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 f,g:(0,+\mathrm{\infty })\to \mathbb{R} are continuous functions with possible singularities at zero and h:[0,\omega ]\times \mathbb{R}\to \mathbb{R} is a Carathéodory function. The results obtained are rewritten for the equation of the type

where g_1, g_2, δ are non-negative constants, c, μ, ν, γ are real numbers, and h_0\in L([0,\omega ];\mathbb{R}). 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 \nu >\gamma, 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 R(t) is the ratio of the bubble at the time t, ρ is the liquid density, p_{\mathrm{\infty }} is the pressure in the liquid at a large distance from the bubble, and p(R) 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 {\mu }_{\ast }, 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 R=u^{\frac{2}{5}} in the previous equation leads to the equation

Consequently, the class of equations

with non-negative constants g_1, g_2, δ, real numbers c, μ, ν, γ, and h_0\in L([0,\omega ];\mathbb{R}), plays an important role in fluid mechanics. Therefore, the equation

subjected to the periodic conditions

is investigated in the presented paper. Here, f,g:(0,+\mathrm{\infty })\to \mathbb{R} are continuous, having possible singularities at zero, and h:[0,\omega ]\times \mathbb{R}\to \mathbb{R} is a Carathéodory function, i.e., h(\cdot ,x):[0,\omega ]\to \mathbb{R} is measurable for all x\in \mathbb{R}, h(t,\cdot ):\mathbb{R}\to \mathbb{R} is continuous for a.e. t\in [0,\omega ], and for every r>0, there exists a non-negative function q_r\in L([0,\omega ];\mathbb{R}) such that |h(t,x)|\le q_r(t) for a.e. t\in [0,\omega ] and all |x|\le r. By a solution to (1.2), (1.3) we understand a function u:[0,\omega ]\to \mathbb{R} which is positive, absolutely continuous together with its first derivative, satisfies (1.2) almost everywhere on [0,\omega ], 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 {\overline{h}}_0>0, provided h_0 is bounded. Recently, we have proved (see [28]) that this result cannot be extended to the case when h_0 is a general integrable (and so unbounded) function unless some additional conditions are introduced. In particular, (1.4), (1.3) is solvable for any h_0\in L([0,\omega ];\mathbb{R}) with {\overline{h}}_0>0 if \nu \ge 1; and, moreover, for any \nu \in (0,1), there exists a function h_0\in L([0,\omega ];\mathbb{R}) with {\overline{h}}_0>0 such that (1.4), (1.3) has no solution. At this point, we would like to emphasize the important fact that the condition \nu \ge 1 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, {\mathbb{R}}^{+}=(0,+\mathrm{\infty }), {[x]}_{+}=max\{x,0\}, {[x]}_{-}=max\{-x,0\}.

C([0,\omega ];\mathbb{R}) is the Banach space of continuous functions u:[0,\omega ]\to \mathbb{R} with the norm

C({\mathbb{R}}^{+};\mathbb{R}), resp. C({\mathbb{R}}^{+};{\mathbb{R}}^{+}), is the set of continuous functions u:{\mathbb{R}}^{+}\to \mathbb{R}, resp. u:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}.

C^1({\mathbb{R}}^{+};{\mathbb{R}}^{+}) is the set of functions u:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+} which are continuous together with their first derivative.

A{C}^1([0,\omega ];\mathbb{R}) is a set of all functions u:[0,\omega ]\to \mathbb{R} such that u and u^{\prime } are absolutely continuous.

L([0,\omega ];\mathbb{R}) is the Banach space of the Lebesgue integrable functions p:[0,\omega ]\to \mathbb{R} endowed with the norm

For a given p\in L([0,\omega ];\mathbb{R}), its mean value is defined by

Given \phi ,\psi \in L([0,\omega ];\mathbb{R}), 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 \alpha \in A{C}^1([0,\omega ];\mathbb{R}) is called a lower function to the problem (1.2), (1.3) if \alpha (t)>0 for every t\in [0,\omega ] and

Definition 1.2 A function \beta \in A{C}^1([0,\omega ];\mathbb{R}) is called an upper function to the problem (1.2), (1.3) if \beta (t)>0 for every t\in [0,\omega ] and

The following theorem is well known in the theory of differential equations (see, e.g., [[12], Theorem 8.12]).

