Periodic and fixed points of rotative random op- op-erators

In this section, we first show an existence of a random fixed point of a nonexpansive rotative random operator which not only provides a random analogue of theorem 17.1 of [11] (see also [12]) but also improves theorem 2.1 of [17] in the sense that it does not require the boundedness of T(ω, F)for anyω ∈Ω. Moreover we replace continuous condensing random operator by nonexpansive rotative random operator.

Periodic point problems were systematically studied since the beginning of fifties (see [9, 10, 13, 14, 19, 20]). We show some results on the existence of random periodic points of random single valuedǫ- contractive operator in the setting of a separable metric space.

Periodic fixed points of random operators 43 Theorem 3.1. Let F be a nonempty closed and convex subset of a separable Ba-nach space X and T: Ω×F → F be a nonexpansive rotative random operator.

Then T has a random fixed point.

Proof. Letξ: Ω→F be any fixed measurable mapping. For0 < α <1and any arbitrary measurable mappingη: Ω→F,define Tα: Ω×F →F as,

Tα(ω, η(ω)) = (1−α)ξ(ω) +αT(ω, η(ω)).

Note that for eachα, the random operatorTα has Lipschitz constantα. we may apply [8] to obtain the sequence of random operators Fα: Ω×F → F such that Tα(ω, Fα(ω, ξ(ω))) =Fα(ω, ξ(ω)),for everyω∈Ω.Consequently, we have

Fα(ω, ξ(ω)) = (1−α)ξ(ω) +αT(ω, Fα(ω, ξ(ω))).

It can be verified that each Fα is nonexpansive random operator. By iteratingFα

we obtain

F_α^k(ω, ξ(ω)) = (1−α)F_α^k−1(ω, ξ(ω)) +αT(ω, F_α^k(ω, ξ(ω))), k∈N. (3.1) Note that,

(1−α)Fα(ω, ξ(ω))

= (1−α)ξ(ω) +αT(ω, Fα(ω, ξ(ω)))−αFα(ω, ξ(ω))

= (1−α)ξ(ω) +α(T(ω, Fα(ω, ξ(ω)))−Fα(ω, ξ(ω))).

Thus for each ω∈Ω

(1−α)(ξ(ω)−Fα(ω, ξ(ω)))

=α(Fα(ω, ξ(ω))−T(ω, Fα(ω, ξ(ω)))). (3.2) Now suppose T is a (a, n)-rotative random operator, that is

kξ(ω)−Tⁿ(ω, ξ(ω))k6akξ(ω)−T(ω, ξ(ω))k, for everyω∈Ω.Now,

Fα(ω, ξ(ω))−F_α²(ω, ξ(ω))

=

(1−α)ξ(ω) +αT(ω, Fα(ω, ξ(ω)))−(1−α)Fα(ω, ξ(ω))

−αT(ω, F_α²(ω, ξ(ω)))

=

(1−α)(ξ(ω)−Fα(ω, ξ(ω))) +α(T(ω, Fα(ω, ξ(ω)))

−αT(ω, F_α²(ω, ξ(ω)))

=

α(Fα(ω, ξ(ω))−T(ω, Fα(ω, ξ(ω)))) +α(T(ω, Fα(ω, ξ(ω)))

−αT(ω, F_α²(ω, ξ(ω)))

=αF_α(ω, ξ(ω))−T(ω, F_α²(ω, ξ(ω)))

44 I. Beg, M. Abbas, A. Azam 6αkFα(ω, ξ(ω))−Tⁿ(ω, Fα(ω, ξ(ω)))k

+αTⁿ(ω, Fα(ω, ξ(ω)))−T(ω, F_α²(ω, ξ(ω))) 6αakFα(ω, ξ(ω))−T(ω, Fα(ω, ξ(ω)))k

+αTⁿ⁻¹(ω, Fα(ω, ξ(ω)))−F_α²(ω, ξ(ω))

= (1−α)akFα(ω, ξ(ω))−ξ(ω)k

+αTⁿ⁻¹(ω, Fα(ω, ξ(ω)))−F_α²(ω, ξ(ω)),

for everyω∈Ω.Now we claim that the following inequality holds for everyω∈Ω andm>2.

