Monotonic Function

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monotonic function

[¦män·ə¦tän·ik ′fəŋk·shən]
(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 = Monotonic Function 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:

and

References in periodicals archive ?
1), we shall use the variational approach and monotone operator theory (see [4,19-22]).
Showalter, Monotone Operators in Banach Space and Nonlinear Differential Equations, American Mathematical Society, Providence, 1997.
where F : D(F) [subset] X [right arrow] X is a monotone operator and X is a Hilbert space.
1) and let F: D(F) [subset] X [right arrow] X be a monotone operator in X.
It is well known that the sub differential [partial derivative][phi] is a maximal monotone operator.
8] Let A be a maximal monotone operator, then the resolvent operator associated with A is defined as
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.
Sadarangani: Fixed point theorems for mixed monotone operators and applications to integral equations, Nonlinear Anal.
He covers topological Lie algebras, Lie groups and their Lie algebras, enlargeability, smooth homogeneous spaces, quasimultiplicative maps, complex structures in homogeneous spaces, equivariant monotone operators, L*-ideals and equivariant monotone operators, homogeneous spaces of pseudo-restricted groups.
1) is equivalent to finding a zero of the sum of two monotone operators (2.