Fixed Point Theorems for Times Reasonable Expansive Mapping

Volume 2008, Article ID 302617,6pages doi:10.1155/2008/302617

Research Article

Fixed Point Theorems for

n

Times Reasonable

Expansive Mapping

Chunfang Chen and Chuanxi Zhu

Institute of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China

Correspondence should be addressed to Chuanxi Zhu,[email protected]

Received 29 February 2008; Revised 3 May 2008; Accepted 16 August 2008

Recommended by Jerzy Jezierski

Based on previous notions of expansive mapping, ntimes reasonable expansive mapping is defined. The existence of fixed point forntimes reasonable expansive mapping is discussed and some new results are obtained.

Copyrightq2008 C. Chen and C. Zhu. 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.

1. Introduction and preliminaries

The research about fixed points of expansive mapping was initiated by Machucasee1. Later, Jungck discussed fixed points for other forms of expansive mappingsee2. In 1982, Wang et al.see3published a paper inAdvances in Mathematicsabout expansive mapping which draws great attention of other scholars. Also, Zhang has done considerable work in this field. In order to generalize the results about fixed point theory, Zhangsee4published his workFixed Point Theory and Its Applications, in which the fixed point problem for expansive mapping is systematically presented in a chapter. As applications, he also investigated the existence of solutions of equations for locally condensing mapping and locally accretive mapping. In 1991, based on the results obtained by others, the author defined several new kinds of expansive-type mappings in5, which expanded the expansive-type mapping from 19 to 23, and gave some new applications. Recently, the study about fixed point theorem for expansive mapping and nonexpansive mapping is deeply explored and has extended too many other directions. Motivated and inspired by the workssee1–13, in this paper, we define ntimes reasonable expansive mapping and discuss the existence of fixed point for ntimes reasonable expansive mapping. For the sake of convenience, we briefly recall some definitions.

LetX, dbe a complete metric space and letT :X→Xbe a mapping.

T : X → X is called an expansive mapping if there exists a constanth >1 such that dTx, Ty≥hdx, y, for allx, y∈X.

T :X→Xis called a two times reasonable expansive mapping if there exists a constant h >1 such thatdx, T2_{x≥hdx, Tx, for allx∈X.}

T : X → X is called a twenty-one type expansive mapping if there exists a constant h >1 such that

dTx, Ty≥hmindx, y, dx, Tx, dy, Ty, dx, Ty, dy, Tx, ∀x, y∈X. 1.1

T :X →X is called a twenty-three type expansive mapping if there exists a constant h >1 such that

d2Tx, Ty≥hmind2x, y, dx, y·dx, Tx, dx, Tx·dy, Ty,

d2x, Tx, dy, Ty·dx, Ty, dy, Ty·dy, Tx, ∀x, y∈X. 1.2

2. Main results

Definition 2.1. LetX, dbe a complete metric space.T :X→Xis called annn≥2, n∈N times reasonable expansive mapping if there exists a constanth >1 such that

dx, Tn_{x≥hdx, Tx, ∀x∈X n≥2, n∈N. 2.1}

Definition 2.2. LetX, dbe a complete metric space.T :X → Xis called anH1-typenn ≥ 2, n∈Ntimes reasonable expansive mapping if there exists a constanth >1 such that

dTn−1x, Tn−1y≥hmindx, y, dx, Tx, dTn−2y, Tn−1y,

dx, Tn−1y, dTn−2y, Tn−1x, ∀x, y∈X n≥2, n∈N. 2.2

Definition 2.3. LetX, dbe a complete metric space.T :X → Xis called anH2-typenn ≥ 2, n∈Ntimes reasonable expansive mapping if there exists a constanth >1 such that

d2_Tn−1_{x, T}n−1_y≥hmind2_{x, y, dx, y·dx, Tx, dx, Tx·dT}n−2_{y, T}n−1_{y, d}2_{x, Tx,}

dTn−2y, Tn−1y·dx, Tn−1y, dTn−2y, Tn−1y·dTn−2y, Tn−1x,

∀x, y∈X n≥2, n∈N.

