Abstract
Purpose
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.
Results
The main finding of this paper is the identification of some random fixed point theorems and the relevant application with appropriate supporting examples.
Conclusion
A random fixed point theorem is useful to determine the existence of a solution in a Banach space of a random nonlinear integral equation.
MSC: 47H10, 60H25.
Keywords:
complete probability measure space; random variable; random solution; random fixed point equation; Bochner integral1 Introduction
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 [3]. Progress in the study on fixed points of nonexpansive 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 [5,6] first proved random fixed point theorems for random contraction mappings on separable complete metric spaces. The article by BharuchaReid [7] 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 [8]. A random version of Schaduer’s fixed point theorem on an atomic probability measure space has been provided by Mukherjee [9]. The results of this work have been generalized by BharuchaReid [1,7] on a general probability measure space. Itoh [8] 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 [10] have been obtained by Sehgal and Waters [11]. Kumam in a series of papers (see [1225]) proved some remarkable results on random fixed point theorems. In a couple of papers [17] and [19], he along with his coauthor proved some random fixed point theorems for multivalued nonexpansive nonself operators in the framework of Banach spaces satisfying inwardness conditions. In another paper, Kumam and Plubtieng [13] 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 singlevalued and multivalued operators. Saha [26], Saha and Debnath [27] in their works established some random fixed point theorems over a separable Banach space and a separable Hilbert space. On the other hand, Padgett [28] applied a random fixed point theorem to prove the existence of a solution in a Banach space of a random nonlinear integral equation. Achari [29], Saha and Dey [30] developed this new area of application.
Banach’s contraction principle [31] 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 [32] or Theorem 1.2.2 [33]).
Theorem 1.1 (Banach’s contraction principle)
Letbe a complete metric space, andbe a mapping such that for each, .
Thenfhas a unique fixed point, and for each, .
On the other hand, Greguš [34] proved the following fixed point theorem.
Theorem 1.2LetXbe a Banach space, Cbe a closed convex subset ofXandbe a mapping satisfying
for all, where, and. ThenThas a unique fixed point.
During the eighties, many theorems which are closely related to the Greguš theorem have appeared in several literatures (see [3538] and [39]). Also, Ćirić [40] 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 [34].
Theorem 1.3LetCbe a closed convex subset of a complete convex metric spaceXandbe a mapping satisfying
where, , for all. ThenThas a unique fixed point.
In light of Theorem 1.3, the following theorem has been proved by Ćirić [40].
Theorem 1.4LetCbe a closed convex metric spaceXandbe a mapping satisfying
for all, where, , , and. ThenThas 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 [41] proved some nonunique fixed point theorems on Ćirićtype contractive operators in cone metric spaces. Also Karapinar et al.[42] proved a fixed point theorem on a metric space for a class of maps that satisfy the Ćirićtype 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 [40]. 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.
In order to make the paper selfcontained, we state some important definitions and an example that are available in Joshi and Bose [3] and Debnath and Mikusinski [2].
2 Preliminaries
Let be a separable Banach space, where is a σalgebra of Borel subsets of X, and let denote a complete probability measure space with measure μ and β be a σalgebra of subsets of Ω. For more details, one can see Joshi and Bose [3].
Definition 2.1 A mapping is said to be an Xvalued 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 finitelyvalued 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 realvalued 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 [3]) 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 [3])
Letbe strong random variables andα, βbe constants. Then the following statements hold:
(a) is a strong random variable.
(b) Ifis a realvalued random variable andis a strong random variable, thenis a strong random variable.
(c) Ifis a sequence of strong random variables converging strongly toalmost surely, i.e., if there exists a setwithsuch thatfor every, thenis 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 [3].
Definition 2.7 A mapping is said to be a random mapping if is a Yvalued 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 demicontinuous at the if implies almost surely.
Theorem 2.10 (see Theorem 6.2.2 of Joshi and Bose [3])
Letbe a demicontinuous random mapping where a Banach spaceYis separable. Then, for anyXvalued random variablex, the functionis aYvalued random variable.
Remark 2.11 (see [3])
Since a continuous random mapping is a demicontinuous 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 [3].
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 Xvalued 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 [3].
Example 2.16 Let X be the set of all real numbers and let E be a nonmeasurable subset of X. Let be a random mapping defined as for all .
In this case, the realvalued function , defined as for all , is a random fixed point of F. However, the realvalued function defined as
is a wide sense solution of the fixed point equation without being a random fixed point of F.
3 Main results
Theorem 3.1LetXbe a separable Banach space andbe a complete probability measure space. Letbe a continuous random operator such that for, Tsatisfies
for all, where, , are realvalued random variables such that, , almost surely.
Then there exists a unique random fixed point ofTinX.
Let S be a countable dense subset of X. We now prove that
Since S is dense in X, given (), there exist such that ; .
