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• W2032831011 abstract "Let E be a real Banach space, C a closed, convex subset of E and f: [0, 11 x E X E E be continuous. Let u0, u1 E C and consider the boundary value problem (*) f(t= u, U'), u(O) = uo (l) = ul. We establish sufficient conditions in order that (*) have a solution U: [0, 11 C. Introduction. Let C be a closed, convex subset of the real Banach space E and let f: [0, 1] x C x E E be a function with the property (1) p E E* (2),x E C, dp(x) maxqECPq) * ift X y)) >0. y E F, ep(y) =0,0?= t S1 In this paper we show that under some additional (sometimes rather restrictive) assumptions the boundary value problem (BVP) (2) Urif(t, u, u'), u(O) = uo, u() = u1, 0 < t S 1, (uo, uI E C) has a solution u: [0, 1] C. We note that (1) describes the behavior of f on the boundary aC of C, for if Vp 4 0, then condition (1) implies x E ac. In case E = RW, n-dimensional Euclidean space, and C is bounded with int C(3) :# 0, various results of this type exist in the literature (see e.g. [5] for a survey of such results). In this finite dimensional situation the general case may easily be obtained by projection methods. On the other hand, if E is infinite dimensional, certain additional assumptions, eithler on E or on f seem to be needed to pass from the case int C # 0 to the general case. The paper is divided into two parts. In the first part we assume f(t, x, y) Received by the editors August 23, 1974. AMS (MOS) subject classifications (1970). Primary 34B135, 34G05." @default.
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• W2032831011 date "1976-01-01" @default.
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• W2032831011 title "Boundary value problems for second order differential equations in convex subsets of a Banach space" @default.
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• W2032831011 doi "https://doi.org/10.1090/s0002-9947-1976-0397110-5" @default.
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