Positive Solutions of Advanced Differential Systems

We study asymptotic behavior of solutions of general advanced differential systems y˙(t)=F(t,yt), where F : Ω → ℝn is a continuous quasi-bounded functional which satisfies a local Lipschitz condition with respect to the second argument and Ω is a subset in ℝ × C r n, C r n : = C([0, r], ℝn), y t∈C r n, and y t(θ) = y(t + θ), θ ∈ [0, r]. A monotone iterative method is proposed to prove the existence of a solution defined for t → ∞ with the graph coordinates lying between graph coordinates of two (lower and upper) auxiliary vector functions. This result is applied to scalar advanced linear differential equations. Criteria of existence of positive solutions are given and their asymptotic behavior is discussed.

Let us consider a system of functional differential equations of advanced type:̇( where : Ω → R is a continuous quasi-bounded functional which satisfies a local Lipschitz condition with respect to the second argument and Ω is a subset in R × . We The paper is organized as follows. In Section 2 necessary iterative technique is formulated. In Section 3 it is applied to general nonlinear advanced differential system. By monotone iterative method, we will prove a general criterion on existence of bounded solutions of the nonlinear system (3) or, more exactly, we give necessary conditions for the existence of a solution defined for → ∞ with the graph coordinates lying between graph coordinates of two (lower and upper) auxiliary vector functions. If the lower function can be taken with positive coordinates, then the statement of theorem concerns positive solutions. Section 4 is devoted to scalar linear cases. Nonlinear result proved in Section 3 is applied to the scalar advanced linear differential equatioṅ 2 The Scientific World Journal where , ∈ R + = (0,∞) are constants and ( ) is a locally Lipschitz continuous function satisfying | ( )| < < for ∈ [ 0 , ∞). We will assume ( + ) < 1/( ) as well. The case when there exist positive solutions is studied. With the aid of two auxiliary equations, constructed using lower and upper estimates of the right-hand side of (4), and under the supposition of the existence of two real (positive) different roots of corresponding transcendental equations, the existence of a positive solution of (4) is proved. Simultaneously its asymptotic behavior is derived. Next, a linear advanced equation, more general than (4), where : [ 0 , ∞) → R + is bounded, locally Lipschitz continuous function and is a positive constant, is considered and a criterion of existence of positive solutions is given.
The fact of existence of positive solutions of an advanced differential equation can be documented, for example, by the equatioṅ( which admits a pair of positive and asymptotically different for → ∞ solutions: In the literature one can find some results on existence of positive solutions of advanced equations. For example, in accordance with [2, page 31] and [3, page 21], the first-order advanced type differential equatioṅ where , ∈ ([ 0 , ∞), R + ), has a positive solution in the case when ( ) ≡ > 0 if and only if there exists a continuous function ( ) such that where Λ( ) = ∫ 0 ( ) and Λ −1 is corresponding inverse function.
As a consequence of our result for nonlinear advanced equations we get a sufficient and necessary criterion of positivity different from the above-mentioned criterion (9).
Concluding remarks and open problems are formulated in Section 5.

Preliminaries
In this section we introduce some definitions and theorems which will be used later.
(1) The order cone K is called normal if and only if there is a number > 0 such that, for all , ∈ L, (2) The operator : ( ) ⊆ L → Y is called monotone increasing if and only if it is true for all , ∈ ( ) that where ( ) denotes domain of definition of the operator . The operator is called strictly or strongly monotone increasing if and only if the symbol "≤" is replaced by "<" or "≪, " respectively. If we replace "≤" by "≥, " then is monotone decreasing. Similarly, we have operators which are strictly or strongly monotone decreasing.

Supersolutions, Subsolutions, and Iterative Methods.
We consider the operator equation together with two iterative methods Definition 3 (see [4, page 282]). The point is called a supersolution, a strict supersolution, or a strong supersolution of (13) if and only if ≥ , > , or ≫ , respectively. The prefix "super" is replaced by "sub" when the respective inequalities are reversed. (ii) maps bounded sets into relatively compact sets.
Let us recall that a set A is bounded if and only if there is a number > 0 such that ‖ ‖ ≤ for all ∈ A and is relatively compact (resp., compact) if and only if every sequence in A contains a convergent subsequence (resp., the limit of which also belongs to A).

