#### Abstract

The main purpose of this paper is to introduce a new class of ΔiriΔ-type contraction and to present some fixed point theorems for this mapping as well as for Caristi-type contraction. Several examples are given to show that our results are proper extension of many known results.

#### 1. Introduction

Probabilistic metric space has been introduced and studied in 1942 by Menger in USA [1], and since then the theory of probabilistic metric spaces has developed in many directions [2]. The idea of Menger was to use distribution functions instead of nonnegative real numbers as values of the metric. The notion of a probabilistic metric space corresponds to the situation when we do not know exactly the distance between two points, we know only probabilistic of metric spaces to be well adapted for the investigation of physiological thresholds and physical quantities particularly in connections with both string and infinity which were introduced and studied by a well-known scientific hero, El Naschie [3β5].

ΔiriΔβs fixed point theorem [6] and Caristiβs fixed point theorem [7] have many applications in nonlinear analysis. These theorems are extended by several authors, see [8β16] and the references therein.

In this paper, we introduce a new class of ΔiriΔ-type contraction and present some fixed point theorems for this mapping as well as for Caristi-type contraction. Several examples are given to show that our results are proper extension of many known results.

#### 2. Preliminaries

Throughout this paper, we denote by the set of all positive integers, by the set of all nonnegative integers, by the set of all real numbers, and by the set of all nonnegative real numbers. We shall recall some definitions and lemmas related to Menger space.

*Definition 2.1. *A mapping is called a distribution if it is nondecreasing left continuous with and . We will denote by the set of all distribution functions. The specific distribution function is defined by

*Definition 2.2 (see [17]). *Probabilistic metric space (PM-space) is an ordered pair , where is an abstract set of elements, and is defined by , where , where the functions satisfied the following:(a) for all if and only if ; (b); (c); (d) and , then .

*Definition 2.3. *A mapping is called a -norm if(e) and for all ; (f) for all ; (g) for all with and ; (h) for all .

*Definition 2.4. *Menger space is a triplet , where is PM space and is a norm such that for all and all ,

*Definition 2.5 (see [17]). *Let be a Menger space.(1)A sequence in is said to converge to a point in (written as if for every and , there exists a positive integer such that for all .(2)A sequence in is said to be Cauchy if for each and , there is a positive integer such that for all with .(3)A Menger space is said to be complete if every Cauchy sequence in is converged to a point in .

*Definition 2.6 (see [18]). *-norm is said to be of type if a family of functions is equicontinuous at , that is, for any , there exists , such that and imply . The -norm is a trivial example of -norm of type, but there are -norms of type with -norm (see, e.g., HadziΔ [19]).

*Definition 2.7. *Let be a Menger space, and let be a selfmapping. For each , and , let
where it is understood that .

A Menger space is said to be orbitally complete if and only if every Cauchy sequence which is contained in for some converges in .

From Definition 2.1β*~* βDefinition 2.5, we can prove easily the following lemmas.

Lemma 2.8 (see [20]). *Let be a metric space, and let be a selfmapping on . Define by
**
for all and , where . Suppose that -norm is defined by for all . Then, *(1)* is a Menger space; *(2)*If is orbitally complete, then is orbitally complete.**Menger space generated by a metric is called the induced Menger space.*

Lemma 2.9. *In a Menger space , if for all , then for all .*

#### 3. ΔiriΔ-Type Fixed Point Theorems

In 2010, ΔiriΔ proved the following theorem.

Theorem A (see ΔiriΔ [9], 2010). * Let be a complete Menger space under a -norm of type. Let be a generalized -probabilistic contraction, that is,
**
for all and , where satisfies the following conditions: , , and for each . Then, has a unique fixed point and converges to for each .*

*Definition 3.1. *Let be a Menger space with for all , and let be a mapping of . We will say that is ΔiriΔ-type-generalized contraction if
for all and , where is a mapping and for all and , is the same as in Definition 2.2.

It is clear that (*1) implies (*2).

The following example shows that a ΔiriΔ-type-generalized contraction need not be a generalized -probabilistic contraction.

*Example 3.2. *Let , be defined by , and let be defined by
For each , let be defined by for all , where is the same as in Definition 2.1, and is a usual metric on . Then, since for all , we have for all and . Thus,
for all and , which satisfies (*2). If and , then and . Thus, , which does not satisfy (*1).

In the next example, we shall show that there exists that does not satisfy (*2) with , ββ.

*Example 3.3. *Let , be defined by and let be defined by , . For each , let be defined by for all , where is the same as in Definition 2.1, and is a usual metric on . If , and , then for simple calculations, and
Therefore, for , , and , the mapping does not satisfy (*2). Thus, we showed that there exists that does not satisfy (*2) with ,.

