Existence Theorems for Generalized Distance on Complete Metric Spaces

Volume 2010, Article ID 397150,21pages doi:10.1155/2010/397150

Research Article

Existence Theorems for Generalized Distance on Complete Metric Spaces

Jeong Sheok Ume

Department of Applied Mathematics, Changwon National University, Changwon 641-773, Republic of Korea

Correspondence should be addressed to Jeong Sheok Ume,jsume@changwon.ac.kr Received 20 September 2009; Revised 7 May 2010; Accepted 20 May 2010

Academic Editor: L. G ´orniewicz

Copyrightq2010 Jeong Sheok Ume. 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 first introduce the new concept of a distance calledu-distance, which generalizesw-distance, Tataru’s distance, andτ-distance. Then we prove a new minimization theorem and a new fixed point theorem by using au-distance on a complete metric space. Our results extend and unify many known results due to Caristi, ´Ciri´c, Ekeland, Kada-Suzuki-Takahashi, Kannan, Ume, and others.

1. Introduction

The Banach contraction principle1, Ekeland’sε-variational principle2, and Caristi’s fixed point theorem3are very useful tools in nonlinear analysis, control theory, economic theory, and global analysis. These theorems are extended by several authors in different directions.

Takahashi4proved the following minimization theorem. LetXbe a complete metric space and letf :X → −∞,∞be a proper lower semicontinuous function, bounded from below. Suppose that, for each u ∈ X with fu > inf_x∈Xfx,there existsv ∈ X such that v /uandfv du, v≤fu.Then there existsx0 ∈Xsuch thatfx0 inf_x∈Xfx.Some authors5–7have generalized and extended this minimization theorem in complete metric spaces.

In 1996, Kada et al. 5 introduced the concept of w-distance on a metric space as follows. LetXbe a metric space with metricd. Then a functionp:X×X → 0,∞is called aw-distance onXif the followings are satisfied.

1px, z≤px, y py, zfor anyx, y, z∈X.

2For anyx∈X, px,·:X → 0,∞is lower semicontinuous.

3For any > 0, there exists δ > 0 such that pz, x ≤ δ and pz, y ≤ δ imply dx, y≤.

They gave some examples of w-distance and improved Caristi’s fixed point theorem 3, Ekeland’s variational principle2, and Takahashi’s nonconvex minimization theorem 4.

The fixed point theorems with respect to aw-distance were proved in8–12.

Throughout this paper we denote byNthe set of all positive integers, byRthe set of all real numbers, and byRthe set of all nonnegative real numbers.

Recently, Suzuki6introduced the concept of τ-distance on a metric space, which generalizes Tataru’s distance13as follows. LetXbe a metric space with metricd.

Then a functionpfromX×XintoRis calledτ-distance onXif there exists a function ηfromX×RintoRand the followings are satisfied:

τ1px, z≤px, y py, zfor allx, y, z∈X;

τ2ηx,0 0 andηx, t≥tfor allx∈Xandt∈R, andηis concave and continuous in its second variable;

τ3lim_nx_n x and lim_nsup{ηzn, pzn, x_m : m ≥ n} 0 imply pw, x ≤ limninfnpw, xnfor allw∈X;

τ4limnsup{pxn, ym:m≥n}0 and limnηxn, tn 0 imply limnηyn, tn 0;

τ5lim_nηzn, pzn, x_n 0 and lim_nηzn, pzn, y_n 0 imply lim_ndxn, y_n 0.

In this paper, we first introduce the new concept of a distance calledu-distance, which generalizesw-distance, Tataru’s distance, andτ-distance. Then we prove a new minimization theorem and a new fixed point theorem by using u-distance on a complete metric space.

Our results extend and unify many known results due to Caristi3, ´Ciri´c14, Ekeland2, Takahashi4, Kada et al.5, Kannan15, Suzuki6, and Ume7,12and others.

2. Preliminaries

Definition 2.1. LetXbe metric space with metricd. Then a functionpfromX×XintoRis calledu-distance onXif there exists a functionθfromX×X×R×RintoRsuch that

u1px, z≤px, y py, zfor allx, y, z∈X;

u2θx, y,0,0 0 andθx, y, s, t ≥ min{s, t}for allx, y ∈ X ands, t ∈ R, and for anyx∈Xand for everyε >0, there existsδ > 0 such that|s−s0|< δ,|t−t0|< δ, s, s₀, t, t₀∈Randy∈Ximply

