Abstract

We study asymptotic behavior of solutions of general advanced differential systems , where is a continuous quasi-bounded functional which satisfies a local Lipschitz condition with respect to the second argument and is a subset in , , , and , . A monotone iterative method is proposed to prove the existence of a solution defined for 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.

1. Introduction

Let , where , , be the Banach space of the continuous mappings from the interval to equipped with the supremum norm where is the maximum norm in . In the case of and , we will denote this space as ; that is,

If , , and , then, for each , we define by , .

Let us consider a system of functional differential equations of advanced type: where is a continuous quasi-bounded functional which satisfies a local Lipschitz condition with respect to the second argument and is a subset in . We recall that the functional is quasi-bounded if is bounded on every set of the form , where , , and is a closed bounded subset of (cf. [1, page  305]).

A function is said to be a solution of (3) on with if , for , and satisfies (3) for .

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 equation where are constants and is a locally Lipschitz continuous function satisfying for . We will assume 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 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 equation 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 equation where , has a positive solution in the case when if and only if there exists a continuous function such that where and 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.

2. Preliminaries

In this section we introduce some definitions and theorems which will be used later.

Definition 1 (see [4, page  276]). Let be a Banach space and let . Then is called an order cone if and only if (i)is closed, nonempty, and ;(ii), , ;(iii) and .

On this basis, we define for

Definition 2 (see [4, page  277]). Let , be Banach spaces and let be an order cone. (1)The order cone is called normal if and only if there is a number such that, for all , (2)The operator 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.

2.1. 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.

Definition 4 (see [4, page  54]). Let , be Banach spaces, and an operator. is called compact if and only if(i) is continuous; (ii) maps bounded sets into relatively compact sets.
Let us recall that a set is bounded if and only if there is a number such that for all and is relatively compact (resp., compact) if and only if every sequence in contains a convergent subsequence (resp., the limit of which also belongs to ).

The following theorem is a part of Theorem in [4, page  283].

Theorem 5. Suppose that is a compact monotone increasing operator on a real Banach space with normal order cone . If is a subsolution of (13) and if is a supersolution of (13) with , then the iterative sequence in (14) converges to a fixed point of , namely, to the greatest fixed point of in , and converges to the smallest fixed point of in . Furthermore, one has the error estimates

3. 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)).

By we denote the set of all component-wise nonnegative (positive) vectors in ; that is, with () for . For , we say that if , if , if , and .

For a fixed we put . In the following, let and .

Now for a given , we consider two systems of the integrofunctional inequalities on , for , , where is defined by where , , .

The next theorem provides conditions under which it is possible to construct (with the aid of auxiliary given functions , satisfying (16)) monotone sequences of functions converging to a solution of the operator equation on with

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 if , for , and satisfies (19) for .

Theorem 6. Let one assumes the following. (i)For any , there are , such that for any and any function with , where the norm is defined by (1) with and . (ii)There is a and functions , satisfying on and inequalities (16) on , that is, the inequalities (iii)For any , with for , , one has
Then there exists a solution of the system (3) on satisfying for .

Proof. We prove that (19) has a continuous solution satisfying for . For every fixed , we introduce the Banach space of the continuous functions taking into equipped with the maximum norm and the normal cone of the continuous functions taking into . By the cone , a partial ordering in is given. For we say that if and only if . We introduce the operator defined by
Let with . Condition (iii) implies that if . Let Condition (ii) implies that In order to show the convergence of the sequences (we set and , ) 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 be a bounded subset of . We have to show that is a relatively compact subset of . Due to the theorem of Arzelà and Ascoli, it suffices to show that is bounded and equicontinuous. That is bounded follows from condition (22), and the equicontinuity of 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 and of with The subsolution and supersolution necessary for application of Theorem 5 are equal to and , respectively. Since it is easy to see that for and , we have for , . Thus the functions defined, for example, by where , , satisfy on and for , . Now choosing, for example, , and (see substitution (21)), the proof is completed since is strictly increasing with respect to the second argument.

Remark 7. Tracing the proof of Theorem 6 one can see that assumptions on functional (continuity, quasi-boundedness, and local Lipschitz condition) are not explicitly mentioned. Nevertheless these assumptions can be viewed as almost necessary for validity of inequalities (22) and (23).

4. 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.

4.1. Equation

Let us consider linear advanced equation (4): where and are positive constants, function is locally Lipschitz continuous, , and .

To apply Theorem 6 we consider two auxiliary scalar linear advanced equations: Looking for solutions in the form or we get corresponding transcendental equations: The following lemma about the real roots of transcendental equations is a simple consequence of Lemma  1 in [5, page  13] (assumption is here substantial).

Lemma 8. The following is true: (a)equation (41) has just two different positive roots , , , and (42) has just two different positive roots , , ,(b) and , (c) and , (d) and .

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 (we omit index due to scalar case).

Using substitution (21) we obtain operator defined by formula (20) in the form

Theorem 9. Let there be continuous functions , , for , satisfying the inequalities on . Then there exists a solution of (4) on such that

Proof. Firstly we verify conditions (22) and (23) of Theorem 6. Due to properties of functions , , inequalities hold for every , where is a suitable constant. The boundedness of operator (condition (22)) is on obvious:
Now, condition (22) holds. Let us verify condition (23). At first we estimate the difference Now we have with the value lying between Since obviously we get finally with
Inequality (23) is verified. Validity of condition (ii) follows from inequalities (45), and (46), and condition (iii) holds in view of the form of the operator (see (44)).