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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 754701, 16 pages
Existence and Asymptotic Behavior of Positive Solutions of Functional Differential Equations of Delayed Type
Department of Mathematics, Faculty of Science, University of Žilina, 010 26 Žilina, Slovakia
Received 30 September 2010; Accepted 14 October 2010
Academic Editor: Elena Braverman
Copyright © 2011 J. Diblík and M. Kúdelčíková. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Solutions of the equation are considered for . The existence of two classes of positive solutions which are asymptotically different is proved using the retract method combined with Razumikhin's technique. With the aid of two auxiliary linear equations, which are constructed using upper and lower linear functional estimates of the right-hand side of the equation considered, inequalities for both types of positive solutions are given as well.
Let , where , , be the Banach space of the continuous mappings from the interval into 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 , .
The present article is devoted to the problem of the existence of two classes of asymptotically different positive solutions of the delayed equation for , where is a continuous quasibounded functional that satisfies a local Lipschitz condition with respect to the second argument and is an open subset in such that conditions which use are well defined.
The main supposition of our investigation is that the right-hand side of (1.3) can be estimated as follows: where , and , are continuous functions satisfying Quite lots of investigations are devoted to existence of positive solutions of different classes of equations (we mention at least monographs [1–6] and papers [7–12]). The investigation of two classes of asymptotically different solutions of (1.3) has been started in the paper  using a monotone iterative technique and a retract principle. Assumptions of results obtained are too cumbersome and are applied to narrow classes of equations. In the presented paper we derive more general statements under weaker conditions. This progress is related to more general inequalities (1.4) for the right-hand side of (1.3) which permit to omit utilization of properties of solutions of transcendental equations used in .
1.1. Ważewski’s Principle
In this section we introduce Ważewski’s principle for a system of retarded functional differential equations where is a continuous quasibounded map which satisfies a local Lipschitz condition with respect to the second argument and is an open subset in . We recall that the functional is quasibounded if is bounded on every set of the form , where , and is a closed bounded subset of (compare [2, page 305]).
In accordance with , a function is said to be a solution of system (1.7) on if there are and such that , , and satisfies the system (1.7) for . For a given , , we say is a solution of the system (1.7) through if there is an such that is a solution of the system (1.7) on and . In view of the above conditions, each element determines a unique solution of the system (1.7) through on its maximal interval of existence , which depends continuously on initial data . A solution of the system (1.7) is said to be positive if on for each . A nontrivial solution of the system (1.7) is said to be oscillatory on (under condition if (1.8) does not hold on any subinterval , .
As a method of proving the existence of positive solutions of (1.3), we use Ważewski's retract principle which was first introduced by Ważewski  for ordinary differential equations and later extended to retarded functional differential equations by Rybakowski  and which is widely applicable to concrete examples. A summary of this principle is given below.
As usual, if a set , then and denote the interior and the boundary of , respectively.
Definition 1.1 (see ). Let the continuously differentiable functions and , , be defined on some open set . The set
is called a regular polyfacial set with respect to the system (1.7), provided that it is nonempty, if to below hold. () For such that for , we have . () For all , all for which , and all for which and , . It follows that , where
() For all , all for which , and all for which and . It follows that , where
The elements in the sequel are assumed to be such that .
Definition 1.2. A system of initial functions with respect to the nonempty sets and , where is defined as a continuous mapping such that and below hold. () If , then for .() If , then for and .
Definition 1.3 (see ). If are subsets of a topological space and is a continuous mapping from onto such that for every , then is said to be a retraction of onto . When a retraction of onto exists, is called a retract of .
The following lemma describes the main result of the paper .
Lemma 1.4. Let be a regular polyfacial set with respect to the system (1.7), and let be defined as follows: Let be a given set such that is a retract of but not a retract of . Then for each fixed system of initial functions , there is a point such that for the corresponding solution of (1.7), one has for each .
Remark 1.5. When Lemma 1.4 is applied, a lot of technical details should be fulfilled. In order to simplify necessary verifications, it is useful, without loss of generality, to vary the first coordinate in definition of the set in (1.9) within a half-open interval open at the right. Then the set is not open, but tracing the proof of Lemma 1.4, it is easy to see that for such sets it remains valid. Such possibility is used below. We will apply similar remark and explanation to sets of the type , which serve as domains of definitions of functionals on the right-hand sides of equations considered.
For continuous vector functions with for (the symbol here and below means that for all , continuously differentiable on , we define the set In the sequel, we employ the following result from [18, Theorem 1], which is proved with the aid of the retract technique combined with Razumikhin's approach.
