FIXED POINT PROBLEMS

FIXED POINT PROBLEMS

SIMEON REICH AND ALEXANDER J. ZASLAVSKI Received 16 October 2004

We establish generic well-posedness of certain null and fixed point problems for ordered Banach space-valued continuous mappings.

The notion of well-posedness is of great importance in many areas of mathematics and its applications. In this note, we consider two complete metric spaces of continuous mappings and establish generic well-posedness of certain null and fixed point problems (Theorems1 and 2, resp.). Our results are a consequence of the variational principle established in [2]. For other recent results concerning the well-posedness of fixed point problems, see [1,3].

Let (X, · ,≥) be a Banach space ordered by a closed convex coneX+= {x∈X:x≥ 0}such thatx ≤ yfor each pair of pointsx,y∈X+satisfyingx≤y. Let (K,ρ) be a complete metric space. Denote byMthe set of all continuous mappingsA:K→X. We equip the setMwith the uniformity determined by the following base:

E()=

(A,B)∈M×M:Ax−Bx ≤∀x∈K, (1) where>0. It is not difficult to see that this uniform space is metrizable (by a metricd) and complete.

Denote byM_pthe set of allA∈Msuch that Ax∈X+ ∀x∈K,

infAx:x∈K=0. (2)

It is not difficult to see thatMpis a closed subset of (M,d).

We can now state and prove our first result.

Theorem1. There exists an everywhere denseG_δsubsetᏲ⊂M_psuch that for eachA∈Ᏺ, the following properties hold.

(1) There is a uniquex¯∈Ksuch thatAx¯=0.

(2) For any>0, there existδ >0and a neighborhoodUofAinM_psuch that ifB∈U and ifx∈KsatisfiesBx ≤δ, thenρ(x, ¯x)≤.

Copyright©2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:2 (2005) 207–211 DOI:10.1155/FPTA.2005.207

Proof. We obtain this theorem as a realization of the variational principle established in [2, Theorem 2.1] with f_A(x)= Ax,x∈K. In order to prove our theorem by using this variational principle, we need to prove the following assertion.

(A) For eachA∈M_pand each>0, there are ¯A∈M_p,δ >0, ¯x∈K, and a neighbor- hoodWof ¯AinM_psuch that

(A, ¯A)∈E(), (3)

and ifB∈Wandz∈KsatisfyBz ≤δ, then

ρ(z, ¯x)≤. (4)

LetA∈Mpand>0. Choose ¯u∈X+such that u¯ =

4, (5)

and ¯x∈Ksuch that

Ax¯ ≤

8. (6)

SinceAis continuous, there is a positive numberrsuch that r <min

1,

16

, (7)

Ax−A¯x ≤

8 for eachx∈Ksatisfyingρ(x, ¯x)≤4r. (8) By Urysohn’s theorem, there is a continuous functionφ:K→[0, 1] such that

φ(x)=1 for eachx∈Ksatisfyingρ(x, ¯x)≤r, (9) φ(x)=0 for eachx∈Ksatisfyingρ(x, ¯x)≥2r. (10) Define

Ax¯ =

1−φ(x)(Ax+ ¯u), x∈K. (11)

It is clear that ¯A:K→Xis continuous. Now (9), (10), and (11) imply that

Ax¯ =0 for eachx∈Ksatisfyingρ(x, ¯x)≤r, (12) Ax¯ ≥u¯ for eachx∈Ksatisfyingρ(x, ¯x)≥2r. (13) It is not difficult to see that ¯A∈M_p. We claim that (A, ¯A)∈E().

Letx∈K. There are two cases: either

ρ(x, ¯x)≥2r (14)

or

ρ(x, ¯x)<2r. (15)

Assume first that (14) holds. Then it follows from (14), (10), (11), and (5) that Ax−Ax¯ = u¯ =

4. (16)

Now assume that (15) holds. Then by (15), (11), and (5), Ax¯ −Ax =1−φ(x)(Ax+ ¯u)−Ax

≤ u¯+Ax ≤

4+Ax. (17)

It follows from this inequality, (15), (8), and (6) that Ax¯ −Ax ≤

4+Ax<

2. (18)

Therefore, in both cases,Ax¯ −Ax ≤/2. Since this inequality holds for anyx∈K, we conclude that

(A, ¯A)∈E(). (19)

Consider now an open neighborhoodUof ¯AinMpsuch that U⊂

B∈M_p: ( ¯A,B)∈E

16 . (20)

Let

B∈U, z∈K, (21)

Bz ≤

16. (22)

Relations (22), (21), (20), and (1) imply that

Az¯ ≤ Bz+Az¯ −Bz ≤ 16+

16. (23)

We claim that

ρ(z, ¯x)≤. (24)

We assume the converse. Then by (7),

ρ(z, ¯x)>≥2r. (25)

When combined with (13), this implies that

Az¯ ≥u.¯ (26)

It follows from this inequality, the monotonicity of the norm, (21), (20), (1), and (5) that Bz ≥ Az¯ −

16^≥ u¯ − 16

= 4⁻

16⁼

3 16.

(27)

This, however, contradicts (22). The contradiction we have reached proves (24) and

Theorem 1itself.

Now assume that the setKis a subset ofXand

ρ(x,y)= x−y, x,y∈K. (28) Denote byM_nthe set of all mappingsA∈Msuch that

Ax≥x ∀x∈K,

infAx−x:x∈K=0. (29)

Clearly,M_nis a closed subset of (M,d). Define a mapJ:M_n→M_pby

J(A)x=Ax−x ∀x∈K (30)

and allA∈M_n. Clearly, there existsJ⁻¹:M_p→M_n, and bothJ and its inverseJ⁻¹are continuous. ThereforeTheorem 1implies the following result regarding the generic well- posedness of the fixed point problem forA∈Mn.

Theorem2. There exists an everywhere denseGδsubsetᏲ⊂Mnsuch that for eachA∈Ᏺ, the following properties hold.

(1) There is a uniquex¯∈Ksuch thatAx¯=x.¯

(2) For any>0, there existδ >0and a neighborhoodUofAinMnsuch that ifB∈U and ifx∈KsatisfiesBx−x ≤δ, thenx−x¯ ≤.

Acknowledgments

The work of the first author was partially supported by the Israel Science Foundation founded by the Israel Academy of Sciences and Humanities (Grant 592/00), by the Fund for the Promotion of Research at the Technion, and by the Technion VPR Fund.

References

[1] F. S. De Blasi and J. Myjak,Sur la porosit´e de l’ensemble des contractions sans point fixe[On the porosity of the set of contractions without fixed points], C. R. Acad. Sci. Paris S´er. I Math.308 (1989), no. 2, 51–54 (French).

[2] A. D. Ioffe and A. J. Zaslavski,Variational principles and well-posedness in optimization and calculus of variations, SIAM J. Control Optim.38(2000), no. 2, 566–581.

[3] S. Reich and A. J. Zaslavski,Well-posedness of fixed point problems, Far East J. Math. Sci. (FJMS), (2001), Special Volume (Functional Analysis and Its Applications), Part III, 393–401.

Simeon Reich: Department of Mathematical and Computing Sciences, Tokyo Institute of Technol- ogy, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan

E-mail address:[email protected]

Alexander J. Zaslavski: Department of Mathematics, Technion – Israel Institute of Technology, 32000 Haifa, Israel

E-mail address:[email protected]