type iterative algorithm with variable coefficients:

x_n1α_nx_nβ_nη_nγ_nu_n, ∃ η_n∈Tx_n, n∈N, 3.17

where{un}is an arbitrary bounded sequence inD,

α_n1−β_n−γ_n, β_n β_n

R²_n, γ_n γ_n

R²_n, R_n2xnη_nun, 3.18 {βn}and{γn}are sequences in0,1 satisfying the following conditions:

∞ n0

βn∞, ^∞

n0

β^p_n<∞, ^∞

n0

γn<∞. 3.19

Then,{xn}converges strongly to the unique fixed point ofT.

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 numberγ₀ in Theorem CC1, Theorem 3.8 is dropped off.

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

Theorem 3.4. LetX be a uniformly smooth real Banach space andDa nonempty convex subset of X. SupposeT : D → 2^D is a multivalued generalized Φ-hemicontractive mapping with bounded range. For any givenx0, u0, v0∈D, let{xn}be the sequence generated by the following Ishikawa-type iterative algorithm with variable coefficients:

y_na_nx_nb_n ξ_nc_n v_n, ∃ξn∈Tx_n, x_n1α_n x_nβ_n η_nγ_n u_n, ∃ηn∈Ty_n,

n∈N, 3.20

where{un}and{vn}are arbitrary bounded sequences inD,

a_n1−b_n−c_n, b_n b_n

r_n², c_n c_n

r_n², r_n2xnξnvn,

α_n1−β_n−γ_n, β_n β_n

R²_n, γ_n γ_n

R²_n, R_nr_nη_nun,

3.21

{βn},{γn},{bn}and{cn}are four sequences in0,1 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,{xn}converges strongly to the unique fixed point ofT.

Proof. FromTheorem 3.1,T has a unique fixed point, sayx^∗. Let{xn},{yn}be the sequences generated by the algorithm3.20. SinceT has a bounded range, we set

d:supξ−η:x, y∈D, ξ∈Tx, η∈Ty

sup{un−x^∗, n∈N}

sup{vn−x^∗ , n∈N}. 3.23

Obviously,d <∞. Next, we will prove that forn≥ 0,xn−x^∗ ≤ dx0−x^∗. In fact, for n0, the above inequality holds. Assume the inequality is true fornk. Then, fornk1, there exists aη_k∈Ty_ksuch that

x_k1−x^∗ ≤α_kxn−x^∗β_kη_k−x^∗γ_kuk−x^∗

≤α_kdx0−x^∗ β_kdγ_kd

≤dx0−x^∗.

3.24

By induction, we have the sequence{xn}bounded. Similarly, we have the sequence{yn}also bounded.

From the proof ofTheorem 3.1, we havexn1−y_n → 0 asn → ∞. SinceXis a real uniformly smooth Banach space, so that the normalized duality mappingJ is single valued and uniformly continuous on any bounded subset ofX, thus

d_n:Jx_n1−x^∗−J y_n−x^∗−→0 3.25 asn → ∞.

Next, following the reasoning in the proof ofTheorem 3.1, we deduce the conclusion ofTheorem 3.4.

Remark 3.5. In view of Example 1.4, the class of Φ-hemicontractive mappings is a proper subset of the class of generalizedΦ-hemicontractive mappings. Hence,Theorem 3.4improves essentially the result of2, Theorem 2 .

As applications, we give the following theorems.

Theorem 3.6. LetXbe a p-uniformly smooth Banach spaceT :X → 2^X, a multivalued generalized Φ-hemiaccretive and bounded mapping. For any givenf∈X, defineS:X → 2^XbySx:x−Txf for allx ∈ X. For any givenx0, u0, v0 ∈ X, let{xn}be the Ishikawa-type iterative sequence with variable coefficients, defined by

y_na_n x_nb_n ξ_nc_nv_n, ∃ξn∈Sx_n, x_n1α_n x_nβ_nη_nγ_n u_n, ∃ηn∈Sy_n,

n∈N, 3.26

where{un},{vn}are bounded sequences inX,

an1−bn−cn, bn bn

r_n², cn cn

r_n², rn2xnξnvn,

αn1−βn−γn, βn βn

R²_n, γn γn

R²_n, Rnrnηnun,

3.27

{βn},{γn},{bn}, and{cn}are four sequences in0,1 satisfying the following conditions:

∞ n0

β_n∞, ^∞

n0

β_n^p<∞, ^∞

n0

γ_n<∞, bn≤O β_n

, c_n≤O β_n

. 3.28

Then, {xn} converges strongly to the unique solution of the generalizedΦ-hemiaccretive mapping equationf ∈Tx.

Theorem 3.7. Let X be a uniformly smooth Banach space andT : X → 2^X a generalized Φ- hemiaccretive with bounded range. For any givenf ∈X, defineS :X → 2^XbySx : x−Txf for allx ∈ X. For any givenx₀, u₀, v₀ ∈ X, let{xn}be the Ishikawa-type iterative sequence with variable coefficients, defined by

y_na_n x_nb_n ξ_nc_n v_n, ∃ξn∈Sx_n, x_n1α_n x_nβ_n η_nγ_n u_n, ∃ηn∈Sy_n,

n0,1,2, . . . , 3.29

where{un},{vn}are bounded sequences inX,

an1−bn−cn, bn bn

r_n², cn cn

r_n², rn2xnξnvn,

αn1−βn−γn, βn βn

R²_n, γn γn

R²_n, Rnrnηnun,

3.30

{βn},{γn},{bn} and{cn}are four sequences in0,1 satisfying the following conditions:

∞ n0

β_n∞, ^∞

n0

β_n²<∞, ^∞

n0

γ_n<∞, b_n≤O β_n

, c_n≤O β_n

. 3.31

Then, {xn} converges strongly to the unique solution of the generalizedΦ-hemiaccretive mapping equationf ∈Tx.

Remark 3.8. 1Theorem 3.6improves some recent results, for example,1, Theorem 3.7 and 2, Theorem 2 inp-uniformly smooth real Banach spaces since the multivalued generalized Φ-hemiaccretive mapping within the equation has no generalized Lipschitz assumption.

2In view ofExample 1.4, the class ofΦ-hemicontractive mappings is a proper subset of the class of generalizedΦ-hemicontractive mappings. Hence,Theorem 3.7improves essentially the result of2, Theorem 2 in uniformly smooth real Banach spaces.

Acknowledgment

The work was partially supported by the Specialized Research Fund 2010 for the Doctoral Program of Anhui University of Architecture and the Natural Science Foundation of Anhui Educational Committee.

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