Oscillatory Solutions of Singular Equations Arising in Hydrodynamics
© Irena Rachůnková et al. 2010
Received: 26 December 2009
Accepted: 29 March 2010
Published: 7 April 2010
We investigate the singular differential equation on the half-line [ ), where satisfies the local Lipschitz condition on and has at least two simple zeros. The function is continuous on [ ) and has a positive continuous derivative on ( ) and . We bring additional conditions for and under which the equation has oscillatory solutions with decreasing amplitudes.
A function which has continuous second derivative on and satisfies (1.1) for all is called a solution of (1.1).
If ( or ), then is called a damped solution (a homoclinic solution or an escape solution) of problem (1.1), (1.7).
In [10, 12] these three types of solutions of problem (1.1), (1.7) have been studied, and the existence of each type has been proved for sublinear or linear asymptotic behaviour of near . In , has been supposed to have a zero . Here we generalize and extend the results of [10–12] concerning damped solutions. We prove their existence under weaker assumptions than in the above papers. Moreover, we bring conditions under which each damped solution is oscillatory; that is, it has an unbounded set of isolated zeros.
We replace assumptions (1.4)–(1.6) by the following ones.
( is possible).
2. Damped Solutions
Theorem 2.1 (Existence and uniqueness).
Hence is a contraction on , and the Banach fixed point theorem yields a unique fixed point of .
Having in mind that , can be (uniquely) extended as a function satisfying (2.3) onto . Since is arbitrary, can be extended onto as a solution of (2.3). We have proved that problem (2.3), (1.7) has a unique solution.
Let for some . Then (2.16) yields which is not possible because is decreasing on by (1.9) and (2.2). Therefore for . Consequently, due to (2.2), is a solution of (1.1).
Assume that there exists another solution of problem (1.1), (1.7). Then we can prove similarly as in Step 3 that for . This implies that is also a solution of problem (2.3), (1.7) and by Step 2, . We have proved that problem (1.1), (1.7) has a unique solution.
Then for all .
We see that the constant function is a solution of (1.1). Let be a solution of (1.1) satisfying (2.17) and let for some . Then the regular initial problem (1.1), (2.17) has two different solutions and , which contradicts (1.2).
Theorem 2.4 (Existence of damped solutions).
Assume that (1.2), (1.3), (1.9), and (1.10) hold. Let be given by (2.19), and assume that is a solution of problem (1.1), (1.7) with . Then is a damped solution.
This contradicts (2.20).
3. Oscillatory Solutions
Then the next lemmas can be proved.
We get which contradicts . The obtained contradictions imply that (3.4) cannot occur and hence satisfying (3.3) must exist.
We can argue as in the proof of Lemma 3.1 working with and instead of and .
We argue similarly as in the proof of Lemma 3.1.
and we derive as in the proof of Lemma 3.1 that (3.10) holds.
We get contrary to . The obtained contradictions imply that (3.24) cannot occur and that satisfying (3.23) must exist.
Assume that (1.2), (1.3), (1.9), (1.10), (3.1), and (3.2) hold. Let be a solution of problem (1.1), (1.7) with . If is a damped solution, then is oscillatory and its amplitudes are decreasing.
which contradicts (3.32). Therefore and there exists such that (3.22) holds. Lemma 3.3 yields satisfying (3.23). Therefore has just one positive local maximum between its first zero and second zero .
which contradicts (3.32). Therefore and there exists such that (3.20) holds. Corollary 3.2 yields satisfying (3.21). Therefore has just one negative minimum between its second zero and third zero .
If , then and hence , that is, .
Theorem 3.6 (Existence of oscillatory solutions).
Assume that (1.2), (1.3), (1.9), (1.10), (3.1), and (3.2) hold. Let be given by (2.19) and let be a solution of problem (1.1), (1.7) with . Then is an oscillatory solution with decreasing amplitudes.
The assertion follows from Theorems 2.4 and 3.4.
The assumption (1.10) in Theorem 3.6 can be omitted, because it has no influence on the existence of oscillatory solutions. It follows from the fact that (1.10) imposes conditions on the function values of the function for arguments greater than ; however, the function values of oscillatory solutions are lower than this constant . This condition (used only in Theorem 2.1) guaranteed the existence of solution of each problem (1.1), (1.7) for each on the whole half-line, which simplified the investigation of the problem.
This work was supported by the Council of Czech Government MSM 6198959214.
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