Theorem 1.1 Let α and β be lower and upper functions to the problem (1.2), (1.3) such that

Then there exists a solution u to the problem (1.2), (1.3) such that

Theorem 2.1 Let {\rho }_0\in C^1({\mathbb{R}}^{+};{\mathbb{R}}^{+}) and {\rho }_1\in C({\mathbb{R}}^{+};{\mathbb{R}}^{+}) be non-decreasing functions, h_0,h_1\in L([0,\omega ];\mathbb{R}), and x_0>0 be such that

and let there exist c_0,c_1\in \mathbb{R} such that

Let, moreover, there exist \lambda \in [0,1] such that

and let either

or

Furthermore, let us suppose that {\rho }_0 fulfills at least one of the following conditions:

1. (a)

   there exists a sequence {\{y_n\}}_{n=1}^{+\mathrm{\infty }} of positive numbers such that

   \underset{n\to +\mathrm{\infty }}{lim}{y}_n=+\mathrm{\infty },\underset{n\to +\mathrm{\infty }}{lim}\frac{{\rho }_0^{1-\lambda }(y_n)}{\sigma (y_n)}=0,

   (2.8)

and there exist {\epsilon }_0>0, {\epsilon }_1\in (0,{\epsilon }_0], and n_0\in \mathbb{N} such that

where \phi (t)=h_0(t)-c_0 for almost every t\in [0,\omega ] and

1. (b)

   the function \frac{{\rho }_0^{1-\lambda }(x)}{\sigma (x)} is non-increasing and

   (2.12)

where \phi (t)=h_0(t)-c_0 for almost every t\in [0,\omega ] and σ is given by (2.11).

Besides, let us suppose that {\rho }_1 fulfills at least one of the following conditions:

1. (c)

   there exists a sequence {\{z_n\}}_{n=1}^{+\mathrm{\infty }} of positive numbers such that

   \underset{n\to +\mathrm{\infty }}{lim}{z}_n=+\mathrm{\infty },\underset{n\to +\mathrm{\infty }}{lim}\frac{{\rho }_1(z_n)}{z_n}=0,

and there exist {\epsilon }_2>0 and n_1\in \mathbb{N} such that

where \psi (t)=h_1(t)-c_1 for almost every t\in [0,\omega ];

1. (d)

   the function \frac{{\rho }_1(x)}{x} is non-increasing and

   \frac{\omega }{4}{\mathrm{\Psi }}_{+}{\mathrm{\Psi }}_{-}\frac{{\rho }_1(x_0)}{x_0}\le {\mathrm{\Psi }}_{+}-{\mathrm{\Psi }}_{-},

where \psi (t)=h_1(t)-c_1 for almost every t\in [0,\omega ].

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

Remark 2.1 Note that there exists a suitable {\epsilon }_1 such that (2.10) holds, e.g., if

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

Corollary 2.1 Let g_1>0, g_2\ge 0, , \nu >\gamma, \nu +\delta >0 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 h_0\in L([0,\omega ];\mathbb{R}) and {\overline{h}}_0>0, has a positive ω-periodic solution if \nu \ge 1. Moreover, there is also an example introduced showing that for any \nu \in (0,1), there exists h_0\in L([0,\omega ];\mathbb{R}) with {\overline{h}}_0>0 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 \nu \ge 1 can be weakened. For example,

has a positive solution satisfying (1.3) for any \nu >0 if \mu \ge 1, provided {\overline{h}}_0>0. Also, the equation

subjected to the boundary conditions (1.3) is solvable for any \nu >0 if \delta \in [1/2,1), provided {\overline{h}}_0>0.

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 h_0\in L([0,\omega ];\mathbb{R}), c, g_1, g_2 are positive constants and \gamma \in \mathbb{R} (see [9, 10]).

The results dealing with the existence of positive ω-periodic solutions of (2.14) were established in [10] provided h_0 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. 1.

