# monotone sequence

## monotone sequence

[′män·ə‚tōn ¦sē·kwəns]
(mathematics)
A sequence of real numbers that is monotone-nondecreasing or monotone-nonincreasing.
A sequence of real-valued functions, defined on the same domain, that is either monotone-nondecreasing or monotone-nonincreasing.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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We say that X has the property (C) whenever for each monotone sequence {[x.sub.n]} in X with [x.sub.n] [right arrow] x for some x [member of] X, there exists a subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of {[x.sub.n]} such that every element of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is comparable with x.
If {[G.sub.xn]} converges in Y whenever {[x.sub.n]} is a monotone sequence in [a, b], then function G has the minimal fixed point [x.sub.*] [member of] [a, b] and the maximal fixed point x* [member of] [a, b].
Vatsala  is exposed the classical theory of the method of lower and upper solutions and the monotone iterative technique, that give us the expression of the solution as the limit of a monotone sequence formed by functions that solve linear problems related with the nonlinear considered equations.
If each sequence {Q[x.sub.n]} [subset] Q([a,b]) converges, whenever {[x.sub.n]} is a monotone sequence in [a,b], then the sequence of Q-iteration of a converges to the least fixed point [x.sub.*] of Q and the sequence of Q-iteration of b converges to the greatest fixed point [x.sup.*] of Q.
We develop the approximation scheme and show that under suitable conditions on f, there exists a bounded monotone sequence of solutions of linear problems that converges uniformly to a solution of the original problem.
and they obtained a monotone sequence of approximate solutions converging uniformly and quadratically to a solution of the semilinear telegraph system.
Finally, we show that the norm statistical convergence and the weakly statistical convergence are equivalent for monotone sequences.
As a second result, we prove that the minimum density of monotone sequences of length k + 1 in an arbitrarily large layered permutation is asymptotically 1/[k.sup.k].
Then there exist monotone sequences, [v.sub.n](t) and [w.sub.n](t), such that [v.sub.n](t) [right arrow] v(t) and [w.sub.n](t) [right arrow] w(t) uniformly and monotonically, where v(t) and w(t) are coupled minimal and maximal solutions of equation (2.6) on J.
In the present work we have shown that even with constant [lambda] monotone sequences can be generated.
Under natural assumptions, we prove quadratic convergence of monotone sequences to a unique solution.
where [[member of].sub.j] = [[mu].sub.j] + 1/ J + 1 The sequence {ej} is the product of two totally monotone sequences, so it is totally monotone, and all of the forward differences are nonnegative and bounded.

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