# Article

Full entry | PDF   (0.2 MB)
Keywords:
Kuratowski convergence; Attouch-Wets convergence; $\tau$-convergence; Kuratowski convergence on compacta and Hausdorff metric convergence on compacta
Summary:
This paper completes and improves results of [10]. Let $(X,d_{_X})$, $(Y,d_{_Y})$ be two metric spaces and $G$ be the space of all $Y$-valued continuous functions whose domain is a closed subset of $X$. If $X$ is a locally compact metric space, then the Kuratowski convergence $\tau_{_K}$ and the Kuratowski convergence on compacta $\tau_{_K}^c$ coincide on $G$. Thus if $X$ and $Y$ are boundedly compact metric spaces we have the equivalence of the convergence in the Attouch-Wets topology $\tau_{_{AW}}$ (generated by the box metric of $d_{_X}$ and $d_{_Y}$) and $\tau_{_K}^c$ convergence on $G$, which improves the main result of [10]. In the second part of paper we extend the definition of Hausdorff metric convergence on compacta for general metric spaces $X$ and $Y$ and we show that if $X$ is locally compact metric space, then also $\tau$-convergence and Hausdorff metric convergence on compacta coincide in $G$.
References:
[1] Beer G.: Topologies on Closed and Closed Convex Sets. Kluwer, 1993. MR 1269778 | Zbl 0792.54008
[2] Brandi P., Ceppitelli R.: Esistenza, unicitá e dipendenza continua per equazioni differenziali in una struttura ereditaria. Atti Sem. Mat. Fis. Univ. Modena 35 (1987), 357-363. MR 0937975
[3] Brandi P., Ceppitelli R.: Existence, uniqueness and continuous dependence for hereditary differential equations. J. Diff. Equations 81 (1989), 317-339. MR 1016086 | Zbl 0709.34062
[4] Brandi P., Ceppitelli R.: A new graph topology. Connections with compact open topology. Appl. Analysis 53 (1994), 185-196. MR 1379407
[5] Brandi P., Ceppitelli R.: A new graph topology intended for functional differential equations. Atti Sem. Mat. Univ. Modena 54 (1996), 43-52. MR 1405228 | Zbl 0890.54010
[6] Brandi P., Ceppitelli R.: A hypertopology intended for functional differential equations. Appl. Analysis 67 (1997), 73-88. MR 1609874 | Zbl 0886.54009
[7] Brandi P., Ceppitelli R., Holá L'.: Topological properties of a new graph topology. J. Convex Anal. 5 (1998), 2 1-12. MR 1713949
[8] Ceppitelli R., Faina L.: Differential equations with hereditary structure induced by a Volterra type property. preprint. MR 1821774 | Zbl 0988.34049
[9] Holá L'.: The Attouch-Wets topology and a characterization of normable linear spaces. Bull. Austral. Math. Soc. 44 (1991), 11-18. MR 1120389
[10] Piccione P., Sampalmieri R.: Attouch-Wets convergence and Kuratowski convergence on compact sets. Comment. Math. Univ. Carolinae 36 (1995), 551-562. MR 1364496 | Zbl 0844.54010
[11] Sampalmieri R.: Kuratowski convergence on compact sets. Atti Sem. Mat. Fis. Univ. Modena 39 (1992), 381-390. MR 1200296 | Zbl 0770.54016

Partner of