On the structure of the set of solutions of nonlinear equations

On the Aronszajn Property for an Integro-Differential Equation in Banach Spaces

“…Since w n uniformly converges to 0 as n -• oo, from (10) we conclude that the sequence (u n ) converges uniformly on Q to a limit u.…”

“…Let us recall that in the case of ordinary differential equations this problem was investigated by many authors. For example see to [10], [11] and [6], [9].…”

On Systems of Differential Equations in Banach Spaces

“…Now, by Th. 2.4 from [11], we conclude that the set T -1 (0) is a compact Rs. It is clear that S = T _1 (0).…”

On the Structure of Solution Set of an Integro-Differential Equation in Banach Spaces

“…where Vn is a compact subset of B defined by (5) and (6) it follows that un(W) is a relatively compact set in C. Since, by (9), un is continuous, it is completely continuous mapping W -» W. Furthermore, analogously as in [4], the inequality (7) …”

“…it is homeomorphic to the intersection of a decreasing sequence of compact absolute retracts. Let us mention that in the case of ordinary differential equations this problem was studied by many authors (for example see to [9], [10] and [5], [6], [8]). Fix r € J and x € W. As the set J 2 x x( J) is compact, from the continuity of g it follows that for each e > 0 there exists 6 > 0 such that s, x(s)) -g(r, s, x(s))|| < £ for t, s E J with |i -r| < S.…”