Note that
We now examine the following cases.
Case I:
Suppose
Now
Using (3.2), (3.3), (3.7), we get by routine calculation
Since for a particular , is a continuous function of x, so for any , there exists () such that
and
Now choose
and
For such a choice of , by (3.8), we get
As is arbitrary, it follows that
Case II:
Suppose
Now
Using (3.2), (3.4), (3.5), (3.14), by routine calculation, we get
Since for a particular , is a continuous function of x, by using (3.9), (3.10), (3.11), (3.12) and for such a choice of , , we get by (3.15)
As is arbitrary, it follows that
Case III:
Suppose
Now
Using (3.3), (3.6), (3.17), by a routine checkup, we get
Since for a particular , is a continuous function of x, by using (3.9), (3.10), (3.11), (3.12) and for such a choice of , , we get from relation (3.18)
As is arbitrary, it follows that
Case IV:
Suppose
Now
Using (3.4), (3.5), (3.6) and (3.20), we get by a routine checkup
Since for a particular , is a continuous function of x, by using (3.9), (3.10), (3.11), (3.12) and for such a choice of , , we get by (3.21)
As is arbitrary, it follows that
Combining (3.13), (3.16), (3.19) and (3.22), we get
Also,
Thus,
So, for each , is a deterministic operator due to Ćirić [40]. Hence, T has a unique fixed point in X. □
Theorem 3.2LetXbe a separable Banach space andbe a complete probability measure space. Letbe a continuous random operator such that for, Tsatisfies
for all, where, , are realvalued random variables such that
and
thenThas a unique random fixed point inX.
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 σalgebra of Lebesgue measurable sets of .
By a routine checkup, 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 σalgebra of Lebesgue measurable sets of ℝ.
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
Here we apply Theorem 3.1 to prove the existence of a solution in a Banach space of a random nonlinear integral equation of the following form:
where
(i) S is a locally compact metric space with a metric d on equipped with a complete σfinite measure defined on the collection of Borel subsets of S;
(ii) , where ω is the supporting element of a set of probability measure space ;
(iii) is the unknown vectorvalued random variable for each ;
(iv) is the stochastic free term defined for ;
(v) is the stochastic kernel defined for t and s in S and
(vi) is a vectorvalued function of and x.
The integral in equation (4.1) is interpreted as a Bochner integral [43].
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 [44]).
Definition 4.1 We define the space to be the space of all continuous functions from S into with the topology of uniform convergence on compact sets of S, that is, for each fixed , is a vectorvalued random variable such that
It may be noted that is a locally convex space (see [43]) whose topology is defined by a countable family of seminorms given by
Moreover, is complete relative to this topology since is complete.
We further define to be the Banach space of all bounded continuous functions from S into with the norm
Here the space is the space of all secondorder vectorvalued functions defined on S which are bounded and continuous in mean square. We will consider the function and to be in the space with respect to the stochastic kernel. We assume that for each pair and denote the norm by
Let us suppose that is such that is integrable with respect to s for each and in , and let there exist a realvalued function G defined a.e. on S, so that is integrable so that for each pair ,
a.e. Further, for almost all , will be continuous in t from S into .
We now define the random integral operator T on by
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, .
Definition 4.2 (see [29] and [45])
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 [28])
The linear operatorTdefined by (4.2) is continuous frominto itself.
IfTis a continuous linear operator frominto itself andare Banach spaces stronger thansuch thatis admissible with respect toT, thenTis continuous fromBintoD.
Remark 4.5 (see [28])
The operator T defined by (4.2) is a bounded linear operator from B into D.
A random solution of equation (4.1) will mean a function in which satisfies equation (4.1) μa.e.
We are now in a position to prove the following theorem.
Theorem 4.6We consider the stochastic integral equation (4.1) subject to the following conditions:
(a) BandDare Banach spaces stronger thansuch thatis admissible with respect to the integral operator defined by (4.2);
(b) is an operator from the setinto the spaceBsatisfying
for, , , are realvalued random variables whereandwithalmost surely, and
Then there exists a unique random solution of (4.1) in, providedand
Proof Define the operator from into D by
Now
Then from (4.3) of this theorem,
Suppose
Hence,
So, by (4.4)
Again, suppose
So,
Therefore, by (4.6)
Also, suppose
Then
Therefore, by (4.8)
Suppose
So,
Therefore, by (4.10)
Then by (4.5), (4.7), (4.9) and (4.11), we get .
Then for . We have by condition (b)
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). □
Example 4.7 Consider the following nonlinear stochastic integral equation:
Comparing with (4.1), we see that
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).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both the authors are jointly responsible for the present research work and have equally contributed to the research work. Both the authors have read and approved the final manuscript.
Acknowledgements
Authors remain grateful to the honorable reviewers for their kind suggestions for improvement of our paper.
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