Abstract

In this paper, we present the random (Stochastic) version of Banach contraction principle and some of its generalizations in setting of rectangular metric spaces.

1. Introduction and Preliminaries

To develop an approach for solving probabilistic models, probabilistic functional analysis has emerged as one of the indispensable mathematical disciplines and tools. The theory of random operators is of interest in its own right as a stochastic generalization of deterministic operator theory. Operator theory is that branch of stochastic which deals with the study of operator-valued random variables with their properties. Stochastic generalizations of classical or deterministic fixed point theorems is Random fixed point theorems and are required for the theory of random equations, random matrices, random partial differential equations, and various classes of random operators arising in physical systems ^{1, 2}. In 1950, the concept of random fixed point was initiated by the Prague school of probabilistic. The interest in this area was enhanced after publication of the survey paper by Bharucha Reid ³. In recent years, the study of random fixed points have attracted much attention, some of the recent literatures in random fixed point may be noted in ^{4, 5}. The classical concept of a metric space has been generalized in different directions by partly changing the conditions of the metric. Among these generalizations, Branciari ⁶ introduced the rectangular metric space, by replacing the sum at the right hand side of the triangle inequality by a three-term expression. Later, Azam and Arshad ⁷ obtained sufficient conditions for existence of a unique fixed point of Kannan type mappings in the framework of rectangular metric spaces. Following this trend, a number of authors focused on rectangular metric spaces and proved the existence and uniqueness of a fixed point for certain type of mappings ^{8, 9, 10, 11, 12, 13}.

Definition 1.1 ³ Let ℳ be a nonempty set and let Δ: ℳ × ℳ → [0,+ ∞) be a mapping such that for all α, β ∈ ℳ and for all distinct points each of them different from α and β, one has

(i) Δ(α, β) = 0 if and only if α = β,

(ii) Δ(α, β) = Δ( β, α),

(iii) Δ(α, β) ≤ Δ(α, 𝑢) + Δ(𝑢, v) + Δ(v, β) (the rectangular inequality).

Then, the map Δ is called rectangular metric. The pair (ℳ, Δ) is called a rectangular metric space. In this paper, we abbreviate a “rectangular metric space” with RMS.

Definition 1.2 ³ Let (, Δ) be a RMS.

(i) A sequence is called RMS convergent to α ∈ if and only if → 0 as n → +∞. In this case, we use the notation

(ii) A sequence in is called a RMS Cauchy if and only if for each 𝜖 > 0, there exists a natural number such that for all

(iii) A RMS (, Δ) is called a RMS complete if every RMS Cauchy sequence is RMS convergent in .

Definition 1.3 Let be a rectangular metric space and be a function, where is a non-empty set. Then function is said to be a random fixed point of the function if for all in

Example 1.4 Let us take Define as

Then clearly, a mapping defined as for every is unique random fixed point of

2. Main Results

In this section we shall prove random version of Banach contraction principle and its generalizations in the framework of rectangular metric spaces.

Theorem 2.1. Let be a complete RMS and let be a function satisfying:

(2.1)

where [0, 1).

Then has a unique random fixed point.

Proof. We define a sequence of functions as where is arbitrary function for ∈ ℝ for n= 0, 1, 2, 3, …

Now we shall show that is a rectangular Cauchy sequence.

Suppose that

(2.2)

If then for some

Now

This implies that

Similarly,

Hence for all

Then is a Cauchy sequence in

Let for all then from inequality (2.1), we get

(2.3)

and

(2.4)

Using inequality (2.3) and (2.4), we have

(2.5)

Now

(2.6)

From equation (2.5) and (2.6), we get

(2.7)

Now

(2.8)

Using equation (2.8), we have

(2.9)

From equation (2.8) and (2.9), we get

(2.10)

For sequence we consider in two cases:

Case 1. If i.e odd for

Then by using rectangular inequality and equation (2.7), we get

Hence

(2.11)

Case 2. If i.e even for

(2.12)

From equation (2.11) and (2.12), we have

(2.13)

Taking for all

Hence is rectangular Cauchy sequence in

Since is complete so such that

Now, we shall show that is fixed point of i.e

For any we have

Since as

This implies that i.e.