θ

x, y, s, t

−θ

x, y, s₀, t₀< ε; 2.1

u3

nlim→ ∞xnx,

nlim→ ∞sup θ

w_n, z_n, pwn, x_m, pzn, x_m

:m≥n

0 2.2

imply

p y, x

≤ lim

n→ ∞infp y, x_n

2.3 for ally∈X;

u4

nlim→ ∞sup

pxn, wm:m≥n 0,

nlim→ ∞sup p

y_n, z_m

:m≥n 0,

nlim→ ∞θxn, w_n, s_n, t_n 0,

nlim→ ∞θ

yn, zn, sn, tn

0

2.4

imply

nlim→ ∞θwn, z_n, s_n, t_n 0 2.5

or

nlim→ ∞sup

pwm, x_n:m≥n 0,

nlim→ ∞sup p

z_m, y_n

:m≥n 0,

nlim→ ∞θxn, wn, sn, tn 0,

nlim→ ∞θ

y_n, z_n, s_n, t_n 0

2.6

imply

nlim→ ∞θwn, z_n, s_n, t_n 0; 2.7

u5

nlim→ ∞θ

w_n, z_n, pwn, x_n, pzn, x_n 0,

nlim→ ∞θ

w_n, z_n, p w_n, y_n

, p

z_n, y_n

0 2.8

imply

nlim→ ∞d xn, yn

0 2.9

or

nlim→ ∞θ

a_n, b_n, pxn, a_n, pxn, b_n 0,

nlim→ ∞θ

a_n, b_n, p y_n, a_n

, p

y_n, b_n

0 2.10

imply

nlim→ ∞d xn, yn

0. 2.11

Remark 2.2. Suppose thatθ:X×X×R×R → Ris a mapping satisfyingu2∼u5. Then there exists a mappingηfromX×X×R×R intoR such thatηis nondecreasing in its third and fourth variable, respectively, satisfyingu2η ∼u5η, whereu2η ∼u5η stand for substitutingηforθinu2∼u5, respectively.

Proof. Suppose thatθ : X×X×R×R → R is a mapping satisfyingu2∼u5. Define a functionη:X×X×R×R → Rby

η

x, y, s, t

stsup θ

x, y, α, β

: 0≤α≤s,0≤β≤t

for all x, y∈X ands, t∈R. 2.12 By2.12, we haveηx, y,0,0 0 andηx, y, s, t ≥min{s, t}for allx, y ∈Xands, t ∈R. Also it follows from2.12thatηis nondecreasing in its third and fourth variable, respectively.

We shall prove the following:

for anyx∈X and for every ε >0, there existsδ >0 such that s−s< δ, t−t< δ, s, s, t, t∈R andy∈X imply

η

x, y, s, t

−η

x, y, s, t< ε.

2.13

Suppose that2.13does not hold. Then

there existsx∈X, ε>0, sequences{sn}, s_n

, {tn},and t_n ofR,and sequence

yn

ofX such thatsn−s_n< 1 n, tn−t_n< 1

n,andη

x, yn, sn, tn

−η

x, yn, sn, tn

≥ε for alln∈N.

2.14

By virtue of2.12and2.14, we have 0< ε≤η

x, yn, sn, tn

−η

x, yn, s_n, t_n sntn sup

θ

x, yn, α, β

|0≤α≤sn,0≤β≤tn

− s_nt_n

sup θ

x, yn, α, β

|0≤α≤s_n,0≤β≤t_n

≤sn−s_ntn−t_n sup

θ

x, y_n, α, β

|0≤α≤s_n,0≤β≤t_n

−sup θ

x, yn, α, β

|0≤α≤s_n,0≤β≤t_n

< 2

nsup θ

x, yn, α, β

|0≤α≤sn 1

n,0≤β≤tn 1 n

−sup θ

x, y_n, α, β

|0≤α≤s_n−1

n,0≤β≤t_n− 1 n

.

2.15

Combiningu2and2.14, we have the following:

for somex∈X and for everyε >0, there exists δ >0, such that s−s< δ, t−t< δ, s, s, t, t∈R andy∈X imply

θ

x, y, s, t

−θ

x, y, s, t< ε 4.

2.16

Due to2.16, we get that

for this δ >0, there exists M∈N such thatn≥Mimplies 2

n< δ. 2.17 From2.16and2.17, we obtain the following.

for everyε >0, there existsM∈Nsuch thatn≥Mimplies sn−δ

2 < sn− 1

n < sn< sn 1

n < snδ 2, tn−δ

2 < tn− 1

n < tn< tn 1

n < tnδ 2.

2.18

For eachn∈N, let l_1,nsup θ

x, y_n, α, β

|0≤α≤s_n− 1

n,0≤β≤t_n− 1 n

. 2.19

For eachn∈N, let l_2,nsup θ

x, y_n, α, β

|0≤α≤s_n 1

n,0≤β≤t_n 1 n

. 2.20

In terms of2.19and2.20, we deduce that

l_1,n≤l_2,n ∀n∈N. 2.21

In view of2.21, we get that

nlim→ ∞infl1,n≤ lim

n→ ∞infl2,n. 2.22

On account of2.20, we know the following:

for eachn∈Nand for everyε >0, there exists α_n∈ 0, s_n1

n

andβ_n∈ 0, t_n 1 n

such that l2,n−ε < θ

x, yn, αn, βn

.

