Optimal retraction problem for proper $k$-ball-contractive mappings in $C^m [0,1]$

Diana Caponetti, Alessandro Trombetta, Giulio Trombetta

DOI: http://dx.doi.org/10.12775/TMNA.2018.041

Abstract


In this paper for any $\eps > 0$ we construct a new proper $k$-ball-contractive retraction of the closed unit ball of the Banach space $C^m [0,1]$ onto its boundary with $k < 1+ \eps$, so that the Wo\'sko constant $W_\gamma (C^m [0,1])$ is equal to $1$.

Keywords


Retraction; measure of noncompactness; proper mapping

Full Text:

PREVIEW FULL TEXT

References


M. Annoni, Retrazioni e minimo spostamento negli spazi di Banach, Degree Thesis, (2005).

M. Annoni and E. Casini, An upper bound for the Lipschitz retraction constant in l1 , Studia Math. 180 (2007), no. 1, 73–76.

J. Appell, N.A. Erzakova, S. Falcon Santana and M. Väth, On some Banach space constants arising in nonlinear fixed point and eigenvalue theory, Fixed Point Theory Appl. 4 (2004), 317–336.

J. Appell, E. De Pascale and A. Vignoli, Nonlinear Spectral Theory, de Gruyter, Berlin, 2004.

Y. Benyamini and Y. Sternfeld, Spheres in infinite-dimensional normed spaces are Lipschitz contractible, Proc. Amer. Math. Soc. 88 (1983), no. 3, 439–445.

K. Bolibok, Minimal displacement and retraction problems for balls in Banach spaces, Mariae Curie-Sklodowska University, Ph.D. Thesis, 1999 (in Polish).

D. Caponetti and G. Trombetta, On proper k-ball contractive retractions in the Banach space BC([0, ∞)), Nonlinear Func. Anal. Appl. 10 (2005), no. 3, 461–467.

D. Caponetti, A.Trombetta and G. Trombetta, A note on boundary conditions for nonlinear operators, Indian J. Pure Appl. Math. 39 (2008), no. 5, 423–433.

D. Caponetti, A.Trombetta and G. Trombetta, On boundary conditions for wedge operators on radial sets, Numer. Funct. Anal. Optim. 29 (2008), no. 9–10, 979–986.

D. Caponetti, A.Trombetta and G. Trombetta, Examples of proper k-ball contractive retractions in F -normed spaces, J. Math. Anal. Appl. 335 (2007), no. 2, 1105–1118.

D. Caponetti, A.Trombetta and G. Trombetta, An extension of Guo’s theorem via k-ψ-contractive retractions, Nonlinear Anal. 64 (2006), no. 9, 1897–1907.

D. Caponetti, A.Trombetta and G. Trombetta, Proper 1-ball contractive retractions in Banach spaces of measurable functions, Bull. Austral. Math. Soc. 72 (2005), no. 2, 299–315.

E. Casini and K. Goebel, Why and how much Brouwer’s fixed point theorem fails in noncompact setting?, Milan J. Math. 78 (2010), no. 2, 371–394.

E. Casini and L. Piasecki, The minimal displacement and optimal retraction problems in some Banach spaces, J. Nonlinear Convex Anal. 18 (2017), no. 1, 61–71.

V. Colao, A. Trombetta and G. Trombetta, Hausdorff norms of retractions in Banach spaces of continuous functions, Taiwanese J. Math. 13 (2009), no. 4, 1139–1158.

W. Feng, A new spectral theory for nonlinear operators and its applications, Abstr. Appl. Anal. 2 (1997), 163–183.

K. Goebel, On the minimal displacement of points under Lipschitzian mappings, Pacific J. Math. 45 (1973), 151–163.

K. Goebel, On the problem of retracting balls onto their boundary, Abstr. Appl. Anal. 2 (2003), 101–110.

K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, 1990.

K. Goebel, G. Marino, L. Muglia and R. Volpe, The retraction constant and minimal displacement characteristic of some Banach spaces, Nonlinear Anal. 67 (2007), no. 3, 735–744.

I.-S. Kim and M. Väth, Some remarks on measures of noncompactness and retractions onto spheres, Topology Appl. 154 (2007), no. 17, 3056–3069.

V. Klee, Convex bodies and periodic homeomorphisms in Hilbert spaces, Trans. Amer. Math. Soc. 74 (1953), 10–43.

V. Klee, Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955), 30–45.

B. Nowak, On the Lipschitz retraction of the unit ball in infinite dimensional Banach spaces onto boundary, Bull. Acad. Polon. Sci. 27 (1979), 861–864.

L. Piasecki, Retracting ball onto sphere in BC0 (R), Topol. Methods Nonlinear Anal. 33 (2009), no. 2, 307–313.

L. Piasecki, Retracting a ball onto a sphere in some Banach spaces, Nonlinear Anal. 74 (2011), no. 2, 396–399.

M. Väth, On the minimal displacement problem of γ-Lipschitz maps and γ-Lipschitz retractions onto the sphere, Z. Anal. Anwend. 21 (2002), no. 4, 901–914.

J. Wośko, An example related to the retraction problem, Ann. Univ. Mariae CurieSklodowska 45 (1991), 127–130.


Refbacks

  • There are currently no refbacks.

Partnerzy platformy czasopism