αT^m−1(ω, Fα(ω, ξ(ω)))−F_α²(ω, ξ(ω)) 6(m−1)−mα+α^mkξ(ω)−Fα(ω, ξ(ω))k

+α^mFα(ω, ξ(ω))−F_α²(ω, ξ(ω)). (3.3) For this consider,

αT(ω, Fα(ω, ξ(ω)))−Fα²(ω, ξ(ω))

=αT(ω, Fα(ω, ξ(ω)))−(1−α)Fα(ω, ξ(ω))−αT(ω, F_α²(ω, ξ(ω)))

=α

(1−α)(T(ω, Fα(ω, ξ(ω)))−Fα(ω, ξ(ω)))−α(T(ω, F_α²(ω, ξ(ω)))

−T(ω, Fα(ω, ξ(ω))))

6(1−α)kα(T(ω, Fα(ω, ξ(ω)))−Fα(ω, ξ(ω)))k

+α²T(ω, F_α²(ω, ξ(ω)))−T(ω, Fα(ω, ξ(ω)))

= (1−α)²kξ(ω)−Fα(ω, ξ(ω))k+α²T(ω, F_α²(ω, ξ(ω)))−T(ω, Fα(ω, ξ(ω))) 6(1−α)²kξ(ω)−Fα(ω, ξ(ω))k+α²F_α²(ω, ξ(ω))−Fα(ω, ξ(ω)).

So (3.3) is valid for m= 2and for anyω∈Ω.

Assuming the validity of (3.3) form=j and for anyω∈Ω,consider αT^j(ω, Fα(ω, ξ(ω)))−F_α²(ω, ξ(ω))

=αT^j(ω, Fα(ω, ξ(ω)))−(1−α)Fα(ω, ξ(ω))−αT(ω, F_α²(ω, ξ(ω)))

=α

(1−α)(T^j(ω, Fα(ω, ξ(ω)))−Fα(ω, ξ(ω))) +α(T^j(ω, Fα(ω, ξ(ω)))

−T(ω, F_α²(ω, ξ(ω))))

6α(1−α)T^j(ω, Fα(ω, ξ(ω)))−Fα(ω, ξ(ω))

+α²T^j(ω, Fα(ω, ξ(ω)))−T(ω, F_α²(ω, ξ(ω))) 6jα(1−α)kFα(ω, ξ(ω))−T(ω, Fα(ω, ξ(ω)))k

+α²T^j−1(ω, Fα(ω, ξ(ω)))−F_α²(ω, ξ(ω)) 6jα(1−α)kFα(ω, ξ(ω))−T(ω, Fα(ω, ξ(ω)))k

+α[(j−1)−jα+α^j]kξ(ω)−Fα(ω, ξ(ω))k +α^j+1Fα(ω, ξ(ω))−F_α²(ω, ξ(ω))

6j(1−α)²+α²[(j−1)−jα+α^j]kξ(ω)−Fα(ω, ξ(ω))k

Periodic fixed points of random operators 45 +α^j+1Fα(ω, ξ(ω))−F_α²(ω, ξ(ω))

6[j−(j+ 1)α+α^j+1]kξ(ω)−Fα(ω, ξ(ω))k +α^j+1Fα(ω, ξ(ω))−Fα²(ω, ξ(ω)).

So by induction inequality(3.3)is valid for everyω∈Ωand m>2.

Now consider, forω∈Ω

F_α(ω, ξ(ω))−F_α²(ω, ξ(ω)) 6(1−α)akFα(ω, ξ(ω))−ξ(ω)k

+αTⁿ⁻¹(ω, Fα(ω, ξ(ω)))−F_α²(ω, ξ(ω)) 6(1−α)akFα(ω, ξ(ω))−ξ(ω)k

+ [(n−1)−nα+αⁿ]kξ(ω)−Fα(ω, ξ(ω))k +αⁿF_α(ω, ξ(ω))−F_α²(ω, ξ(ω)).