2.23

Using2.16,2.18,2.19, and2.23, we have the following:

for everyε >0, there existsM∈Nsuch that l2,n−ε < l1,nε

2, for alln∈Nwith M≤n. 2.24

By2.24, we have

nlim→ ∞infl_2,n≤ lim

n→ ∞infl_1,n. 2.25

By virtue of2.15,2.19,2.20,2.22, and2.25, we have 0< ε≤0 which is a contradiction.

Henceu2ηholds. From2.12andu2∼u5, it follows thatu3η∼u5ηare satisfied.

Remark 2.3. FromRemark 2.2, we may assume thatθis nondecreasing in its third and fourth variables, respectively, for a functionθ:X×X×R×R → Rsatisfyingu2∼u5.

We give some examples ofu-distance.

Example 2.4. Let X 0,∞ be the set of real numbers with the usual metric and let p : X×X → Rbe defined bypx, y 1/4x². Thenpis au-distance onXbut not aτ-distance onX.

Proof. Defineθ:X×X×R×R → Rbyθx, y, s, t sfor allx, y∈Xands, t∈R. Thenp andθsatisfyu1∼u5. But for an arbitrary functionη :X×R → Rand for all sequences {zn},{xn},and{yn}ofXsuch that

0 lim

n→ ∞η

z_n, pzn, x_n lim

n→ ∞η

z_n,1 4zn²

, 0 lim

n→ ∞η z_n, p

z_n, y_n lim

n→ ∞η

z_n,1 4zn²

,

2.26

since the limit of the sequence {ηzn, pzn, xn}^∞_n1 and the limit of the sequence {ηzn, pzn, y_n}^∞_n1do not depend on{xn}and{yn}, the limit of the sequence{dxn, y_n}^∞_n1 may not be 0. This does not satisfyτ5. Hencepis not aτ-distance on X. Thereforepis a u-distance onXbut not aτ-distance onX.

Example 2.5. Letpbe aτ-distance on a metric spaceX, d. Thenpis also au-distance onX.

Proof. Sincepis aτ-distance, there exists a functionη : X×R → R satisfyingτ1∼τ5.

Defineθ:X×X×R×R → Rby

θ

x, y, s, t

2η x, p

x, y 1η

x, p x, y

·s ∀x, y∈X, s, t∈R. 2.27

Then it is easy to see thatpandθsatisfyu2∼u5. Thuspis au-distance onX.

Example 2.6. LetXbe a normed space with norm·. Then a functionp:X×X → Rdefined bypx, y xfor everyx, y∈Xis au-distance onXbut not aτ-distance.

Proof. Letθ:X×X×R×R → Rbe as in the proof ofExample 2.4. Then it is clear thatp satisfiesu1andθsatisfiesu2∼u5onXbutpdoes not satisfyτ5. Thuspis au-distance onXbut not aτ-distance.

Example 2.7. LetXbe a normed space with norm·. Then a functionp:X×X → Rdefined bypx, y yfor everyx, y∈Xis au-distance onX.

Proof. Defineθ:X×X×R×R → R byθx, y, s, t stfor allx, y∈X ands, t ∈R. Thenpsatisfiesu1andθsatisfiesu2∼u5. Thuspis au-distance onX.

Example 2.8. Letpbe au-distance on a metric spaceX, dand letcbe a positive real number.

Then a functionqfromX×XintoRdefined byqx, y c·px, yfor everyx, y∈Xis also au-distance onX.

Proof. Sincepis au-distance onX, there exists a functionη:X×X×R×R → Rsatisfying u2η∼u5ηandpsatisfiesu1. Defineθ:X×X×R×R → Rbyθx, y, s, t c·ηx, y, s, t for allx, y ∈ X ands, t ∈ R. Then it is clear thatqsatisfiesu1andθsatisfiesu2∼u5.

Thusqis au-distance onX.

The following examples can be easily obtained fromRemark 2.3.

Example 2.9. LetXbe a metric space with metricdand letpbe au-distance onX such that pis a lower semicontinuous in its first variable. Then a functionq:X×X → Rdefined by qx, y max{px, y, py, x}for allx, y∈Xis au-distance onX.

Example 2.10. LetXbe a metric space with metricd. Letpbe au-distance onXand letαbe a function fromXintoR. Then a functionq:X×X → Rdefined by

q x, y

max αx, p

x, y

, for everyx, y∈X 2.28

is au-distance onX.

Remark 2.11. It follows fromExample 2.4toExample 2.10thatu-distance is a proper extension ofτ-distance.