Theorem 1.6. Let there be a such that (i)if , and for any , then
for any , (If , this condition is omitted.) (ii)if , and for any then
for any . (If , this condition is omitted.)
Then, there exists an uncountable set of solutions of (1.7) on such that each satisfies
1.2. Structure of Solutions of a Linear Equation
In this section we focus our attention to structure of solutions of scalar linear differential equation of the type (1.3) with variable bounded delay of the form with continuous functions and .
In accordance with above definitions of positive or oscillatory solutions, we call a solution of (1.19) oscillatory if it has arbitrary large zeros, otherwise it is called nonoscillatory (positive or negative).
Theorem 1.9. Suppose the existence of a positive solution of (1.19) on . Then there exist two positive solutions and of (1.19) on satisfying the relation such that every solution of (1.19) on can be represented by the formula where the constant depends on .
The symbol , applied in (1.23) and below, is the Landau order symbol frequently used in asymptotic analysis.
Definition 1.11 (see [20, Definition 2]). Suppose that the positive solutions and of (1.19) on satisfy the relation (1.22). Then, we call the solution a dominant solution and the solution a subdominant solution.
2. Main Results
Let us consider two auxiliary linear equations: where and , are positive continuous functions on , . According to the Theorems 1.7 and 1.9, both (2.1) and (2.2) have two types of positive solutions (subdominant and dominant). Let us denote them , for (2.1) and , for (2.2), respectively, such that Without loss of generality, we can suppose that and on .
2.1. Auxiliary Linear Result
Lemma 2.1. Let (1.5) be valid. Let , be dominant and subdominant solutions for (2.2). Then there are positive solutions , of (2.1) on such that: (a), , (b), , (c) and are dominant and subdominant solutions for (2.1).
Proof. (a) To prove the part (a), we employ Theorem 1.6 with ; that is, we apply the case (i). Consider (2.1), set , , , and assume (see the case (i)):
Now we have to verify the inequalities (1.16), that is, in our case:
and if . Further, we have
and if . Since both inequalities are fulfilled and all assumptions of Theorem 1.6 are satisfied for the case in question, there exists a solution of (2.1) on such that for .
(b) To prove the part (b), we consider a solution of the following initial problem: Now, let us define a function We find the sign of the full derivative of along the trajectories of (2.7) if : It means that function is nonincreasing and it holds and hence . It will be showed that this inequality holds also for every .
On the contrary, let us suppose that the inequality is not true, that is, there exists a point such that . Then there exists a point such that , otherwise on . Without loss of generality, we can suppose that on with a and on . Then, there exists a point such that for a constant and Hence, for a function defined as , , we get It means that on a right-hand neighborhood of . This is a contradiction with inequality hence it is proved that the existence of a solution satisfies on .
(c) To prove the part (c), we consider . Due to (a) and (b), we get and and are (by Definition 1.11) dominant and subdominant solutions for (2.1).
2.2. Existence of Positive Solutions of (1.3)
The next theorems state that there exist two classes of positive solutions of (1.3) such that graphs of each solution of the first class are between graphs of dominant solutions of (2.1) and (2.2), and graphs of each solution of the second class are between graphs of subdominant solutions of (2.1) and (2.2), respectively. It means that we prove there are two classes of asymptotically different positive solutions of (1.3). Without loss of generality (see Remark 1.5), we put . In the following, we will use our main supposition (1.4); that is, we assume that for inequalities, hold, where is supposed to be positive.
Theorem 2.2. Let be a continuous quasibounded functional. Let inequality (1.5) be valid, and (2.16) holds for any with , . Let be a positive solution of (2.1) on , and let be a positive solution of (2.2) on such that on . Then there exists an uncountable set of positive solutions of (1.3) on such that each solution satisfies for .
Proof. To prove this theorem, we employ Theorem 1.6 with ; that is, we apply the case (i). Set , , ; hence, the set will be defined as Now, we have to verify the inequalities (1.16). In our case Therefore, Both inequalities (1.16) are fulfilled, and all assumptions of Theorem 1.6 are satisfied for the case in question. There exists class of positive solutions of (1.3) on that for each solution from this class it is satisfied that for .
