Skip to main content

Weak contractive inequalities and compatible mixed monotone random operators in ordered metric spaces

Abstract

The aim of this work is to define the notion of compatible random operators in a partially ordered metric space and prove some coupled random coincidence theorems for a pair of compatible mixed monotone random operators satisfying (ϕ,φ)-weak contractive conditions. These results present random versions and extensions of recent results of Ćirić and Lakshmikantham (Stoch. Anal. Appl. 27:1246-1259, 2009), Choudhury and Kundu (Nonlinear Anal. 73:2524-2531, 2010), Alotaibi and Alsulami (Fixed Point Theory Appl. 2011:44, 2011) and many others.

1 Introduction

Random coincidence point theorems are stochastic generalizations of classical coincidence point theorems. Some random fixed point theorems play an important role in the theory of random differential and random integral equations (see [1, 2]). Random fixed point theorems for contractive mappings on separable complete metric spaces have been proved by several authors [38]. Sehgal and Singh [9] have proved different stochastic versions of the well-known Schauder fixed point theorem. Fixed point theorems for monotone operators in ordered Banach spaces have been investigated and have found various applications in differential and integral equations (see [1012] and references therein). Fixed point theorems for mixed monotone mappings in partially ordered metric spaces are of great importance and have been utilized for matrix equations, ordinary differential equations, and for the existence and uniqueness of solutions for some boundary value problems (see [1319]).

Recently Ćirić and Lakshmikantham [20] and Zhu and Xiao [21] proved some coupled random fixed point and coupled random coincidence results in partially ordered complete metric spaces. The purpose of this article is to improve these results for a pair of compatible mixed monotone random mappings F:Ω×(X×X)X and g:Ω×XX, where F and g satisfy the (ϕ,φ)-weak contractive conditions. Presented results are also the extensions and improvements of the corresponding results in [2224] and many others.

2 Preliminaries

Recall that if (X,) is a partially ordered set and F:XX is such that for x,yX, xy implies F(x)F(y), then a mapping F is said to be non-decreasing. Similarly, a non-increasing map may be defined. Bhaskar and Lakshmikantham [25] introduced the following notions of a mixed monotone mapping and a coupled fixed point.

Definition 2.1 ([25])

Let (X,) be a partially ordered set and F:X×XX. The mapping F is said to have the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any x,yX,

x 1 , x 2 X; x 1 x 2 F( x 1 ,y)F( x 2 ,y)

and

y 1 , y 2 X; y 1 y 2 F(x, y 1 )F(x, y 2 ).

Definition 2.2 ([25])

An element (x,y)X×X is called a coupled fixed point of the mapping F:X×XX if

F(x,y)=x,F(y,x)=y.

The concept of the mixed monotone property is generalized in [24].

Definition 2.3 ([24])

Let (X,) be a partially ordered set and F:X×XX and g:XX. The mapping F is said to have the mixed g-monotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument, that is, for any x,yX,

x 1 , x 2 X;g( x 1 )g( x 2 )F( x 1 ,y)F( x 2 ,y)

and

y 1 , y 2 X;g( y 1 )g( y 2 )F(x, y 1 )F(x, y 2 ).

Definition 2.4 An element (x,y)X×X is called a coupled fixed point of the mapping F:X×XX and g:XX if

F(x,y)=g(x),F(y,x)=g(y).

Definition 2.5 The mappings F and g, where F:X×XX and g:XX, are said to be compatible if

lim n d ( g ( F ( x n , y n ) ) , F ( g ( x n ) , g ( y n ) ) ) =0

and

lim n d ( g ( F ( y n , x n ) ) , F ( g ( y n ) , g ( x n ) ) ) =0,

whenever { x n }, { y n } are sequences in X such that lim n F( x n , y n )= lim n g( x n )=x and lim n F( y n , x n )= lim n g( y n )=y for all x,yX are satisfied.

Using the concept of compatible maps and the mixed g-monotone property, Choudhury and Kundu [23] proved the following theorem.

Theorem 2.6 Let (X,) be a partially ordered set, and let there be a metric d on X such that (X,d) is a complete metric space. Let φ:[0,)[0,) be such that φ(t)<t and limi t r t + φ(r)<t for all t>0. Let F:X×XX and g:XX be two mappings such that F has the mixed g-monotone property and satisfy

d ( F ( x , y ) , F ( u , v ) ) φ ( d ( g x , g u ) + d ( g y , g v ) 2 )

for all x,y,u,vX, for which gxgu and gygv. Let F(X×X)g(X), g be continuous and monotone increasing and F and g be compatible mappings. Also, suppose either

  1. (a)

    F is continuous or

  2. (b)

    X has the following properties:

  3. (i)

    if a non-decreasing sequence { x n }x, then x n x for all n0,

  4. (ii)

    if a non-increasing sequence { y n }y, then y n y for all n0.

If there exist x 0 , y 0 X, such that g( x 0 )F( x 0 , y 0 ) and g( y 0 )F( y 0 , x 0 ), then there exist x,yX such that g(x)=F(x,y) and g(y)=F(y,x), that is, F and g have a coupled coincidence point in X.

As in [17], let Φ denote all functions ϕ:[0,)[0,) which satisfy

  1. 1.

    ϕ is continuous and non-decreasing,

  2. 2.

    ϕ(t)=0 if and only if t=0,

  3. 3.

    ϕ(t+s)ϕ(t)+ϕ(s), t,s[0,),

and let Ψ denote all the functions ψ:[0,)(0,) which satisfy lim t r ψ(t)>0 for all r>0 and lim t 0 + ψ(t)=0.

Alotaibi and Alsulami in [22] proved the following coupled coincidence result for monotone operators in partially ordered metric spaces.

Theorem 2.7 Let (X,) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Let F:X×XX be a mapping having the mixed g-monotone property on X such that there exist two elements x 0 , y 0 X with

g x 0 F( x 0 , y 0 ) and g y 0 F( y 0 , x 0 ).

Suppose there exist ϕΦ and φΨ such that

ϕ ( d ( F ( x , y ) , F ( u , v ) ) ) 1 2 ϕ ( d ( g x , g u ) + d ( g y , g v ) ) ψ ( d ( g x , g u ) + d ( g y , g v ) 2 )

for all x,y,u,vX with gxgu and gygv. Suppose F(X×X)g(X), g is continuous and compatible with F and also suppose either

  1. (a)

    F is continuous or

  2. (b)

    X has the following property:

  3. (i)

    if a non-decreasing sequence { x n }x, then x n x for all n,

  4. (ii)

    if a non-increasing sequence { y n }y, then y y n for all n.

