We say that X has the property (C) whenever for each

monotone sequence {[x.

n]} is a

monotone sequence in [a, b], then function G has the minimal fixed point [x.

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.

assuming a one-sided Lipschitz condition in funcion f, two monotone sequences that start at the lower solution a and the upper solution 0 and converge to a solutions [phi] and [PHI], are constructed; moreover every solution U [member of] [[alpha],[beta]] [a, /0] of problem (3.

n]} is a

monotone sequence in [a,b], then the sequence of Q-iteration of a converges to the least fixed point [x.

We show that under suitable conditions on f, there exists a monotone sequence of solutions of linear problems that converges uniformly and rapidly to unique solution of the original nonlinear problem.

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.

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.

Under natural assumptions, we prove quadratic convergence of

monotone sequences to a unique solution.

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.

X] is called order closed if for

monotone sequences {[u.

As follows from [6], the

monotone sequences {[[bar.