# On Quadruple Random Fixed Point Theorems in Partially Ordered Metric Spaces

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## Ordered Metric Spaces

Rajesh Shrivastava 1,Ramakant Bhardwaj 2, Manish Sharma 3

Professor, Department of Mathematics, Institute for Excellence in Higher Education, Bhopal 2Department of Mathematics,TIT Group of Institutes, (TIT-Excellence) Bhopal

3Truba Institute of Engg. and Information Technology Bhopal, M.P., India

1. Introduction

Bhaskar and Lakshmikantham in [15] introduced the concept of coupled fixed point of a mapping

and investigated the existence and uniquencess of a coupled fixed point theorem in partially ordered complete metric space. Lakshmikantham and Ciric [16] defined mixed g-monotone property and coincidence point in partially ordered metric space. V. Berinde and M. Borcut[18] introduced the concept of triple fixed point and proved some related theorems. Following this trand, Karapinar[19] introduced the nation of quadruple fixed point. The object of this note is to prove quadruple random fixed point theorem in partially ordered metric spaces.

2. Preliminaries

Definition 2.1[19]. Let be a partially ordered set and . The map has the mixed monotone property if is monotone nondecreasing in and and is monotone nonincreasing in ; that is, for

any ,

Definition 2.2[19]. An element is called a quadruple fixed point of a mapping if

Definition 2.3[20]. Let be a partially ordered set and and Then the map has the mixed g-monotone property if is monotone decreasing in and and is monotone g-non-increasing in and ; that is, for any .

g(

Definition 4[20]. An element is called a quadruple coincidence point of a mappings and if

Definition 5[20]. Let and be mappings. We say and are commutative if for all

Let denote the all functions which are continuous and satisfy that

(i)

(ii) for each

Let be a measurable space with a sigma algebra of subsets of and let be a metric space. A mapping is called measurable if for open subset of A mapping

is said to be random mapping if for each fixed the mapping is measurable. A measurable mapping is called a random fixed point of the random mapping if

for each A measurable mapping is called a random coincidence of

and if foe each

3. Main Result

Theorem: Let be a complete separable metric space, and let be a measurable space and . Let and be mappings such that

(2)

(2) are measurable for all and espectively,

(3) and are such that has the mixed g-monotone property and (1)

For all for which , and

for all Suppose for each And is continuous and commutes with and also suppose either

(a) is continuous or

(b) has the following property:

(i) If a non decreasing sequence then for all , (ii) If a non increasing sequence then for all If there exist measurable mappings such that

For all .

Then there are measurable mappings such that

For all

that is, and have a quadruple random coincidence point .

Proof . Let be a family of measurable mappings. Define a function as follows: Since is continuous for all we conclude that is

continuous for all Also, since is measurable for all we conclude that is measurable for all (see Wagner [11],page 868).Thus, is the Caratheodory function. Therefore, if

is a measurable mapping, then is also measurable (see [9]). Also, for each the function defined by is measurable; that is,

Now, we will construct four sequences of measurable mappings and and four

sequences and in X as follows:

Let such that

(2)

Since then by a sort of filippov measurable implicit function theorem [1,5,6,24] , we can choose such that

(3)

Again taking into account that and continuing this process, we can construct sequences { and in such that

(4)

We shall show that

(5)

For this purpose, we will use mathematical induction. By using (2) and (3), we obtain

For all .

Therefore (5) hold for

Suppose that (5) hold for some Then since has the mixed g-monotone property and by (4) we have

(4)

Thus (5) holds for all

Assume, for some that

Then, by (4), is a quadruple coincidence point of and From now on, assume for any that at least

Due to (1) and (4), we have

(6) (7) (8) (9) Having in mind that for all , so from (6 )-(9 ) we obtain that

(5)

(10)

It follows that

Thus, is a positive decreasing

sequence. Hence there exist such that

Suppose that Letting in (10),we obtain that

It is contraction. We deduce that

(11)

We shall show that there exists and are

Cauchy sequences. Assume the contrary, that is one of the sequence , or is not a cauchy sequence, that is,

or

Or

or

This means that there exist , for which we can find subsequences of integers and with such that

(12)

Further, corresponding to we can choose in such a way that it is the smallest integer with and satisfying (12). Then

(13)

(6)

Letting and using (11), we get

Similarly , we have

Again by (13) ,we have

Letting and using (11), we get

(14)

Similarly , we have

(15)

(16)

(17)

Using (12) and (14)-(17), we have

(18) Now using inequality (1) we obtain

(7)

(20) (21) (22) From (19) – (22) we deduce that

(23)

Letting in (23) and having in mind (18) we get that

It is contraction. Thus and are Cauchy sequences in

(8)

(24)

Since and are measurable, then the function

and , defined by

(25) Are measurable too. Thus

(26)

Since is continuous, (26) implies that

(27) by using the fact that and are commutative, From (4)

(28) (29) (30) (31) Now we will show that if the assumption (a) and (b) hold, then

For all .

