Let (X,d) be a boundedly compact metric space and C be the set of all closed nonempty subsets of X. For a closed subset E, let G_E denote the set of all graphs of continuous functions in C(E,R^m); and let G = UG_E. A new topology, called the \tau-topology, is defined on G, and it is shown that the topological space (G,\tau) is homeomorphic to the quotient space [(C,\tau)x(X,R^m)]/W with a suitable equivalence relation W. The topological subspace (G_E,\tau) is homeomorphic to the space C(E,R^m) for every fixed closed subset E . The obtained results are useful for proving existence and continuous dependence results for solutions of ordinary and partial differential equations with hereditary structure.
A new graph topology intended for functional differential equations
BRANDI, Primo;CEPPITELLI, Rita
1996
Abstract
Let (X,d) be a boundedly compact metric space and C be the set of all closed nonempty subsets of X. For a closed subset E, let G_E denote the set of all graphs of continuous functions in C(E,R^m); and let G = UG_E. A new topology, called the \tau-topology, is defined on G, and it is shown that the topological space (G,\tau) is homeomorphic to the quotient space [(C,\tau)x(X,R^m)]/W with a suitable equivalence relation W. The topological subspace (G_E,\tau) is homeomorphic to the space C(E,R^m) for every fixed closed subset E . The obtained results are useful for proving existence and continuous dependence results for solutions of ordinary and partial differential equations with hereditary structure.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
