Boundary Value Problems for Second Order Differential Equations in Convex Subsets of a Banach Space

Abstract

ABSTRACT. Let E be a real Banach space, C a closed, convex subset of E and /: [0, 1 ] XEXÎ -► E be continuous. Let u0, ux £ C and consider the boundary value problem (») u" = f(t, u, u'), u(0) = u0, u(l) = I/j.We establish sufficient conditions in order that (*) have a solution u: [0,1) -► C.

Invariant sets and the hokohara-kneser property for systems of parabolic partial differential equations

“…Under somewhat weaker assumptions than in [ 15] we not only prove invariance of that convex region but also demonstrate existence of solutions. Using some ideas suggested by [11] we further show that essentially the same type of result holds for convex sets with empty interior.…”

“…It easily follows that Theorem 2 may be applied to conclude that for every e G (0,1], (11) has a solution u € G C 2+s ' 1+s/2 (fï r ) such that u € : n r -» S and | Vw f (x, £)| ^ M, where M is a constant independent of €, not necessarily equal to the constant M of the proof of Theorem 2. Thus {w € }o<€^i is precompact in C 10 (n T ).…”

“…We next consider the case where S has possibly empty interior. The idea of the proof was suggested by the results of [11]. THEOREM 4.…”

Boundary value problems with solutions in convex sets

“…It is known [9,10,14] that property (1.1), in tandem with certain compactness or Lipschitz conditions, leads to the existence of solutions u : [0, 1] → C of the boundary value problem u (t) + f (t, u(t), u (t)) = 0, u(0) = x 0 , u(1) = x 1 , (…”

An Invariance Property of Solutions to Second Order Differential Inequalities in Ordered Banach Spaces

“…The approach used in these results on flow invariant sets has also recently been applied to show existence of solutions to boundary value problems for second order differential equations in a Banach space (of. [8]). Some general studies of the role of differential inequalities in obtaining upper and lower bounds on solutions to boundary value problems have been made recently by J. Schr6der; see [9], 10], [11] for further references in this area.…”