Definition 2.12. LetXbe a metric space with a metricdand letpbe au-distance onX. Then a sequence{xn}ofX is calledp-Cauchy if there exists a functionθ:X×X×R×R → R satisfyingu2∼u5and a sequence{zn}ofXsuch that

nlim→ ∞sup θ

z_n, z_n, pzn, x_m, pzn, x_m

:m≥n

0, 2.29

or

nlim→ ∞sup θ

z_n, z_n, pxm, z_n, pxm, z_n

:m≥n

0. 2.30

The following lemmas play an important role in proving our theorems.

Lemma 2.13. LetX be a metric space with a metricdand letpbe au-distance onX. If{xn}is a p-Cauchy sequence, then{xn}is a Cauchy sequence.

Proof. By assumption, there exists a functionθfromX×X×R×RintoRsatisfyingu2∼u5 and a sequence{zn}ofXsuch that

nlim→ ∞sup θ

z_n, z_n, pzn, x_m, pzn, x_m

:m≥n

0, 2.31

or

nlim→ ∞sup θ

z_n, z_n, pxm, z_n, pxm, z_n

:m≥n

0. 2.32

Then fromu5, we have limn→ ∞sup{dxi, xj : j > i ≥ n} 0. This means that {xn} is a Cauchy sequence.

Lemma 2.14. LetXbe a metric space with a metricdand letpbe au-distance onX.

1If sequences{xn}and{yn}ofXsatisfy lim_{n→ ∞}pz, xn 0 and lim_{n→ ∞}pz, yn 0 for somez∈X, then lim_{n→ ∞}dxn, y_n 0.

2Ifpz, x 0 andpz, y 0, thenxy.

3Suppose that sequences {xn} and {yn} of X satisfy lim_{n→ ∞}pxn, z 0 and lim_{n→ ∞}pyn, z 0 for somez∈X, then lim_{n→ ∞}dxn, y_n 0.

4Ifpx, z 0 andpy, z 0, thenxy.

Proof. 1 Let θ be a function from X × X ×R × R into R satisfying u2∼u5. From Remark 2.3and hypotheses,

nlim→ ∞θ

z, z, pz, xn, pz, xn 0,

nlim→ ∞θ z, z, p

z, yn

, p z, yn

0.

2.33

Byu5, limn→ ∞dxn, yn 0.

2In1, puttingxnxandynyfor alln∈N,2holds.

By method similar to1and2, results of3and4follow.

Lemma 2.15. LetXbe a metric space with a metricdand letpbe au-distance onX. Suppose that a sequence{xn}ofXsatisfies

nlim→ ∞sup

pxn, xm:m > n

0 2.34

or

nlim→ ∞sup

pxm, x_n:m > n

0. 2.35

Then{xn}is ap-Cauchy sequence and{xn}is a Cauchy sequence.

Proof. Sincepis au-distance onX, there exists a functionθ:X×X×R×R → Rsatisfying u2∼u5. Suppose limn→ ∞sup{pxn, x_m :m > n} 0. Letα_n sup{pxi, x_j :j > i≥n}.

Then we have lim_{n→ ∞}α_n 0. Let{xfn}be an arbitrary subsequence of{xn}. By assumption andu2, there exists a subsequence{xfgn}of{xfn}such that

nlim→ ∞θ

x_fgn, x_fgn1, α_fgn1, α_fgn1 0,

nlim→ ∞sup

sup

m≥np

x_fgn, x_fgm1

≤ lim

n→ ∞α_fgn0.

2.36

Fromu4, we obtain

nlim→ ∞θ

x_fgn, xfgn, αfgn, α_fgn lim

n→ ∞θ

x_fgn1, x_fgn1, α_fgn1, α_fgn1

0. 2.37

Since{xfn}is an arbitrary sequence of{xn},{xfgn}is also an arbitrary sequence of{xn}.

Hence

nlim→ ∞θxn, xn, αn, αn 0. 2.38

Therefore we get

nlim→ ∞sup

m≥nθ

x_n−1, x_n−1, px_n−1, x_m, px_n−1, x_m

≤ lim

n→ ∞θxn−1, x_n−1, α_n−1, α_n−1 0.

2.39

This implies that{xn}is ap-Cauchy sequence. ByLemma 2.13,{xn}is a Cauchy sequence.

Similarly, if limn→ ∞sup{pxm, xn : m > n} 0,we can prove that {xn}is also a Cauchy sequence.

3. Minimization Theorems and Fixed Point Theorems

The following theorem is a generalization of Takahashi’s minimization theorem4.

