Since F is a compact multivalued map, it is easy to check that Ω ( x ) is a bounded set for all x ∈ X.
Since F is a compact multivalued map, it is easy to check that N ( x ) is a bounded set for all x ∈ X.
The resulting structure will satisfy all the axioms of Zermelo set theory except Separation, and will satisfy all instances of Separation in which each quantifier is bounded in a set (this boundedness comes in because each instance of Comprehension in TST has each quantifier bounded in a type, which becomes a bounding set for that quantifier in the interpretation of Mac Lane set theory).
We establish a general existence result for solutions lying in an admissible bounding set for the system of ordinary differential equations (1) satisfying boundary conditions (2) and impulses (3).
A family of processes, is said to be uniformly -limit compact if for any and bounded set, the set is bounded for every and is precompact set as.
For any given bounded set, there exists such that for all.
Step 5. We show that for every bounded set.
We say that G is bounded on bounded sets if G ( B ) is bounded in X for each bounded set B of X.
(H11) The function is continuous and completely continuous, and for every bounded set, the set is equicontinuous in.
From ((f_{1})), (3.1) and (F x, - u) = F x,u)), it is easy to see that (varphi_{rho,lambda} (u)) maps a bounded set into bounded sets uniformly for (lambdain[1,2]), and varphi_{rho,lambda} ( - u) = varphi_{rho,lambda} (u quadmbox{for all } (lambda,u) in[1,2] times X, which shows ((A_{1})) of Lemma 2.4 is satisfied.
