Periodic solutions to the Liénard type equations with phase attractive singularities

Sufficient conditions are established guaranteeing the existence of a positive ω-periodic solution to the equation

where are continuous functions with possible singularities at zero and is a Carathéodory function. The results obtained are rewritten for the equation of the type

where , , δ are non-negative constants, c, μ, ν, γ are real numbers, and . The last equation also covers the so-called Rayleigh-Plesset equation, frequently used in fluid mechanics to model the bubble dynamics in liquid. In the paper, the case when , i.e., the case which covers the attractive singularity of the function g, is studied. The results obtained assure that there exists a positive ω-periodic solution to the above-mentioned equation if the power μ or ν is sufficiently large.

MSC: 34C25, 34B16, 34B18, 76N15.

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 is the ratio of the bubble at the time t, ρ is the liquid density, is the pressure in the liquid at a large distance from the bubble, and is the pressure in the liquid at the bubble boundary. In 1949, Plesset proposed to use a more exact equation involving the surface-tension constant S and the coefficient of the liquid viscosity , which was finally improved by adding a term with polytropic coefficient k in 1977, nowadays known as a Rayleigh-Plesset equation (see [1])

The transformation in the previous equation leads to the equation

Consequently, the class of equations

with non-negative constants , , δ, real numbers c, μ, ν, γ, and , plays an important role in fluid mechanics. Therefore, the equation

subjected to the periodic conditions

is investigated in the presented paper. Here, are continuous, having possible singularities at zero, and is a Carathéodory function, i.e., is measurable for all , is continuous for a.e. , and for every , there exists a non-negative function such that for a.e. and all . By a solution to (1.2), (1.3) we understand a function which is positive, absolutely continuous together with its first derivative, satisfies (1.2) almost everywhere on , and verifies (1.3). In spite of the fact that the problem (1.2), (1.3) was investigated by many mathematicians (see, e.g., [2-27]), most of the mentioned works deal with the repulsive case and/or when f has no singularity. However, the physical model, covered by equation (1.1), justifies considering the types of equations with a singular friction-like term.

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 , provided is bounded. Recently, we have proved (see [28]) that this result cannot be extended to the case when is a general integrable (and so unbounded) function unless some additional conditions are introduced. In particular, (1.4), (1.3) is solvable for any with if ; and, moreover, for any , there exists a function with such that (1.4), (1.3) has no solution. At this point, we would like to emphasize the important fact that the condition can be weakened if (1.4) is generalized to equation (1.1), see Remark 2.2 below.

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 well-ordered 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, , , .

is the Banach space of continuous functions with the norm

, resp. , is the set of continuous functions , resp. .

is the set of functions which are continuous together with their first derivative.

is a set of all functions such that u and are absolutely continuous.

is the Banach space of the Lebesgue integrable functions endowed with the norm

For a given , its mean value is defined by

Given , then

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 is called a lower function to the problem (1.2), (1.3) if for every and

Definition 1.2 A function is called an upper function to the problem (1.2), (1.3) if for every and

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

Theorem 2.1Letandbe non-decreasing functions, , andbe such that

and let there existsuch that

Let, moreover, there existsuch that

and let either

or

Furthermore, let us suppose thatfulfills at least one of the following conditions:

(a) there exists a sequenceof positive numbers such that

and there exist, , andsuch that

wherefor almost everyand

(b) the functionis non-increasing and

wherefor almost everyandσis given by (2.11).

Besides, let us suppose thatfulfills at least one of the following conditions:

(c) there exists a sequenceof positive numbers such that

and there existandsuch that

wherefor almost every;

(d) the functionis non-increasing and

wherefor almost every.

Then there exists at least one solution to the problem (1.2), (1.3).

Remark 2.1 Note that there exists a suitable such that (2.10) holds, e.g., if

For equation (1.1), from Theorem 2.1 we get the following assertion.

Corollary 2.1Let, , , , and

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 and , has a positive ω-periodic solution if . Moreover, there is also an example introduced showing that for any , there exists with such that (2.13), (1.3) has no positive solution.

Corollary 2.1 says that if a friction-like term or sub-linear term are added to (2.13), the condition can be weakened. For example,

has a positive solution satisfying (1.3) for any if , provided . Also, the equation

subjected to the boundary conditions (1.3) is solvable for any if , provided .

Example 2.1 As it was mentioned in the introduction, the particular case of (1.1) is the so-called Rayleigh-Plesset equation frequently used in fluid mechanics. This equation has the following form:

where , c, , are positive constants and (see [9,10]).