Then there exists x,yX such that

gx=F(x,y)andgy=F(y,x)

that is, F and g have a coupled coincidence point in X.

3 Main results

Let (Ω,Σ) be a measurable space with Σ being a sigma algebra of subsets of Ω, and let (X,d) be a metric space. A mapping T:ΩX is called Σ-measurable if for any open subset U of X, T 1 (U)={ω:T(ω)U}Σ. In what follows, when we speak of measurability, we will mean Σ-measurability. A mapping T:Ω×XX is called a random operator if for any xX, T(,x) is measurable. A measurable mapping ζ:ΩX is called a random fixed point of a random function T:Ω×XX if ζ(ω)=T(ω,ζ(ω)) for every ωΩ. A measurable mapping ζ:ΩX is called a random coincidence of T:Ω×XX and g:Ω×XX if g(ω,ζ(ω))=T(ω,ζ(ω)) for every ωΩ.

Definition 3.1 Let (X,d) be a separable metric space and (Ω,Σ) be a measurable space. Then F:Ω×(X×X)X and g:Ω×XX are said to be compatible random operators if

lim n d ( g ( ω , F ( ω , ( x n , y n ) ) ) , F ( ω , ( g ( ω , x n ) , g ( ω , y n ) ) ) ) =0

and

lim n d ( g ( ω , F ( ω , ( y n , x n ) ) ) , F ( ω , ( g ( ω , y n ) , g ( ω , x n ) ) ) ) =0

whenever { x n }, { y n } are sequences in X, such that lim n F(ω,( x n , y n ))= lim n g(ω, x n )=x and lim n F(ω,( y n , x n ))= lim n g(ω, y n )=y for all ωΩ and x,yX are satisfied.

As in [23], let φ:[0,)[0,) be such that φ(t)<t and limi t r t + φ(r)<t for all t>0.

Now, we state our main result.

Theorem 3.2 Let (X,,d) be a complete separable partially ordered metric space, (Ω,Σ) be a measurable space, and F:Ω×(X×X)X and g:Ω×XX be mappings such that

  1. (i)

    g(ω,) is continuous for all ωΩ;

  2. (ii)

    F(,v), g(,x) are measurable for all vX×X and xX respectively;

  3. (iii)

    F(ω,) has the mixed g(ω,)-monotone property for each ωΩ and

    d ( F ( ω , ( x , y ) ) , F ( ω , ( u , v ) ) ) φ ( d ( g ( ω , x ) , g ( ω , u ) ) + d ( g ( ω , y ) , g ( ω , v ) ) 2 )
    (1)

for all x,y,u,vX, for which g(ω,x)g(ω,u) and g(ω,y)g(ω,v) for all ωΩ.

Suppose g(ω×X)=X for each ωΩ, g is monotone increasing, and F and g are compatible random operators. Also suppose either

  1. (a)

    F(ω,) is continuous for all ωΩ or

  2. (b)

    X has the following property:

    (2)
    (3)

If there exist measurable mappings ζ 0 , η 0 :ΩX such that g(ω, ζ 0 (ω))F(ω,( ζ 0 (ω), η 0 (ω))) and g(ω, η 0 (ω))F(ω,( η 0 (ω), ζ 0 (ω))) for all ωΩ, then there are measurable mappings ζ,θ:ΩX such that F(ω,(ζ(ω),θ(ω)))=g(ω,ζ(ω)) and F(ω,(θ(ω),ζ(ω)))=g(ω,θ(ω)) for all ωΩ, that is, F and g have a coupled random coincidence point.

Proof Let Θ={ζ:ΩX} be a family of measurable mappings. Define a function h:Ω×X R + as follows:

h(ω,x)=d ( x , g ( ω , x ) ) .

Since xg(ω,x) is continuous for all ωΩ, we conclude that h(ω,) is continuous for all ωΩ. Also, since xg(ω,x) is measurable for all xX, we conclude that h(,x) is measurable for all ωΩ (see [26], p.868). Thus, h(ω,x) is the Caratheodory function. Therefore, if ζ:ΩX is a measurable mapping, then ωh(ω,ζ(ω)) is also measurable (see [27]). Also, for each ζΘ, the function η:ΩX defined by η(ω)=g(ω,ζ(ω)) is measurable, that is, ηΘ.

Now, we shall construct two sequences of measurable mappings { ζ n } and { η n } in Θ, and two sequences {g(ω, ζ n (ω))} and {g(ω, η n (ω))} in X as follows. Let ζ 0 , η 0 Θ be such that g(ω, ζ 0 (ω))F(ω,( ζ 0 (ω), η 0 (ω))) and g(ω, η 0 (ω))F(ω,( η 0 (ω), ζ 0 (ω))) for all ωΩ. Since F(ω,( ζ 0 (ω), η 0 (ω)))X=g(ω×X) by an appropriate Filippov measurable implicit function theorem [1, 20, 28, 29], there is ζ 1 Θ such that g(ω, ζ 1 (ω))=F(ω,( ζ 0 (ω), η 0 (ω))). Similarly, as F(ω,( η 0 (ω), ζ 0 (ω)))g(ω×X), there is η 1 (ω)Θ such that g(ω, η 1 (ω))=F(ω,( η 0 (ω), ζ 0 (ω))). Now F(ω,( ζ 1 (ω), η 1 (ω))) and F(ω,( η 1 (ω), ζ 1 (ω))) are well defined. Again from F(ω,( ζ 1 (ω), η 1 (ω))),F(ω,( η 1 (ω), ζ 1 (ω)))g(ω×X), there are ζ 2 , η 2 Θ such that g(ω, ζ 2 (ω))=F(ω,( ζ 1 (ω), η 1 (ω))) and g(ω, η 2 (ω))=F(ω,( η 1 (ω), ζ 1 (ω))). Continuing this process, we can construct sequences { ζ n (ω)} and { η n (ω)} in X such that

g ( ω , ζ n + 1 ( ω ) ) = F ( ω , ( ζ n ( ω ) , η n ( ω ) ) ) and  g ( ω , η n + 1 ( ω ) ) = F ( ω , ( η n ( ω ) , ζ n ( ω ) ) )
(4)

for all n0.

We shall prove that

g ( ω , ζ n ( ω ) ) g ( ω , ζ n + 1 ( ω ) ) for all n0
(5)

and

g ( ω , η n ( ω ) ) g ( ω , η n + 1 ( ω ) ) for all n0.
(6)

The proof will be given by mathematical induction. Let n=0. By assumption we have g(ω, ζ 0 (ω))F(ω,( ζ 0 (ω), η 0 (ω))) and g(ω, η 0 (ω))F(ω,( η 0 (ω), ζ 0 (ω))). Since g(ω, ζ 1 (ω))=F(ω,( ζ 0 (ω), η 0 (ω))) and g(ω, η 1 (ω))=F(ω,( η 0 (ω), ζ 0 (ω))), we have

g ( ω , ζ 0 ( ω ) ) g ( ω , ζ 1 ( ω ) ) andg ( ω , η 0 ( ω ) ) g ( ω , η 1 ( ω ) ) .

