A NOTE ON WELL-POSED NULL AND FIXED POINT PROBLEMS

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1 A NOTE ON WELL-POSED NULL AND 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 (Theorems 1 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 cone X + ={x X : x 0} such that x y for each pair of points x, y X + satisfying x y. Let(K,ρ) bea complete metric space. Denote by M the set of all continuous mappings A : K X. We equip the set M with the uniformity determined by the following base: E(ɛ) = { (A,B) M M : Ax Bx ɛ x K }, (1) where ɛ > 0. It is not difficult to see that this uniform space is metrizable (by a metric d) and complete. Denote by M p the set of all A M such that Ax X + x K, inf { Ax : x K } = 0. (2) It is not difficult to see that M p is a closed subset of (M,d). We can now state and prove our first result. Theorem 1. There exists an everywhere dense G δ subset M p such that for each A, the following properties hold. (1)Thereisaunique x K such that A x = 0. (2) For any ɛ > 0, there exist δ>0 and a neighborhood U of A in M p such that if B U and if x K satisfies Bx δ, then ρ(x, x) ɛ. Copyright 2005 Hindawi Publishing Corporation Fixed Point Theory and Applications 2005:2 (2005) DOI: /FPTA

2 208 Well-posed problems 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 each A M p and each ɛ > 0, there are Ā M p, δ>0, x K,andaneighborhood W of Ā in M p such that and if B W and z K satisfy Bz δ,then Let A M p and ɛ > 0. Choose ū X + such that (A,Ā) E(ɛ), (3) ρ(z, x) ɛ. (4) ū = ɛ 4, (5) and x K such that A x ɛ 8. (6) Since A is continuous, there is a positive number r such that { r<min 1, ɛ }, (7) 16 Ax A x ɛ 8 for each x K satisfying ρ(x, x) 4r. (8) By Urysohn s theorem, there is a continuous function φ : K [0,1] such that φ(x) = 1 foreachx K satisfying ρ(x, x) r, (9) φ(x) = 0 foreachx K satisfying ρ(x, x) 2r. (10) Define Āx = ( 1 φ(x) ) (Ax + ū), x K. (11) It is clear that Ā : K X is continuous. Now (9), (10), and (11)implythat Āx = 0 foreachx K satisfying ρ(x, x) r, (12) Āx ū for each x K satisfying ρ(x, x) 2r. (13) It is not difficult to see that Ā M p.weclaimthat(a,ā) E(ɛ).

3 S. Reich and A. J. Zaslavski 209 Let x 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 Now assume that (15)holds.Thenby(15), (11), and (5), Ax Āx = ū = ɛ 4. (16) Āx Ax = ( 1 φ(x) ) (Ax + ū) Ax ū + Ax ɛ 4 + Ax. (17) It follows from this inequality, (15), (8), and (6) that Āx Ax ɛ 4 + Ax < ɛ 2. (18) Therefore, in both cases, Āx Ax ɛ/2. Since this inequality holds for any x K, we conclude that (A,Ā) E(ɛ). (19) Consider now an open neighborhood U of Ā in M p such that { ( )} ɛ U B M p :(Ā,B) E. (20) 16 Let Relations (22), (21), (20), and (1)imply that We claim that B U, z K, (21) Bz ɛ 16. (22) Āz Bz + Āz Bz ɛ 16 + ɛ 16. (23) ρ(z, x) ɛ. (24)

4 210 Well-posed problems We assume the converse. Then by (7), ρ(z, x) > ɛ 2r. (25) When combined with (13), this implies that Āz ū. (26) It follows from this inequality, the monotonicity of the norm, (21), (20), (1), and (5) that Bz Āz ɛ 16 ū ɛ 16 = ɛ 4 ɛ 16 = 3ɛ 16. (27) This, however, contradicts (22). The contradiction we have reached proves (24) and Theorem 1 itself. Now assume that the set K is a subset of X and Denote by M n the set of all mappings A M such that ρ(x, y) = x y, x, y K. (28) Ax x x K, inf { Ax x : x K } = 0. (29) Clearly, M n is a closed subset of (M,d). Define a map J : M n M p by J(A)x = Ax x x K (30) and all A M n. Clearly, there exists J 1 : M p M n, and both J and its inverse J 1 are continuous. Therefore Theorem 1 implies the following result regarding the generic wellposedness of the fixed point problem for A M n. Theorem 2. There exists an everywhere dense G δ subset M n such that for each A, the following properties hold. (1)Thereisaunique x K such that A x = x. (2) For any ɛ > 0, there exist δ>0 and a neighborhood U of A in M n such that if B U and if x K satisfies Bx x δ, then x x ɛ. 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.

5 S. Reich and A. J. Zaslavski 211 References [1] F.S.DeBlasiandJ.Myjak,Sur la porosité de l ensemble des contractions sans point fixe [On the porosity of the set of contractions without fixed points],c.r.acad.sci.parissér. I Math. 308 (1989), no. 2, (French). [2] A.D.Ioffe and A. J. Zaslavski, Variational principles and well-posedness in optimization and calculus of variations, SIAMJ. ControlOptim.38 (2000), no. 2, [3] S.ReichandA.J.Zaslavski,Well-posedness of fixed point problems,fareastj.math.sci.(fjms), (2001), Special Volume (Functional Analysis and Its Applications), Part III, Simeon Reich: Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, O-okayama, Meguro-ku, Tokyo , Japan address: sreich@tx.technion.ac.il Alexander J. Zaslavski: Department of Mathematics, Technion Israel Institute of Technology, Haifa, Israel address: ajzasl@tx.technion.ac.il

6 Mathematical Problems in Engineering Special Issue on Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios Call for Papers Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from Qualitative Theory of Differential Equations, allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers. This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment. Authors should follow the Mathematical Problems in Engineering manuscript format described at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at mts.hindawi.com/ according to the following timetable: Guest Editors José Roberto Castilho Piqueira, Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, São Paulo, Brazil; piqueira@lac.usp.br Elbert E. Neher Macau, Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, São Paulo, Brazil ; elbert@lac.inpe.br Celso Grebogi, Department of Physics, King s College, University of Aberdeen, Aberdeen AB24 3UE, UK; grebogi@abdn.ac.uk Manuscript Due February 1, 2009 First Round of Reviews May 1, 2009 Publication Date August 1, 2009 Hindawi Publishing Corporation