Theorem 3.1. LetX be a metric space with metricd, let f : X → −∞,∞be a proper function which is bounded from below, and let L : X ×X × X × X → R be a function such that,

one has the following.

iLx, y, y, x≤Lx, z, z, x Lz, y, y, zfor allx, y, z∈X.

iiFor any sequence{vn}^∞_n1inXsatisfying

nlim→ ∞sup{Lvn, v_m, v_m, v_n:m > n}0, 3.1

there existsx0∈Xsuch that limn→ ∞vnx0, fx0≤ lim

n→ ∞supfvn, Lvn, x0, x0, vn≤ lim

m→ ∞infLvn, vm, vm, vn. 3.2

iiiLx, y, y, x Lx, z, z, x 0 implyyz.

ivFor everyx∈Xwith inf_v∈Xfv< fx, there existsy∈X− {x}such that h

x, y

≤fx−f y

, 3.3

where a functionh:X×X → Ris defined by

hv, w Lv, w, w, v 3.4

for allv, w∈X. Then, there existsx0∈Xsuch that fx0 inf

v∈Xfv. 3.5

Proof. Suppose inf_v∈Xfv< fxfor allx∈X. For eachx∈X, let

Sx

v∈X|hx, v≤fx−fv

. 3.6

Then, by conditionivand3.6,Sxis nonempty for eachx∈X. From conditioniand 3.6, we obtain

Sv⊆Sx, for eachv∈Sx. 3.7

For eachx∈X, let

cx inf

fv|v∈Sx

. 3.8

Choose x ∈ X with fx < ∞. Then, from 3.7 and 3.8, there exists a sequence {xn}^∞_n1inXsuch that

x1 x, x_n1∈Sxn, Sxn⊆Sx, fx_n1< cxn

1 n

3.9

for alln∈N.

From3.6,3.8and3.9, we have

hxn, x_n1≤fxn−fxn1, 3.10

fx_n1− 1

n < cxn≤fx_n1. 3.11

By 3.10, {fxn}^∞_n1 is a nonincreasing sequence of real numbers and so it converges.

Therefore, from3.11there is someβ∈Rsuch that

β lim

n→ ∞cxn lim

n→ ∞fxn. 3.12

From conditioniand3.10, we get

hxn, xm≤fxn−fxm 3.13

for allm > n. From3.12and3.13, we have

nlim→ ∞sup{Lxn, x_m, x_m, x_n:m > n}0. 3.14

Thus, by conditionii,3.12, and3.13, there existsx0∈Xsuch that

nlim→ ∞x_nx₀, 3.15

fx0≤ lim

n→ ∞fxn β, 3.16

hxn, x0≤ lim

m→ ∞infhxn, xm. 3.17

From3.13,3.16, and3.17, we have fx0≤β lim

m→ ∞supfxm

≤ lim

m→ ∞sup

fxn−hxn, xm fxn lim

m→ ∞sup{−hxn, x_m} fxn− lim

m→ ∞infhxn, x_m

≤fxn−hxn, x0.

3.18

From3.6,3.8, and3.18, it follows that

x₀∈Sxnand hencecxn≤fx0, ∀n∈N. 3.19 Taking the limit in inequality3.19whenntends to infinity, we have

nlim→ ∞cxn≤fx0. 3.20

From3.12,3.16, and3.20, we have

βfx0. 3.21

On the other hand, by conditionivand3.6, we have the following property:

there existsv1∈X− {x0}, satisfyingv1∈Sx0. 3.22 From3.7,3.8,3.19, and3.22, we have

v1 ∈Sxn, ∀n∈N,

cxn≤fv1. 3.23

From3.6,3.12,3.21,3.22,3.23, it follows that

βfv1. 3.24

From3.21,3.22, and3.24, we have

Lx0, v1, v1, x0 0. 3.25

By method similar to3.22∼3.25,

there exists v2∈X− {v1}, such thatLv1, v2, v2, v1 0. 3.26

From3.25,3.26, and conditioni, we obtain

Lx0, v₂, v₂, x₀ 0. 3.27

From3.25,3.27, and conditioniii, we obtain

v1v2. 3.28

This is a contradiction from3.26.

Corollary 3.2. LetXbe a complete metric space with metricd, and letf:X → −∞,∞be a proper lower semicontinuous function which is bounded from below. Assume that there exists au-distancep onX such that for eachu∈ Xwithfu >inf{fx|x ∈X}, there existsv∈ Xwithv /uand fv pu, v≤fu. Then there existsx₀∈Xsuch thatfx0 inf{fx|x∈X}.

Proof. LetL:X×X×X×X → Rbe a mapping such that

L

x, y, v, w

max

px, v, p w, y

3.29 for allx, y, v, w,∈X. It follows easily fromDefinition 2.12, Lemmas2.13,2.14, and2.15, and u3 that conditions ofCorollary 3.2satisfy all conditions ofTheorem 3.1. Thus, we obtain result ofCorollary 3.2.

Remark 3.3. Corollary 3.2 is a generalization of Kadaet al. 5, Theorem 1 and Suzuki 6, Theorem 5.

From Lemmas2.13,2.14, and2.15, we have the following fixed point theorem.

