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Abstract
The purpose of this paper is to introduce the random Jungck–Mann-type and the random Jungck–Ishikawa-type iterative processes. We prove some convergence and stability results for these random iterative processes for certain random operators. Furthermore, we apply our results to study random non-linear integral equation of the Hammerstein type. Our results generalize, extend and unify several well-known deterministic results in the literature. Moreover, our results generalize recent results of Okeke and Abbas and Okeke and Kim .
Public Interest Statement
Probabilistic functional analysis has attracted the attention of several well-known mathematicians due to its applications in pure mathematics and applied sciences. Studies in random methods have revolutionized the financial markets. Moreover, random fixed point theorems are required for the theory of random equations, random matrices, random partial differential equations and many classes of random operators. In this paper, we introduce the random Jungck–Mann-type and the random Jungck–Ishikawa-type iterative processes. We establish some random fixed point theorems. Furthermore, we prove the existence of a solution of a random non-linear integral equation of the Hammerstein type in a Banach space.
1. Introduction
Real-world problems are embedded with uncertainties and ambiguities. To deal with probabilistic models, probabilistic functional analysis has emerged as one of the momentous mathematical discipline and attracted the attention of several mathematicians over the years in view of its applications in diverse areas from pure mathematics to applied sciences. Random non-linear analysis, an important branch of probabilistic functional analysis, deals with the solution of various classes of random operator equations and related problems. Of course, the development of random methods has revolutionized the financial markets. Random fixed point theorems are stochastic generalizations of classical or deterministic 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 (see, Joshi & Bose, Citation1985; Okeke & Abbas, Citation2015; Okeke & Kim, Citation2015; Zhang, Citation1984). Random fixed point theory was initiated in 1950s by Prague school of probabilists. Spacek (Citation1955) and Hans (Citation1961) established a stochastic analogue of the Banach fixed point theorem in a separable complete metric space. Itoh (Citation1979) generalized and extended Spacek and Han’s theorem to a multi-valued contraction random operator. The survey article by Bharucha-Reid (Citation1976), where he studied sufficient conditions for a stochastic analogue of Schauder’s fixed point theorem for random operators, gave wings to random fixed point theory. Now this area has become a full-fledged research area and many interesting techniques to obtain the solution of non-linear random system have appeared in the literature (see, Arunchai & Plubtieng, Citation2013, Beg & Abbas, Citation2006,Citation2007,Citation2010; Chang, Cho, Kim, & Zhou, Citation2005; Itoh, Citation1979; Joshi & Bose, Citation1985; Papageorgiou, Citation1986; Shahzad & Latif, Citation1999; Spacek, Citation1955; Xu, Citation1990; Zhang, Citation1984; Zhang, Wang, & Liu, Citation2011).
Papageorgiou (Citation1986) established an existence of random fixed point of measurable closed and non-closed-valued multifunctions satisfying general continuity conditions and hence improved the results in Engl (Citation1976), Itoh (Citation1979) and Reich (Citation1978). Xu (Citation1990) extended the results of Itoh to a non-self-random operator T, where T satisfies weakly inward or the Leray–Schauder condition. Shahzad and Latif (Citation1999) proved a general random fixed point theorem for continuous random operators. As applications, they derived a number of random fixed point theorems for various classes of 1-set and 1-ball contractive random operators. Arunchai and Plubtieng (Citation2013) obtained some random fixed point results for the sum of a weakly strongly continuous random operator and a non-expansive random operator in Banach spaces.
Mann (Citation1953) introduced an iterative scheme and employed it to approximate the solution of a fixed point problem defined by non-expansive mapping where Picard iterative scheme fails to converge. Later in 1974, Ishikawa (Citation1974) introduced an iterative scheme to obtain the convergence of a Lipschitzian pseudocontractive operator when Mann’s iterative scheme is not applicable. Jungck (Citation1976) introduced the Jungck iterative process and used it to approximate the common fixed points of the mappings S and T satisfying the Jungck contraction. Singh et al. (Citation2005) introduced the Jungck–Mann iterative process for a pair of Jungck–Osilike-type maps on an arbitrary set with values in a metric or linear metric space. Khan et al. (Citation2014) introduced the Jungck–Khan iterative scheme for a pair of non-self-mappings and studied its strong convergence, stability and data dependence. Alotaibi, Kumar and Hussain (Citation2013) introduced the Jungck–Kirk–SP and Jungck–Kirk–CR iterative schemes, and proved convergence and stability results for these iterative schemes using certain quasi-contractive operators. Sen and Karapinar (Citation2014) investigated some convergence properties of quasi-cyclic Jungck-modified TS-iterative schemes in complete metric spaces and Banach spaces.