It further implies that

(1−αⁿ)Fα(ω, ξ(ω))−F_α²(ω, ξ(ω))

6[(1−α)a+ (n−1)−nα+αⁿ]kξ(ω)−Fα(ω, ξ(ω))k, for everyω∈Ω.Now we arrive at

Fα(ω, ξ(ω))−F_α²(ω, ξ(ω))

6(1−αⁿ)⁻¹[(1−α)a+ (n−1)−nα+αⁿ]kξ(ω)−Fα(ω, ξ(ω))k 6(a+n)(1−α)(1−αⁿ)⁻¹−1kξ(ω)−Fα(ω, ξ(ω))k

= [(a+n)(

nX−1 i=0

αⁱ)⁻¹−1]kξ(ω)−Fα(ω, ξ(ω))k

=g(α)kξ(ω)−Fα(ω, ξ(ω))k,

for every ω ∈ Ω, where g(α) = [(a+n)(Pn−1

i=0 αⁱ)⁻¹−1]. Since g is continuous and decreasing for α∈(0,1]with g(1) = ^a_n < 1, there exists b ∈(0,1]such that g(1)<1 for α∈(b,1]. For suchα, the sequence of measurable mappings defined byηn(ω) =F_αⁿ(ω, ξ(ω))→η(ω), for eachω∈Ω, η: Ω→F,being the limit of the sequence of measurable functions, is also measurable (see remark 2.6).From (3.1)

it follows that η is a random fixed point ofT.

Example 3.2. LetΩ = [0,1]andΣbe the sigma algebra of Lebesgue’s measurable subsets of Ω. Take X = R with d(x, y) = |x−y|, for x, y ∈ R. Define random operatorT from Ω×X toX as,T(ω, x) =ω−x.

Define a fixed measurable mapping ξ: Ω→ X as ξ(ω) = ^ω₃, for every ω ∈Ω.

Note that T is nonexpansive random operator. Since random operator equation T²(ω, ξ(ω)) = ξ(ω)holds for every ω ∈ Ω, therefore it is (2,1)−rotative random operator. Thus the conditions of Theorem 3.1 are satisfied. Moreover a measurable mappingη: Ω→X defined asη(ω) =^ω₂,for everyω∈Ω,serve as a unique random fixed point ofT.

46 I. Beg, M. Abbas, A. Azam Theorem 3.3. Let X be a separable metric space and T: Ω×X → X be a ǫ-contractive random operator. Letξ0: Ω→X be any measurable mapping such that a sequence {Tⁿ(ω, ξ0(ω))} has a point wise convergent subsequence of measurable mappings. Then T has a random periodic point.

Proof. Let{Tⁿⁱ(ω, ξ0(ω))}be a subsequence of{Tⁿ(ω, ξ0(ω))}such thatTⁿⁱ(ω, ξ0

(ω))→ξ(ω)for eachω∈Ωasni → ∞where{ni}is a strictly increasing sequence of positive integers. The mapping ξ: Ω → X being point wise limit of sequence of measurable mappings is measurable. Define sequence of measurable mappings ξi: Ω →X as ξi(ω) =Tⁿⁱ(ω, ξ0(ω)).Given ǫ >0, there exists an integern0 such that

d(ξi(ω), ξ(ω))< ǫ

4, fori>n0 andω∈Ω.

Put k=ni+1−ni.Consider,

d(ξi+1(ω), T^k(ω, ξ(ω))) =d(T^k(ω, ξi(ω)), T^k(ω, ξ(ω)))

< d(ξi(ω), ξ(ω))< ǫ

4, for eachω∈Ω.

Now,

d(ξ(ω), T^k(ω, ξ(ω)))

6d(ξi+1(ω), T^k(ω, ξ(ω))) +d(ξi+1(ω), ξ(ω))

< ǫ 4+ ǫ

4 = ǫ

2, for everyω∈Ω.

Now we claim that ξ is a random periodic point ofT. To prove this, assume that η: Ω→X be any measurable mapping such thatη(ω) =T^k(ω, ξ(ω))but

η(ω)6=ξ(ω), for some ω∈Ω. (3.4) Which implies that0< d(η(ω), ξ(ω))< ǫ. AsT is aǫ- contractive random operator therefore forω∈Ωfor which (3.4) holds, we have

d(T(ω, ξ(ω)), T(ω, η(ω)))< d(ξ(ω), η(ω)).

Define h: Ω×X² →R as,h(ω, x(ω), y(ω)) = ^d(T(ω,x(ω)),T(ω,y(ω)))

d(x(ω),y(ω)) ,where x(ω)6= y(ω) ∈ X for each ω ∈ Ω. Now h(ω, ., .) is continuous at (ξ(ω), η(ω)) for every ω∈Ωfor which (3.4) is valid.