Therefore, (5) and (6) hold for n=0.

Suppose now that (5) and (6) hold for some fixed n0. Then, since g(ω, ζ n (ω))g(ω, ζ n + 1 (ω)) and g(ω, η n (ω))g(ω, η n + 1 (ω)) and as F is monotone g-non-decreasing in its first argument, from (2) and (4), we have

F ( ω , ( ζ n ( ω ) , η n ( ω ) ) ) F ( ω , ( ζ n + 1 ( ω ) , η n ( ω ) ) ) and F ( ω , ( η n + 1 ( ω ) , ζ n ( ω ) ) ) F ( ω , ( η n ( ω ) , ζ n ( ω ) ) ) .
(7)

Similarly, from (3) and (4), as g(ω, η n + 1 (ω))g(ω, η n (ω)) and g(ω, ζ n (ω))g(ω, ζ n + 1 (ω)),

F ( ω , ( ζ n + 1 ( ω ) , η n + 1 ( ω ) ) ) F ( ω , ( ζ n + 1 ( ω ) , η n ( ω ) ) ) and F ( ω , ( η n + 1 ( ω ) , ζ n ( ω ) ) ) F ( ω , ( η n + 1 ( ω ) , ζ n + 1 ( ω ) ) ) .
(8)

Now from (7), (8), and (4), we get

g ( ω , ζ n + 1 ( ω ) ) g ( ω , ζ n + 2 ( ω ) )
(9)

and

g ( ω , η n + 1 ( ω ) ) g ( ω , η n + 2 ( ω ) ) .
(10)

Thus, by mathematical induction we conclude that (5) and (6) hold for all n0.

Denote for each ωΩ

δ n =d ( g ( ω , ζ n ( ω ) ) , g ( ω , ζ n + 1 ( ω ) ) ) +d ( g ( ω , η n ( ω ) ) , g ( ω , η n + 1 ( ω ) ) ) .

We show that

δ n 2 φ ( δ n 1 2 ) for all n1.
(11)

Since from (5) and (6) we have g(ω, ζ n 1 (ω))g(ω, ζ n (ω)) and g(ω, η n 1 (ω))g(ω, η n (ω)), therefore from (4) and (1), we get

(12)

Similarly, from (4) and (1), as g(ω, η n (ω))g(ω, η n 1 (ω)) and g(ω, ζ n (ω))g(ω, ζ n 1 (ω)),

(13)

By adding (12) and (13), and dividing by 2, we obtain (11).

From (11), since φ(t)<t for t>0, it follows that { δ n } is the monotone decreasing sequence of positive reals. Therefore, there is some δ0 such that

lim n δ n =δ+.

We show that δ=0. Suppose, to the contrary, that δ>0. Taking the limit in (11) when δ n δ+ and having in mind that we assume that lim r t + φ(t)<t for all t>0, we have

δ 2 = lim n δ n 2 lim n φ ( δ n 1 2 ) = lim δ n 1 δ + φ ( δ n 1 2 ) < δ 2 ,

a contradiction. Thus, δ=0.

Now we prove that for each ωΩ, {g(ω, ζ n (ω))} and {g(ω, η n (ω))} are Cauchy sequences. Suppose, to the contrary, that at least one, {g(ω, ζ n (ω))} or {g(ω, η n (ω))}, is not a Cauchy sequence. Then there exist an ϵ0 and two subsequences of positive integers {l(k)}, {m(k)}, m(k)>l(k)k with

r k = d ( g ( ω , ζ l ( k ) ( ω ) ) , g ( ω , ζ m ( k ) ( ω ) ) ) + d ( g ( ω , η l ( k ) ( ω ) ) , g ( ω , η m ( k ) ( ω ) ) ) ϵ
(14)

for k{1,2,}.

We may also assume

d ( g ( ω , ζ l ( k ) ( ω ) ) , g ( ω , ζ m ( k ) 1 ( ω ) ) ) +d ( g ( ω , η l ( k ) ( ω ) ) , g ( ω , η m ( k ) 1 ( ω ) ) ) <ϵ.
(15)

By choosing m(k) to be the smallest number exceeding l(k) for which (14) holds, such m(k) for which (15) holds exists, because δ n 0. From (14), (15) and by the triangle inequality, we have

ϵ r k d ( g ( ω , ζ l ( k ) ( ω ) ) , g ( ω , ζ m ( k ) 1 ( ω ) ) ) + d ( g ( ω , ζ m ( k ) 1 ( ω ) ) , g ( ω , ζ m ( k ) ( ω ) ) ) + d ( g ( ω , η l ( k ) ( ω ) ) , g ( ω , η m ( k ) 1 ( ω ) ) ) + d ( g ( ω , η m ( k ) 1 ( ω ) ) , g ( ω , η m ( k ) ( ω ) ) ) = d ( g ( ω , ζ l ( k ) ( ω ) ) , g ( ω , ζ m ( k ) 1 ( ω ) ) ) + d ( g ( ω , η l ( k ) ( ω ) ) , g ( ω , η m ( k ) 1 ( ω ) ) ) + δ m ( k ) 1 < ϵ + δ m ( k ) 1 .

Taking the limit as k, we get

lim k r k =ϵ+.
(16)

Inequality (14) and the triangle inequality imply now

r k = d ( g ( ω , ζ l ( k ) ( ω ) ) , g ( ω , ζ m ( k ) ( ω ) ) ) + d ( g ( ω , η l ( k ) ( ω ) ) , g ( ω , η m ( k ) ( ω ) ) ) d ( g ( ω , ζ l ( k ) ( ω ) ) , g ( ω , ζ l ( k ) + 1 ( ω ) ) ) + d ( g ( ω , ζ l ( k ) + 1 ( ω ) ) , g ( ω , ζ m ( k ) + 1 ( ω ) ) ) + d ( g ( ω , ζ m ( k ) + 1 ( ω ) ) , g ( ω , ζ m ( k ) ( ω ) ) ) + d ( g ( ω , η l ( k ) ( ω ) ) , g ( ω , η l ( k ) + 1 ( ω ) ) ) + d ( g ( ω , η l ( k ) + 1 ( ω ) ) , g ( ω , η m ( k ) + 1 ( ω ) ) ) + d ( g ( ω , η m ( k ) + 1 ( ω ) ) , g ( ω , η m ( k ) ( ω ) ) ) = d ( g ( ω , ζ l ( k ) ( ω ) ) , g ( ω , ζ l ( k ) + 1 ( ω ) ) ) + d ( g ( ω , η l ( k ) ( ω ) ) , g ( ω , η l ( k ) + 1 ( ω ) ) ) + [ d ( g ( ω , ζ m ( k ) + 1 ( ω ) ) , g ( ω , ζ m ( k ) ( ω ) ) ) + d ( g ( ω , η m ( k ) ( ω ) ) , g ( ω , η m ( k ) + 1 ( ω ) ) ) ] + d ( g ( ω , ζ l ( k ) + 1 ( ω ) ) , g ( ω , ζ m ( k ) + 1 ( ω ) ) ) + d ( g ( ω , η l ( k ) + 1 ( ω ) ) , g ( ω , η m ( k ) + 1 ( ω ) ) ) .