Suppose (a) hold from (26), (27), (28) and the continuity of we obtain

(9)

and similarly

Thus , we proved that is a quadruple random coincidence of and Suppose, now the assumption (b) holds. Since

Therefore, by the triangle inequality

(10)

Letting and by (27), we get

But

Hence

Hence

Similarly, we can show that

.

For all

Thus we showed that is a quadruple random coincidence of and

References

[1] I. Beg and N. Shahzad, “Random fixed point theorems for non expansive and contractive-type random operators on Banach spaces,” Journal of Applied Mathematics and Stochastic Analysis,

vol. 7, no. 4, pp. 569–580,1994.

[2] A. T. Bharucha-Reid, “Fixed point theorems in probabilistic analysis,” Bulletin of the American Mathematical Society, vol. 82, no. 5, pp. 641–657, 1976.

[3] O. Hans, “Reduzierende zulliallige transformaten,” Czechoslovak Mathematical Journal, vol. 7, no. 82, pp. 154–158, 1957.

[4] O. Hans, “Random operator equations,” in Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, vol. 2, pp. 185–202, University California Press, Berkeley, Calif, USA, 1961.

[5] C. J. Himmelberg, “Measurable relations,” Fundamenta Mathematical ,vol. 87, pp. 53–72, 1975.

[6] S. Itoh, “A random fixed point theorem for a multi valued contraction mapping,” Pacific Journal of Mathematics, vol. 68, no. 1, pp. 85–90, 1977.

[7] S. Itoh, “Random fixed-point theorems with an application to random differential equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 261–273, 1979.

[8] N. S. Papageorgiou, “Random fixed point theorems for measurable multifunctions in Banach spaces,” Proceedings of the American Mathematical Society, vol. 97, no. 3, pp. 507–514, 1986.

[9] R. T. Rockafellar, “Measurable dependence of convex sets and functions on parameters,” Journal of Mathematical Analysis and Applications, vol. 28, pp. 4–25, 1969.

[10] A. ˇSpaˇcek, “Zuf¨allige gleichungen,” Czechoslovak Mathematical Journal, vol. 5, no. 80, pp. 462–466, 1955.

[11] D. H.Wagner, “Survey of measurable selection theorems,” SIAM Journal on Control and Optimization, vol. 15, no. 5, pp. 859–903,1977.

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[13] M. Turinici, “Abstract comparison principles and multivariable Gronwall-Bellman inequalities,” Journal of Mathematical Analysis and Applications, vol. 117, no. 1, pp. 100–127, 1986.

[14] A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society,vol. 132, no. 5, pp. 1435–1443, 2004. [15] T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis: Theory, Methods &Applications, vol. 65,no. 7, pp. 1379–

1393, 2006.

[16] V. Lakshmikantham and L. B. ´ Ciric, “Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces,”NonlinearAnalysis:Theory,Methods&Applications, vol. 70, no. 12, pp. 4341–4349, 2009. [17] L. B. ´ Ciri´c and V. Lakshmikantham, “Coupled random fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Stochastic Analysis and Applications, vol. 27, no.

6, pp. 1246–1259, 2009.

[18] V. Berinde and M. Borcut, “Tripled fixed point theorems for contractive type mappings in partially ordered metric spaces,” Nonlinear Analysis: Theory,Methods & Applications, vol. 74, no.

15, pp. 4889–4897, 2011.

[19] E. Karapinar, “Quartet fixed point for nonlinear contraction,”submitted to General Topology, 2011,

http://arxiv.org/pdf/1106 .5472.pdf.

[20] E. Karapinar and V. Berinde, “Quadruple fixed point theorems for nonlinear contractions in partially ordered metric spaces,” Banach Journal of Mathematical Analysis, vol. 6, no. 1, pp. 74–89, 2012.

[21] E. Karapınar andN. V. Luong, “Quadruple fixed point theorems for nonlinear contractions,” Computers & Mathematics with Applications, vol. 64, no. 6, pp. 1839–1848, 2012.

[22] Z. Mustafa, H. Aydi, and E. Karapınar, “Mixed 𝑔-monotone property and quadruple fixed point theorems in partially orderedmetric spaces,” Fixed Point Theory andApplications, vol.2012, article 71, 2012.

[23] E. Karapınar, W. Shatanawi, and Z. Mustafa, “Quadruple fixed point theorems under nonlinear contractive conditions in partially ordered metric spaces,” Journal of Applied Mathematics,vol. 2012, Article ID951912, 17 pages, 2012.

[24] E. J. McShane and R. B. Warfield, Jr., “On Filippov’s implicit functions lemma,” Proceedings of the American Mathematical Society, vol. 18, pp. 41–47, 1967.

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