SOME REMARKS ON THE MONOTONE ITERATIVE TECHNIQUE

. We consider an abstract operator equation in coincidence form Lu = N ( u ) and establish some comparison results and existence results via the monotone iterative technique. We use a generalized iteration method developed by Carl-Heikkila (1999). An application to a boundary value problem for a second-order functional diﬀerential equation is considered.


INTRODUCTION
Let X be a nonempty set and Z be an ordered metric space.Let us consider the operator equation of the form (1.1) Lu = N u, and the iterative scheme where L, N : X → Z.
In our work the operators L and N will satisfy some extended monotonicity conditions, which are described exactly in the following definition.
Definition 1.1.N is monotone increasing with respect to L if u 1 , u 2 ∈ X and Lu 1 ≤ Lu 2 imply that N u 1 ≤ N u 2 .
If in the last relation the reversed inequality holds, then N is monotone decreasing with respect to L.
Let X be an ordered set.If Lu 1 ≤ Lu 2 implies u 1 ≤ u 2 then L is said to be inverse-monotone (see [8]) or of monotone-type (see [9]).
The plan of our paper is as follows.In Section 2 we deal with operator inequalities corresponding with (1.1) and extend the abstract Gronwall lemma of Rus [6].Let us mention that the result from [6] generalize some results from [9] and [11].In Section 3 we generalize some known existence results for equation (1.1) ( [5,10,4,1,9]) involving monotone increasing or monotone decreasing operators.We shall use a generalized iteration method developed in [2].In Section 4 we shall apply some of our results to implicit second order functional-differential equations.For another treatment of this type of functional-differential equations it can be seen [7].

OPERATOR INEQUALITIES IN ORDERED METRIC SPACES
In this section we extend the notion of Picard operator [6], in Definition 2.1, and the abstract Gronwall lemma of Rus [6], in Theorem 2.3.The above mentioned notion and result correspond, in our setting, with the case when X = Z and L is the identity mapping of Z.As a consequence of Theorem 2.3, we shall find a condition in Corollary 2.4, which assure the existence of ordered lower and upper solutions for equation (1.1).
Definition 2.1.N is Picard with respect to L if there exists a unique v * ∈ Z with the following properties.
(i) there exists u * ∈ X such that Then N is Picard with respect to L. Let us mention that, also, N is monotone increasing with respect to L.
Example 2.4.If Z is also a complete metric space, L is surjective and N is contraction with respect to L then N is Picard with respect to L. Let us mention that N is contraction with respect to L if there exists 0 < a < 1 such that for all For the proof of this result, also known as the Coincidence Theorem of Goebel, we refer to [3].Lemma 2.2.If N is monotone increasing (or monotone decreasing or contraction) with respect to L then If L is inverse-monotone then L is injective.
Proof.Let us consider only that N is monotone increasing with respect to L.
This obviously implies the conclusion.
For the last statement we have to prove that, if L is of monotone type, then Lu 1 = Lu 2 implies u 1 = u 2 .This can be done like above.
Proof.Let us consider u 0 ∈ X such that Lu 0 ≤ N u 0 and the sequence defined by (1.2) starting from u 0 .The following relations hold, and, passing to the limit when n → ∞, The next relation can be proved similarly.
If, in addition, L is inverse-monotone, then, by Lemma 2.2, u * given by Definition 2.1 is unique and, of course, We apply now Theorem 2.3 for N and deduce that Lu ¯≤ Lū.The last part of the conclusion follows in an obvious way.

