Random fixed point theorem on a Ćirić-type contractive mapping and its consequence
© Saha and Ganguly; licensee Springer 2012
Received: 25 June 2012
Accepted: 6 November 2012
Published: 22 November 2012
The main purpose of this paper is to prove a random fixed point theorem in a separable Banach space equipped with a complete probability measure for a certain class of contractive mappings.
The main finding of this paper is the identification of some random fixed point theorems and the relevant application with appropriate supporting examples.
A random fixed point theorem is useful to determine the existence of a solution in a Banach space of a random nonlinear integral equation.
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, 2] and . 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  and Hanš [5, 6] first proved random fixed point theorems for random contraction mappings on separable complete metric spaces. The article by Bharucha-Reid  in 1976 attracted the attention of several mathematicians and led to the development of this theory. Špaček’s and Hanš’s theorems have been extended to multivalued contraction mappings by Itoh . A random version of Schaduer’s fixed point theorem on an atomic probability measure space has been provided by Mukherjee . The results of this work have been generalized by Bharucha-Reid [1, 7] on a general probability measure space. Itoh  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 based on Rothe  have been obtained by Sehgal and Waters . Kumam in a series of papers (see [12–25]) proved some remarkable results on random fixed point theorems. In a couple of papers  and , he along with his coauthor proved some random fixed point theorems for multivalued non-expansive non-self operators in the framework of Banach spaces satisfying inwardness conditions. In another paper, Kumam and Plubtieng  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. Saha , Saha and Debnath  in their works established some random fixed point theorems over a separable Banach space and a separable Hilbert space. On the other hand, Padgett  applied a random fixed point theorem to prove the existence of a solution in a Banach space of a random nonlinear integral equation. Achari , Saha and Dey  developed this new area of application.
Banach’s contraction principle  is one of the pivotal results of nonlinear analysis. It has been the source of metric fixed point theory and its significance rests in its vast applicability in different branches of mathematics. In the general setting of a complete metric space, this theorem runs as follows (see Theorem 2.1  or Theorem 1.2.2 ).
Theorem 1.1 (Banach’s contraction principle)
Let be a complete metric space, and be a mapping such that for each , .
Then f has a unique fixed point , and for each , .
On the other hand, Greguš  proved the following fixed point theorem.
for all , where , and . Then T has a unique fixed point.
During the eighties, many theorems which are closely related to the Greguš theorem have appeared in several literatures (see [35–38] and ). Also, Ćirić  dealt with a class of mappings (not necessarily continuous) which are defined on a metric space and proved the following fixed point theorem which is a double generalization of Greguš .
where , , for all . Then T has a unique fixed point.
In light of Theorem 1.3, the following theorem has been proved by Ćirić .
for all , where , , , and . Then T has a unique fixed point.
Ćirić also introduced several contractive operators on metric spaces and proved many fixed point theorems on such operators. Inspired by Ćirić’s contractive operators, many researchers have obtained fixed point theorems on Ćirić’s operators in different settings. In this context, Karapinar  proved some non-unique fixed point theorems on Ćirić-type contractive operators in cone metric spaces. Also Karapinar et al.  proved a fixed point theorem on a metric space for a class of maps that satisfy the Ćirić-type contractive condition.
In this paper, our main objective is to prove some random fixed point theorems in a separable Banach space equipped with a complete probability measure for a certain class of contractive mappings. The results are stochastic generalizations of deterministic fixed point theorems of Ćirić . The result obtained in this paper will also be useful in application to a random nonlinear integral equation. Also, we have introduced some appropriate supporting examples.
Let be a separable Banach space, where is a σ-algebra of Borel subsets of X, and let denote a complete probability measure space with measure μ and β be a σ-algebra of subsets of Ω. For more details, one can see Joshi and Bose .
Definition 2.1 A mapping is said to be an X-valued random variable if the inverse image under the mapping x of every Borel set B of X belongs to β, that is, for all .
Definition 2.2 A mapping is said to be a finitely-valued random variable if it is constant on each finite number of disjoint sets and is equal to 0 on . x is called a simple random variable if it is finitely valued and .
Definition 2.3 A mapping is said to be a strong random variable if there exists a sequence of simple random variables which converges to almost surely, that is, there exists a set with such that , .
Definition 2.4 A mapping is said to be a weak random variable if the function is a real-valued random variable for each , the space denoting the first normed dual space of X.
In a separable Banach space X, the notions of strong and weak random variables (see Corollary 1 of Joshi and Bose ) coincide, and in respect of such a space X, x is termed as a random variable.
We recall the following results.
Theorem 2.5 (see Theorem 6.1.2(a) of Joshi and Bose )
is a strong random variable.
If is a real-valued random variable and is a strong random variable, then is a strong random variable.
If is a sequence of strong random variables converging strongly to almost surely, i.e., if there exists a set with such that for every , then is a strong random variable.
