The Banach contraction principle [1], Ekeland's -variational principle [2], and Caristi's fixed point theorem [3] are very useful tools in nonlinear analysis, control theory, economic theory, and global analysis. These theorems are extended by several authors in different directions.

Takahashi [4] proved the following minimization theorem. Let be a complete metric space and let be a proper lower semicontinuous function, bounded from below. Suppose that, for each with there exists such that and Then there exists such that Some authors [5–7] have generalized and extended this minimization theorem in complete metric spaces.

In 1996, Kada et al. [5] introduced the concept of -distance on a metric space as follows. Let be a metric space with metric . Then a function is called a -distance on if the followings are satisfied.

(1) for any .

(2)For any is lower semicontinuous.

(3)For any , there exists such that and imply

They gave some examples of -distance and improved Caristi's fixed point theorem [3], Ekeland's variational principle [2], and Takahashi's nonconvex minimization theorem [4]. The fixed point theorems with respect to a -distance were proved in [8–12].

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

Recently, Suzuki [6] introduced the concept of -distance on a metric space, which generalizes Tataru's distance [13] as follows. Let be a metric space with metric .

Then a function from into is called -distance on if there exists a function from into and the followings are satisfied:

(1) for all ;

(2) and for all and , and is concave and continuous in its second variable;

(3) and imply for all ;

(4) and imply ;

(5) and imply .

In this paper, we first introduce the new concept of a distance called -distance, which generalizes -distance, Tataru's distance, and -distance. Then we prove a new minimization theorem and a new fixed point theorem by using -distance on a complete metric space. Our results extend and unify many known results due to Caristi [3], Ćirić [14], Ekeland [2], Takahashi [4], Kada et al. [5], Kannan [15], Suzuki [6], and Ume [7, 12] and others.

Definition 2.1.

Let be metric space with metric . Then a function from into is called -distance on if there exists a function from into such that

(u) for all ;

(u) and for all and , and for any and for every , there exists such that , , and imply

(u)

imply

for all ;

(u)

imply

or

imply

(u)

imply

or

imply

Remark 2.2.

Suppose that is a mapping satisfying (u2)(u5). Then there exists a mapping from into such that is nondecreasing in its third and fourth variable, respectively, satisfying (u2)(u5), where (u2)(u5) stand for substituting for in (u2)(u5), respectively.

Proof.

Suppose that is a mapping satisfying (u2)(u5). Define a function by

By (2.12), we have and for all and . Also it follows from (2.12) that is nondecreasing in its third and fourth variable, respectively.

We shall prove the following:

Suppose that (2.13) does not hold. Then

By virtue of (2.12) and (2.14), we have

Combining (u2) and (2.14), we have the following:

Due to (2.16), we get that

From (2.16) and (2.17), we obtain the following.

In terms of (2.19) and (2.20), we deduce that

In view of (2.21), we get that

On account of (2.20), we know the following:

Using (2.16), (2.18), (2.19), and (2.23), we have the following:

By (2.24), we have

By virtue of (2.15), (2.19), (2.20), (2.22), and (2.25), we have which is a contradiction. Hence (u2) holds. From (2.12) and (u2)~(u5), it follows that (u3)~(u5) are satisfied.

Remark 2.3.

From Remark 2.2, we may assume that is nondecreasing in its third and fourth variables, respectively, for a function satisfying (u2)(u5).

We give some examples of -distance.

Example 2.4.

Let be the set of real numbers with the usual metric and let be defined by . Then is a -distance on but not a -distance on .

Proof.

Define by for all and . Then and satisfy (u1)~(u5). But for an arbitrary function and for all sequences , and of such that

since the limit of the sequence and the limit of the sequence do not depend on and , the limit of the sequence may not be . This does not satisfy (5). Hence is not a -distance on . Therefore is a -distance on but not a -distance on .

Example 2.5.

Let be a -distance on a metric space . Then is also a -distance on .

Proof.

Since is a -distance, there exists a function satisfying (1)~(5). Define by

Then it is easy to see that and satisfy (u2)~(u5). Thus is a -distance on .

Example 2.6.

Let be a normed space with norm . Then a function defined by for every is a -distance on but not a -distance.

Proof.

Let be as in the proof of Example 2.4. Then it is clear that satisfies and satisfies (u2)~(u5) on but does not satisfy . Thus is a -distance on but not a -distance.

Example 2.7.

Let be a normed space with norm . Then a function defined by for every is a -distance on .

Proof.

Define by for all and . Then satisfies and satisfies (u2)~(u5). Thus is a -distance on .