*Definition 3.4. *Let be a Menger space with for all and let be a self mapping of . We will say that is a mapping of type if there exists such that
where is a mapping, and is identity mapping.

The following example shows that has no fixed point, even though satisfies (*2) and (*3).

*Example 3.5. *Let , be defined by , and let be defined by
For each , let be defined by for all , where is the same as in Definition 2.1, and is a usual metric on . Then, since for all , we have for all and . Thus,
for all and , which implies (*2). It is easy to see that there exists such that
which implies (*3). But has no fixed point.

*Remark 3.6. *It follows from Example 3.5 that must satisfy (*2), (*3), and other conditions in order to have fixed point of .

The following is ΔiriΔ-type fixed point theorem which is generalization of ΔiriΔβs fixed point theorems [6, 9].

Theorem 3.7. *Let be a Menger space with continuous norm and for all , let be a self-mapping on satisfying (*2) and (*3). Let be orbitally complete. Suppose that is a mapping such that *(i)* for all and , where is identity mapping, *(ii)* and are strictly increasing and onto mappings, *(iii)* for each , where is -time repeated composition of with itself.** Then, *(a)* for all , , and , *(b)* for all , and , *(c)* is Cauchy sequence for each , where*(d)* has a unique fixed point in .*

*Proof. *Let , and be arbitrary. By Definition 2.2 and Definition 2.7, clearly, we have which implies (a). From (i), (ii), and (*2), we have
By virtue of (i), (ii), (3.8), and (a), we obtain
By repeating application of (3.9), we have
Since converges to 1 when tends to infinity, it follows that
On account of (a) and (3.11), we have for , and , which implies (b). To prove (c), let and be two positive integers with , , and let be any positive real number. By (*2) and (b), we have
In terms of (i), (ii), and (3.13), we get
By repeating the same method as in (3.13) and (3.14), we have
On account of (iii), (*3), (b), (3.15), and Definition 2.2, we have
It follows from (3.15) and (3.16) that
This implies that is a Cauchy sequence for . This is the proof of (c). Since is orbitally complete, and is a Cauchy sequence for , has a limit in . To prove (d), let us consider the following inequality;
Since , from (3.18), we get
In terms of (3.19), (i), (ii), (iii), and Definition 2.2, we deduce that , that is, is a fixed point of . To prove uniqueness of a fixed point of , let be another fixed point of . Then . Putting and in (*2), we get
which gives . Thus, is a unique fixed point of , which implies (d).

Corollary 3.8 (see [6]). *let be a quasicontraction on a metric space , that is, there exists such that
**
Suppose that is orbitally complete. Then, has a unique fixed point in .*

*Proof. *Define by for all and for all and , where and are the same as in Definition 2.1. Let be defined by for all . Let be defined by
Then from Lemma 2.8, is a orbitally complete Menger space. It follows from (3.21), Lemma 2.8, Lemma 2.9, and [6, lemma 2] that all conditions of Theorem 3.7 are satisfied. Therefore, has a unique fixed point in .

Now we shall present an example to show that all conditions of Theorem 3.7 are satisfied but condition (3.21) in Corollary 3.8 and condition (*1) in Theorem A are not satisfied.

*Example 3.9. *Let be the closed interval with the usual metric and and be mappings defined as follows:
Define by
for all and , where and are the same as in Definition 2.1 and Definition 2.2. Let be defined by for all . Then from Lemma 2.8, is a orbitally complete Menger space, and is continuous function on which satisfy (i), (ii), and (iii). Clearly satisfies (*3). To show that condition (*2) is satisfied, we need to consider several possible cases.

*Case 1. *Let . Then

*Case 2. *Let and . Then

*Case 3. *Let and . Then

*Case 4. *Let . Then, by simple calculation,

*Case 5. *Let and . Then
Thus,

*Case 6. *Let . Then,
Hence, we obtain
where
From (3.25) and (3.34), we have
for all and , which implies (*2). Therefore, all hypotheses of Example 3.9 satisfy that of Theorem 3.7. Hence, has a unique fixed point 0 in . On the other hand, let be any fixed number. Then, for and with , we have
Thus,
which shows that does not satisfy (3.21).

Finally, in above Example 3.9, we shall show that does not satisfy (*1). In fact, we need to show that there are and such that . Let , , and Then, , β, and . Hence, and . Thus, . Therefore, Theorem 3.7 is a proper extension of Theorem A and Corollary 3.8.

#### 4. Caristi-Type Fixed Point Theorems

The following Lemma plays an important role to prove Caristi-type fixed point theorem which is generalization of Caristiβs fixed point theorem [7].