Take 0 < α < 1, then there exists δ > 0 such that x(ω) ∈ B(ξ(ω), δ) and y(ω)∈B(η(ω), δ)gives

d(T(ω, x(ω)), T(ω, y(ω)))< αd(x(ω), y(ω)).

As, lim

r→∞T^k(ω, ξr(ω)) = T^k(ω, ξ(ω)) = η(ω), for every ω ∈ Ω. So there exists n1>n0 such that

d(ξr(ω), ξ(ω))< δ

Periodic fixed points of random operators 47 and

d(T^k(ω, ξr(ω)), η(ω))< δ, forr>n1 andω∈Ω.Hence we have

d(T(ω, ξr(ω)), T(ω, T^k(ω, ξr(ω))))< αd(ξr(ω), T^k(ω, ξr(ω))). (3.5) Consider,

d(ξr(ω), T^k(ω, ξr(ω)))

6d(ξr(ω), ξ(ω)) +d(ξ(ω), T^k(ω, ξ(ω))) +d(T^k(ω, ξ(ω)), T^k(ω, ξr(ω)))

< ǫ 4 +ǫ

2+ ǫ

4 =ǫ, (3.6)

for r>n1 >n0 andω ∈Ωfor which (3.4) holds. Now using (3.5) and (3.6), we have

d(T(ω, ξr(ω)), T(ω, T^k(ω, ξr(ω))))

< αd(ξr(ω), T^k(ω, ξr(ω)))< d(ξr(ω), T^k(ω, ξr(ω)))< ǫ,

for r>n1. Since T is a ǫ- contractive random operator so forr >n1 andq >0, we have

d(T^q(ω, ξr(ω)), T^q(ω, T^k(ω, ξr(ω))))

< d(ξr(ω), T^k(ω, ξr(ω)))< ǫ α.

Put q=nr+1−nr,we haved(ξr+1(ω), T^k(ω, ξr+1(ω)))<_α^ǫ.Hence, d(ξs(ω), T^k(ω, ξs(ω)))< ǫα^s−r.

Now,

d(ξ(ω), η(ω)) 6 d(ξ(ω), ξs(ω)) +d(ξs(ω), T^k(ω, ξs(ω))) +d(T^k(ω, ξs(ω)), η(ω))→0, ass→ ∞.

for thoseω∈Ωfor which (3.4) holds. This contradiction concludes the result.

Corollary 3.4. If in theorem 3.2, the random periodic pointξ(say) ofT satisfies d(ξ(ω), T(ω, ξ(ω)))< ǫ, for every ω∈Ω. (3.7) Then ξ is a random fixed point ofT.

Proof. Letkbe the positive integer such thatT^k(ω, ξ(ω)) =ξ(ω),for everyω∈Ω.

Ifξis not a random fixed point ofT,thenξ(ω)6=T(ω, ξ(ω)for someω∈Ω.Since T isǫ- contractive random operator, using (3.7) we have

d(ξ(ω), T(ω, ξ(ω))) =d(T^k(ω, ξ(ω)), T^k+1(ω, ξ(ω)))

< d(ξ(ω), T(ω, ξ(ω))).

This contradiction concludes the proof.

48 I. Beg, M. Abbas, A. Azam Remark 3.5. IfX is a separable compact metric space andT: Ω×X →X is an ǫ- contractive random operator. Then applying theorem 3.3, we conclude that T has a random periodic point.

Theorem 3.6. Let X be a separable compact metric space and T: Ω×X → X be an ǫ- contractive random operator. Then T has finitely many random periodic points.

Proof. Let ξ, ζ: Ω → X be two random periodic points ofT with ξ(ω) 6=ζ(ω) and d(ξ(ω), ζ(ω)) < ǫ for some ω ∈ Ω. Let m, n > 1 be two integers such that T^m(ω, ξ(ω)) = ξ(ω) and Tⁿ(ω, ζ(ω)) = ζ(ω) for every ω ∈ Ω. Obviously T^mn(ω, ξ(ω)) =ξ(ω)andT^mn(ω, ζ(ω)) =ζ(ω)for eachω ∈Ω. Now consider,

d(ξ(ω), ζ(ω)) =d(T^mn(ω, ξ(ω)), T^mn(ω, ζ(ω)))

< d(ξ(ω), ζ(ω)),

which is contradiction. Therefore any two random periodic point of T must be at leastǫ- apart. Compactness ofX prevents us defining infinitely many random

periodic points from Ω×X to X.