Theorem 3.4. LetXbe a complete metric space with metricd, letpbe au-distance onX,and letT be a selfmapping ofX. Suppose that there existsr ∈0,1such that

p

Tx, Ty

≤r·max p

x, y

, px, Tx, p y, Ty

, p x, Ty

, p y, Tx

, p

y, x

, pTx, x, p Ty, y

, p Ty, x

, p

Tx, y 3.30

for allx, y∈Xand

inf p

x, y

px, Tx:x∈X

>0 3.31

for every y ∈ X with y /Ty. Then there exists x0 ∈ X such that Tx0 x0 and px0, x0 0.

Moreover, ifvTv, thenx₀v,pv, v 0.

Proof. By method similar to12, Lemma 2.4, for everyx∈X, αx:sup

p

Tⁱx, T^jx

|i, j∈N∪ {0}

<∞. 3.32

Defineq:X×X → Rby

q x, y

max

αx, p x, y

3.33

for everyx, y∈X. ByExample 2.10,qis au-distance onX. Then we get q

Tx, T²x

max

αTx, p

Tx, T²x αTx≤r·αx r·qx, Tx, q

T²x, Tx

max α

T²x , p

T²x, Tx

≤αTx≤r·αx r·qx, Tx, qTx, Tx max

αTx, pTx, Tx αTx≤r·αx r·qx, x

3.34

for allx∈X. Thus we have

qTⁿx, T^mx≤^m−1

kn

q

T^kx, T^k1x

≤^m−1

kn

r^k·qx, Tx≤ rⁿ

1−rqx, Tx

3.35

for allm > n. Now we have

nlim→ ∞sup

qTⁿx, T^mx:m > n

≤ lim

n→ ∞

rⁿ

1−rqx, Tx 0. 3.36

Thus

nlim→ ∞sup

qTⁿx, T^mx:m > n

0. 3.37

ByLemma 2.15,{Tⁿx}is aq-Cauchy and hence{Tⁿx}is a Cauchy fromLemma 2.13. SinceX is complete and{Tⁿx}is aq-Cauchy, there existsx₀∈Xsuch that

nlim→ ∞Tⁿxx₀, qTⁿx, x0≤ lim

m→ ∞infqTⁿx, T^mx≤ rⁿ

1−rqx, Tx.

3.38

Supposex0/Tx0. Then, by hypothesis, we have 0<inf

px, x0 px, Tx:x∈X

≤inf

qx, x0 qx, Tx:x∈X

≤inf

qTⁿx, x0 q

Tⁿx, Tⁿ¹x

:n∈N

≤inf 2rⁿ

1−rqx, Tx:n∈N 0.

3.39

This is a contradiction. Therefore we havex₀Tx₀. IfvTv, we havepv, v pTv, Tv≤ rpv, vand hence pv, v 0. To prove unique fixed point ofT, letx0 Tx0 andv Tv.

Then, by hypothesis, we have

px0, v pTx0, Tv≤r·max

px0, v, pv, x0, px0, x₀, pv, v , pv, x0 pTv, Tx0≤r·max

px0, v, pv, x0, px0, x0, pv, v , px0, x₀ pTx0, Tx₀≤r·max

px0, v, pv, x0, px0, x₀, pv, v , pv, v pTv, Tv≤r·max

px0, v, pv, x0, px0, x₀, pv, v .

3.40

Thus

px0, v pv, x0 px0, x0 pv, v 0. 3.41

ByLemma 2.14, we havex0v.

FromTheorem 3.4, we have the following corollary which generalizes the results of Ciri´c´ 14, Kannan15, and Ume12.

Corollary 3.5. LetXbe a complete metric space with metricd, letpbe aτ-distance onX,and letT be a selfmapping ofX. Suppose that there existsr ∈0,1such that

p

Tx, Ty

≤r·max p

x, y

, px, Tx, p y, Ty

, p x, Ty

, p y, Tx

, p

y, x

, pTx, x, p Ty, y

, p Ty, x

, p

Tx, y 3.42

for allx, y∈Xand

inf p

x, y

px, Tx:x∈X

>0 3.43

for every y ∈ X with y /Ty. Then there exists x0 ∈ X such that Tx0 x0 and px0, x0 0.

Moreover, ifvTv, thenvx₀andpv, v 0.

Proof. Since aτ-distance is au-distance,Corollary 3.5follows fromTheorem 3.4.

The following corollary is a generalization of Suzuki’s fixed point theorem6.

Corollary 3.6. LetX, T,andpbe as inCorollary 3.5. Suppose that there existsr ∈0,1such that

p

Tx, T²x

≤r·max

px, x, px, Tx, pTx, x

3.44

for allx, y∈X. Assume that if

nlim→ ∞sup

pxn, x_m:m > n 0,

nlim→ ∞pxn, Tx_n 0,

nlim→ ∞pxn, z 0,

3.45

thenTzz. Then there existsx0 ∈Xsuch thatTx0 x0andpx0, x0 0. Moreover, ifTv v, thenvx0andpv, v 0.

