# Monotonic Function

(redirected from Monotone operator)

## monotonic function

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(mathematics)

## Monotonic Function

(or monotone function), a function whose increments Δf(x) = f(x′) − f(x) do not change sign when Δx = x′ − x > 0; that is, the increments are either always nonnegative or always nonpositive. Somewhat inaccurately, a monotonic function can be defined as a function that always varies in the same direction. Different types of monotonic functions are represented in Figure 1. For example, the function y = x3 is an increasing function. If a function f(x) has a derivative f′(x) that is nonnegative at every point and that vanishes only at a finite number of individual points, then f(x) is an increasing function. Similarly, if f′(x) ≤ 0 and vanishes only at a finite number of points, then f(x) is a decreasing function.

Figure 1

A monotonicity condition can hold either for all x or for x on a given interval. In the latter case, the function is said to be monotonic on this interval. For example, the function y = increases on the interval [−1,0] and decreases on the interval [0, +1]. A monotonic function is one of the simplest classes of functions and is continually encountered in mathematical analysis and the theory of functions. If f(x) is a monotonic function, then the following limits exist for any X0:

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References in periodicals archive ?
It is a well known result that if [phi] is proper, convex and lower semicontinuous, then [partial derivative][phi] is a maximal monotone operator.
2]m([OMEGA]), it turns out that [PHI]' is a strongly monotone operator.
Let A : D(A) [subset] H [right arrow] H be a (possibly multivalued) maximal monotone operator.
1), we shall use the variational approach and monotone operator theory (see [4,19-22]).
where F : D(F) [subset] X [right arrow] X is a monotone operator and X is a Hilbert space.
X*] is a maximal monotone operator such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], a theorem of Rockafellar (see [5, Th.
It is well known that the sub differential [partial derivative][phi] is a maximal monotone operator.
Wu: New fixed point theorems and applications of mixed monotone operator, J.
Let T be a strongly monotone operator with constant [alpha] > 0 and Lipschitz continuous with constant [beta] > 0.
Verma [11] studied a class of variational inequalities with relaxed monotone operators.

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