The application of fixed point theory in different branches of mathematics, statistics, engineering and economics relating to problems associated with approximation theory, theory of differential equations, theory of integral equations etc. has been recognized in the existing literature [1]-[3]. Progress in the study on fixed points of non-expansive mappings, contractive mappings in various spaces like a metric space, a Banach space, a fuzzy metric space, a cone metric space etc. has been saturated at large. After the initial impetus given by the Prague school of Probability in 1950s, considerable attention has been given to the study of random fixed point theorems. This arises because of the significance of fixed point theorems in probabilistic functional analysis and probabilistic models along with several applications. Issues relating to measurability of solutions, probabilistic and statistical aspects of random solutions have arisen due to the introduction of randomness. It is no denying the fact that random fixed point theorems are stochastic generalizations of classical fixed point theorems that have been described as deterministic results.

Špaček [4] and Hanš [5] first proved random fixed point theorems for random contraction mappings on separable complete metric spaces. The article by Fierro et al. [6] in 2009 attracted the attention of several mathematicians and led to the development of this theory. Špaček and Hanš’s theorems have been extended to multivalued contraction mappings by Itoh [7]. A random version of Schaduer’s fixed point theorem on an atomic probability measure space has been provided by Mukherjee [8]. Itoh [7] obtained random fixed point theorems with an application to random differential equations in Banach spaces. Several random fixed point theorems including random analogue of the classical results have been obtained by many authors [9]-[20]. Kumam [19] proved some random coincidence points and random common fixed point theorems for nonlinear multivalued random operators. They also proved the existence of a random coincidence point for a pair of reciprocally continuous and compatible single-valued and multivalued operators. Later, many scholars have obtained a lot of multi-valued and single valued operator of fixed point results [21]-[27].

Motivated and inspired by the above results, our main objective is to prove some random fixed point theorems in a separable complete metric space for a certain class of random multivalued contractive mappings. The results are stochastic generalizations of deterministic fixed point theorems of single value mappings. The result obtained in this paper will also be useful in application to a random nonlinear integral equation.

Remark 1.1 (Motivation for the contraction condition). The rational expressions m 1 , m 2 , m 3 in (3.2) are introduced to handle situations where the mappings A , B do not necessarily commute with S , T , and where the distances between image sets may vanish. These terms are inspired by the classical rational contraction inequalities of Ćirić and others, and they allow for a unified treatment of both single-valued and multivalued mappings under weak compatibility. The term m 3 captures the product of distances between cross-images, which is essential when the usual triangle inequality fails to provide a contraction. These expressions are not arbitrary; they represent a natural generalization of the standard rational contractive conditions in the random multivalued setting.

In order to make the paper self-contained, we state some important definitions and Lemmas relate to our paper.

Let ( Ω , Σ ) be a measurable space with Σ being a σ -algebra of subsets of Ω, and let ( X , d ) be a complete separable metric space. We denote by C B ( X ) the family of all nonempty closed bounded subset of X , and by H the Hausdorff metric on C B ( X ) induced by d , that is

for A , B ∈ C B ( X ) , where d ( x , E ) = inf { d ( x , y ) | y ∈ E } is the distance from x to E ⊂ X .

Definition 2.1 ([18]).A measurable operator ξ : Ω → X is said to be a random fixed point of an operator F : Ω × X → C B ( X ) if ξ ( ω ) ∈ F ( ω , ξ ( ω ) ) ,for all ω ∈ Ω .

Definition 2.2 ([18]).A measurable operator ξ : Ω → X is said to be a random coincidence point of g : Ω × X → X and F : Ω × X → C B ( X ) if

Definition 2.3 ([18]).A measurable operator ξ : Ω → X is said to be a random common fixed point of g : Ω × X → X and F : Ω × X → C B ( X ) if

Definition 2.4.The random mapping F : Ω × X → C B ( X ) and g : Ω × X → X are said to be weakly random compatible,if for all ω ∈ Ω and x ∈ X ,the condition g ( ω , x ) ∈ F ( ω , x ) implies

Remark 2.1.The notion of “weakly random compatible” is the random analogue of the deterministic “weakly compatible” condition. In this paper, both terms are used interchangeably in the context of random mappings, and they refer to the same compatibility condition.