Hence,

r k δ l ( k ) + δ m ( k ) + d ( g ( ω , ζ l ( k ) + 1 ( ω ) ) , g ( ω , ζ m ( k ) + 1 ( ω ) ) ) + d ( g ( ω , η l ( k ) + 1 ( ω ) ) , g ( ω , η m ( k ) + 1 ( ω ) ) ) .
(17)

From (5) and (6), we conclude that g(ω, ζ l ( k ) (ω))g(ω, ζ m ( k ) (ω)) and g(ω, η l ( k ) (ω))g(ω, η m ( k ) (ω)).

Now (1) and (4) imply that

(18)

Also, from (1) and (4), as g(ω, η m ( k ) (ω))g(ω, η l ( k ) (ω)) and g(ω, ζ m ( k ) (ω))g(ω, ζ l ( k ) (ω)),

(19)

Inserting (18) and (19) in (17), we obtain

r k δ l ( k ) + δ m ( k ) +2φ ( r k 2 ) .

Letting k, we get by (16)

ϵ2 lim k φ ( r k 2 ) =2 lim r k ϵ + φ ( r k 2 ) <2 ϵ 2 =ϵ,
(20)

a contradiction. Therefore, our supposition (14) was wrong. Thus, we proved that {g(ω, ζ n (ω))} and {g(ω, η n (ω))} are Cauchy sequences in X.

Since X is complete and g(ω×X)=X, there exist ζ 0 , θ 0 Θ such that lim n g(ω, ζ n (ω))=g(ω, ζ 0 (ω)) and lim n g(ω, η n (ω))=g(ω, θ 0 (ω)). Since g(ω, ζ 0 (ω)) and g(ω, θ 0 (ω)) are measurable, therefore the functions ζ(ω) and θ(ω), defined by ζ(ω)=g(ω, ζ 0 (ω)) and θ(ω)=g(ω, θ 0 (ω)) are measurable. Thus,

lim n F ( ω , ( ζ n ( ω ) , η n ( ω ) ) ) = lim n g ( ω , ζ n ( ω ) ) = ζ ( ω ) and lim n F ( ω , ( η n ( ω ) , ζ n ( ω ) ) ) = lim n g ( ω , η n ( ω ) ) = θ ( ω ) .
(21)

Since F and g are compatible mappings, we have by (21)

(22)
(23)

Next, we prove that

g ( ω , ζ ( ω ) ) =F ( ω , ( ζ ( ω ) , θ ( ω ) ) )

and

g ( ω , θ ( ω ) ) =F ( ω , ( θ ( ω ) , ζ ( ω ) ) ) .

Let (a) hold. We have

Taking the limit as n, using (4), (21), and (22) and the fact that F and g are continuous, we have

d ( g ( ω , ζ ( ω ) ) , F ( ω , ( ζ ( ω ) , θ ( ω ) ) ) ) =0.

Similarly, from (4), (21), and (23) and the continuity of F and g, we have

d ( g ( ω , θ ( ω ) ) , F ( ω , ( θ ( ω ) , ζ ( ω ) ) ) ) =0.

Combining the above two results, we obtain

g ( ω , ζ ( ω ) ) =F ( ω , ( ζ ( ω ) , θ ( ω ) ) )

and

g ( ω , θ ( ω ) ) =F ( ω , ( θ ( ω ) , ζ ( ω ) ) )

for each ωΩ.

Next, suppose that (b) holds. From (5), (6), and (21), we have {g(ω, ζ n (ω))} is non-decreasing and {g(ω, η n (ω))} is non-increasing sequence and

g ( ω , ζ n ( ω ) ) g ( ω , ζ ( ω ) ) ,g ( ω , η n ( ω ) ) g ( ω , θ ( ω ) ) .

So, from (2) and (3), we have for all n0

g ( ω , ζ n ( ω ) ) g ( ω , ζ ( ω ) ) andg ( ω , η n ( ω ) ) g ( ω , θ ( ω ) ) .
(24)

Since F and g are compatible mappings and g is continuous, by (22) and (23) we have

(25)

and

(26)

Now, we have

Taking the limit as n in the above inequality, using (4) and (25), we have

Since the mapping g is monotone increasing, by (1), (24), and the above inequality, we have for all n0

Using (21) and the property of a φ-function, we obtain

d ( g ( ω , ζ ( ω ) ) , F ( ω , ( ζ ( ω ) , θ ( ω ) ) ) ) 0.

That is,

g ( ω , ζ ( ω ) ) =F ( ω , ( ζ ( ω ) , θ ( ω ) ) ) .

And similarly, by the virtue of (4), (21), and (26), we obtain

g ( ω , θ ( ω ) ) =F ( ω , ( θ ( ω ) , ζ ( ω ) ) ) .

This proves that F and g have a coupled random coincidence point. □

Corollary 3.3 Let (X,,d) be a complete separable partially ordered metric space, (Ω,Σ) be a measurable space, and F:Ω×(X×X)X and g:Ω×XX be mappings such that

  1. (i)

    g(ω,) is continuous for all ωΩ;

  2. (ii)

    F(,v), g(,x) are measurable for all vX×X and xX respectively;

  3. (iii)

    F(ω,) has the mixed g(ω,)-monotone property for each ωΩ and for some k[0,1)

    d ( F ( ω , ( x , y ) ) , F ( ω , ( u , v ) ) ) k 2 ( d ( g ( ω , x ) , g ( ω , u ) ) + d ( g ( ω , y ) , g ( ω , v ) ) )

for all x,y,u,vX, for which g(ω,x)g(ω,u) and g(ω,y)g(ω,v) for all ωΩ.