2.3

Lemma 2.4 see6. Let X, d be a complete metric space, let A be a subset ofX, and let the mappingsf, g:A→Xsatisfy the following conditions:

ifis a surjective mappingfA X;

iithere exists a functionalϕ :X → Rwhich is lower semicontinuous bounded from below such thatdfx, gx≤ϕfx−ϕgx, for allx∈A.

Then, f and g have a coincidence point, that is, there exists at least an x ∈ A such that fx gx.

Theorem 2.5. LetX, dbe a complete metric space and letT :X→Xbe a continuous and surjective mapping if there exists a constanth >1such that

dTn−1x, Tnx≥hdx, Tx, ∀x∈X n≥2, n∈N. 2.4

Then,Thas a fixed point inX.

Proof. By2.4, we have

dTn−1_{x, T}n_{x−dx, Tx≥hdx, Tx−dx, Tx, ∀x∈X. 2.5}

Thus,

dx, Tx≤ 1 h−1

dTn−1_{x, T}n_{x−dx, Tx, ∀x∈X. 2.6}

Letϕx 1/h−1dTn−1_{x, T}n−2_{x dT}n−2_{x, T}n−3_{x · · ·dT}2_{x, Tx dTx, x.} Then we havedx, Tx≤ϕTx−ϕx, for allx∈X. From the continuity ofd,we know thatϕxis continuous. Thusϕxis lower semicontinuous bounded from below. Therefore the conclusion follows immediately fromLemma 2.4.

Theorem 2.6. LetX, dbe a complete metric space and letT :X→Xbe a continuous and surjective nn≥2, n∈Ntimes reasonable expansive mapping. Assume that either (i) or (ii) holds:

iT is anH1-typentimes reasonable expansive mapping;

iiT is anH2-typentimes reasonable expansive mapping.

Then,Thas a fixed point inX.

Proof. In the case ofi, takingyTxin2.2, we have

dTn−1x, Tnx≥hmindx, Tx, dx, Tx, dTn−1x, Tnx, dx, Tnx, dTn−1x, Tn−1x

hmindx, Tx, dTn−1x, Tnx, dx, Tnx.

2.7

BecauseT is anntimes reasonable expansive mapping, we have

dx, Tnx≥hdx, Tx > dx, Tx. 2.8

Thus, we obtain

dTn−1x, Tnx≥hmindx, Tx, dTn−1x, Tnx. 2.9

IfdTn−1_{x, T}n_{x min{dx, Tx, dT}n−1_{x, T}n_x},thendTn−1_{x, T}n_x≥hdTn−1_{x, T}n_x. Hence, dTn−1_{x, T}n_{x 0 otherwise, dT}n−1_{x, T}n_{x > dT}n−1_{x, T}n_{x, which is a} contradiction. Therefore,Tn−1_{x T}n_{x,that isT}n−1_{x TT}n−1_{x,which implies thatT}n−1_x is a fixed point ofTinX.

In the case ofii, takingyTxin2.3, we have

d2Tn−1x, Tnx≥hmind2x, Tx, dx, Tx·dx, Tx, dx, Tx·dTn−1x, Tnx,

d2x, Tx, dTn−1x, Tnx·dx, Tnx, dTn−1x, Tnx·dTn−1x, Tn−1x

hmind2x, Tx, dx, Tx·dTn−1x, Tnx, dTn−1x, Tnx·dx, Tnx.