The study of convergence of different random iterative processes constructed for various random operators is a recent development (see, Beg & Abbas, Citation2006,Citation2007,Citation2010; Chang et al., Citation2005; Okeke & Abbas, Citation2015; Okeke & Kim, Citation2015 and references mentioned therein). Recently, Zhang et al. (Citation2011) studied the almost sure T-stability and convergence of Ishikawa-type and Mann-type random algorithms for certain -weakly contractive-type random operators in the set-up of separable Banach space. They also established the Bochner integrability of random fixed point for such random operators. In this paper, we introduce the random Jungck–Mann-type and the random Jungck–Ishikawa-type iterative processes. We prove some convergence and stability results for these random iterative processes for certain random operators. Furthermore, we apply our results to study random non-linear integral equation of the Hammerstein type. Our results generalize, extend and unify several well-known deterministic results in the literature, including the results of Hussain et al. (Citation2013), Khan et al. (Citation2014), Olatinwo (Citation2008a), Singh et al. (Citation2005), Akewe and Okeke (Citation2012), Akewe, Okeke, and Olayiwola (Citation2014) and the references therein. Moreover, our results are a generalization of recent results of Okeke and Abbas (Citation2015) and Okeke and Kim (Citation2015).
2. Preliminaries
Let be a complete probability measure space and (E, B(E)) measurable space, where E a separable Banach space, B(E) is Borel sigma algebra of E,
is a measurable space (
—sigma algebra) and
a probability measure on
, that is a measure with total measure one. A mapping
is called (a) E-valued random variable if
is
- measurable (b) strongly
-measurable if there exists a sequence
of
-simple functions converging to
- almost everywhere. Due to the separability of a Banach space E, the sum of two E-valued random variables is E-valued random variable. A mapping
is called a random operator if for each fixed e in E, the mapping
is measurable.
The following definitions and results will be needed in the sequel.
Definition 2.1
(Zhang et al., Citation2011) Let be a complete probability measure space and E a nonempty subset of a separable Banach space X and
a random operator. Denote by
such that
for each
( the random fixed point set of T ).
Suppose that X is a Banach space, Y an arbitrary set and such that
For
consider the iterative scheme:
(2.1)
(2.1)
The iterative process (2.1) is called Jungck iterative process, introduced by Jungck (Citation1976). Clearly, this iterative process reduces to the Picard iteration when (identity mapping) and
Let be random operator, where E is a nonempty convex subset of a separable Banach space X.
For Singh et al. (Citation2005) defined the Jungck–Mann iterative process as
(2.2)
(2.2)
For Olatinwo (Citation2008a) defined the Jungck–Ishikawa and Jungck–Noor iterative processes as follows:
(2.3)
(2.3)
(2.4)
(2.4)
Motivated by the above results, we now introduce the following random Jungck-type iterative processes. The random Jungck–Ishikwa-type iterative process is a sequence of function defined by
(2.5)
(2.5)
The random Jungck–Mann-type iterative process is a sequence of functions defined by
(2.6)
(2.6)
where and
is an arbitrary measurable mapping.
Jungck (Citation1976) used the iterative process (2.1) to approximate the common fixed points of the mappings S and T satisfying the following Jungck contraction:(2.7)
(2.7)
Olatinwo (Citation2008a) used a more general contractive condition than (2.7) to prove the stability and strong convergence results for the Jungck–Ishikawa iteration process. The contractive conditions used by Olatinwo (Citation2008a) are as follows:
Definition 2.2
(Olatinwo, Citation2008a) For two non-self mappings with
where S(Y) is a complete subspace of E,
(a) | there exist a real number | ||||
(b) | there exist real numbers |
Hussain et al. (Citation2013) introduced certain Jungck-type iterative process and used the contractive condition (2.8) to establish some stability and strong convergence results in arbitrary Banach spaces. Their results generalize and improve several known results in the literature, including the results of Olatinwo (Citation2008a).
Zhang et al. (Citation2011) studied the almost sure T-stability and convergence of Ishikawa-type and Mann-type random iterative processes for certain -weakly contractive-type random operators in a separable Banach space. The following is the contractive condition studied by Zhang et al. (Citation2011).