Suppose g(ω×X)=X for each ωΩ, g is monotone increasing, and F and g are compatible random operators. Also suppose either

  1. (a)

    F(ω,) is continuous for all ωΩ or

  2. (b)

    X has the following property:

  3. (i)

    if a non-decreasing sequence { x n }x, then x n x for all n,

  4. (ii)

    if a non-increasing sequence { y n }y, then y n y for all n.

If there exist measurable mappings ζ 0 , η 0 :ΩX such that g(ω, ζ 0 (ω))F(ω,( ζ 0 (ω), η 0 (ω))) and g(ω, η 0 (ω))F(ω,( η 0 (ω), ζ 0 (ω))) for all ωΩ, then there are measurable mappings ζ,θ:ΩX such that F(ω,(ζ(ω),θ(ω)))=g(ω,ζ(ω)) and F(ω,(θ(ω),ζ(ω)))=g(ω,θ(ω)) for all ωΩ, that is, F and g have a coupled random coincidence point.

Proof Taking ϕ(t)=kt with k[0,1) in Theorem 3.2, we obtain the result. □

The following theorem presents the stochastic version of Theorem 2.7 and generalizes the recent results in [20].

Theorem 3.4 Let (X,,d) be a complete separable partially ordered metric space, (Ω,Σ) be a measurable space, and F:Ω×(X×X)X and g:Ω×XX be mappings such that

  1. (i)

    g(ω,) is continuous for all ωΩ;

  2. (ii)

    F(,v), g(,x) are measurable for all vX×X and xX, respectively;

  3. (iii)

    F:Ω×(X×X)X and g:Ω×XX are such that F(ω,) has the mixed g(ω,)-monotone property for each ωΩ; and suppose there exist ϕΦ and ψΨ, satisfying conditions of Theorem  2.7, such that

    (27)

for all x,y,u,vX, for which g(ω,x)g(ω,u) and g(ω,y)g(ω,v) for all ωΩ.

Suppose g(ω×X)=X for each ωΩ, g is monotone increasing, and F and g are compatible random operators. Also suppose either

  1. (a)

    F(ω,) is continuous for all ωΩ or

  2. (b)

    X has the following property:

    (28)
    (29)

If there exist measurable mappings ζ 0 , η 0 :ΩX such that g(ω, ζ 0 (ω))F(ω,( ζ 0 (ω), η 0 (ω))) and g(ω, η 0 (ω))F(ω,( η 0 (ω), ζ 0 (ω))) for all ωΩ, then there are measurable mappings ζ,θ:ΩX such that F(ω,(ζ(ω),θ(ω)))=g(ω,ζ(ω)) and F(ω,(θ(ω),ζ(ω)))=g(ω,θ(ω)) for all ωΩ, that is, F and g have a coupled random coincidence.

Proof Let ={ζ:ΩX} be a family of measurable mappings. Define a function h:Ω×X R + as follows:

h(ω,x)=d ( x , g ( ω , x ) ) .

Since Xg(ω,x) is continuous for all ωΩ, we conclude that h(ω,) is continuous for all ωΩ. Also, since xg(ω,x) is measurable for all xX, we conclude that h(,x) is measurable for all ωΩ (see [26], p.868). Thus, h(ω,x) is the Caratheodory function. Therefore, if ζ:ΩX is a measurable mapping, then ωh(ω,ζ(ω)) is also measurable (see [27]). Also, for each ζ, the function η:ΩX defined by η(ω)=g(ω,ζ(ω)) is measurable, that is, η. Now, we shall construct two sequences of measurable mappings { ζ n } and { η n } in , and two sequences {g(ω, ζ n (ω))} and {g(ω, η n (ω))} in X as follows. Let ζ 0 , η 0 such that g(ω, ζ 0 (ω))F(ω,( ζ 0 (ω), η 0 (ω))) and g(ω, η 0 (ω))F(ω,( η 0 (ω), ζ 0 (ω))) for all ωΩ. Since F(ω,( ζ 0 (ω), η 0 (ω)))X=g(ω×X), by an appropriate Filippov measurable implicit function theorem [1, 20, 28, 29], there is ζ 1 such that g(ω, ζ 1 (ω))=F(ω,( ζ 0 (ω), η 0 (ω))). Similarly, as F(ω,( η 0 (ω), ζ 0 (ω)))X=g(ω×X), there is η 1 (ω) such that g(ω, η 1 (ω))=F(ω,( η 0 (ω), ζ 0 (ω))). Now, F(ω,( ζ 1 (ω), η 1 (ω))) and F(ω,( η 1 (ω), ζ 1 (ω))) are well defined. Again, from F(ω,( ζ 1 (ω), η 1 (ω))),F(ω,( η 1 (ω), ζ 1 (ω)))g(ω×X), there are ζ 2 , η 2 such that g(ω, ζ 2 (ω))=F(ω,( ζ 1 (ω), η 1 (ω))) and g(ω, η 2 (ω))=F(ω,( η 1 (ω), ζ 1 (ω))). Continuing this process, we can construct sequences { ζ n (ω)} and { η n (ω)} in X such that

g ( ω , ζ n + 1 ( ω ) ) = F ( ω , ( ζ n ( ω ) , η n ( ω ) ) ) and  g ( ω , η n + 1 ( ω ) ) = F ( ω , η n ( ω ) , ζ n ( ω ) )
(30)

for all n0.

We shall prove that

g ( ω , ζ n ( ω ) ) g ( ω , ζ n + 1 ( ω ) ) for all n0
(31)

and

g ( ω , η n ( ω ) ) g ( ω , η n + 1 ( ω ) ) for all n0.
(32)

The proof will be given by mathematical induction. Let n=0. By assumption, we have g(ω, ζ 0 (ω))F(ω,( ζ 0 (ω), η 0 (ω))) and g(ω, η 0 (ω))F(ω,( η 0 (ω), ζ 0 (ω))). Since g(ω, ζ 1 (ω))=F(ω,( ζ 0 (ω), η 0 (ω))) and g(ω, η 1 (ω))=F(ω,( η 0 (ω), ζ 0 (ω))), we have g(ω, ζ 0 (ω))g(ω, ζ 1 (ω)) and g(ω, η 0 (ω))g(ω, η 1 (ω)). Therefore, (31) and (32) hold for n=0. Suppose now that (31) and (32) hold for some fixed n0. Then g(ω, ζ n (ω))g(ω, ζ n + 1 (ω)) and g(ω, η n (ω))g(ω, η n + 1 (ω)) as F is monotone g-non-decreasing in its first argument, from (28) and (30),

