Generalized Caristi's Fixed Point Theorems

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Volume 2009, Article ID 170140,7pages doi:10.1155/2009/170140

Research Article

Generalized Caristi’s Fixed Point Theorems

Abdul Latif

Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Abdul Latif,[email protected]

Received 27 December 2008; Accepted 9 February 2009

Recommended by Mohamed A. Khamsi

We present generalized versions of Caristi’s fixed point theorem for multivalued maps. Our results either improve or generalize the corresponding generalized Caristi’s fixed point theorems due to Bae2003, Suzuki2005, Khamsi2008, and others.

Copyrightq2009 Abdul Latif. 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

A number of extensions of the Banach contraction principle have appeared in literature. One of its most important extensions is known as Caristi’s fixed point theorem. It is well known that Caristi’s fixed point theorem is equivalent to Ekland variational principle 1, which is nowadays an important tool in nonlinear analysis. Many authors have studied and generalized Caristi’s fixed point theorem to various directions. For example, see 2–

8. Kada et al. 9 and Suzuki 10 introduced the concepts ofw-distance and τ-distance on metric spaces, respectively. Using these generalized distances, they improved Caristi’s fixed point theorem and Ekland variational principle for single-valued maps. In this paper, using the concepts of w-distance and τ-distance, we present some generalizations of the Caristi’s fixed point theorem for multivalued maps. Our results either improve or generalize the corresponding results due to Bae4,11, Kada et al.9, Suzuki8,10, Khamsi5, and many of others.

Let X be a metric space with metric d. We use 2X _{to denote the collection of all}

nonempty subsets of X. A point x ∈ X is called a fixed point of a map f : X → X

T :X → 2X_if_x_f_x _x_∈_T_x_.

In 1976, Caristi12 obtained the following fixed point theorem on complete metric spaces, known as Caristi’s fixed point theorem.

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x∈X,

dx, fx≤ψx−ψfx. 1.1

Thenfhas a fixed point.

To generalizeTheorem 1.1, one may consider the weakening of one or more of the following hypotheses: i the metric d; ii the lower semicontinuity of the real-valued functionψ;iiithe inequality1.1;ivthe functionf.

In9, Kada et al. introduced a concept ofw-distance on a metric space as follows. A function ω : X ×X → 0,∞ is a w-distance on X if it satisfies the following conditions for anyx, y, z∈X:

w1ωx, z≤ωx, y ωy, z;

w2the mapωx,·:X → 0,∞is lower semicontinuous;

w3for any > 0, there exists δ > 0 such thatωz, x ≤ δ and ωz, y ≤ δ imply

dx, y≤.

Clearly, the metricdis aw-distance onX. LetY, · be a normed space. Then the functionsω1, ω2:Y ×Y → 0,∞defined byω1x, y yandω2x, y xyfor all

x, y∈Y arew-distances. Many other examples ofw-distance are given in9,13. Note that, in general, forx, y∈X,ωx, y/ωy, x, and neither of the implicationsωx, y 0⇔xy

necessarily holds.

In the sequel, otherwise specified, we shall assume thatX is a complete metric space with metricd,ψ :X → 0,∞is a lower semicontinuous function andωis aw-distance on

X.

Using the concept of w-distance, Kada et al. 9 generalized Caristi’s fixed point theorem as follows.

Theorem 1.2. Letfbe a single-valued self map onXsuch that for everyx∈X,

ψfx ωx, fx≤ψx. 1.2

Then, there existsx0∈Xsuch thatfx0 x0andωx0, x0 0.

2. The Results

ApplyingTheorem 1.2, first we prove the following generalization ofTheorem 1.1.

Theorem 2.1. Letg:X → 0,∞be any function such that for somer >0,

supgx:x∈X, ψx≤inf

z∈Xψz r

<∞. 2.1

LetT :X → 2Xbe a multivalued map such that for eachx∈X,there existsy∈Txsatisfying

ωx, y≤gxψx−ψy. 2.2

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Proof. Define a functionf :X → X byfx y ∈ Tx ⊆X.Note that for eachx∈ X,we have

ωx, fx≤gxψx−ψfx. 2.3

Now, sincegx>0,it follows thatψfx≤ψx.Put

Mx∈X:ψx≤ inf

z∈Xψz r

, αsup

z∈M

gz<∞. 2.4

Note that Mis nonempty, and by the lower semicontinuity of ψ and ωx,·,M is closed subset of a complete metric spaceX, and hence it is complete. Now, we show thatfM⊆M.

Letu∈M, andfu v∈Tu, then we have

ψfu≤ψu≤inf

z∈Xψz r, 2.5

and thusfu∈M, and hencef is a self map onM. Note thatαψis lower semicontinuous and for eachx∈M, we have

ωx, fx≤αψx−αψfx. 2.6

ByTheorem 1.2, there existsx0∈Msuch thatfx0 x0 ∈Tx0andωx0, x0 0.

Now, applyingTheorem 2.1, we obtain generalized Caristi’s fixed point results.

Theorem 2.2. LetT:X → 2X_{be a multivalued map such that for each}_x_∈_X,_{there exists}_y_∈_T_x satisfying

ωx, y≤max{cψx, cψy}ψx−ψy, 2.7

wherec: 0,∞ → 0,∞is an upper semicontinuous function from the right. ThenT has a fixed pointx0∈Xsuch thatωx0, x0 0.

