Fixed Point Theory

By Z. Ésik. 2009

2,002 citations recorded by the source corpus.

Fixed points and fixed point computations occur in just about every field of Computer Science. It has been noticed that several fundamental theorems are consequences of just a few equational properties of fixed point operations. This chapter gives an introduction to that part of the theory of fixed points that has applications to weighted automata and languages.

Subfields: Artificial Intelligence, Computational Theory and Mathematics

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Citations in the corpus

Modelling transition dynamics in MDPs with RKHS embeddings (2012, background): “Since b B deﬁnes a sup-norm contraction mapping on a complete metric space, by Banach’s ﬁxed point theorem (e.g. Granas & Dugundji, 2003) there exists a unique ﬁxed point V ∗ of B , such that choosing V 0 arbitrarily and iterating b b b b Suppose we perform κ iterations, obtaining the estimate V κ…”
Existence results for fractional order functional differential equations with infinite delay (2008, methodology): “As a consequence of the nonlinear alternative of Leray–Schauder type [8], we deduce that P has a fixed point z in U .”
Leverage, Moral Hazard and Liquidity (2010, positive): “Existence and uniqueness of solution to the fixed-point problem: Granas and Dugundji (2003), Theorem 2.1, shows a general approach to existence of Volterra integral equations of the second kind.”
Solvability of Langevin equations with two Hadamard fractional derivatives via Mittag–Leffler functions (2020, background): “(The Schauder fixed point theorem [12]) Let E be a Banach space and Q be a nonempty bounded convex and closed subset of E, and N : Q→ Q is a compact and continuous map.”
Remarks on asymptotic regularity and fixed points (2019, background): “For such mappings one of the fundamental results in fixed-point theory is the classical Knaster–Tarski theorem (also known as the Abian–Brown theorem), see [26], [38].”
Approximate Controllability of Second-Order Evolution Differential Inclusions in Hilbert Spaces (2015, background): “Due to its importance, several researchers have studied the problems represented by evolution equations using different kinds of fixed point theorems [19,39].”
REMARKS ON ULAM STABILITY OF THE OPERATORIAL EQUATIONS (2009, background): “in C(R+) endowed with the generalized metric (d(x, y) ∈ R+ ∪ {+∞}), (see for example [32], pp.”
Iterative methods for approximating fixed points of Bregman nonexpansive operators (2012, background): “Banach space, Bregman distance, Bregman firmly nonexpansive operator, Bregman strongly nonexpansive operator, Bregman projection, fixed point, iterative algorithm, Legendre function, totally convex function.”
Generalizing the Krein–Rutman theorem, measures of noncompactness and the fixed point index (2010, background): “” We refer the reader to [6], [9], [12], [19], [25] and [30] for descriptions of the classical fixed point index and some of its generalizations.”
Common fixed points of mappings satisfying implicit contractive conditions (2012, neutral): “These results unify, extend, generalize related results (see [2,3,7-16,21,25-38]).”
Existence results for boundary-value problems with nonlinear fractional differential inclusions and integral conditions. (2010, methodology): “We shall show that N satisfies the assumptions of the nonlinear alternative of Leray-Schauder type [32].”
APPROXIMATING COMMON FIXED POINTS OF NONCOMMUTING ALMOST CONTRACTIONS IN METRIC SPACES (2010, background): “This explains why the study of fixed and common fixed points of mapping satisfying certain contractive conditions attracted more researchers and stimulated an impressive research work in the last three decades, see for example [25] and the very recent monograph [26].”
On fractional order derivatives and Darboux problem for implicit differential equations (2012, background): “[12] (Nonlinear alternative of Leray-Schauder type) Let X be a Banach space and C a nonempty convex subset of X.”