Research Article | Open Access
M. Bartušek, M. Cecchi, Z. Došlá, M. Marini, "Fourth-Order Differential Equation with Deviating Argument", Abstract and Applied Analysis, vol. 2012, Article ID 185242, 17 pages, 2012. https://doi.org/10.1155/2012/185242
Fourth-Order Differential Equation with Deviating Argument
We consider the fourth-order differential equation with middle-term and deviating argument , in case when the corresponding second-order equation is oscillatory. Necessary and sufficient conditions for the existence of bounded and unbounded asymptotically linear solutions are given. The roles of the deviating argument and the nonlinearity are explained, too.
The aim of this paper is to investigate the fourth-order nonlinear differential equation with middle-term and deviating argument
The following assumptions will be made.(i) is a continuously differentiable bounded away from zero function, that is, for large such that (ii) are continuous functions for , is not identically zero for large , , and , .(iii) is a continuous function such that for .
Observe that (i) implies that there exists a positive constant such that and the linear second-order equation is oscillatory. Moreover, solutions of (1.3) are bounded together with their derivatives, see for example, [1, Theorem 2].
A solution of (1.1) is said to be asymptotically linear (AL-solution) if either or for some constants .
Fourth-order nonlinear differential equations naturally appear in models concerning physical, biological, and chemical phenomena, such as, for instance, problems of elasticity, deformation of structures, or soil settlement, see, for example, [2, 3].
When (1.3) is nonoscillatory and is its eventually positive solution, it is known that (1.1) can be written as the two-term equation In this case, the question of oscillation and asymptotics of such class of equations has been investigated with sufficient thoroughness, see, for example, the papers [3–10] or the monographs [11, 12] and references therein.
Nevertheless, as far we known, there are only few results concerning (1.1) when (1.3) is oscillatory. For instance, the equation without deviating argument has been investigated by Kiguradze in  in case and by the authors in [14, 15] when satisfies (i). In particular, in  the oscillation of (1.1) in the case is studied. In , the existence of positive bounded and unbounded solutions as well as of oscillatory solutions for (1.7) has been considered and the case has been analyzed in detail. Other results can be found in  and references therein, in which the existence and uniqueness of almost periodic solutions for equations of type (1.1) with almost periodic coefficients are studied.
Motivated by [14, 15], here we study the existence of AL-solutions for (1.1). The approach is completely different from the one used in , in which an iteration process, jointly with a comparison with the linear equation , is employed. Our tools are based on a topological method, certain integral inequalities, and some auxiliary functions. In particular, for proving the continuity in the Fréchet space of the fixed point operators here considered, we use a similar argument to that in the Vitali convergence theorem.
Our results extend to the case with deviating argument analogues ones stated in  for (1.7) when . We obtain sharper conditions for the existence of unbounded AL-solutions of (1.1), and, in addition, we show that under additional assumptions on , these conditions become also necessary for the existence of AL-solutions, in both the bounded and unbounded cases. In the final part, we consider the particular case and we study the possible coexistence of bounded and unbounded AL-solutions. The role of deviating argument and the one of the growth of the nonlinearity are also discussed and illustrated by some examples.
2. Unbounded Solutions
Here we study the existence of unbounded AL-solutions of (1.1). Our first main result is the following.
Theorem 2.1. For any , , there exists an unbounded solution of (1.1) such that provided where for
Proof. Without loss of generality, we prove the existence of solutions of (1.1) satisfying (2.1) for .
Let and be two linearly independent solutions of (1.3) with Wronskian . Denote As claimed by the assumptions on , all solutions of (1.3) and their derivatives are bounded. Thus, put Let be such that for . Define and choose large so that
Denote by the Fréchet space of all continuous functions on , endowed with the topology of uniform convergence on compact subintervals of , and consider the set given by Let and define on the function where and . Then, Moreover, , and it holds for that Integrating, we obtain From here and (1.3), we get Thus, and so or, in view of (2.7),
Thus, from (2.10), as , we get Hence, the operator given by is well defined for any . Moreover, in view of (2.19), we have From here, in virtue of (2.7) we get Hence, . From (2.5) and (2.11), we have and so . Similarly, and thus, , too. In addition, Hence, any fixed point of is a solution of (1.1) for large .