Acknowledgement. The authors are thankful to referee for precise remarks to improve the presentation of the paper.

References

[1] Aubin, J. P., and Frankowska, H., Set-Valued Analysis, Birkhauser, Boston, 2009.

[2] Beg, I., Random fixed points of random operators satisfying semicontractivity con-ditions,Math. Japan., 46 (1) (1997), 151–155.

[3] Beg, I., Random Edelstein theorem,Bull.Greek Math. Soc., 45 (2001), 31–41.

[4] Beg, I., Minimal displacement of random variables under Lipschitz random maps, Topological Methods in Nonlinear Anal., 19 (2002), 391–397.

[5] Beg, I., and Abbas, M., Iterative procedures for solutions of random operator equations in Banach spaces,J. Math. Anal. Appl., 315(1)(2006), 181–201.

[6] Beg, I., andThakur, B.S., Solution of random operator equations using general composite implicit iteration process,Int. J. Modern Math., 4(1)(2009), 19–34.

[7] Bharucha-Reid, A.T.,Random Integral Equations, Academic Press, New York and London, 1972.

[8] Bharucha- Reid, A.T., Fixed point theorems in probabilistic analysis,Bull. Amer.

Math. Soc., 82 (1976), 641–645.

[9] Edelstein, M., On fixed and periodic points under contractive mapping,J. London Math. Soc., 37 (1962), 74–79.

[10] Fournier, G., and Górniewicz, L., The Lefschetz fixed point theorem for some non compact multivalued maps,Fund. Math., 94 (1977), 245–254.

Periodic fixed points of random operators 49 [11] Goebel, K., andKirk, W.A.,Topics in Metric Fixed Point Theory, Cambridge

University Press, Cambridge 1990.

[12] Goebel, K., and Koter, M., A remark on nonexpansive mappings, Canadian Math. Bull., 24 (1981), 113–115.

[13] Górniewicz, L.,Topological Fixed Point Theory of Multivalued Mappings, Kluwer, Dordrecht, 1999.

[14] Halpern, B., Periodic points on tori,Pacific J. Math., 83 (1979), 117–133.

[15] Hanš, O., Reduzierende zulliällige transformaten, Czechoslovak Math. J., 7 (1957), 154–158.

[16] Hanš, O., Random operator equations, in: Proceedings of the Fourth Berkeley Sym-posium on Mathematical Statistics and Probability, Vol. II, Part I, University of Cal-ifornia Press, Berkeley, 1961, 185–202.

[17] Itoh, S., Random fixed point theorems with an application to random differential equations in Banach spaces,J. Math. Anal. Appl., 67 (1979), 261–273.

[18] Kalmoun, E.M., Some deterministic and random vector equilibrium problems, J.

Math. Anal. Appl., 267 (2002), 62–75.

[19] Kampen, J., On fixed points of maps and iterated maps and applications,Nonlinear Anal., 42(3) (2000), 509–532.

[20] Matsuoka, T., The number of periodic points of smooth maps, Ergod. Th. and Dynam. Sys., 8 (1989), 153–163.

[21] O’Regan, D., A continuation type result for random operators,Proc. Amer. Math.

Soc., 126, 7(1998), 1963–1971.

[22] Papageorgiou, N.S., Random fixed point theorems for measurable multifunctions in Banach spaces,Proc. Amer. Math. Soc., 97 (1986), 507–514.

[23] Papageorgiou, N.S., On measurable multifunctions with stochastic domain, J.

Austral. Math. Soc.(Ser. A), 45 (1988), 204–216.

[24] Spacek, A., Zufällige Gleichungen,Czechoslovak Math. J., 5 (1955), 462–466.

Ismat Beg, Mujahid Abbas

Center for Advanced Studies in Mathematics, Lahore University of Management Sciences, Lahore-54792, Pakistan

Phone: 0092-42-35608229 Fax: 0092-42-35722591 e-mail: ibeg@lums.edu.pk

mujahid@lums.edu.pk Akbar Azam

Department of Mathematics,

COMSATS Institute of Information Technology, Islamabad, Pakistan

Annales Mathematicae et Informaticae 37(2010) pp. 51–75

http://ami.ektf.hu

Evaluating a probabilistic model checker for

In document Annales Mathematicae et Informaticae (37.) (Pldal 44-53)