Now we will prove uniqueness of random fixed point.

Let be another random fixed point of

So and Now

This implies that i.e. is a unique random fixed point.

Theorem 2.2. Let be a complete RMS and be a function satisfying:

(2.14)

for all where and

Then has a unique random fixed point.

Proof. Choose

Put …,

We shall show that is a Cauchy rectangular sequence.

Suppose that

(2.15)

If then for some Now

This implies that i.e.

Similarly,

Hence for all

Therefore is a Cauchy rectangular sequence in

Now assume for all then from (2.14), we have

for all

which implies that

(2.16)

where

Also,

(2.17)

for all

which implies that

where

Using (2.16) and (2.17), we get

(2.18)

for all

and

(2.19)

From (2.18) and (2.19), we deduce that

(2.20)

for all

From (2.20) and the fact that

for all

Now

(2.21)

for all

From (2.20) and (2.21), we have

(2.22)

For sequence we consider in two cases:

Case 1. If i.e odd for

Then by rectangular property and (2.20), we have

Hence

(2.23)

for all

Case 2. If i.e even for

Then by rectangular property, (2.20) and (2.23), we have

(2.24)

for all

From equation (2.23) and (2.24), we get

(2.25)

for all

Taking we get for all

Hence is rectangular Cauchy sequence in

Since is complete so there exists such that

Now, we shall show that is fixed point of i.e

For any we have

Since as

Suppose be another random fixed point of

So and

This implies that

i.e. is a unique random fixed point of

Theorem 2.3. Let be a complete RMS and be a function satisfying the following condition:

(2.26)

for all where and

Then has a unique random fixed point.

Proof. Choose

Put ,…,

If for all then

i. For and then there exists such that

where

ii. whenever n ≠ m.

iii. Δ(O( n)) ≤ [Δ(( ) + Δ].

iv. Δ(O(∞)) ≤ [Δ(( ) + Δ].

v. Where Δ(O(∞))=

vi. is a Cauchy sequence.

Proof. (i) Let and

Using (2.1), for any with we have that

This implies that

Since

there exists with such that

(2.27)

(ii) Suppose that for some

Then, by (2.26) we obtain that

This implies that Δ(O) = 0.

However, this is impossible because

Therefore, whenever

(iii) Let Then, using equation (2.26) and (2.27), we get

This implies that

(2.28)

(iv) Note that =Δ(O).

Thus, from (2.28) we see that

(v) For any,

which tends to

(vi) Therefore, is a rectangular Cauchy sequence.

Since is complete therefore, tends to some

Suppose then

Taking

which is a contradiction with < 1 and

Thus, we prove that that is is fixed point of

Suppose that and are two fixed point of then

Now

This implies that i.e. has unique random fixed point.

Corollary 2.4. Let be a complete RMS and be a function satisfying:

(2.29)

for all where and

Then has a unique random fixed point.

Example 2.5. Let where and L = [1,2].

Define the rectangular metric on as follows:

and (α, β) = |α – β| if α, β ∈ L or α ∈ K, β ∈ L, or α ∈ L, β ∈ K. It is clear that is a RM.

Define by

Then satisfies the condition of Theorem 2.1 and defined by for all is random unique fixed point of

3. Conclusions

If we take ℝ to be singleton set in Theorem 2.1, 2.2 and 2.3 we get Banach, Kannan and Chatterjea type contraction in RMS. In this paper, we give a complete solution to the Problem given in ¹⁴ if we take ℝ to be singleton set.

Acknowledgments

The authors are highly appreciated the referees efforts of this paper who helped us to improve it in several places.

Conflicts of Interest

The authors declare no conflict of interest.

References