Let us show that is relatively compact, that is, consists of functions equibounded and equicontinuous on every compact interval of . Because , the equiboundedness follows. Moreover, in view of (2.7), is bounded for any , which yields the equicontinuity of the elements in .
Now we prove the continuity of in . Let , , be a sequence in , which uniformly converges to on every compact interval of . Fixing , in virtue of (2.23), the dominated convergence Lebesgue theorem gives Moreover, In view of (2.19), we have Thus, choosing sufficiently large, we get from (2.27) and so the continuity of in follows. By the Tychonoff fixed point theorem, the operator has a fixed point , which is an unbounded solution of (1.1) satisfying (2.1).
Remark 2.2. With minor modifications, Theorem 2.1 gives also the existence of eventually negative unbounded AL-solutions. The details are omitted.
Remark 2.3. When , Theorem 2.1 is related with Theorem 1 in , from which the existence of unbounded AL-solutions of (1.1) can be obtained under stronger assumptions. A comparison between Theorem 1 in  and Theorem 2.1 is given in Section 4.
Our next result gives a necessary condition for the existence of unbounded solutions of (1.1) satisfying for large and some and
Proof. Assume , and let be an eventually positive solution of (1.1) satisfying (2.30). Then, there exists such that Consequently, in view of (2.31), we have Thus, integrating (1.1), we get Furthermore, where . Hence , which gives a contradiction with the boundedness of . Finally, if , the argument is similar and the details are left to the reader.
3. Bounded Solutions
In this section we study the existence of bounded AL-solutions of (1.1). The following holds.
Theorem 3.1. If then, for any , there exists a solution of (1.1) satisfying
Proof. Without loss of generality, we prove the existence of solutions of (1.1) satisfying (3.2) for .
We proceed by a similar way to that in the proof of Theorem 2.1, and we sketch the proof.
Let be the constant given in (2.5), and let Choose large so that and define as in (2.6). Denote by the Fréchet space of all continuous functions on , endowed with the topology of uniform convergence on compact subintervals of , and consider the set given by Let , and, for any , consider again the function given in (2.10). Reasoning as in the proof of Theorem 2.1, with minor changes, we obtain Hence, in virtue of (3.1), the operator given by is well defined and . In view of (3.6), we get A similar estimation holds for . Thus, . In view of (3.4), from (3.8), we obtain that is, . Moreover, a standard calculation gives and so any fixed point of is, for large , a solution of (1.1). Proceeding by a similar way to that in the proof of Theorem 2.1, we obtain that is relatively compact.
Now we prove the continuity of in . Let , , be a sequence in , which uniformly converges to on every compact interval of . Since in virtue of (3.1), the dominated convergence Lebesgue theorem gives Moreover, fixing , we have In view of (3.9), we have and so, choosing sufficiently large, from (3.13) we obtain the continuity of in . Hence, by the Tychonoff fixed point theorem, the operator has a fixed point , which is a bounded solution of (1.1) satisfying (3.2).
Remark 3.2. When , Theorem 3.1 extends to equations with deviating argument of a similar result stated in  for (1.7). Observe that our approach used here is completely different from that in .
The following lemmas are needed for proving Theorem 3.3.
Proof. Suppose on . We have
from (3.22), we get
where is a suitable constant. Since are bounded, , in view of (1.1), inequality (3.20) follows.
Moreover, is nondecreasing for . Because is a positive bounded function, then on . Thus, inequality (3.21) follows integrating by parts and using (1.1). Finally, if on , the argument is similar.
Proof. Since , there exists such that for
Without loss of generality, let . Thus, .