F ( ω , ( ζ n ( ω ) , η n ( ω ) ) ) F ( ω , ( ζ n + 1 ( ω ) , η n ( ω ) ) ) and F ( ω , ( η n + 1 ( ω ) , ζ n ( ω ) ) ) F ( ω , ( η n ( ω ) , ζ n ( ω ) ) ) .
(33)

Similarly, from (29) and (30), as g(ω, ζ n (ω))g(ω, ζ n + 1 (ω)) and g(ω, η n (ω))g(ω, η n + 1 (ω)), we have

F ( ω , ( ζ n + 1 ( ω ) , η n + 1 ( ω ) ) ) F ( ω , ( ζ n + 1 ( ω ) , η n ( ω ) ) ) and F ( ω , ( η n + 1 ( ω ) , ζ n ( ω ) ) ) F ( ω , ( η n + 1 ( ω ) , ζ n + 1 ( ω ) ) ) .
(34)

Now, from (30), (33), and (34), we get

g ( ω , ζ n + 1 ( ω ) ) g ( ω , ζ n + 2 ( ω ) )
(35)

and

g ( ω , η n + 1 ( ω ) ) g ( ω , η n + 2 ( ω ) ) .
(36)

Thus, by mathematical induction we conclude that (31) and (32) hold for all n0.

Therefore,

g ( ω , ζ 0 ( ω ) ) g ( ω , ζ 1 ( ω ) ) g ( ω , ζ n ( ω ) ) g(ω, ζ n + 1 )
(37)

and

g ( ω , η 0 ( ω ) ) g ( ω , η 1 ( ω ) ) g ( ω , η n ( ω ) ) g(ω, η n + 1 ).
(38)

Since g(ω, ζ n 1 (ω))g(ω, ζ n (ω)) and g(ω, η n (ω))g(ω, η n 1 (ω)), using (27) and (30), we have

(39)

Similarly, since g(ω, ζ n 1 (ω))g(ω, ζ n (ω)) and g(ω, η n 1 (ω))g(ω, η n (ω)), using (27) and (30), we also have

(40)

Using (39) and (40), we have

(41)

From the property (iii) of ϕ, we have

(42)

Using (41) and (42), we have

(43)

Since ψ is a non-negative function, therefore we have

Using the fact that ϕ is non-decreasing, we get

Let

δ n =d ( g ( ω , ζ n + 1 ( ω ) ) , g ( ω , ζ n ( ω ) ) ) +d ( g ( ω , η n + 1 ( ω ) ) , g ( ω , η n ( ω ) ) ) .

Now, we show that δ n 0 as n. It is clear that the sequence { δ n }is decreasing; therefore, there is some δ0 such that

lim n δ n = lim n [ d ( g ( ω , ζ n + 1 ( ω ) ) , g ( ω , ζ n ( ω ) ) ) + d ( g ( ω , η n + 1 ( ω ) ) , g ( ω , η n ( ω ) ) ) ] = δ .
(44)

We shall show that δ=0. Suppose, to the contrary, that δ>0. Then taking the limit as n on both sides of (43) and as lim t r ψ(t)>0 for all r>0 and ϕ is continuous, we have

ϕ ( δ ) = lim n ϕ ( δ n ) lim n [ ϕ ( δ n 1 ) 2 ψ ( δ n 1 2 ) ] = ϕ ( δ ) 2 lim δ n 1 δ ψ ( δ n 1 2 ) < ϕ ( δ ) ,

a contradiction. Thus, δ=0, that is

lim n δ n = lim n [ d ( g ( ω , ζ n + 1 ( ω ) ) , g ( ω , ζ n ( ω ) ) ) + d ( g ( ω , η n + 1 ( ω ) ) , g ( ω , η n ( ω ) ) ) ] = 0 .
(45)

Now, we will prove that {g(ω, ζ n )}, {g(ω, η n )} are Cauchy sequences. Suppose, to the contrary, that at least one of {g(ω, ζ n )} or {g(ω, η n )} is not a Cauchy sequence. Then there exists an ϵ>0 for which we can find subsequences of positive integers { m k }, { n k } with n(k)>m(k)k such that

r k : = d ( g ( ω , ζ n ( k ) ( ω ) ) , g ( ω , ζ m ( k ) ( ω ) ) ) + d ( g ( ω , η n ( k ) ( ω ) ) , g ( ω , η m ( k ) ( ω ) ) ) ϵ
(46)

for k={1,2,3,}. We may also assume

d ( g ( ω , ζ n ( k ) 1 ( ω ) ) , g ( ω , ζ m ( k ) ( ω ) ) ) +d ( g ( ω , η n ( k ) 1 ( ω ) ) , g ( ω , η m ( k ) ( ω ) ) ) <ϵ
(47)

by choosing n(k) in such a way that it is the smallest integer with n(k)>m(k) and satisfying (46). Using (46), (47), and the triangle inequality, we have

ϵ r k : = d ( g ( ω , ζ n ( k ) ( ω ) ) , g ( ω , ζ m ( k ) ( ω ) ) ) + d ( g ( ω , η n ( k ) ( ω ) ) , g ( ω , η m ( k ) ( ω ) ) ) d ( g ( ω , ζ n ( k ) ( ω ) ) , g ( ω , ζ n ( k ) 1 ( ω ) ) ) + d ( g ( ω , ζ n ( k ) 1 ( ω ) ) , g ( ω , ζ m ( k ) ( ω ) ) ) + d ( g ( ω , η n ( k ) ( ω ) ) , g ( ω , η n ( k ) 1 ( ω ) ) ) + d ( g ( ω , η n ( k ) 1 ( ω ) ) , g ( ω , η m ( k ) ( ω ) ) ) d ( g ( ω , ζ n ( k ) ( ω ) ) , g ( ω , ζ n ( k ) 1 ( ω ) ) ) + d ( g ( ω , η n ( k ) ( ω ) ) , g ( ω , η n ( k ) 1 ( ω ) ) ) + d ( g ( ω , ζ n ( k ) 1 ( ω ) ) , g ( ω , ζ m ( k ) ( ω ) ) ) + d ( g ( ω , η n ( k ) 1 ( ω ) ) , g ( ω , η m ( k ) ( ω ) ) ) d ( g ( ω , ζ n ( k ) ( ω ) ) , g ( ω , ζ n ( k ) 1 ( ω ) ) ) + d ( g ( ω , η n ( k ) ( ω ) ) , g ( ω , η n ( k ) 1 ( ω ) ) ) + ϵ .