Definition 2.3
(Zhang et al., Citation2011) Let be a complete probability measure space and E be a nonempty subset of a separable Banach space X. A random operator
is the
-weakly contractive type if there exists a non-decreasing continuous function
with
and
such that
,
(2.10)
(2.10)
Recently, Okeke and Abbas (Citation2015) introduced the concept of generalized -weakly contraction random operators and then proved the convergence and almost sure T-stability of Mann-type and Ishikawa-type random iterative schemes. Their results generalize the results of Zhang et al. (Citation2011), Olatinwo (Citation2008b) and several known deterministic results in the literature. Furthermore, Okeke and Kim (Citation2015) introduced the random Picard–Mann hybrid iterative process. They established strong convergence theorems and summable almost T-stability of the random Picard–Mann hybrid iterative process and the random Mann-type iterative process generated by a generalized class of random operators in separable Banach spaces. Their results improve and generalize several well-known deterministic stability results in a stochastic version.
Next, we introduce the following contractive condition, which will be used to prove the main results of this paper. This contractive condition could be seen as the stochastic verse of those introduced by Olatinwo (Citation2008a).
Definition 2.4
Let be a complete probability measure space, E and Y be nonempty subsets of a separable Banach space X; and
random operators such that
Then, the random operators
are said to be generalized
-contractive type if there exists a monotone increasing function
such that
for all
,
and
we have
(2.11)
(2.11)
The following definitions will be needed in this study.
Definition 2.5
Let be a complete probability measure space. Let
be two random self-maps. A measurable map
is called a common random fixed point of the pair (f, g) if
for each
and some
. If
for each
and some
then the random variable
is called a random point of coincidence of f and g. A pair (f, g) is said to be weakly compatible if f and g commute at their random coincidence points.
The following definition of (S, T)-stability can be found in Singh et al. (Citation2005).
Definition 2.6
(Singh et al., Citation2005) Let be non-self operators for an arbitrary set Y such that
and p a point of coincidence of S and T. Let
be the sequence generated by an iterative procedure
(2.12)
(2.12)
where is the initial approximation and f is some function. Suppose that
converges to p. Let
be an arbitrary sequence and set
. Then, the iterative procedure (2.12) is said to be (S, T)-stable or stable if and only if
implies
Motivated by the results of Singh et al. (Citation2005), we now give the following definition which could be seen as the stochastic verse of Definition 2.6 of Singh et al. (Citation2005).
Definition 2.7
Let be a complete probability measure space, E and Y be nonempty subsets of a separable Banach space X; and
random operators such that
and
a random point of coincidence of S and T. For any given random variable
define a random iterative scheme with the help of functions
as follows:
(2.13)
(2.13)
where f is some function measurable in the second variable. Suppose that converges to
Let
be an arbitrary sequence of a random variable. Denote
by
(2.14)
(2.14)
Then, the iterative scheme (2.13) is (S, T)-stable almost surely (or the iterative scheme (2.13) is stable with respect to (S, T) almost surely) if and only if as
implies that
almost surely.
The following lemma will be needed in the sequel.
Lemma 2.1
(Berinde, Citation2004) If is a real number such that
and
is a sequence of positive numbers such that
then for any sequence of positive numbers
satisfying
(2.15)
(2.15)
one has
3. Convergence theorems
We begin this section by proving the following convergence results.
Theorem 3.1
Let be a complete probability measure space, let Y be nonempty subset of a separable Banach space X; and let
be continuous random non-self-operators satisfying contractive condition (2.11). Assume that
; S(Y) is a subset of X and
(say). For
let
be the random Jungck–Ishikawa-type iterative process defined by (2.5), where
are sequences of positive numbers in [0,1] with
satisfying
Then, the random Jungck–Ishikawa-type iterative process
converges strongly to
almost surely. Also,
will be the unique random common fixed point of the random operators S, T provided that
and S and T are weakly compatible.
