text: # Monotonic Function
  
  (redirected from Monotone operator)
  
  ## 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 = 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 ?
  Convex analysis and monotone operator theory in Hilbert Spaces.
  It is well-known that a monotone operator T is maximal monotone if and only if R(T + [lambda]J) = [X.sup.*] for every [lambda] > 0 (cf.
  (A) If H(q) is a diagonally monotone operator family in H with compact resolvent, then all eigenvalues [E.sup.H(q).sub.i] of H(q) are also diagonally monotone.
  Kumam, "A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping," Journal of Applied Mathematics and Computing, vol.
  where A is a monotone operator and the set of zero point of A is denoted by [A.sup.-1] (0).
  This problem has been investigated by use of a variety of nonlinear analyses such as fixed point theorem for mixed monotone operator [7, 15, 22-25], maximal principle [6], Banach's contraction mapping principle [26-29], and the linear operator theory [27, 30, 31].
  It is a well known result that if [phi] is proper, convex and lower semicontinuous, then [partial derivative][phi] is a maximal monotone operator. We refer the reader to the books by Barbu [4], Brezis [5] and Morosanu [17] for further details on the properties of monotone operators and subdifferentials of convex functions in Hilbert spaces.
  for all v, w [member of] X and that a monotone operator on a reflexive Banach space is hemicontinuous if and only if it is demicontinuous (see e.g.
  By the assumption L < 1/[K.sup.2]m([OMEGA]), it turns out that [PHI]' is a strongly monotone operator. So, by applying Minty-Browder theorem [29, Theorem 26.A], [PHI]': X [right arrow] [X.sup.*] admits a Lipschitz continuous inverse.
  where B is k-Lipschitzian and [eta]-strongly monotone operator. Then he proved that if the sequence {[[alpha].sub.n]} satisfies appropriate conditions, the sequence {[x.sub.n]} generated by (6) converges strongly to the unique solution [x.sup.*] [member of] [F.sub.ix] (T) of the variational inequality
  Let A : D(A) [subset] H [right arrow] H be a (possibly multivalued) maximal monotone operator. Consider the difference equation
  To study solvability of the nonlinear problem (2.1), we shall use the variational approach and monotone operator theory (see [4,19-22]).
  
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