OPERATOR EQUATIONS IN ORDERED BANACH SPACES
In this section we shall establish two existence results for equation (1.1), involving an operator N which is increasing with respect to L, in Theorem 3.2, respectively monotone decreasing with respect to L, in Theorem 3. 3 We shall use a generalized iteration method developed in [2].As it is mentioned in [2], this method enlarges the range of applications since neither L nor N need be continuous.In this spirit, Theorem 3.2 generalizes Theorem 3.1 in [4], and Theorem 3.3 generalizes Theorem 3 in [10] and Theorem 2 in [5] (these are given in the case X = Z and L = I).
The following result is Proposition 3.4 from [2] and we shall use it to derive Theorem 3.2.Proposition 3.1.Assume that the following conditions hold.
(i) There exists u ¯a lower solution of (1.1), u ¯∈ W ⊂ X; (ii) N is monotone increasing with respect to L; (iii) L(W ) is an ordered metric space and if (u n ) is a sequence in W such that the sequences (Lu n ) and (N u n ) are increasing, then (N u n ) converges in L(W ).Then (1.1) has a solution u * with the property ¯≤ Lw and Lw ≥ N w }.If, in addition, W is an ordered space and L is of monotone type, then u * is the minimal solution of (1.1) in W 0 = {u ∈ W | Lu ¯≤ Lu}.We notice that the dual result is valid.In the following results, i.e.Theorem 3.2 and Theorem 3.3, Z will be an ordered Banach space (OBS) with a normal cone K.
For v ≤ w the order interval [v, w] is the set of all u ∈ Z such that v ≤ u ≤ w.Every order interval for an OBS is bounded if and only if the cone K is normal.In an OBS with a normal cone, every monotone increasing sequence which has a convergent subsequence, is convergent.
A cone K is said to be regular if every monotone increasing sequence contained in some order interval, is convergent.Theorem 3.2.Assume that the following conditions hold.
(i) u ¯is a lower solution and ū is a super-solution of (1.1) with Lu ¯≤ Lū; (ii) N is monotone increasing with respect to L; which is a closed subset of Z, thus is an ordered metric space.
Let (u n ) be a sequence in W such that (Lu n ) and (N u n ) are increasing.Using (i) and (ii), All the hypotheses of Proposition 3.1 are fulfilled.Hence, the conclusion follows.
Remark.If, in addition to the hypotheses of Theorem 3.2, N is continuous with respect to L then u * can be obtained by (1.2) starting from u ¯, in the sense that a sequence defined by (1.2) with u 0 = u ¯is such that (Lu n ) converges to Lu * .
Let us mention that N is said to be continuous with respect to L if for every sequence (Lu n ) from L(X) convergent to Lu * ∈ L(X), the sequence (N u n ) converges to N u * .Theorem 3.3.Assume that the following conditions hold.
(i) Lu ≥ 0 implies N u ≥ 0; (ii) N is monotone decreasing with respect to L; , for u µ given by Lu µ = µLu, and for all µ ∈ (0, 1); Proof.For every u ∈ X, if ũ is such that N u = Lũ, let us define Ñ u = N ũ.By Lemma 2.2, Ñ u does not depend on the choice of ũ, thus the operator Ñ : X → Z is well-defined.
Ñ is monotone increasing with respect to L. Indeed, Let us consider also u 2 , u 3 such that N u 1 = Lu 2 and N u 2 = Lu 3 .By (i) and (iii), Lu 1 ≥ 0 = Lu 0 , which implies, by (ii), that N u 1 ≤ N u 0 .Using the definitions of u 2 and u 1 , the following relation holds. (3.1) We shall focus our attention to the equation We shall prove that u 2 is a lower solution and u 1 is an upper solution of (3.2).This follows by the following implications. and We use Theorem 3.2 and deduce that equation (3.2) has a solution u * , i.e.

APPLICATION
In this section we shall establish a weak maximum principle for the functional-differential operator and an existence result for the following boundary value problem for a second order implicit functional-differential equation.
Then equation (4.3) is equivalent to By o straightforward calculation, the following relation can be proved . Thus, A w is a contraction on the Banach space C[0, 1], so it has a unique fixed point.Hence, L is surjective.
In order to prove that L is inverse-monotone, because L is linear it is sufficient to prove that Lu ≤ 0 implies u ≤ 0.
The operator A w * is Picard and monotone increasing and, in this case, it is easy to see that A(0) ≤ 0.Then, by Theorem 2.3 (or Theorem 4.1 in [6]) u * ≤ 0.
The following theorem is an existence result for the BVP considered at the beginning of this section.Proof.This follows easily by Theorem 3.2.Let us omit the details and notice only some useful facts.Z = L 2 (0, 1) is an ordered Banach space with a regular cone (see [1]).
[Lu ¯, Lū] ⊂ L(X) because L is surjective.The condition (f2) and that L is inverse-monotone imply that N is monotone increasing with respect to L.
is a compact subset of Z. Then (1.1) has a solution u * with the property Lu * = min{Lw ∈ [Lu ¯, Lū] | Lw ≥ N w} and a solution u * with the property Lu * = max{Lw ∈ [Lu ¯, Lū] | Lw ≤ N w}.If, in addition, L is of monotone type, then u * is the minimal solution, and u * the maximal solution of (1.1)