Abstract
Sufficient conditions are established guaranteeing the existence of a positive ωperiodic solution to the equation
where
where
MSC: 34C25, 34B16, 34B18, 76N15.
Keywords:
RayleighPlesset equation; singular equation; periodic solution; upper and lower function1 Introduction
The topic of singular boundary value problems has been of substantial and rapidly growing interest for many scientists and engineers. The importance of such investigation is emphasized by the fact that numerical simulations of solutions to such problems usually break down near singular points.
On the other hand, problems of this type arise frequently in applied science. Namely, in fluid mechanics, since 1917 the physicists have used the Rayleigh equation,
to model the bubble dynamics in liquid, where
The transformation
Consequently, the class of equations
with nonnegative constants
subjected to the periodic conditions
is investigated in the presented paper. Here,
A particular case of (1.1) is the equation
studied by Lazer and Solimini. Their results were published in 1987 (see [11]) and they proved, among others, that (1.4), (1.3) has at least one solution if and
only if
The structure of the paper is as follows. After the introduction and basic notation, we recall the definition of lower and upper functions to the problem (1.2), (1.3), and we formulate the classical theorem on the existence of a solution to (1.2), (1.3) in the case when there exists a couple of wellordered lower and upper functions. In Section 2, we establish our main results and their consequences. Sections 3 and 4 are devoted to auxiliary propositions and proofs of the main results, respectively.
For convenience, we finish the introduction with a list of notations which are used throughout the paper:
ℕ is the set of all natural numbers, ℝ is the set of all real numbers,
For a given
Given
The following definitions of lower and upper functions are suitable for us. For more general definitions, one can see, e.g., [[12], Definition 8.2].
Definition 1.1 A function
Definition 1.2 A function
The following theorem is well known in the theory of differential equations (see, e.g., [[12], Theorem 8.12]).
Theorem 1.1Letαandβbe lower and upper functions to the problem (1.2), (1.3) such that
Then there exists a solutionuto the problem (1.2), (1.3) such that
2 Main results
Theorem 2.1Let
and let there exist
Let, moreover, there exist
and let either
or
Furthermore, let us suppose that
(a) there exists a sequence
and there exist
where
(b) the function
where
Besides, let us suppose that
(c) there exists a sequence
and there exist
where
(d) the function
where
Then there exists at least one solution to the problem (1.2), (1.3).
Remark 2.1 Note that there exists a suitable
For equation (1.1), from Theorem 2.1 we get the following assertion.
Corollary 2.1Let
If
then (1.1), (1.3) has at least one solution.
Remark 2.2 In [28], it is proved, among others, that the equation
with
Corollary 2.1 says that if a frictionlike term or sublinear term are added to (2.13),
the condition
has a positive solution satisfying (1.3) for any
subjected to the boundary conditions (1.3) is solvable for any
Example 2.1 As it was mentioned in the introduction, the particular case of (1.1) is the socalled RayleighPlesset equation frequently used in fluid mechanics. This equation has the following form:
where
The results dealing with the existence of positive ωperiodic solutions of (2.14) were established in [10] provided
1.
2.
3.
Thus, Corollary 2.1 assures that the boundedness of
Corollary 2.2Let
where
Then the problem (1.1), (1.3) with
Remark 2.3 According to [29] and Theorem 1.1, it can be easily verified that the problem
with
Indeed, according to [[29], Definition 1.1], the inclusion
Therefore there exist constants
one can easily realize that α and β are, respectively, lower and upper functions to (2.16) satisfying (1.5). Now, the existence of a positive solution to (2.16) follows from Theorem 1.1.
On the other hand, the existence of a positive solution to (2.16) implies the inclusion
However, one of the optimal effective conditions guaranteeing such an inclusion is
(see [[29], Corollary 2.5]). Therefore, the condition (2.15) is natural in a certain sense.
When the righthand side of equation (1.3) does not depend on u, i.e., when
From Theorem 2.1, for equation (2.17) we get the following assertion.
Corollary 2.3Let there exist
and let
Let, moreover, either
or
Then there exists at least one solution to the problem (2.17), (1.3).
In the following result, the assumptions do not depend on the frictionlike term.
On the other hand, a certain smallness of oscillation of the primitive to
Theorem 2.2Let
Let, moreover,
and let there exist
Besides, let us suppose that
In the particular case, when equation (1.2) has the form (1.1), the following assertion immediately follows from Theorem 2.2.
Corollary 2.4Let
Let, moreover,
Then the problem (1.1), (1.3) has at least one solution.
Remark 2.4 The consequence of Theorem 2.2 for the problem (2.17), (1.3) coincides with the result obtained in [[10], Theorem 3.6].
3 Auxiliary propositions
In what follows, we will show the existence of a solution to the equation
satisfying the boundary conditions (1.3). Here,
where
Obviously,
and
The following three results can be found in [10].