   \gamma >-1/5 and {\overline{h}}_0>0;

2. 2.

   \gamma =-1/5 and {\overline{h}}_0>-g_2;

3. 3.

   .

Thus, Corollary 2.1 assures that the boundedness of h_0 is not necessary.

Corollary 2.2 Let g_1>0, g_2\ge 0, \nu >\gamma, \nu +1>0. Let, moreover, either g^{\ast }=-\mathrm{\infty } or

where \phi (t)=h_0(t)-g^{\ast } for almost every t\in [0,\omega ], and

Then the problem (1.1), (1.3) with \delta =1 has at least one solution.

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

with g_1>0 and \nu >0, has a positive solution if and only if the inclusion \mathcal{L}[0,-h_0]\in V^{-} holds (see notation in [29]).

Indeed, according to [[29], Definition 1.1], the inclusion \mathcal{L}[0,-h_0]\in V^{-} implies the existence of a positive solution v to the problem

Therefore there exist constants x>0 and y>0 such that x^{1+\nu }\le v(t)\le y^{1+\nu } for t\in [0,\omega ]. 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 \mathcal{L}[0,-h_0]\in V^{-} (see [[29], Theorem 2.1]).

However, one of the optimal effective conditions guaranteeing such an inclusion is h_0\not\equiv 0 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 h(t,x)\equiv h_0(t), then (1.3) has the form

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

Corollary 2.3 Let there exist x_0>0 and c_0\in \mathbb{R} such 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 h_0 is supposed. Clearly, Theorems 2.1 and 2.2 are independent.

Theorem 2.2 Let {\rho }_0\in C^1({\mathbb{R}}^{+};{\mathbb{R}}^{+}) and {\rho }_1\in C({\mathbb{R}}^{+};{\mathbb{R}}^{+}) be non-decreasing functions, h_0,h_1\in L([0,\omega ];\mathbb{R}), and be such that

Let, moreover,

and let there exist c_1\in \mathbb{R} such that

Besides, let us suppose that {\rho }_1 fulfills 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 2.4 Let , and let be 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, {\rho }_0\in C({\mathbb{R}}^{+};{\mathbb{R}}^{+}) is a non-decreasing function, h_0\in L([0,\omega ];\mathbb{R}), and f,g\in C({\mathbb{R}}^{+};\mathbb{R}). Together with (3.1), for every k\in \mathbb{N}, consider the auxiliary equation

where

Obviously,

and

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

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

Let x_0>0 and c\in \mathbb{R} be such that

Let, moreover, there exist a sequence {\{y_n\}}_{n=1}^{+\mathrm{\infty }} of positive numbers such that

and let there exist \epsilon >0 and n_0\in \mathbb{N} such that

where \phi (t)=h_0(t)-c for almost every t\in [0,\omega ]. Then there exists an upper function β to the problem (3.1), (1.3) satisfying

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

Let x_0>0 and c\in \mathbb{R} be such that (3.6) holds. If \frac{{\rho }_0(x)}{x} is a non-increasing function such that

where \phi (t)=h_0(t)-c for almost every t\in [0,\omega ], then there exists an upper function β to the problem (3.1), (1.3) satisfying (3.8).

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

Let x_0>\frac{\omega }{8}{\parallel h_0-{\overline{h}}_0\parallel }_1 be such that

Then there exists a lower function α to the problem (2.17), (1.3) with

Lemma 3.4 Let x_0>0 and c_0\in \mathbb{R} be such that (2.2) holds. Let, moreover, there exist a sequence {\{y_n\}}_{n=1}^{+\mathrm{\infty }} of positive numbers such that (3.7) is fulfilled, and let there exist \epsilon >0 and n_0\in \mathbb{N} such that

where \phi (t)=h_0(t)-c_0 for almost every t\in [0,\omega ]. Then there exist k_0\in \mathbb{N} and an upper function β to the problems (3.2), (1.3) for k\ge k_0 satisfying (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 k_0\in \mathbb{N} such that