Let and be two linearly independent solutions of (1.3) with Wronskian . By assumptions on , all solutions of (1.3) and their derivatives are bounded. Thus, by the variation constant formula, there exist constants and such that and, in view of (3.25), is bounded.
Let us prove (3.26), and suppose on . Multiplying (1.1) by and integrating from to , we get or, in view of Lemma 3.4, where is a suitable constant. Since is bounded and from (3.17) and (3.30), we have which gives the assertion. The case on can be treated in a similar way.
Proof of Theorem 3.3. Suppose , for . Without loss of generality, assume also that (3.27) holds for . Define
Since for , we have
Case I. Assume Since for we have and, as claimed, , we get Thus, from (3.36), we obtain , that is, is unbounded. Hence, in view of (3.33), we obtain a contradiction with the boundedness of .Case II. Now assume (3.17) and (3.25). In view of Lemma 3.5, without loss of generality, we can suppose that (3.26) holds for . Then, Hence, . Integrating (3.36), we get and, in view of Lemma 3.4, we have Thus, , that is, as before, a contradiction. Finally, the case , for large follows in a similar way.
Here we present some applications of our results to a particular case of (1.1), namely, the equation jointly with some suggestions for future research.
4.1. Coexistence of Both Types of AL-Solutions
Corollary 4.1 shows also that the deviating argument can produce a different situation concerning the unboundedness of solutions with respect to the corresponding equation without delay, as the following example illustrates.
Example 4.2. In view of Corollary 4.1(a), the equation where satisfies (i), has unbounded AL-solutions, while the corresponding ordinary equation in view of Theorem 2.4, does not have unbounded AL-solutions. Moreover, if in addition for large , then from Corollary 4.1(b) (4.3) does not have bounded AL-solutions.
The following example shows that the opposite situation to the one described in Example 4.2 can occur.
Example 4.3. Consider the equation where satisfies (i). From Theorem 3.1, (4.5) has bounded AL-solutions and the same occurs for the corresponding ordinary equation. Nevertheless, in view of Corollary 4.1(a), (4.5) has no unbounded AL-solutions.
Corollary 4.4. Let for large .
(a) Assume for large If (4.1) has unbounded AL-solutions, then (4.1) also has AS bounded solutions.
(b) Assume for large If (4.1) has bounded AL-solutions, then (4.1) also has unbounded AL-solutions.
For the equation without deviating argument from Corollary 4.4 we get the following.
Corollary 4.5. Let for large .
(a) Assume . If (4.8) has unbounded AL-solutions, then (4.8) has also bounded AL-solutions.
(b) Assume and for large . If (4.8) has bounded AL-solutions, then (4.8) has also unbounded AL-solutions.
4.2. Comparison with Some Results in 
Theorem A. If then there exists a solution of (4.8) such that where are functions of bounded variation for large and .
Therefore, when , Theorem 2.1 ensures the existence of unbounded AL-solutions of (4.8) under a weaker condition than (4.9), namely, On the other hand, Theorem A gives an asymptotic formula for such solutions.
4.3. An Open Problem
Equation (1.1) can admit also other types of nonoscillatory solutions, as the following examples show.
Example 4.6. Consider the equation In virtue of Corollary 4.1(b), (4.12) has no bounded AL-solutions. Nevertheless, this equation admits nonoscillatory bounded solutions because is a solution of (4.12).
Example 4.7. Consider the equation Thus, (3.1) holds, while . Hence, in virtue of Corollary 4.1, (4.13) has bounded AL-solutions, but no unbounded AL-solutions. Nevertheless, this equation admits nonoscillatory unbounded solutions because is a solution of (4.13).
The existence of nonoscillatory solutions satisfying either or will be a subject of our next research.
The first and third authors are supported by Grant GAP 201/11/0768 of the Czech Grant Agency. The fourth author is supported by the research Project PRIN09-Area 01, Ordinary Differential Equations, of the Italian Ministry of Education.
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