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A theory of interval iteration

Author: L. B. Rall
Journal: Proc. Amer. Math. Soc. 86 (1982), 625-631
MSC: Primary 65G10; Secondary 65J15
MathSciNet review: 674094
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Abstract: A theory of interval iteration, based on a few simple assumptions, is given for the fixed point problem for operators in partially ordered topological spaces. A comparison of interval with ordinary iteration is made which shows that their properties are converse in a certain sense with respect to existence or nonexistence of fixed points. The theory of interval iteration is shown to hold without modification if the computation is restricted to a finite set of points, as in actual practice. In this latter case, interval iteration is shown to converge or diverge in a finite number of steps, for which an upper bound is given. By the introduction of a suitable iteration operator, the method of interval iteration is extended to the problem of solution of equations in linear spaces.

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Keywords: Fixed point problems, interval iteration, convergence and divergence, existence and nonexistence of solutions, lower and upper bounds, finite convergence, solution of equations
Article copyright: © Copyright 1982 American Mathematical Society