2.10

BecauseT is annn≥2, n∈Ntimes reasonable expansive mapping, we have

dx, Tnx≥hdx, Tx> dx, Tx. 2.11

Hence,dx, Tn_x·dTn−1_{x, T}n_{x> dx, Tx·dT}n−1_{x, T}n_x. Therefore, we have

d2Tn−1x, Tnx≥hmind2x, Tx, dx, Tx·dTn−1x, Tnx. 2.12

Ifd2_{x, Tx min{d}2_{x, Tx, dx, Tx·dT}n−1_{x, T}n_{x}, then}

d2Tn−1x, Tnx≥hd2x, Tx ∀x∈X, 2.13

that is,dTn−1_{x, T}n_x≥√_{hdx, Tx.}

Because√h >1,byTheorem 2.5, we obtain thatThas a fixed point inX.

If dx, Tx·dTn−1_{x, T}n_{x min{d}2_{x, Tx, dx, Tx·dT}n−1_{x, T}n_{x}, then d}2_Tn−1_x, Tn_{x≥hdx, Tx·dT}n−1_{x, T}n_{x, that is}

dTn−1x, Tnx·dTn−1x, Tnx−hdx, Tx≥0. 2.14

IfdTn−1_{x, T}n_{x 0,thenT}n−1_{x T}n_{x,that isT}n−1_{x TT}n−1_{x,which implies that} Tn−1_{xis a fixed point ofTinX.}

IfdTn−1_{x, T}n_x/0,thendTn−1_{x, T}n_{x≥hdx, Tx.ByTheorem 2.5, we obtain thatT} has a fixed point inX.

Therefore, by induction we derive thatThas a fixed point inX.

Corollary 2.7. LetX, dbe a complete metric space. IfT :X → X is a continuous and surjective twenty-one type expansive mapping andT : X → X is a two times reasonable expansive mapping, thenT has a fixed point inX.

Proof. We denoteyTo_{y; takingn2 under the conditioniofTheorem 2.6,Corollary 2.7} is proved immediately.

Similarly, we denoteyToy; takingn2 under the conditioniiofTheorem 2.6, we can obtain the followingCorollary 2.8.

Corollary 2.8. LetX, dbe a complete metric space. IfT :X → X is a continuous and surjective twenty-three type expansive mapping andT :X →X is a two times reasonable expansive mapping, thenT has a fixed point inX.

Theorem 2.10. Let X, d be a complete metric space and letT : X → X be a continuous and surjectivenn≥2, n∈Ntimes reasonable expansive mapping. If there exists a constanth >1such that

dTnx, Tny≥hmindx, y, dy, Tny, ∀x, y∈X n≥2, n∈N, 2.15

thenT has a fixed point.

Proof. LettingxTyin2.15, we have

dTn1y, Tny≥hmindTy, y, dy, Tny, ∀y∈X. 2.16

SinceT is annn≥2, n∈Ntimes reasonable expansive mapping, then

dy, Tny≥hdy, Ty> dy, Ty, ∀y∈X. 2.17

By2.16and2.17, we havedTn1_{y, T}n_{y≥hdTy, yfor ally∈X.} It follows fromTheorem 2.5thatThas a fixed point inX.

Remark 2.11. Generally speaking,nn≥2, n∈Ntimes reasonable expansive mapping does not necessarily have a fixed point. This can be illustrated by the following examples.

Example 2.12. We denote byB1the unit circle which takes the original point as its center and 1 as its radius on the complex plane, that is,B1 {Z| |Z|1, Z∈C}.B1can also be written as{eiθ_|eiθ_{∈C,−∞< θ <∞}.Suppose thatT :B}

1→B1is a mapping defined as follows: TZTeiθeiθ2π/3n. 2.18

For everyZ∈B1,that is,Zeiθ,we have

TZTeiθeiθ2π/3n,

T2ZTTZ TTeiθTeiθ2π/3n eiθ22π/3n,

· · ·

TkZeiθk2π/3n,

· · ·

TnZeiθn2π/3n eiθ2π/3.