Proof
Let
Let K be a countable subset of X and let We need to show that
(3.1)
(3.1)
Let Then, for all
we have
(3.2)
(3.2)
Let We have
(3.3)
(3.3)
Since for a particular is a continuous function of x, hence for arbitrary
there exists
such that
(3.4)
(3.4)
and(3.5)
(3.5)
Now choose(3.6)
(3.6)
and(3.7)
(3.7)
For all such choice of in (3.6) and (3.7), we have
(3.8)
(3.8)
Since is arbitrary, we obtain
(3.9)
(3.9)
Hence, This implies that
(3.10)
(3.10)
Clearly,(3.11)
(3.11)
Using (3.10) and (3.11), we have:(3.12)
(3.12)
Let(3.13)
(3.13)
Then, Take
and
Using (2.5) and (2.11), we have
(3.14)
(3.14)
Next, we have the following estimate:(3.15)
(3.15)
Using (3.15) in (3.14), we obtain:(3.16)
(3.16)
Since relation (3.16) becomes
(3.17)
(3.17)
Since and
we see that
as
Hence, by (3.17), we have that
(3.18)
(3.18)
Therefore, converges strongly to
Next, we prove that is a unique common random fixed point of T and S. Suppose that there exists another random point of coincidence say
Then, there exists
such that
(3.19)
(3.19)
From (2.11), we obtain(3.20)
(3.20)
this implies that since
Since S and T are weakly compatible and we have
hence
Hence,
is a random point of coincidence of S and T. Since the random point of coincidence is unique, then
Thus,
Therefore,
is a unique common random fixed point of S and T. The proof of Theorem 3.1 is completed.
Remark 3.1
Theorem 3.1 generalizes, extends and unifies several deterministic results in the literature, including the results of Hussain et al. (Citation2013), Olatinwo (Citation2008a), Proinov and Nikolova (Citation2015), Singh et al. (Citation2005) and the references therein. Moreover, it generalizes the recent results of Okeke and Abbas (Citation2015) and Okeke and Kim (Citation2015).
Next, we consider the following corollary, which is a special case of Theorem 3.1.
Corollary 3.1
Let be a complete probability measure space, let Y be nonempty subset of a separable Banach space X; and let
be continuous random non-self-operators satisfying contractive condition (2.11). Assume that
; S(Y) is a subset of X and
(say). For
let
be the random Jungck–Mann-type iterative process defined by (2.6), where
is a sequence of positive number in [0, 1] with
satisfying
Then, the random Jungck–Mann-type iterative process
converges strongly to
almost surely. Also,
will be the unique random common fixed point of the random operators S, T provided that
and S and T are weakly compatible.
Proof
Put in the random Jungck–Ishikawa-type iterative process (2.5); then, the convergence of the random Jungck–Mann-type iterative process (2.6) can be proved on the same lines as in Theorem 3.1. The proof of Corollary 3.2 is completed.
4. Stability results
In this section, we prove some stability results for the random Jungck–Ishikawa-type and the random Jungck–Mann-type iterative processes introduced in this paper.
Theorem 4.1
Let be a complete probability measure space, let Y be nonempty subset of a separable Banach space X; and let
be random non-self-operators satisfying contractive condition (2.11). Assume that
S(Y) is a subset of X and
(say). For
and
let
be the random Jungck–Ishikawa-type iterative process converging to
where
are sequences of positive numbers in [0,1] with
satisfying
for each
Then, the random Jungck–Ishikawa-type iterative process defined by (2.5) is (S, T)-stable almost surely.
Proof
Let be an arbitrary sequence and set
(4.1)
(4.1)
where(4.2)
(4.2)
and let Hence, using the random Jungck–Ishikawa-type iterative process (2.5), we have
(4.3)
(4.3)
We now obtain the following estimate.(4.4)
(4.4)
Using (4.4) in (4.3), we obtain(4.5)
(4.5)
Using the fact that and
we have that
Using Lemma 2.1, then we see in (4.5) that
as
Conversely, let as
Using the contractive condition (2.11) and the triangle inequality, we have:
(4.6)
(4.6)
Using (4.4) in (4.6), we have(4.7)
(4.7)
Hence, we see that as
Therefore, the random Jungck–Ishikawa-type iteration scheme is (S, T)-stable almost surely. The proof of Theorem 4.1 is completed.
Remark 4.1
Theorem 4.1 generalizes, extends and unifies several deterministic results in the literature, including the results of Hussain et al. (Citation2013), Olatinwo (Citation2008a), Proinov and Nikolova (Citation2015), Singh et al. (Citation2005() and the references therein. Moreover, it generalizes the recent results of Okeke and Abbas (Citation2015) and Okeke and Kim (Citation2015).
Next, we consider the following corollary which is a special case of Theorem 4.1.
Corollary 4.1
Let be a complete probability measure space, let Y be nonempty subset of a separable Banach space X; and let
be random non-self-operators satisfying contractive condition (2.11). Assume that
S(Y) is a subset of X and
(say). For
and
let
be the random Jungck–Mann-type iterative process converging to
where
are sequences of positive numbers in [0,1] with
satisfying
for each
Then, the random Jungck–Mann-type iterative process defined by (2.6) is (S, T)-stable almost surely.