Proof. Let q and T be as in Theorem 3.4. Then from Theorem 3.4 and hypotheses of Corollary 3.6, we have the following properties.

1{Tⁿx}is a Cauchy sequence.

2There existsx0∈Xsuch that limn→ ∞Tⁿxx0. 3One has

nlim→ ∞pTⁿx, x0≤ lim

n→ ∞qTⁿx, x0

≤ lim

n→ ∞

rⁿ 1−rmax

qx, Tx, qx, x .

3.46

4There exists

n→ ∞lim p

Tⁿx, Tⁿ¹x lim

n→ ∞p

Tⁿ¹x, Tⁿx

0. 3.47

5One has

nlim→ ∞sup

pTⁿx, T^mx:m > n

0. 3.48

By1∼5and hypotheses, we haveTx0x0. The remainders are same asTheorem 3.4.

The following theorem is a generalization of Caristi’s fixed point theorem3.

Theorem 3.7. LetX be a metric space with metricd, let f : X → −∞,∞be a proper function which is bounded from below, and letL:X×X×X×X → Rbe a function satisfying (i), (ii), and (iii) ofTheorem 3.1. LetT be a selfmapping ofXsuch that

fTx hx, Tx≤fx, ∀x∈X, 3.49

where a functionh:X×X → Ris defined by

hv, w Lv, w, w, v 3.50

for allv, w∈X. Then, there existsx0∈Xsuch that

Tx₀x₀, Lx0, Tx₀, Tx₀, x₀ 0. 3.51 Proof. Supposex /Txfor allx∈X. Then, byTheorem 3.1, there existsx0∈Xsuch that

fx0 inf

v∈Xfv. 3.52

Since

fTx0 hx0, Tx0≤fx0, 3.53

we have

fTx0 fx0 inf

v∈Xfv, Lx0, Tx₀, Tx₀, x₀ 0.

3.54

By hypothesis, we obtain f

T²x0

h

Tx0, T²x0

≤fTx0. 3.55

Hence

f T²x0

fTx0,

L

Tx0, T²x0, T²x0, Tx0

0.

3.56

By conditionsiandiiiofTheorem 3.1, it follows that

Tx₀T²x₀. 3.57

This is a contradiction.

Corollary 3.8. LetXbe a complete metric space with metricdand letf:X → −∞,∞be a proper lower semicontinuous function which is bounded from below. Letpbe au-distance onX. Suppose that Tis a selfmapping ofXsuch that

fTx px, Tx≤fx 3.58

for allx∈X. Then there existsx₀∈Xsuch that

Tx₀x₀, px0, x₀ 0. 3.59

Proof. DefineL:X×X×X×X → Rby L

v, w, x, y

max

pv, x, p y, w

3.60 for allv, w, x, y∈X. Then, byDefinition 2.12and Lemmas2.13,2.14, and2.15, we can easily show that conditions ofCorollary 3.8satisfy all conditions ofTheorem 3.7. Thus,Corollary 3.8 follows fromTheorem 3.7.

Remark 3.9. Since a w-distance and a τ-distance are a u-distance, Corollary 3.8 is a generalization of Kada-Suzuki-Takahashi5, Theorem 2and Suzuki6, Theorem 3.

The following theorem is a generalization of Ekeland’sε-variational principle2.

Theorem 3.10. LetXbe a complete metric space with metricd, letf :X → −∞,∞be a proper lower semicontinuous function which is bounded from below, and letL:X×X×X×X → Rbe a function satisfying (i), (ii), and (iii) ofTheorem 3.1. Then the following (1) and (2) hold.

1For eachx∈Xwithfx<∞, there existsv∈Xsuch thatfv≤fxand

fm> fv−hv, m 3.61

for allm∈Xwithm /v,where a functionh:X×X → Ris defined by

hv, w Lv, w, w, v 3.62

for allv, w∈X.

2For eachε >0 andx∈Xwithhx, x 0, and fx<inf

a∈Xfa ε, 3.63

there existsv∈Xsuch thatfv≤fx, hx, v≤1,

fm> fv−ε·hv, m 3.64

for allm∈Xwithm /v.