Remark 2.6 If X is a separable Banach space, then every strong and also weak random variable is measurable in the sense of Definition 2.1.
Let Y be another Banach space. We also need the following definitions as cited in Joshi and Bose .
Definition 2.7 A mapping is said to be a random mapping if is a Y-valued random variable for every .
Definition 2.8 A mapping is said to be a continuous random mapping if the set of all for which is a continuous function of x has measure one.
Definition 2.9 A mapping is said to be demi-continuous at the if implies almost surely.
Theorem 2.10 (see Theorem 6.2.2 of Joshi and Bose )
Let be a demi-continuous random mapping where a Banach space Y is separable. Then, for any X-valued random variable x, the function is a Y-valued random variable.
Remark 2.11 (see )
Since a continuous random mapping is a demi-continuous random mapping, Theorem 2.5 is also true for a continuous random mapping.
We shall also recall the following definitions as seen in Joshi and Bose .
Definition 2.12 An equation of the type , where is a random mapping, is called a random fixed point equation.
Definition 2.13 Any mapping which satisfies the random fixed point equation almost surely is said to be a wide sense solution of the fixed point equation.
Definition 2.14 Any X-valued random variable which satisfies is said to be a random solution of the fixed point equation or a random fixed point of F.
Remark 2.15 A random solution is a wide sense solution of the fixed point equation. But the converse is not necessarily true. This is evident from the following example as found under Remark 1 in the work of Joshi and Bose .
Example 2.16 Let X be the set of all real numbers and let E be a non-measurable subset of X. Let be a random mapping defined as for all .
is a wide sense solution of the fixed point equation without being a random fixed point of F.
3 Main results
for all , where , , are real-valued random variables such that , , almost surely.
Then there exists a unique random fixed point of T in X.
Since S is dense in X, given (), there exist such that ; .
We now examine the following cases.
Let , then .
So, for each , is a deterministic operator due to Ćirić . Hence, T has a unique fixed point in X. □
then T has a unique random fixed point in X.
Proof Set .
Since , the relation (3.23), (3.24) and (3.25) would imply Theorem 3.1 with . Therefore, we can apply Theorem 3.1 and consequently T has a unique random fixed point in X. □
We now give a couple of examples in support of Theorem 3.1 and Theorem 3.2.
Example 3.3 Let with the usual norm of reals.
Consider and let β be a σ-algebra of Lebesgue measurable sets of .
Define by , where and .
By a routine check-up, we see that the condition of Theorem 3.1 is satisfied whenever , and . The function with is a unique random fixed point of T.
By considering E and Ω as above, we take , and . We see that condition (3.23) of Theorem 3.2 is satisfied and the function with is the unique random fixed point of T.
Example 3.4 Let with the usual norm of reals. Let . β be a σ-algebra of Lebesgue measurable sets of ℝ.
Define by .
All the conditions of Theorem 3.1 and Theorem 3.2 are satisfied. In both of the cases, we see that the function with is the unique random fixed point of T.
4 Application to a random nonlinear integral equation
S is a locally compact metric space with a metric d on equipped with a complete σ-finite measure defined on the collection of Borel subsets of S;
, where ω is the supporting element of a set of probability measure space ;
is the unknown vector-valued random variable for each ;
is the stochastic free term defined for ;
is the stochastic kernel defined for t and s in S and
is a vector-valued function of and x.
We shall further assume that S is the union of a decreasing sequence of countable family of compact sets such that for any other compact set in S there is a which contains it (see ).
Moreover, is complete relative to this topology since is complete.
-a.e. Further, for almost all , will be continuous in t from S into .
where the integral is a Bochner integral. Moreover, we have that for each , and that is continuous in mean square by the Lebesgue dominated convergence theorem. So, .
Let B and D be two Banach spaces. The pair is said to be admissible with respect to a random operator if .
Lemma 4.3 (see )
The linear operator T defined by (4.2) is continuous from into itself.
If T is a continuous linear operator from into itself and are Banach spaces stronger than such that is admissible with respect to T, then T is continuous from B into D.
Remark 4.5 (see )
The operator T defined by (4.2) is a bounded linear operator from B into D.
We are now in a position to prove the following theorem.
B and D are Banach spaces stronger than such that is admissible with respect to the integral operator defined by (4.2);
Then by (4.5), (4.7), (4.9) and (4.11), we get .
Therefore, is a random contractive nonlinear operator on . Hence, by Theorem 3.1, there exists a random fixed point of , which is the random solution of equation (4.1). □
By routine calculation, it is easy to show that (4.3) is satisfied with , and .
Comparing with integral operator equation (4.2), we see that the norm of is .
Also, we see that . So, all the conditions of Theorem 4.6 are satisfied and hence there exists a random fixed point of the integral operator T satisfying (4.2).
Authors remain grateful to the honorable reviewers for their kind suggestions for improvement of our paper.
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