Lemma 2.1 ([28]).A mapping T : Ω × X → C B ( X ) is called a multi-valued random operator if for every x ∈ X , T ( ⋅ , x ) is measurable. A mapping s : Ω → X is called a measurable selector of a measurable multifunction T : Ω × X → C B ( X ) if s is measurable and s ( ω ) ∈ T ( ω ) , for all ω ∈ Ω .

Lemma 2.2 ([28]).Let S , T : Ω → C B ( X ) be measurable mappings, there is a measurable selector u : Ω → X of S , then for any measurable real valued function α : Ω → ( 1 , ∞ ) , there exists measurable selector v : Ω → X of T such that d ( u ( ω ) , v ( ω ) ) ≤ α ( ω ) H ( S ( ω ) , T ( ω ) ) .

Lemma 2.3 ([6]).A multivalued operator T : Ω × X → C B ( X ) is called (Σ)-measurable, if for any open subset B of X , T − 1 ( B ) = { ω ∈ Ω : T ( ω ) ∩ B ≠ ∅ } ∈ Σ . The set F ( ω ) be the fixed point set of T ( ω , ⋅ ) , if F ( ω ) : = { x ∈ T ( ω , x ) } ≠ ∅ . Then T has a random fixed point.

Recall that T : Ω × X → C B ( X ) is continuous, if for each fixed ω ∈ Ω , the operator T : ( ω , ⋅ ) → C B ( X ) is continuous.

Throughout this paper, we assume that R + = [ 0 , + ∞ ) , R = ( − ∞ , + ∞ ) , N 0 = { 0 } ∪ N , where N denotes the set of all positive integers and we give the following definition.

Definition 2.5. Ψ = { ψ : ℝ + → ℝ + is semi-continuous upper on ℝ + \ { 0 } , if and only if lim n → ∞ r n = 0 and ψ ( t ) < t for each t > 0 } , Ψ 1 = { ψ : ℝ + → ℝ + does not decrease on ℝ + , ψ ( t ) < t and ∑ n = 1 ∞ ψ n ( t ) < ∞ for each t > 0 } .

Now we show two random common fixed point theorems for random multivalued contractive mappings in metric spaces.

Theorem 3.1.Let ( X , d ) be a complete separable metric space, let ( Ω , Σ ) be a measurable space, and let S , T : Ω × X → C B ( X ) and A , B : Ω × X → X be mappings such that

(i) A ( ω , ⋅ ) , B ( ω , ⋅ ) , T ( ω , ⋅ ) , S ( ω , ⋅ ) are continuous for all ω ∈ Ω ;

(ii) A ( ⋅ , x ) , B ( ⋅ , x ) , T ( ⋅ , x ) , S ( ⋅ , x ) are measurable for all x ∈ X ;

(iii) S ( Ω × X ) ⊆ B ( Ω × X ) and T ( Ω × X ) ⊆ A ( Ω × X ) ;

(iv) the pairs { A , S } and { B , T } are weakly random compatible;

(v) one of A ( Ω × X ) , B ( Ω × X ) , S ( Ω × X ) and T ( Ω × X ) is complete and

Then A , B , S and T have a unique random common fixed point in X .

Proof. Let Θ = { ξ : Ω → X } be a family of measurable mappings. Define a function g : Ω × X → R + as follows:

Since x → T ( ω , x ) is continuous for all ω ∈ Ω , we conclude that g ( ω , ⋅ ) is continuous for all ω ∈ Ω . Also, since ω → T ( ω , x ) is measurable for all x ∈ X , we conclude that g ( ⋅ , x ) is measurable (see Wagner ([29], page 868, 31, 32)) for all ω ∈ Ω . Thus g ( ω , x ) is the Caratheodory function. Therefore, if ξ : Ω → X is a measurable mapping, then ω → g ( ω , ξ ( ω ) ) is also measurable (see [29]). Let ξ 0 ∈ Θ be arbitrary. Then the multifunction F , G : Ω → C B ( X ) defined by F ( ω ) = B ( ω , ξ 0 ( ω ) ) , G ( ω ) = A ( ω , ξ 0 ( ω ) ) is measurable [30].