Corollary 2.3. Let, in accordance with Lemma 2.1, be the subdominant solution of (2.1), and let be the subdominant solution of (2.2), that is, on . Then, there exists an uncountable set of positive solutions of (1.3) on such that each solution satisfies
If inequality (1.6) holds, then dominant solutions of (2.1) and of (2.2) have finite positive limits This is a simple consequence of positivity of solutions , and properties of dominant and subdominant solutions (see Theorem 1.7, Remark 1.8, Theorem 1.9, formulas (1.22)–(1.25) and (2.3)). Then, due to linearity of (2.1) and (2.2), it is clear that there are dominant solutions , of both equations such that on . In the following lemma, we without loss of generality suppose that and are such solutions and their initial functions are nonincreasing on initial interval . We will need constants and satisfying
Lemma 2.4. Let be a continuous quasibounded functional. Let inequalities (1.5) and (1.6) be valid, and (2.16) holds for any with , . Let , be a dominant solution of (2.1), nonincreasing on , and let , be a dominant solution of (2.2), nonincreasing on , such that , . Then there exists another dominant solution of (2.2) and a positive solution of (1.3) on such that it holds that for and .
Proof. Both dominant solutions and , of (2.1) and (2.2), respectively, have nonzero positive limits and . From linearity of (2.1) and (2.2), it follows that solutions multiplied by an arbitrary constant are also solutions of (2.1) and (2.2), respectively. It holds that
Now, we define the set in the same way as (2.18) in the proof of Theorem 2.2, but with instead of and with instead of , that is, According to the Theorem 2.2 (with instead of and with instead of ), it is visible that there exists a positive solution of (1.3) satisfying where ; that is, inequalities (2.24) hold.
2.3. Asymptotically Different Behavior of Positive Solutions of (1.3)
Somewhat reformulating the statement of Theorem 2.5, we can define a class of positive solutions of (1.3) such that every solution is defined on and satisfies where for a positive constant and, for every positive constant , there exists a solution satisfying (2.31) on .
The following theorem states that positive solutions and of (1.3) have a different order of vanishing.
Theorem 2.6. Let all the assumptions of Corollary 2.3 and Theorem 2.5 be met. Then there exist two classes and of positive solutions of (1.3) described by inequalities (2.21) and (2.31). Every two solutions , , such that and , have asymptotically different behavior, that is,
Proof. Let the solution be the one specified in Corollary 2.3 and the solution specified by (2.31) with a positive constant . Now let us verify that (2.32) holds. With the aid of inequalities (2.21) and (2.31), we get in accordance with (1.22), since and are positive (subdominant and dominant) solutions of linear equation (2.2).
Theorem 2.7. Let be a continuous quasibounded functional. Let inequalities (1.5) and (1.6) be valid, and (2.16) holds for any with , . Then on there exist (a)dominant and subdominant solutions , of (2.1), (b)dominant and subdominant solutions , of (2.2), (c)solutions , of (1.3) such that
Example 2.8. Let (1.3) be reduced to and let auxiliary linear equations (2.1) and (2.2) be reduced to that is, Let be sufficiently large. Inequalities (1.5), (1.6), and (2.16) hold. In view of linearity and by Remark 1.8, we conclude that there exist dominant solutions of (2.37) and of (2.38) such that Moreover, there exist subdominant solutions of (2.37) and of (2.38) such that , which are defined as By Theorem 2.7, we conclude that there exist solutions and of (2.36) satisfying inequalities (2.34), and (without loss of generality) inequalities hold on .
3. Conclusions and Open Problems
The following problems were not answered in the paper and present interesting topics for investigation.
Open Problem 3.1. In Lemma 2.4 and Theorems 2.5–2.7 we used the convergence assumption (1.6) being, without loss of generality, equivalent to It is an open question whether similar results could be proved if the integral is divergent, that is, if
Open Problem 3.2. Dominant and subdominant solutions are used for representation of family of all solutions of scalar linear differential delayed equation, for example, by formula (1.25). Investigation in this line of the role of solutions and of (1.3) (see Theorems 2.6 and 2.7) is an important question. Namely, it seems to be an interesting question to establish sufficient conditions for the right-hand side of (1.3) such that its every solution can be represented on by the formula where the constant depends only on .
Open Problem 3.3. The notions dominant and subdominant solutions are in the cited papers defined for scalar differential delayed equations only. It is a rather interesting question if the results presented can be enlarged to systems of differential delayed equations.
Remark 3.4. Except for papers and books mentioned in this paper we refer, for example, to sources [21–23], treating related problems as well. Note that the topic is connected with similar questions for discrete equations (e.g., [24–27]).
This research was supported by the Grant no. 1/0090/09 of the Grant Agency of Slovak Republic (VEGA) and by the Project APVV-0700-07 of Slovak Research and Development Agency.
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