Fixed-point property

Numerical methods in rate-type viscoplasticity

Published in N.D. Cristescu, H.R. Hardy, R.O. Simionescu, Basic and Applied Salt Mechanics, 2020

In this section a numerical algorithm, which can be directly run on a computer is considered, in order to approximate the solution (u, σ) of the variational problem (20)–(22). This algorithm is based on the approximation of the unknowns in space and time. As it follows from above, the existence and uniqueness of the solution for the problem (20)–(22) may be obtained in two steps: i) the study of the elliptic problem (35)–(36) defined for every element η of C(0, T,H); ii) the fixed point property of the operator R defined by (33). For this reason, in order to approximate (20)–(22) we consider the discret version of (30)–(32) and the discret version of the above fixed point property. These considerations lead us to consider the following algorithm:

Deterministic and random fixed points for maps on extension type spaces

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Journal Information

Published in Applicable Analysis, 2018

Donal O’Regan

Now one can put conditions on the space X and the map F so that F has an -fixed point for each would guarantee F has a fixed point; for examples we refer the reader to [7, Lemma 1.2], [8, Theorem 1.4 and Remark 1.6]. We say (X, F) has the -fixed point property if F having an -fixed point for each guarantees that F has a fixed point. Thus we have: Suppose , is a compact map, and (X, F) has the -fixed point property. Then F has a fixed point.

Fixed-point property

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Numerical methods in rate-type viscoplasticity

Deterministic and random fixed points for maps on extension type spaces