Now we will construct sequences of measurable mappings { y n ( ω ) } , { x n ( ω ) } ∈ Θ , the sequence

By the Kuratowski-Ryll-Nardzewski selector theorem [28], there exists a measurable selector y 1 ( ω ) ∈ S ( ω , x 0 ( ω ) ) . Then, using Lemma 2.2 with α ( ω ) ≡ 2 , we obtain a measurable selector y 2 ( ω ) ∈ T ( ω , x 1 ( ω ) ) such that

Since S ( Ω × X ) ⊆ B ( Ω × X ) and T ( Ω × X ) ⊆ A ( Ω × X ) , there exist measurable mappings x 1 , x 2 ∈ Θ such that

Continuing this process inductively, and at each step applying Lemma 2.2 to select the next point with a controlled distance estimate, we construct sequences of measurable mappings { y n ( ω ) } , { x n ( ω ) } ⊂ Θ satisfying

The distance d ( y n , y n + 1 ) is then controlled by the Hausdorff metric via Lemma 2.2, which justifies the subsequent inequalities involving H ( S , T ) . Put d n = d ( y n ( ω ) , y n + 1 ( ω ) ) for each n ∈ ℕ .

Firstly, we show that A ( ω , x ) , B ( ω , x ) , S ( ω , x ) and T ( ω , x ) have at most a random common fixed point in X , for all ω ∈ Ω , x ∈ X . Suppose that u ( ω ) and v ( ω ) are two different random common fixed points of A ( ω , x ) , B ( ω , x ) , S ( ω , x ) and T ( ω , x ) . It follows from (3.1), (3.2), we have

and

which is a contradiction. Hence A ( ω , x ( ω ) ) , B ( ω , x ( ω ) ) , S ( ω , x ( ω ) ) and T ( ω , x ( ω ) ) have at most a random common fixed point in X .

Secondly we show that A ( ω , x ( ω ) ) , B ( ω , x ( ω ) ) , S ( ω , x ( ω ) ) and T ( ω , x ( ω ) ) have a random common fixed point A ( ω , a ( ω ) ) ∈ X , if there exist a ( ω ) , b ( ω ) ∈ X satisfying

Assume that (3.4) holds for some measurable mappings a ( ω ) , b ( ω ) ∈ Θ , i.e.,

Put c ( ω ) = A ( ω , a ( ω ) ) ∈ X . Note that (iv) implies that

and

Suppose that c ( ω ) ∉ T ( ω , c ( ω ) ) . In view of (3.1), (3.2), (3.4), (3.5) and ψ ∈ Ψ , we infer that

and

which is impossible. Consequently, c ( ω ) = B ( ω , c ( ω ) ) ∈ T ( ω , c ( ω ) ) . Similarly we conclude that c ( ω ) = A ( ω , c ( ω ) ) ∈ S ( ω , c ( ω ) ) . That is, c ( ω ) is a random common fixed point of A ( ω , c ( ω ) ) , B ( ω , c ( ω ) ) , S ( ω , c ( ω ) ) and T ( ω , c ( ω ) ) .

Thirdly we show that (3.4) holds for some a ( ω ) , b ( ω ) ∈ X . In order to prove (3.4), we have to consider three possible cases as follows.

Case1. There exists n ∈ N satisfying d 2 n 0 = 0 . We claim that d 2 n 0 + 1 > 0 , otherwise d 2 n 0 + 1 = 0 . Using (3.1)-(3.3) and ψ ∈ Ψ , we deduce that

(10)

and

which is a contradiction. Hence d 2 n 0 + 1 = 0 . It follows that

and

Put a ( ω ) = x 2 n 0 ( ω ) and b ( ω ) = x 2 n 0 + 1 ( ω ) . It is easy to see that (3.4) holds and y 2 n 0 ( ω ) is a random common fixed point of A ( ω , x ( ω ) ) , B ( ω , x ( ω ) ) , S ( ω , x ( ω ) ) and T ( ω , x ( ω ) ) .

Case2. There exists n ∈ N satisfying d 2 n 0 − 1 = 0 . As in the proof of Case1, we infer similarly that (3.4) holds for a ( ω ) = x 2 n 0 ( ω ) and b ( ω ) = x 2 n 0 − 1 ( ω ) and y 2 n 0 − 1 ( ω ) is a random common fixed point of A ( ω , x ( ω ) ) , B ( ω , x ( ω ) ) , S ( ω , x ( ω ) ) and T ( ω , x ( ω ) ) .

Case3. y n ( ω ) ≠ y n + 1 ( ω ) for all n ∈ N . Now we claim that d 2 n ≤ d 2 n − 1 for all n ∈ N . Suppose that d 2 n ≤ d 2 n − 1 for some n ∈ N . By virtue of (3.1), (3.2) and ψ ∈ Ψ , we arrive at