Example 2.8.

Let be a -distance on a metric space and let be a positive real number. Then a function from into defined by for every is also a -distance on .

Proof.

Since is a -distance on , there exists a function satisfying (u2)~(u5) and satisfies (u1). Define by for all and . Then it is clear that satisfies and satisfies (u2)~(u5). Thus is a -distance on .

The following examples can be easily obtained from Remark 2.3.

Example 2.9.

Let be a metric space with metric and let be a -distance on such that is a lower semicontinuous in its first variable. Then a function defined by for all is a -distance on .

Example 2.10.

Let be a metric space with metric . Let be a -distance on and let be a function from into . Then a function defined by

is a -distance on .

Remark 2.11.

It follows from Example 2.4 to Example 2.10 that -distance is a proper extension of -distance.

Definition 2.12.

Let be a metric space with a metric and let be a -distance on . Then a sequence of is called -Cauchy if there exists a function satisfying (u2)~(u5) and a sequence of such that

or

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

Lemma 2.13.

Let be a metric space with a metric and let be a -distance on . If is a -Cauchy sequence, then is a Cauchy sequence.

Proof.

By assumption, there exists a function from into satisfying (u2)~(u5) and a sequence of such that

or

Then from (u5), we have . This means that is a Cauchy sequence.

Lemma 2.14.

Let be a metric space with a metric and let be a -distance on .

(1)If sequences and of satisfy and for some , then .

(2)If and , then .

(3)Suppose that sequences and of satisfy and for some , then .

(4)If and , then .

Proof.

(1) Let be a function from into satisfying (u2)~(u5). From Remark 2.3 and hypotheses,

By (u5), .

(2) In (1), putting and for all , (2) holds.

By method similar to (1) and (2), results of (3) and (4) follow.

Lemma 2.15.

Let be a metric space with a metric and let be a -distance on . Suppose that a sequence of satisfies

or

Then is a -Cauchy sequence and is a Cauchy sequence.

Proof.

Since is a -distance on , there exists a function satisfying (u2)(u5). Suppose . Let . Then we have . Let be an arbitrary subsequence of . By assumption and (u2), there exists a subsequence of such that

From (u4), we obtain

Since is an arbitrary sequence of , is also an arbitrary sequence of . Hence

Therefore we get

This implies that is a -Cauchy sequence. By Lemma 2.13, is a Cauchy sequence. Similarly, if we can prove that is also a Cauchy sequence.

The following theorem is a generalization of Takahashi's minimization theorem [4].

Theorem 3.1.

Let be a metric space with metric , let be a proper function which is bounded from below, and let be a function such that, one has the following.

(i) for all .

(ii)For any sequence in satisfying

there exists such that

(iii) imply .

(iv)For every with , there exists such that

where a function is defined by

for all . Then, there exists such that

Proof.

Suppose for all . For each , let

Then, by condition (iv) and (3.6), is nonempty for each . From condition (i) and (3.6), we obtain

For each , let

Choose with . Then, from (3.7) and (3.8), there exists a sequence in such that

for all .

From (3.6), (3.8) and (3.9), we have

By (3.10), is a nonincreasing sequence of real numbers and so it converges. Therefore, from (3.11) there is some such that

From condition (i) and (3.10), we get

for all . From (3.12) and (3.13), we have

Thus, by condition (ii), (3.12), and (3.13), there exists such that

From (3.13), (3.16), and (3.17), we have

From (3.6), (3.8), and (3.18), it follows that

Taking the limit in inequality (3.19) when tends to infinity, we have

From (3.12), (3.16), and (3.20), we have

On the other hand, by condition (iv) and (3.6), we have the following property:

From (3.7), (3.8), (3.19), and (3.22), we have

From (3.6), (3.12), (3.21), (3.22), (3.23), it follows that

From (3.21), (3.22), and (3.24), we have

By method similar to (3.22)(3.25),

From (3.25), (3.26), and condition (i), we obtain

From (3.25), (3.27), and condition (iii), we obtain

This is a contradiction from (3.26).

Corollary 3.2.

Let be a complete metric space with metric , and let be a proper lower semicontinuous function which is bounded from below. Assume that there exists a -distance on such that for each with , there exists with and . Then there exists such that .

Proof.

Let be a mapping such that

for all . It follows easily from Definition 2.12, Lemmas 2.13, 2.14, and 2.15, and (u3) that conditions of Corollary 3.2 satisfy all conditions of Theorem 3.1. Thus, we obtain result of Corollary 3.2.