Proof
Put in the random Jungck–Ishikawa-type iterative process (2.5); then, the (S, T)-stability almost surely of the random Jungck–Mann-type iterative process (2.6) can be proved on the same lines as in Theorem 4.1. The proof of Corollary 4.2 is completed.
5. Application to random non-linear integral equation of the Hammerstein type
In this section, we shall use our results to prove the existence of a solution in a Banach space of a random non-linear integral equation of the form:(5.1)
(5.1)
where
(i) | S is a locally compact metric space with a metric d on | ||||
(ii) |
| ||||
(iii) |
| ||||
(iv) |
| ||||
(v) |
| ||||
(vi) |
|
Definition 5.1
We define the space to be the space of all continuous functions from S into
with the topology of uniform convergence on compacta, i.e. for each fixed
,
is a vector-valued random variable such that
Note that is a locally convex space, whose topology is defined by a countable family of semi-norms (see, Yosida, Citation1965) given by
Moreover, is complete relative to this topology since
is complete.
We define to be the Banach space of all bounded continuous functions from S into
with norm
The space is the space of all second-order vector-valued stochastic processes defined on S, which is 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 (t, s),
and denote the norm by
Suppose that is such that
is
-integrable with respect to s for each
and
in
and there exists a real-valued function G defined
-a.e. on S, so that
is
-integrable and for each pair
-a.e. Furthermore, for almost all
will be continuous in t from S into
Now, we define the random integral operator T on by
(5.2)
(5.2)
where the integral is a Bochner integral. Moreover, we have that for each and that
is continuous in mean square by Lebesgue-dominated convergence theorem. So
Definition 5.2
(Achari, Citation1983, Lee & Padgett, Citation1977) Let B and D be two Banach spaces. The pair (B, D) is said to be admissible with respect to a random operator if
Lemma 5.1
(Joshi & Bose, Citation1985) The linear operator T defined by (5.2) is continuous from into itself.
Lemma 5.2
(Joshi & Bose, Citation1985, Lee & Padgett, Citation1977) If T is a continuous linear operator from into itself and
are Banach spaces stronger than
such that (B, D) is admissible with respect to T, then T is continuous from B into D.
Remark 5.1
(Padgett, Citation1973) The operator T defined by (5.2) is a bounded linear operator from B into D. It is to be noted that a random solution of Equation (5.1) will mean a function in
which satisfies the Equation (5.1)
- a.e.
We now prove the following theorem.
Theorem 5.1
We consider the stochastic integral Equation (5.1) subject to the following conditions
(a) | B and D are Banach spaces stronger than | ||||
(b) |
| ||||
(c) |
| ||||
(d) |
|
Proof
Define the operator from
into D by
(5.5)
(5.5)
Now,(5.6)
(5.6)
Using condition (5.3) in (5.6), we obtain(5.7)
(5.7)
Hence, using (5.7) in (5.6), we have(5.8)
(5.8)
Hence, Then, for each
we have by condition (b)
(5.9)
(5.9)
Since almost surely, we have by Theorem 3.1 that there exists a unique
which is a fixed point of U, i.e.
is the unique random solution of the non-linear stochastic integral equation of the Hammerstein type (5.1). The proof of Theorem 5.1 is completed.
Remark 5.2
Theorem 5.1 extends and generalizes several well-known results in the literature, including the results of Dey and Saha (Citation2013) and Padgett (Citation1973), among others.
Acknowledgements
The authors wish to thank the anonymous referees for their useful comments and suggestions.
Additional information
Funding
Notes on contributors
Godwin Amechi Okeke
Godwin Amechi Okeke is a faculty member of the Department of Mathematics, College of Physical and Applied Sciences, Michael Okpara University of Agriculture, Umudike, Nigeria. His areas of research interests are Functional Analysis and Non-linear Optimization. He obtained his PhD degree in Mathematics in 2014 from the Department of Mathematics, University of Lagos, Akoka, Nigeria.
Jong Kyu Kim
Jong Kyu Kim received his PhD degree in Mathematics from the Busan National University, Korea in 1988. Now he is a full professor at the Department of Mathematics Education in Kyungnam University. He is the chief editor of the journals: Nonlinear Functional Analysis and Applications (NFAA), International Journal of Mathematical Sciences (IJMS) and East Asian Mathematical Journal (EAMJ). He is an associate editor for many international mathematical journals. He has also delivered invited talks in many international conferences held in European, American and Asian countries. He has published about 320 research papers in journals of international repute.
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