A decomposition relative to convex sets

Nashed, M. Z.

doi:10.1090/s0002-9939-1968-0234268-3

Cited by 11 publications

(2 citation statements)

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“…Now, with additional conditions on A, we consider the problem of approximating the critical point x* of (1) when the suppositions of Theorem 2 or Theorem 3 are fulfilled. In the case that V(x, y) = Ix-yl and E is a Hilbert space, similar problems are considered by Nashed [13] and Zarantonello [15]. The general Banach-space is treated by Aalto [1].…”

Section: (T)-uw(t)l ~ < V(u~(t) Uw(t)) < %(T V(z W)) < %(T M(r)2"mentioning

confidence: 99%

Lyapunov functions and autonomous differential equations in a Banach space

Martin

1973

Math. Systems Theory

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Let R denote the space of real numbers, let E be a Banach space over the real or complex field, and let [. [ denote the norm on E. In this paper we study the existence and behavior of solutions to the autonomous differential equationwhere A is a continuous function from E into E. In particular, sufficient conditions are established to ensure that (1) has a unique critical point which is globally asymptotically stable and, with additional conditions on A, an iterative method is developed which converges to this critical point. The main purpose of this paper is to point out that many of the techniques used in the theory of monotone operators can be applied in a more general situation (see, for example, Hartman [8]). Instead of using the norm on E to study the solutions to (1), we assume the existence of a function V from E× E into [0, or) which has the following basic properties: 66MATHEMATICAL SYSTEMS THEORY t VOI. 7, NO. i .

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Section: (T)-uw(t)l ~ < V(u~(t) Uw(t)) < %(T V(z W)) < %(T M(r)2"mentioning

confidence: 99%

Lyapunov functions and autonomous differential equations in a Banach space

Martin

1973

Math. Systems Theory

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“…The class of firmly nonexpansive mappings is very interesting in optimization and nonlinear analysis, and contains, in particular, projections on closed convex sets, proximal mappings of proper convex lower semicontinuous functions and resolvents of maximal monotone operators, see for example [4] and [10]. This class has been extensively studied up until now.…”

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confidence: 99%