Remark 3.3.

Corollary 3.2 is a generalization of Kadaet al. [5, Theorem ] and Suzuki [6, Theorem ].

From Lemmas 2.13, 2.14, and 2.15, we have the following fixed point theorem.

Theorem 3.4.

Let be a complete metric space with metric , let be a -distance on and let be a selfmapping of . Suppose that there exists such that

for all and

for every with . Then there exists such that and . Moreover, if , then , .

Proof.

By method similar to [12, Lemma ], for every ,

Define by

for every . By Example 2.10, is a -distance on . Then we get

for all . Thus we have

for all . Now we have

Thus

By Lemma 2.15, is a -Cauchy and hence is a Cauchy from Lemma 2.13. Since is complete and is a -Cauchy, there exists such that

Suppose . Then, by hypothesis, we have

This is a contradiction. Therefore we have . If , we have and hence . To prove unique fixed point of , let and . Then, by hypothesis, we have

Thus

By Lemma 2.14, we have .

From Theorem 3.4, we have the following corollary which generalizes the results of Ćirić [14], Kannan [15], and Ume [12].

Corollary 3.5.

Let be a complete metric space with metric , let be a -distance on and let be a selfmapping of . Suppose that there exists such that

for all and

for every with . Then there exists such that and . Moreover, if , then and .

Proof.

Since a -distance is a -distance, Corollary 3.5 follows from Theorem 3.4.

The following corollary is a generalization of Suzuki's fixed point theorem [6].

Corollary 3.6.

Let and be as in Corollary 3.5. Suppose that there exists such that

for all . Assume that if

then . Then there exists such that and . Moreover, if , then and .

Proof.

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

(1) is a Cauchy sequence.

(2)There exists such that .

(3)One has

(4)There exists

(5)One has

By (1)~(5) and hypotheses, we have . The remainders are same as Theorem 3.4.

The following theorem is a generalization of Caristi's fixed point theorem [3].

Theorem 3.7.

Let be a metric space with metric , let be a proper function which is bounded from below, and let be a function satisfying (i), (ii), and (iii) of Theorem 3.1. Let be a selfmapping of such that

where a function is defined by

for all . Then, there exists such that

Proof.

Suppose for all . Then, by Theorem 3.1, there exists such that

Since

we have

By hypothesis, we obtain

Hence

By conditions (i) and (iii) of Theorem 3.1, it follows that

This is a contradiction.

Corollary 3.8.

Let be a complete metric space with metric and let be a proper lower semicontinuous function which is bounded from below. Let be a -distance on . Suppose that is a selfmapping of such that

for all . Then there exists such that

Proof.

Define by

for all . Then, by Definition 2.12 and Lemmas 2.13, 2.14, and 2.15, we can easily show that conditions of Corollary 3.8 satisfy all conditions of Theorem 3.7. Thus, Corollary 3.8 follows from Theorem 3.7.

Remark 3.9.

Since a -distance and a -distance are a -distance, Corollary 3.8 is a generalization of Kada-Suzuki-Takahashi [5, Theorem ] and Suzuki [6, Theorem ].

The following theorem is a generalization of Ekeland's -variational principle [2].

Theorem 3.10.

Let be a complete metric space with metric , let be a proper lower semicontinuous function which is bounded from below, and let be a function satisfying (i), (ii), and (iii) of Theorem 3.1. Then the following (1) and (2) hold.

(1)For each with , there exists such that and

for all with where a function is defined by

for all .

(2)For each and with , and

there exists such that ,

for all with

Proof.

() Let be such that , and let

Then, by hypotheses, is nonempty and closed. Thus is a complete metric space. Hence we may prove that there exists an element such that for all with Suppose not. Then, for every , there exists such that and By Theorem 3.1, there exists such that

Again for , there exists such that and

Hence we have and Similarly, there exists such that and

Thus we have and From conditions (i) and (iii) of Theorem 3.1, we obtain

This is a contradiction. The proof of (1) is complete.

(2) Let

Then is nonempty and closed. Hence is complete. As in the proof of (1), we have that there exists such that

for every with On the other hand, since , we have

This completes the proof of (2).

Corollary 3.11.

Let and be as in Corollary 3.8. Then the following (1) and (2) hold.

(1)For each with , there exists such that and

for all with

(2)For each and with , and

there exists such that ,

for all with

Proof.

By method similar to Corollary 3.8, Corollary 3.11 follows from Theorem 3.10.

Remark 3.12.

Corollary 3.11 is a generalization of Suzuki [6, Theorem ].