Lemma 3.1 (see [[10], Corollary 2.17])
Let
Let, moreover, there exist a sequence
and let there exist
where
Lemma 3.2 (see [[10], Corollary 2.18])
Let
where
Lemma 3.3 (see [[10], Corollary 2.11])
Let
Then there exists a lower functionαto the problem (2.17), (1.3) with
Lemma 3.4Let
where
Proof
Put
Then, obviously, in view of (3.3), we have
and, consequently, on account of (2.2), (3.5), (3.10), and (3.13), there exists
Therefore, according to Lemma 3.1, there exists an upper function β to (3.2), (1.3) with
Lemma 3.5Let
where
Proof Define
Therefore, according to Lemma 3.2, there exists an upper function β to (3.2), (1.3) with
Lemma 3.6Let
and let either
or
Then, for every
we have the estimate
Proof Assume that (3.20) is fulfilled. Let u be a positive solution to (3.2), (1.3) satisfying (3.21). Then there exist
Define the operator ϑ of ωperiodic prolongation by
Then, obviously, from (3.2) and (1.3) it follows that
The integration of (3.25) from
From (3.21), (3.23), and (3.24) it follows that
Put
According to (3.18), we have
Thus, using (3.3), (3.20), (3.21), and (3.27)(3.29) in (3.26), we arrive at
Put
Then, on account of (3.24) and (3.30), we have
On the other hand, the integration of (3.25) from t to
Now, using (3.3), (3.20), (3.21), and (3.27)(3.29) in (3.32), we obtain
Therefore, in view of (3.24), from (3.33) we get
Consequently, (3.31) and (3.34) result in (3.22).
Now, suppose that (3.19) is fulfilled. Put
Then, according to (3.2), we have
where
Analogously to the aboveproved, using (3.19) instead of (3.20), we obtain
with
Thus, (3.35) and (3.37) yield (3.22). □
Lemma 3.7Let
and let either
or
Then, for every
Proof Let u be a positive solution to (3.2), (1.3) satisfying (3.21). Thus, the integration of (3.2) from 0 to ω, in view of (1.3) and (3.4), yields
On the other hand, (3.38) implies the existence of
Let
Obviously, either
or
Obviously, it is sufficient to show the estimate (3.41) is valid just in the case when (3.45) is fulfilled. Let, therefore, (3.45) hold.
If
where ϑ is the operator defined by (3.24). Obviously, (3.25) holds.
Assume that (3.39) holds. Then, according to Lemma 3.6, there exists
Note that in view of (3.46), we have
where
Note that
If (3.40) holds, we integrate (3.25) from
with
Therefore, also in this case, there exists a constant
Lemma 3.8Let
Proof According to Lemma 3.4, there exist
Thus, if we put
Moreover, according to Lemmas 3.6 and 3.7, in view of (3.50), there exist constants
where
Therefore, according to the ArzelàAscoli theorem, there exist
Moreover, since
The following assertion can be proved analogously to Lemma 3.8, just Lemma 3.5 is used instead of Lemma 3.4.
Lemma 3.9Let
Lemma 3.10Let
Proof Because
Consider the auxiliary equation
Put
and, in view of (3.55), from (2.5) we have
Furthermore, the substitution
resp.
Moreover, put
Finally, (2.10) results in
and so, since
Therefore, applying Lemma 3.8, according to (3.57)(3.62), there exists a positive solution u to the problem (3.56), (1.3).
Now, we put
Obviously,
Thus, it can be easily seen that α is a lower function to the problem (3.1), (1.3). □
Analogously to the proof of Lemma 3.10, one can prove the following assertion applying Lemma 3.9 instead of Lemma 3.8.
Lemma 3.11Let
4 Proofs of the main results
Proof of Theorem 2.1 According to Lemmas 3.1, 3.2, 3.10, and 3.11, the conditions of the theorem guarantee a wellordered couple of lower and upper functions, therefore the result is a direct consequence of Theorem 1.1. □
Proof of Corollary 2.1 It follows from Theorem 2.1 with
Then items (a) and (c) of Theorem 2.1 are fulfilled. □
Proof of Corollary 2.2 It follows from Theorem 2.1 with
Proof of Corollary 2.3 It immediately follows from Theorem 2.1 with
Proof of Theorem 2.2
Put
Because
Consider the auxiliary equation
Put
Therefore, according to Lemma 3.3, there exists a lower function w to the problem (4.2), (1.3) satisfying
Now, we put
Obviously, with respect to (4.3),
Thus, on account of (2.18), (3.9), and (4.2), it can be easily seen that α is a lower function to the problem (1.2), (1.3).
The existence of an upper function β to (1.2), (1.3) satisfying
follows from (2.19) and Lemma 3.1, resp. 3.2.
Obviously, in view of (3.9) and (4.4), we have that (1.5) holds. Thus the theorem follows from Theorem 1.1. □
Proof of Corollary 2.4 It follows from Theorem 2.2 with
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
RH and MZ obtained the results in a joint research. Both authors read and approved the final manuscript.
Acknowledgements
The first author was supported by RVO: 67985840; the second author was supported by Ministerio de Educación y Ciencia, Spain, project MTM201123652.
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