The results dealing with the existence of positive ω-periodic solutions of (2.14) were established in [10] provided is bounded from above (see [[10], Theorems 4.4, 4.6, 4.7]). However, Corollary 2.1 says that in the case when , the problem (2.14), (1.3) is solvable if one of the following items is fulfilled:

1. and ;

2. and ;

3. .

Thus, Corollary 2.1 assures that the boundedness of is not necessary.

Corollary 2.2Let, , , . Let, moreover, eitheror

wherefor almost every, and

Then the problem (1.1), (1.3) withhas at least one solution.

Remark 2.3 According to [29] and Theorem 1.1, it can be easily verified that the problem

with and , has a positive solution if and only if the inclusion holds (see notation in [29]).

Indeed, according to [[29], Definition 1.1], the inclusion implies the existence of a positive solution v to the problem

Therefore there exist constants and such that for . By setting

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 (see [[29], Theorem 2.1]).

However, one of the optimal effective conditions guaranteeing such an inclusion is and

(see [[29], Corollary 2.5]). Therefore, the condition (2.15) is natural in a certain sense.

When the right-hand side of equation (1.3) does not depend on u, i.e., when , then (1.3) has the form

From Theorem 2.1, for equation (2.17) we get the following assertion.

Corollary 2.3Let there existandsuch that

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 friction-like term. On the other hand, a certain smallness of oscillation of the primitive to is supposed. Clearly, Theorems 2.1 and 2.2 are independent.

Theorem 2.2Letandbe non-decreasing functions, , andbe such that

Let, moreover,

and let there existsuch that

Besides, let us suppose thatfulfills at least one of the conditions (c) or (d) of Theorem 2.1. Then there exists at least one solution to the problem (1.2), (1.3).

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, and letbe such that

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].

In what follows, we will show the existence of a solution to the equation

satisfying the boundary conditions (1.3). Here, is a non-decreasing function, , and . Together with (3.1), for every , consider the auxiliary equation

where

Obviously,

and

The following three results can be found in [10].

Lemma 3.1 (see [[10], Corollary 2.17])

Letandbe such that

Let, moreover, there exist a sequenceof positive numbers such that

and let there existandsuch that

wherefor almost every. Then there exists an upper functionβto the problem (3.1), (1.3) satisfying

Lemma 3.2 (see [[10], Corollary 2.18])

Letandbe such that (3.6) holds. Ifis a non-increasing function such that

wherefor almost every, then there exists an upper functionβto the problem (3.1), (1.3) satisfying (3.8).

Lemma 3.3 (see [[10], Corollary 2.11])

Letbe such that

Then there exists a lower functionαto the problem (2.17), (1.3) with

Lemma 3.4Letandbe such that (2.2) holds. Let, moreover, there exist a sequenceof positive numbers such that (3.7) is fulfilled, and let there existandsuch that

wherefor almost every. Then there existand an upper functionβto the problems (3.2), (1.3) forsatisfying (3.8).

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 such that

Therefore, according to Lemma 3.1, there exists an upper function β to (3.2), (1.3) with satisfying (3.8). Obviously, in view of (3.4) and the non-negativity of , it follows that β is also an upper function to (3.2), (1.3) for . □

Lemma 3.5Letandbe such that (2.2) holds. Ifis a non-increasing function such that

wherefor almost every, then there existand an upper functionβto the problems (3.2), (1.3) forsatisfying (3.8).

Proof Define , , and by (3.11) and (3.12). Then, obviously, in view of (3.3), we have that (3.13) holds and, consequently, on account of (2.2), (3.5), (3.13), and (3.16), there exists such that (3.14) is valid and

Therefore, according to Lemma 3.2, there exists an upper function β to (3.2), (1.3) with satisfying (3.8). Obviously, in view of (3.4) and the non-negativity of , it follows that β is also an upper function to (3.2), (1.3) for . □

Lemma 3.6Let

and let either

or

Then, for every, there exists a constantsuch that for anyand any positive solutionuof (3.2), (1.3) with

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 such that

Define the operator ϑ of ω-periodic prolongation by

Then, obviously, from (3.2) and (1.3) it follows that

The integration of (3.25) from to t, on account of (3.23), yields

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 , with respect to (3.23), results in

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 above-proved, 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, there exists a constantsuch that for anyand any positive solutionuof (3.2), (1.3) satisfying (3.21), we have the estimate

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 such that

Let be such that

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 for , then applying (3.43) in (3.42) we obtain a contradiction. Thus, there exist points such that

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 such that (3.22) holds. The integration of (3.25) from to , in view of (3.4), (3.21), (3.22), (3.39), (3.43), (3.44), and (3.46), results in

Note that in view of (3.46), we have . Consequently, from (3.48) we obtain

where

Note that does not depend on k. Therefore, if we apply (3.39) in (3.49), it can be easily seen, with respect to (3.44), that there exists a constant such that (3.41) holds.