Lemma 4.1. *Let be a Menger space with continuous norm and for all , and let be the same as in Definition 2.1. Suppose that and are mappings satisfying the following conditions: *(1)* for all , *(2)* is a proper function which is bounded from below, *(3)*for any sequence in satisfying **there exists such that *(4)*for any with , there exists such that **where is the set of all distribution functions,**is defined by for all . Then, there exists such that .*

*Proof. *Suppose that
For each , let
Then, by (4), (4.6), and (4.7) is nonempty for each . From (1) and (4.7), we obtain
For each , let
Choose with . Then from (4.8) and (4.9), there exists a sequence in such that for all
In virtue of (4.7), (4.9), and (4.10), we have
for all . In view of (4.11), is a nonincreasing sequence of real numbers, and so it converges to some . Therefore, due to (4.12),
Combining (1) and (4.11), we get
On account of (4.13) and (4.14), we have
Thus, by virtue of (3), (4.13), (4.14), and (4.15), there exists such that
Using (4.14), (4.17), and (4.18), we obtain
Combining (4.7), (4.9), and (4.19), it follows that and, hence,
Taking the limit in inequality (4.20) when tends to infinity, we have
In terms of (4.13), (4.17), and (4.21), we deduce that
On the other hand, from (4), (4.6), (4.7), and (4.16), we have the following property:
In terms of (4.7), (4.8), (4.9), (4.20), and (4.23), we deduce that
In view of (4.7), (4.13), (4.22), (4.23), and (4.25), we have
Due to (4), (4.22), (4.23), and (4.26), we have the following:
By virtue of (4.27), we obtain
By repeating the application of inequality (4.28), we get
In terms of (4.29), we deduce that converges to 1 as and, hence,
From (4.30) and Definition 2.2, we have . This is a contradiction from (4.23). Therefore, there exists such that

Theorem 4.2. * Let be a metric space and let , and satisfy conditions (1), (2), and (3) in Lemma 4.1. Suppose that for any with , there exists such that
**
for all and some . Thus, there exists such that
*

*Proof. *The proof follows from Lemma 4.1 by considering the induced Menger space , where and

Corollary 4.3 (see [12]). *Let be a complete metric space, and let be a proper lower semicontinuous function, bounded from below. Assume that for any with , there exists with and . Then there exists such that .*

*Proof. *Let be a complete metric space, and let ,, , and be mappings such that
Then, all conditions of Corollary 4.3 satisfy all conditions of Lemma 4.1. Therefore, result of Corollary 4.3 follows from Lemma 4.1.

The following example shows that Theorem 4.2 is more general than Corollary 4.3.

*Example 4.4. *Let , and be the same as in Theorem 4.2. Let be the closed interval with the usual metric,ββ, and let ,ββ, and be mappings defined as follows:
Then, for any with , there exists such that
Let be a sequence of such that . Then, clearly conditions (1), (2), (3), and (4) in Lemma 4.1 are satisfied. Thus there exists such that
Therefore, all conditions of Theorem 4.2 are satisfied. Since is not lower semicontinuous at , and is not metric, Corollary 4.3 cannot be applicable.

Theorem 4.5. *Suppose that condition (4) in Lemma 4.1 is replaced with the following conditions. **For self-mapping on ,
**
Then, has a fixed point in .*

*Proof. *Suppose for all . Then by Lemma 4.1, there exists such that
Since , we have
By the same method as in proof of Lemma 4.1, it follows that . But this contradicts our assumption that for all . The proof of Theorem 4.5 is complete.

Theorem 4.6. *Let be a metric space, , and let , and be satisfied conditions (1), (2), and (3) in Lemma 4.1. Suppose that
**
Then, has a fixed point in .*

*Proof. *By method similar to Theorem 4.2, the result of Theorem 4.6 follows.

Corollary 4.7 (see [7]). *Let be a complete metric space, and let is a proper lower semicontinuous function bounded from below. Let be a mapping from into itself such that
**
Then, has a fixed point in .*

*Proof. *By the same method as in Corollary 4.3, the result of Corollary 4.7 follows.

The following example shows that all conditions of Theorem 4.6 are satisfied but not that of Corollary 4.7.

*Example 4.8. *Let , and be the same as in Example 4.4. Suppose that and are mappings defined as follows:
Then clearly, all conditions of Theorem 4.6 are satisfied but not that of Corollary 4.7, since is not lower semicontinuous at .

Natural question arises from Example 3.5.

*Question 1. *Whether Theorem 3.7 would remain true if (i), (ii), and (iii) in Theorem 3.7 are substituted by some suitable conditions?