Nonlinear Case
In this part the advanced system of differential equations (3) is considered. Using the method of monotone sequences we prove that under certain additional assumptions there exists a solution of (3) satisfying half-infinity interval inequalities formulated in Theorem 6 (inequalities (26)).
For a fixed * ∈ R we put Ω := ( * , ∞) × . In the following, let 0 ∈ R and 0 > * . Now for a given ∈ R >0 , we consider two systems of the integrofunctional inequalities is defined by The next theorem provides conditions under which it is possible to construct (with the aid of auxiliary given functions 1 , 2 satisfying (16)) monotone sequences of functions converging to a solution of the operator equation It is easy to verify that the system (3) is related to operator equation (19) through the substitution A function is said to be a solution of operator equation (19) on [ , + ) with > 0 if ∈ ([ , + + ), R ), ( , ) ∈ Ω for ∈ [ , + ), and ( ) satisfies (19) for ∈ [ , + ).
Theorem 6. Let one assumes the following.
Proof. We prove that (19) has a continuous solution : [ 0 , ∞) → R satisfying 1 ( ) ≤ ( ) ≤ 2 ( ) for ≥ 0 . For every fixed ∈ ( 0 + , ∞), we introduce the Banach space of the continuous functions taking [ 0 , ] into R equipped with the maximum norm and the normal cone of the continuous functions taking [ 0 , ] into R ≥0 . By the cone K , a partial ordering ≤ in L is given. For , ∈ L we say that ≤ if and only if − ∈ K . We introduce the operator : L → L defined by Let , ∈ L with ≤ . Condition (iii) implies that The Scientific World Journal In order to show the convergence of the sequences (we set 0 , = , and 1 , = , , = 1,2) to the fixed points of operator , we need to prove that is continuous and compact. The first property is obvious (due to continuity of and ). Let us prove compactness. To this end, let L be a bounded subset of L . We have to show that L is a relatively compact subset of L . Due to the theorem of Arzelà and Ascoli, it suffices to show that L is bounded and equicontinuous. That L is bounded follows from condition (22), and the equicontinuity of L is assumed by condition (23). Now we are in a position to apply Theorem 5 (about the monotone iterative method) in order to show the existence of the fixed points ,1 and ,2 of with The subsolution and supersolution necessary for application of Theorem 5 are equal to 0 := ,1 and V 0 := ,2 , respectively. Since it is easy to see that where = 1, 2, ∈ [ 0 , ∞), satisfy

Linear Cases
In this section we apply the nonlinear result given in Section 3 to linear advanced equations. We will prove existence of positive solutions, and we derive asymptotic behavior of positive solutions.
Lemma 8. The following is true: As a tool for detecting a positive solution of (4), a linear corollary of Theorem 6 is used. We will reformulate the theorem with respect to the case in question; that is, we put Since obviously we get finally where 1 , 1 are defined in Lemma 8.
Proof. Without loss of generality we may assume that 0 is large enough for the asymptotic relations and inequalities to be valid. Now we employ Theorem 9. Let * First we show that * 1 ( ), * 2 ( ) satisfy inequalities (45), (46). We have to check whether Let us simplify and estimate the right-hand side: That is, inequality (45) is fulfilled. Similarly Thus inequality (46) is valid too. Since all the assumptions of Theorem 9 are fulfilled, for solution ( ) inequalities (56) hold.

Equation( ) = ( ) ( + ).
Let us consider linear advanced equation (5): where : [ 0 , ∞) → R + is bounded, locally Lipschitz continuous and is a positive constant. This equation can be viewed in a sense as a generalization of (4) (with = 0 and ( ) = ( )). In the following criterion of existence of positive solutions of (5) it is important that only one of inequalities similar to inequalities (45) and (46) must be valid, namely, the inequality similar to (46).
Necessity. If a positive solution = ( ) of (5) on [ 0 , ∞) exists, we set * Taking ( ) in a suitable way we can get sufficient conditions of existence of positive solutions of (5) on [ 0 , ∞). The following corollary illustrates such possibility.

Concluding Remarks
Note that criterion (62) of existence of positive solution of advanced equation (5) given by Theorem 11, as a particular case of result for nonlinear system, seems to be independent of the mentioned criterion (9). In recent book [6] there are considered some linear classes of advanced differential equations in Chapter 5.
This occurs, for example, for ( ) = 1/ . It means that inequality (69) in terms of integral average of function is the best possible. In connection with the statement of Theorem 10 a problem about the role played by all real roots 1 , 2 , 1 , and 2 of transcendental equations (41) and (42) arises. Roots 1 , 1 , 1 < 1 with properties given in Lemma 8 were used in Theorem 10 to detect a positive solution of (4) satisfying inequalities (56). The role played by the roots 2 , 2 , 2 > 2 with properties given in Lemma 8 in the discussion on existence of positive solutions of (4) is not clarified yet. Obviously 2 and 2 cannot be used in Theorem 9 to replace * , = 1, 2, in inequalities (45) and (46) (i.e., it is not possible to set * 1 = 2 , * 2 = 2 ). Nevertheless, as it was demonstrated by advanced equation (6), it has two classes of asymptotically different positive solutions given by inequalities (7). This is the reason why we formulate the following claim.
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