Therefore, according to Lemma 3.1, there exists an upper function β to (3.2), (1.3) with k=k_0 satisfying (3.8). Obviously, in view of (3.4) and the non-negativity of {\rho }_0, it follows that β is also an upper function to (3.2), (1.3) for k\ge k_0. □

Lemma 3.5 Let x_0>0 and c_0\in \mathbb{R} be such that (2.2) holds. If \frac{{\rho }_0(x)}{x} is a non-increasing function such that

where \phi (t)=h_0(t)-c_0 for almost every t\in [0,\omega ], then there exist k_0\in \mathbb{N} and an upper function β to the problems (3.2), (1.3) for k\ge k_0 satisfying (3.8).

Proof Define {\phi }_k, {\mathrm{\Phi }}_{k+}, and {\mathrm{\Phi }}_{k-} 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 k_0\in \mathbb{N} 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 k=k_0 satisfying (3.8). Obviously, in view of (3.4) and the non-negativity of {\rho }_0, it follows that β is also an upper function to (3.2), (1.3) for k\ge k_0. □

Lemma 3.6 Let

and let either

or

Then, for every K>0, there exists a constant K_1>0 such that for any k\in \mathbb{N} and any positive solution u of (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 t_0,t_1\in [0,\omega ] 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 t_0 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 t_1+\omega, 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 3.7 Let

and let either

or

Then, for every K>0, there exists a constant a>0 such that for any k\in \mathbb{N} and any positive solution u of (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 x_0\in (0,+\mathrm{\infty }) such that

Let t_m\in [0,\omega ] 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 t\in [0,\omega ], then applying (3.43) in (3.42) we obtain a contradiction. Thus, there exist points t_1,t_2\in (t_m,t_m+\omega ) 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 K_1>0 such that (3.22) holds. The integration of (3.25) from t_m to t_1, 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 \vartheta {(u)}^{\prime }(t_1)\ge 0. Consequently, from (3.48) we obtain

where

Note that K_2 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 a>0 such that (3.41) holds.

If (3.40) holds, we integrate (3.25) from t_2 to t_m+\omega 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 a>0 such that (3.41) holds. □

Lemma 3.8 Let x_0>0 and c_0\in \mathbb{R} be 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 sequence {\{y_n\}}_{n=1}^{+\mathrm{\infty }} of positive numbers such that (3.7) holds, and let there exist \epsilon >0 and n_0\in \mathbb{N} such that (3.10) is fulfilled, where \phi (t)=h_0(t)-c_0 for almost every t\in [0,\omega ]. Then there exists a positive solution u to (3.1), (1.3).

Proof According to Lemma 3.4, there exist k_0\in \mathbb{N} and an upper function β to the problems (3.2), (1.3) for k\ge k_0 satisfying (3.8). On the other hand, in view of (3.4) and (3.38), there exists x_k\in (0,x_0] for k\ge k_0 such that

Thus, if we put {\alpha }_k(t)=x_k for t\in [0,\omega ], according to Theorem 1.1, there exists a solution u_k to (3.2), (1.3) for k\ge k_0 satisfying

Moreover, according to Lemmas 3.6 and 3.7, in view of (3.50), there exist constants K>0, K_1>0, and a>0, not depending on k, such that

where

Therefore, according to the Arzelà-Ascoli theorem, there exist u_0\in C([0,\omega ];\mathbb{R}) and v_0\in C([0,\omega ];\mathbb{R}) such that

Moreover, since u_k are solutions to (3.2), (1.3), in view of (3.3), (3.52), and (3.54), we have u_0\in A{C}^1([0,\omega ];\mathbb{R}), u_0^{\prime }\equiv v_0, and u_0 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 3.9 Let x_0>0 and c_0\in \mathbb{R} be such that (2.2) holds. Let, moreover, (3.38) be fulfilled, and let either (3.39) or (3.40) be valid. Let, in addition, \frac{{\rho }_0(x)}{x} be a non-increasing function and let (3.16) be fulfilled, where \phi (t)=h_0(t)-c_0 for almost every t\in [0,\omega ]. Then there exists a positive solution u to (3.1), (1.3).