If (3.40) holds, we integrate (3.25) from to and apply similar steps as above, just using (3.47) instead of (3.46). Finally, we arrive at

with

Therefore, also in this case, there exists a constant such that (3.41) holds. □

Lemma 3.8Letandbe such that (2.2) holds. Let, moreover, (3.38) be fulfilled, and let either (3.39) or (3.40) be valid. Let, in addition, there exist a sequenceof positive numbers such that (3.7) holds, and let there existandsuch that (3.10) is fulfilled, wherefor almost every. Then there exists a positive solutionuto (3.1), (1.3).

Proof According to Lemma 3.4, there exist and an upper function β to the problems (3.2), (1.3) for satisfying (3.8). On the other hand, in view of (3.4) and (3.38), there exists for such that

Thus, if we put for , according to Theorem 1.1, there exists a solution to (3.2), (1.3) for satisfying

Moreover, according to Lemmas 3.6 and 3.7, in view of (3.50), there exist constants , , and , not depending on k, such that

where

Therefore, according to the Arzelà-Ascoli theorem, there exist and such that

Moreover, since are solutions to (3.2), (1.3), in view of (3.3), (3.52), and (3.54), we have , , and is a positive solution to (3.1), (1.3). □

The following assertion can be proved analogously to Lemma 3.8, just Lemma 3.5 is used instead of Lemma 3.4.

Lemma 3.9Letandbe such that (2.2) holds. Let, moreover, (3.38) be fulfilled, and let either (3.39) or (3.40) be valid. Let, in addition, be a non-increasing function and let (3.16) be fulfilled, wherefor almost every. Then there exists a positive solutionuto (3.1), (1.3).

Lemma 3.10Letbe non-decreasing, , andbe such that (2.2) holds. Let, moreover, there existsuch that (2.4) and (2.5) are valid, and let either (2.6) or (2.7) be fulfilled. Let, in addition, there exist a sequenceof positive numbers such that (2.8) holds and let there exist, , andsuch that (2.9) and (2.10) are fulfilled, wherefor almost everyandσis given by (2.11). Then there exists a lower functionαto the problem (3.1), (1.3).

Proof Because is a positive function, from (2.4) and (2.11) we obtain that σ is a positive increasing function. Therefore, there exists an inverse function to σ which is also increasing. Moreover, in view of (2.4) and (2.11), it follows that

Consider the auxiliary equation

Put , . Then from (2.2) we get

and, in view of (3.55), from (2.5) we have

Furthermore, the substitution in (2.6), resp (2.7), with respect to (2.11) yields

resp.

Moreover, put for . Then from (2.8), in view of (3.55), we get

Finally, (2.10) results in

and so, since is a non-decreasing function, from (2.9) we obtain

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 for , i.e., in view of (2.11),

Obviously, is a positive function and

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 3.11Letbe non-decreasing, , andbe such that (2.2) holds. Let, moreover, there existsuch that (2.4) and (2.5) are valid, and let either (2.6) or (2.7) be fulfilled. Let, in addition, be a non-increasing function and let (2.12) be fulfilled, wherefor almost everyandσis given by (2.11). Then there exists a lower functionαto the problem (3.1), (1.3).

Proof of Theorem 2.1 According to Lemmas 3.1, 3.2, 3.10, and 3.11, the conditions of the theorem guarantee a well-ordered 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 , , , and such that

Then items (a) and (c) of Theorem 2.1 are fulfilled. □

Proof of Corollary 2.2 It follows from Theorem 2.1 with , , and such that , . Then items (b) and (d) of Theorem 2.1 are fulfilled. □

Proof of Corollary 2.3 It immediately follows from Theorem 2.1 with , (). □

Proof of Theorem 2.2

Put

Because is a positive function, from (2.20) and (4.1) we obtain that σ is an increasing function. Therefore, there exists an inverse function to σ which is also increasing.

Consider the auxiliary equation

Put , . Then from (2.20) and (2.21), in view of (4.1), we get

Therefore, according to Lemma 3.3, there exists a lower function w to the problem (4.2), (1.3) satisfying

Now, we put for , i.e., in view of (4.1),

Obviously, with respect to (4.3), is a positive function satisfying (3.9), and

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 , , and . □

The authors declare that they have no competing interests.

RH and MZ obtained the results in a joint research. Both authors read and approved the final manuscript.

The first author was supported by RVO: 67985840; the second author was supported by Ministerio de Educación y Ciencia, Spain, project MTM2011-23652.

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