Proof. 1Letx∈Xbe such thatfx<∞, and let

Z

s∈X|fs≤fx

. 3.65

Then, by hypotheses,Zis nonempty and closed. ThusZis a complete metric space. Hence we may prove that there exists an elementv ∈ Z such thatfm > fv−hv, mfor all m∈Xwithm /v.Suppose not. Then, for everyv∈Z, there existsm∈Zsuch thatm /vand fm hv, m≤fv.ByTheorem 3.1, there existsx0∈Zsuch that

fx0 inf

a∈Zfa. 3.66

Again forx0∈Z, there existsx1∈Zsuch thatx1/x0and

fx1 hx0, x₁≤fx0. 3.67

Hence we havefx1 fx0andLx0, x1, x1, x0 0.Similarly, there existsx2∈Zsuch that x₂/x₁and

fx2 hx1, x₂≤fx1. 3.68

Thus we have fx2 fx1 and Lx1, x₂, x₂, x₁ 0. From conditions i and iii of Theorem 3.1, we obtain

x1x2. 3.69

This is a contradiction. The proof of1is complete.

2Let

Y

a∈X|fa≤fx−ε·hx, a

. 3.70

ThenY is nonempty and closed. HenceY is complete. As in the proof of1, we have that there existsv∈Ysuch that

fm> fv−ε·hv, m 3.71

for everym∈Xwithm /v.On the other hand, sincev∈Y, we have fv≤fx−ε·hx, v≤fx, hx, v≤ 1

ε

fx−fv

≤ 1 ε

fx−inf

a∈Xfa

≤ 1 ε·ε1.

3.72

This completes the proof of2.

Corollary 3.11. LetX, f,andpbe as inCorollary 3.8. Then the following (1) and (2) hold.

1For eachx∈Xwithfx<∞, there existsv∈Xsuch thatfv≤fxand

fm> fv−pv, m 3.73

for allm∈Xwithm /v.

2For eachε >0 andx∈Xwithpx, x 0, and fx<inf

a∈Xfa ε, 3.74

there existsv∈Xsuch thatfv≤fx,

px, v≤1, fm> fv−ε·pv, m 3.75

for allm∈Xwithm /v.

Proof. By method similar toCorollary 3.8,Corollary 3.11follows fromTheorem 3.10.

Remark 3.12. Corollary 3.11is a generalization of Suzuki6, Theorem 4.

Acknowledgments

The author would like to thank the referees for useful comments and suggestions. This work was supported by the Korea Research FoundationKRFGrant funded by the Korea governmentMEST 2009-0073655.

References

1 S. Banach, “Sur les op´erations dans les ensembles abstraits et leur application aux ´equations int´egrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1992.

2 I. Ekeland, “Nonconvex minimization problems,” Bulletin of the American Mathematical Society, vol. 1, no. 3, pp. 443–474, 1979.

3 J. Caristi, “Fixed point theorems for mappings satisfying inwardness conditions,” Transactions of the American Mathematical Society, vol. 215, pp. 241–251, 1976.

4 W. Takahashi, “Minimization theorems and fixed point theorems,” Nonlinear Analysis and Mathemati- cal Economics, RIMS Kokyuroku, vol. 829, pp. 175–191, 1993.

5 O. Kada, T. Suzuki, and W. Takahashi, “Nonconvex minimization theorems and fixed point theorems in complete metric spaces,” Mathematica Japonica, vol. 44, no. 2, pp. 381–391, 1996.

6 T. Suzuki, “Generalized distance and existence theorems in complete metric spaces,” Journal of Mathematical Analysis and Applications, vol. 253, no. 2, pp. 440–458, 2001.

7 J.-S. Ume, “Some existence theorems generalizing fixed point theorems on complete metric spaces,”

Mathematica Japonica, vol. 40, no. 1, pp. 109–114, 1994.

8 N. Shioji, T. Suzuki, and W. Takahashi, “Contractive mappings, Kannan mappings and metric completeness,” Proceedings of the American Mathematical Society, vol. 126, no. 10, pp. 3117–3124, 1998.

9 T. Suzuki, “Fixed point theorems in complete metric spaces,” in Nonlinear Analysis and Convex Analysis, W. Takahashi, Ed., vol. 939, pp. 173–182, RIMS Kokyuroku, 1996.

10 T. Suzuki, “Several fixed point theorems in complete metric spaces,” Yokohama Mathematical Journal, vol. 44, no. 1, pp. 61–72, 1997.

11 T. Suzuki and W. Takahashi, “Fixed point theorems and characterizations of metric completeness,”

Topological Methods in Nonlinear Analysis, vol. 8, no. 2, pp. 371–382, 1996.

12 J.-S. Ume, “Fixed point theorems related to ´Ciri´c’s contraction principle,” Journal of Mathematical Analysis and Applications, vol. 225, no. 2, pp. 630–640, 1998.

13 D. Tataru, “Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms,”

Journal of Mathematical Analysis and Applications, vol. 163, no. 2, pp. 345–392, 1992.

14 Lj. B. ´Ciri´c, “A generalization of Banach’s contraction principle,” Proceedings of the American Mathematical Society, vol. 45, pp. 267–273, 1974.

15 R. Kannan, “Some results on fixed points. II,” The American Mathematical Monthly, vol. 76, pp. 405–408, 1969.