2.19

From the above equations, we obtain

dZ, Tn_ZTn_Z−Zeiθ2π/3_−eiθ_eiθ_·ei2π/3₋₁

cos2π 3 isin

2π 3 −1

−1

2

√

3 2 i−1

√3,

dZ, TZ |TZ−Z|eiθ2π/3n−eiθeiθ·ei2π/3n−1cos2π 3n isin

2π 3n −1

2−2 cos2π 3n 2

sin2π

3n 2 sin π

3n n≥2, n∈N.

Sincen≥2, then sinπ/3n≤1/2. ThusdZ, Tn_{Z/dZ, TZ≥}√_{3, for allZ∈B} 1, that is,dZ, Tn_Z≥√_{3dZ, TZ, for allZ∈B}

1. We can take a constanth

√

3,which means that there exists a constanth >1 such thatdZ, Tn_{Z≥hdZ, TZ, for allZ ∈B}

1n≥2, n∈N. Therefore,T is anntimes reasonable expansive mapping. Sinceeiθ_/eiθ2π/3_{,thenTZ /Z,}

for allZ∈B1. It implies thatTdoes not have a fixed point.

Example 2.13. Suppose thatT :R→Ris a mapping defined asTxx1.

It is obvious thatTis continuous and surjective andTdoes not have a fixed point. Now, we proveT is anntimes reasonable expansive mapping.

In fact, by the definition ofT,we haveTn_{xxn n≥2, n∈N.}

Becausedx, Tn_{x |xn−x| n ≥ 2 and dx, Tx |x1−x| 1, we have} dx, Tnx ≥ 2dx, Tx.Thus, we can take a constanth 2,which means that there exists a constanth >1 such thatdx, Tn_{x≥hdx, Tx, for allx∈R n≥2, n∈N.}

Therefore,Tis anntimes reasonable expansive mapping.

Acknowledgments

This work was supported by the National Natural Science Foundation of China10461007 and 10761007and the Provincial Natural Science Foundation of Jiangxi, China0411043 and 2007GZS2051.

References

1 R. Machuca, “A coincidence theorem,”The American Mathematical Monthly, vol. 74, no. 5, p. 569, 1967. 2 G. Jungck, “Commuting mappings and fixed points,”The American Mathematical Monthly, vol. 83, no.

4, pp. 261–263, 1976.

3 S. Wang, B. Li, and Z. Gao, “Expansive mappings and fixed-point theorems for them,”Advances in Mathematics, vol. 11, no. 2, pp. 149–153, 1982Chinese.

4 S. Zhang,Fixed Point Theory and Its Applications, Chongqing Press, Chongqing, China, 1984.

5 C. Zhu, “Several new mappings of expansion and fixed point theorems,”Journal of Jiangxi Normal University, vol. 15, no. 8, pp. 244–248, 1991Chinese.

6 S. Park, “On extensions of the Caristi-Kirk fixed point theorem,”Journal of the Korean Mathematical Society, vol. 19, no. 2, pp. 143–151, 1983.

7 C. Zhu, “Several nonlinear operator problems in the Menger PN space,”Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 7, pp. 1281–1284, 2006.

8 D. Guo,Nonlinear Functional Analysis, Shandong Science and Technology Press, Jinan, China, 1985. 9 C. Zhu and Z. Xu, “Several problems of random nonlinear equations,” Acta Analysis Functionalis

Applicata, vol. 5, no. 3, pp. 217–219, 2003Chinese.

10 C. Zhu, “Generalizations of Krasnoselskii’s theorem and Petryshyn’s theorem,”Applied Mathematics Letters, vol. 19, no. 7, pp. 628–632, 2006.

11 C. Zhu and Z. Xu, “Random ambiguous point of randomkω-set-contractive operator,”Journal of Mathematical Analysis and Applications, vol. 328, no. 1, pp. 2–6, 2007.

12 C. Zhu and X. Huang, “The topological degree of A-proper mapping in the Menger PN-space—II,” Bulletin of the Australian Mathematical Society, vol. 73, no. 2, pp. 169–173, 2006.