Letting k and using (45), we get

lim k r k = lim k [ d ( g ( ω , ζ n ( k ) ( ω ) ) , g ( ω , ζ m ( k ) ( ω ) ) ) + d ( g ( ω , η n ( k ) ( ω ) ) , g ( ω , η m ( k ) ( ω ) ) ) ] = ϵ .
(48)

By the triangle inequality,

(49)

Using the property of ϕ, we have

ϕ ( r k ) = ϕ ( δ n ( k ) + δ m ( k ) + d ( g ( ω , ζ n ( k ) + 1 ( ω ) ) , g ( ω , ζ m ( k ) + 1 ( ω ) ) ) + d ( g ( ω , η n ( k ) + 1 ( ω ) ) , g ( ω , η m ( k ) + 1 ( ω ) ) ) ) ϕ ( δ n ( k ) + δ m ( k ) ) + ϕ ( d ( g ( ω , ζ n ( k ) + 1 ( ω ) ) , g ( ω , ζ m ( k ) + 1 ( ω ) ) ) ) + ϕ ( d ( g ( ω , η n ( k ) + 1 ( ω ) ) , g ( ω , η m ( k ) + 1 ( ω ) ) ) ) .

Since n(k)>m(k), hence g(ω, ζ n ( k ) (ω))g(ω, ζ m ( k ) (ω)) and g(ω, η n ( k ) (ω))g(ω, η m ( k ) (ω)). Using (27) and (30), we get

(50)

By the same way, we also have

(51)

Putting (50) and (51) in (49), we have

ϕ ( r k ) ϕ ( δ n ( k ) + δ m ( k ) ) + 1 2 ϕ ( r k ) ψ ( r k 2 ) + 1 2 ϕ ( r k ) ψ ( r k 2 ) = ϕ ( δ n ( k ) + δ m ( k ) ) + ϕ ( r k ) 2 ψ ( r k 2 ) .

Taking k and using (45) and (48), we get

ϕ ( ϵ ) ϕ ( 0 ) + ϕ ( ϵ ) 2 lim k ψ ( r k 2 ) = ϕ ( ϵ ) 2 lim r k ϵ ψ ( r k 2 ) < ϕ ( ϵ ) ,

a contradiction. This shows that {g(ω, ζ n )} and {g(ω, η n )} are Cauchy sequences.

Since X is complete and g(ω×X)=X, there exist ζ 0 , θ 0 Θ such that lim n g(ω, ζ n (ω))=g(ω, ζ 0 (ω)) and lim n g(ω, η n (ω))=g(ω, θ 0 (ω)). Since g(ω, ζ 0 (ω)) and g(ω, θ 0 (ω)) are measurable, then the functions ζ(ω) and θ(ω), defined by ζ(ω)=g(ω, ζ 0 (ω)) and θ(ω)=g(ω, θ 0 (ω)), are measurable. Thus,

lim n F ( ω , ( ζ n ( ω ) , η n ( ω ) ) ) = lim n g ( ω , ζ n ( ω ) ) = ζ ( ω ) and lim n F ( ω , ( η n ( ω ) , ζ n ( ω ) ) ) = lim n g ( ω , η n ( ω ) ) = θ ( ω ) .
(52)

Using the compatibility of F and g and the technique of the proof of Theorem 3.2, we obtain the required conclusion. □

Corollary 3.5 Let (X,,d) be a complete separable partially ordered metric space, (Ω,Σ) be a measurable space, and F:Ω×(X×X)X and g:Ω×XX be mappings such that

  1. (i)

    g(ω,) is continuous for all ωΩ;

  2. (ii)

    F(,v), g(,x) are measurable for all vX×X and xX respectively;

  3. (iii)

    F(ω,) has the mixed g(ω,)-monotone property for each ωΩ; and suppose there exist ϕΦ and ψΨ such that

    d ( F ( ω , ( x , y ) ) , F ( ω , ( u , v ) ) ) 1 2 ( d ( g ( ω , x ) , g ( ω , u ) ) + d ( g ( ω , y ) , g ( ω , v ) ) ) ψ ( d ( g ( ω , x ) , g ( ω , u ) ) + d ( g ( ω , y ) , g ( ω , v ) ) 2 )

for all x,y,u,vX, for which g(ω,x)g(ω,u) and g(ω,y)g(ω,v) for all ωΩ.

Suppose g(ω×X)=X for each ωΩ, g is continuous and monotone increasing, and F and g are compatible mappings. Also suppose either

  1. (a)

    F(ω,) is continuous for all ωΩ or

  2. (b)

    X has the following property:

  3. (i)

    if a non-decreasing sequence { x n }X, then x n x for all n,

  4. (ii)

    if a non-increasing sequence { x n }X, then x n x for all n.

If there exist measurable mappings ζ 0 , η 0 :ΩX such that g(ω, ζ 0 (ω))F(ω,( ζ 0 (ω), η 0 (ω))) and g(ω, η 0 (ω))F(ω,( η 0 (ω), ζ 0 (ω))) for all ωΩ, then there are measurable mappings ζ,θ:ΩX such that F(ω,(ζ(ω),θ(ω)))=g(ω,ζ(ω)) and F(ω,(θ(ω),ζ(ω)))=g(ω,θ(ω)) for all ωΩ, that is, F and g have a coupled random coincidence.

Proof Take ϕ(t)=t in Theorem 3.4. □

Corollary 3.6 Let (X,,d) be a complete separable partially ordered metric space, (Ω,Σ) be a measurable space, and F:Ω×(X×X)X and g:Ω×XX be mappings such that

  1. (i)

    g(ω,) is continuous for all ωΩ;

  2. (ii)

    F(,v), g(,x) are measurable for all vX×X and xX respectively;

  3. (iii)

    F(ω,) has the mixed g(ω,)-monotone property for each ωΩ; and suppose there exists k[0,1) such that

    ( d ( F ( ω , ( x , y ) ) , F ( ω , ( u , v ) ) ) ) k 2 ( d ( g ( ω , x ) , g ( ω , u ) ) + d ( g ( ω , y ) , g ( ω , v ) ) )

for all x,y,u,vX, for which g(ω,x)g(ω,u) and g(ω,y)g(ω,v) for all ωΩ.

Suppose g(ω×X)=X for each ωΩ, g is monotone increasing, and F and g are compatible random operators. Also suppose either

  1. (a)

    F(ω,) is continuous for all ωΩ or

  2. (b)

    X has the following property:

  3. (i)

    if a non-decreasing sequence { x n }X, then x n x for all n,

  4. (ii)

    if a non-increasing sequence { x n }X, then x n x for all n.

If there exist measurable mappings ζ 0 , η 0 :ΩX such that g(ω, ζ 0 (ω))F(ω,( ζ 0 (ω), η 0 (ω))) and g(ω, η 0 (ω))F(ω,( η 0 (ω), ζ 0 (ω))) for all ωΩ, then there are measurable mappings ζ,θ:ΩX such that F(ω,(ζ(ω),θ(ω)))=g(ω,ζ(ω)) and F(ω,(θ(ω),ζ(ω)))=g(ω,θ(ω)) for all ωΩ, that is, F and g have a coupled random coincidence point.