Proof. Putt0 infx∈Xψx. By the definition of the functionc, there exist some positive real

numbersr,r0such thatct≤r0for allt∈t0, t0r.Now, for allx∈X, we define

gx max{cψx, cψy}. 2.8

Clearly,gmapsxinto0,∞. Note that for allx ∈X, we getψy≤ ψx, and thus for any

x∈Xwithψx≤t0r, we have

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Now, clearly,gx≤r0<∞and hence we obtain

z∈Xψz r

<∞. 2.10

ByTheorem 2.1,T has a fixed pointx0∈Xsuch thatωx0, x0 0.

ωx, y≤cψxψx−ψy, 2.11

wherec :0,∞ → 0,∞is nondecreasing function. ThenT has a fixed point x0 ∈ X such that

ωx0, x0 0.

Proof. For eachx∈X, definegx cψx. Clearly,g does carryxinto0,∞. Now, since the functioncis nondecreasing, for any real numberr >0 we have

z∈Xψz r

≤cinf

z∈Xψz r

<∞. 2.12

Thus, byTheorem 2.1, the result follows.

Corollary 2.4. LetT :X → 2X_{be a multivalued map such that for each}_x_∈_X,_{there exists}_y_∈_T_x satisfying

ωx, y≤cψyψx−ψy, 2.13

wherec:0,∞ → 0,∞is a nondecreasing function. ThenT has a fixed pointx0 ∈X such that

ωx0, x0 0.

Proof. Since for eachx ∈ X there isy ∈ Txsuch thatψy ≤ ψx and the functioncis nondecreasing, we havecψy≤cψx. Thus the result follows fromTheorem 2.3.

ApplyingTheorem 2.3, we prove the following fixed point result.

Theorem 2.5. LetT:X → 2X_{be a multivalued map such that for each}_x_∈_X,_{there exists}_y_∈_T_x satisfyingωx, y≤ψxand

ωx, y≤ηωx, yψx−ψy, 2.14

whereη :0,∞ → 0,∞is an upper semicontinuous function. ThenT has a fixed pointx0 ∈ X such thatωx0, x0 0.

Proof. Define a functioncfrom0,∞into0,∞by

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Clearly, c is nondecreasing function. Now, since ωx, y ≤ ψx, we have cωx, y ≤ cψx. Thus byTheorem 2.3, the result follows.

The following result can be seen as a generalization of5, Theorem 4.

Corollary 2.6. Letφ:0,∞ → 0,∞be a lower semicontinuous function such that

lim sup

t→0

t

φt <∞. 2.16

LetT : X → 2X _{be a multivalued map such that for each}_x _∈ _X,_{there exists}_y _∈_T_x_satisfying

ωx, y≤ψxand

φωx, y≤ψx−ψy. 2.17

ThenT has a fixed pointx0∈Xsuch thatωx0, x0 0.

Proof. Define a functionη:0,∞ → 0,∞by

η0 lim sup

t→0

t

φt, ηt t

φt, t >0. 2.18

Thenηis upper semicontinuous. Also note that

ωx, y≤ηωx, yψx−ψy. 2.19

Thus byTheorem 2.5,T has a fixed pointx0 ∈Xsuch thatωx0, x0 0.

Now, let p be a τ distance on X 8, using the same technique as in the proof of

Theorem 2.1, and applying8, Theorem 3, we can obtain the following result.

Theorem 2.7. Letg:X → 0,∞be any function such that for somer >0,

z∈Xψz r

<∞. 2.20

LetT :X → 2X_{be a multivalued map such that for each}_x_∈_X,_{there exists}_y_∈_T_x_satisfying

px, y≤gxψx−ψy. 2.21

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px, y≤max{cψx, cψy}ψx−ψy, 2.22

where c : 0,∞ → 0,∞is an upper semicontinuous from the right. ThenT has a fixed point x0∈Xsuch thatωx0, x0 0.

px, y≤cψxψx−ψy, 2.23

ωx0, x0 0.

Corollary 2.10. Let T : X → 2X _{be a multivalued map such that for each} _x _∈ _X,_{there exists}

y∈Txsatisfying

px, y≤cψyψx−ψy, 2.24

ωx0, x0 0.

Theorem 2.11. Let T : X → 2X _{be a multivalued map such that for each} _x _∈ _X, _{there exists}

y∈Txsatisfyingpx, y≤ψxand

px, y≤ηpx, yψx−ψy, 2.25

whereη :0,∞ → 0,∞is an upper semicontinuous function. ThenT has a fixed pointx0 ∈ X such thatωx0, x0 0.

Corollary 2.12. Letφ:0,∞ → 0,∞be a lower semicontinuous function such that

lim sup

t→0

t

φt <∞. 2.26

LetT : X → 2X _{be a multivalued map such that for each}_x _∈ _X,_{there exists}_y _∈_T_x_satisfying

px, y≤ψxand

φpx, y≤ψx−ψy. 2.27

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Acknowledgment

The author is thankful to the referees for their valuable comments and suggestions.

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