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.
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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 [22] 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.