Proof Take ψ(t)= 1 k 2 t in Corollary 3.5. □

Remark 3.7 By defining g:Ω×XX as g(ω,x)=x for all ωΩ in Theorem 3.2-Corollary 3.6, we obtain corresponding coupled random fixed point results.

References

  1. Itoh S: A random fixed point theorem for a multi-valued contraction mapping. Pac. J. Math. 1977, 68: 85–90. 10.2140/pjm.1977.68.85

    Article  Google Scholar 

  2. Lin TC: Random approximations and random fixed point theorems for non-self maps. Proc. Am. Math. Soc. 1988, 103: 1129–1135. 10.1090/S0002-9939-1988-0954994-0

    Article  Google Scholar 

  3. Abbas M, Hussain N, Rhoades BE: Coincidence point theorems for multivalued f -weak contraction mappings and applications. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A, Mat. 2011, 105: 261–272.

    MathSciNet  Article  Google Scholar 

  4. Agarwal RP, O’Regan D, Sambandham M: Random and deterministic fixed point theory for generalized contractive maps. Appl. Anal. 2004, 83: 711–725. 10.1080/00036810410001657206

    MathSciNet  Article  Google Scholar 

  5. Beg I, Khan AR, Hussain N: Approximation of -nonexpansive random multivalued operators on Banach spaces. J. Aust. Math. Soc. 2004, 76: 51–66. 10.1017/S1446788700008697

    MathSciNet  Article  Google Scholar 

  6. Ćirić LB, Ješić SN, Ume JS: On random coincidence for a pair of measurable mappings. J. Inequal. Appl. 2006., 2006: Article ID 81045

    Google Scholar 

  7. Huang NJ: A principle of randomization of coincidence points with applications. Appl. Math. Lett. 1999, 12: 107–113.

    Article  Google Scholar 

  8. Khan AR, Hussain N: Random coincidence point theorem in Frechet spaces with applications. Stoch. Anal. Appl. 2004, 22: 155–167.

    MathSciNet  Article  Google Scholar 

  9. Sehgal VM, Singh SP: On random approximations and a random fixed point theorem for set valued mappings. Proc. Am. Math. Soc. 1985, 95: 91–94. 10.1090/S0002-9939-1985-0796453-1

    MathSciNet  Article  Google Scholar 

  10. Agarwal RP, Hussain N, Taoudi MA: Fixed point theorems in ordered Banach spaces and applications to nonlinear integral equations. Abstr. Appl. Anal. 2012., 2012: Article ID 245872

    Google Scholar 

  11. Chen Y-Z: Fixed points for discontinuous monotone operators. J. Math. Anal. Appl. 2004, 291: 282–291. 10.1016/j.jmaa.2003.11.003

    MathSciNet  Article  Google Scholar 

  12. Hussain N, Khan AR, Agarwal RP: Krasnosel’skii and Ky Fan type fixed point theorems in ordered Banach spaces. J. Nonlinear Convex Anal. 2010, 11(3):475–489.

    MathSciNet  Google Scholar 

  13. Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151

    MathSciNet  Article  Google Scholar 

  14. Hussain N, Alotaibi A: Coupled coincidences for multi-valued nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 82. doi:10.1186/1687–1812–2011–82

    Google Scholar 

  15. Hussain N, Latif A, Shah MH: Coupled and tripled coincidence point results without compatibility. Fixed Point Theory Appl. 2012., 2012: Article ID 77

    Google Scholar 

  16. Lakshimkantham V, Vatsala AS: General uniqueness and monotone iterative technique for fractional differential equations. Appl. Math. Lett. 2008, 21: 828–834. 10.1016/j.aml.2007.09.006

    MathSciNet  Article  Google Scholar 

  17. Luong NV, Thuan NX: Coupled fixed point in partially ordered metric spaces and applications. Nonlinear Anal. 2011, 74: 983–992. 10.1016/j.na.2010.09.055

    MathSciNet  Article  Google Scholar 

  18. Nieto JJ, Rodriguez-Lopez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5

    MathSciNet  Article  Google Scholar 

  19. Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435–1443. 10.1090/S0002-9939-03-07220-4

    MathSciNet  Article  Google Scholar 

  20. Ćirić LB, Lakshmikantham V: Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces. Stoch. Anal. Appl. 2009, 27: 1246–1259. 10.1080/07362990903259967

    MathSciNet  Article  Google Scholar 

  21. Zhu X-H, Xiao J-Z: Random periodic point and fixed point results for random monotone mappings in ordered Polish spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 723216

    Google Scholar 

  22. Alotaibi A, Alsulami S: Coupled coincidence points for monotone operators in partially ordered metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 44

    Google Scholar 

  23. Choudhury BS, Kundu A: A coupled coincidence point in partially ordered metric spaces for compatible mappings. Nonlinear Anal. 2010, 73: 2524–2531. 10.1016/j.na.2010.06.025

    MathSciNet  Article  Google Scholar 

  24. Lakshimkantham V, Ćirić LB: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020

    MathSciNet  Article  Google Scholar 

  25. Bhaskar TG, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017

    MathSciNet  Article  Google Scholar 

  26. Wagner DH: Survey of measurable selection theorems. SIAM J. Control Optim. 1977, 15: 859–903. 10.1137/0315056

    Article  Google Scholar 

  27. Rockafellar RT: Measurable dependence of convex sets and functions in parameters. J. Math. Anal. Appl. 1969, 28: 4–25. 10.1016/0022-247X(69)90104-8

    MathSciNet  Article  Google Scholar 

  28. Himmelberg CJ: Measurable relations. Fundam. Math. 1975, 87: 53–72.

    MathSciNet  Google Scholar 

  29. McShane EJ, Warified RB Jr.: On Filippov’s implicit functions lemma. Proc. Am. Math. Soc. 1967, 18: 41–47.

    Google Scholar 

Download references

Acknowledgements

The first and second author gratefully acknowledge the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.

Author information

Affiliations

Authors

Corresponding author

Correspondence to A Latif.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Hussain, N., Latif, A. & Shafqat, N. Weak contractive inequalities and compatible mixed monotone random operators in ordered metric spaces. J Inequal Appl 2012, 257 (2012). https://doi.org/10.1186/1029-242X-2012-257

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2012-257

Keywords

  • Measurable Mapping
  • Measurable Space
  • Cauchy Sequence
  • Monotone Property
  • Coincidence Point