1. Introduction

We study the equation

(1.1)

on the half-line , where

(1.2)
(1.3)

Equation (1.1) is singular at because . If in (1.1) fulfils moreover assumptions

(1.4)
(1.5)
(1.6)

then (1.1) generalizes equations which appear in hydrodynamics or in the nonlinear field theory [1–5].

Definition 1.1.

A function which has continuous second derivative on and satisfies (1.1) for all is called a solution of (1.1).

Consider and the initial conditions

(1.7)

The initial value problem (1.1), (1.7) has been investigated, for example, in [6–12]. In particular in [10] it was proved that for each negative there exists a unique solution of problem (1.1), (1.7) under the assumptions (1.2)–(1.6). Consider such solution and denote

(1.8)

Definition 1.2.

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 [11], 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.

There exist , , such that

(1.9)
(1.10)

( is possible).

2. Damped Solutions

Theorem 2.1 (Existence and uniqueness).

Assume that (1.2), (1.3), (1.9), and (1.10) hold and let . Then problem (1.1), (1.7) has a unique solution , and moreover the solution satisfies

(2.1)

Proof.

Step 1.

Put

(2.2)

We will study the auxiliary differential equation:

(2.3)

By virtue of (1.2) we find the Lipschitz constant for on , and due to (1.2), (1.10), and (2.2), we find such that

(2.4)

Put for . Having in mind (1.3), we see that is increasing and so

(2.5)

Consequently we can choose such that

(2.6)

Consider the Banach space (with the maximum norm) and define an operator by

(2.7)

Using (2.4) and (2.6), we have

(2.8)

that is maps the ball to itself. Due to (2.2) and the choice of , we have for ,

(2.9)

Hence is a contraction on , and the Banach fixed point theorem yields a unique fixed point of .

Step 2.

The fixed point of Step 1 fulfils

(2.10)

Hence satisfies (2.3) on . Finally, (2.4) and (2.5) yield

(2.11)

Consequently fulfils (1.7). Choose an arbitrary . Then, by (2.5) and (2.10),

(2.12)

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.

Step 3.

According to Step 2 we have

(2.13)

Multiplying (2.13) by and integrating between and , we get

(2.14)

Put

(2.15)

So,(2.14) has the form

(2.16)

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).

Step 4.

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.

Lemma 2.2.

Let and let be a solution of (1.1). Assume that there exists such that

(2.17)

Then for all .

Proof.

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).

Remark 2.3.

Let us put

(2.18)

Due to (1.2) and (1.9) we see that is continuous on , decreasing and positive on , increasing and positive on . Therefore we can define by

(2.19)

(.

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.

Proof.

Since , we can find such that

(2.20)

Assume on the contrary that is not damped, that is,

(2.21)

Then, according to Lemma 2.2, there exists such that

(2.22)

By (1.1), (1.3), and (1.9) we have on . So, is increasing and positive on and hence on . Assumption (2.21) implies that there exists such that

(2.23)

Since fulfils (1.1), we have

(2.24)

Multiplying (2.24) by and integrating between and we get

(2.25)

This contradicts (2.20).

3. Oscillatory Solutions

In this section we assume that, in addition to our basic assumptions (1.2), (1.3), (1.9), and (1.10), the following conditions are fulfilled:

(3.1)
(3.2)

Then the next lemmas can be proved.

Lemma 3.1.

Let be a solution of problem (1.1), (1.7) with . Then there exists such that

(3.3)

Proof.

Step 1.

Assume that such does not exist. Then

(3.4)

Hence (1.1), (1.7), and (1.9) yield and on . Therefore is increasing on and

(3.5)

Multiplying (2.24) by and integrating between and , we get due to (2.18)

(3.6)

Letting , we get

(3.7)

Since the function is positive and increasing, it follows that there exists . If , then contrary to (3.5). Consequently,

(3.8)

Letting in (2.24), we get by (1.3), (1.9), and (3.5)

(3.9)

Due to (3.8), we conclude that and hence . We have proved that if fulfilling (3.3) does not exist, then

(3.10)

Step 2.

We define a function

(3.11)

By (1.3) and (3.2), we have ,

(3.12)
(3.13)
(3.14)

Due to (1.3), (3.1), (3.10) and (3.14) there exist and such that

(3.15)

Due to (3.4), (3.11), (3.13), and (3.15), we get

(3.16)

Thus, is increasing on and has the limit

(3.17)

If , then , which contradicts (3.4) and (3.11). If , then on and

(3.18)

In view of (3.16) we can see that

(3.19)

We get which contradicts . The obtained contradictions imply that (3.4) cannot occur and hence satisfying (3.3) must exist.

Corollary 3.2.

Let be a solution of problem (1.1), (1.7) with . Further assume that there exist and such that

(3.20)

Then there exists such that

(3.21)

Proof.

We can argue as in the proof of Lemma 3.1 working with and instead of and .

Lemma 3.3.

Let be a solution of problem (1.1), (1.7) with . Further assume that there exist and such that

(3.22)

Then there exists such that

(3.23)

Proof.

We argue similarly as in the proof of Lemma 3.1.

Step 1.

Assume that such does not exist. Then

(3.24)

By (1.1), (1.7), and (1.9) we deduce on and

(3.25)

Multiplying (2.24) by , integrating between and , and using (2.18), we obtain

(3.26)

and we derive as in the proof of Lemma 3.1 that (3.10) holds.

Step 2.

We define by (3.11) and get (3.13) for . As in the proof of Lemma 3.1 we find and satisfying (3.15). Due to (3.24), (3.11), (3.13), and (3.15) we get

(3.27)

So, is decreasing on and . If , then which contradicts (3.24) and (3.11). If , then on and

(3.28)

In view of (3.27) we can see that

(3.29)

We get contrary to . The obtained contradictions imply that (3.24) cannot occur and that satisfying (3.23) must exist.

Theorem 3.4.

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.

Proof.

Let be a damped solution. By (2.1) and Definition 1.2, we can find such that

(3.30)

Step 1.

Lemma 3.1 yields satisfying (3.3). Hence there exists a maximal interval such that on . Let . Then, by (3.30), we get , on and

(3.31)

By (1.1), (1.3), and (1.9), we have on . So and are decreasing on and, due to (3.31),

(3.32)

Letting in (2.24) and using (1.3), (1.9), and (3.31), we get

(3.33)

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 .

Step 2.

By (3.23) there exists a maximal interval , where . Let . Then, by (3.30), we have , on , and

(3.34)

By (1.1), (1.3), and (1.9), we get on and so is increasing on . Since , we deduce that is increasing on and, by (3.34), we get (3.32). Letting in (1.1) and using (1.3), (1.9), and (3.34), we get

(3.35)

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 .

Step 3.

We can continue as in Step 1 and Step 2 and get the sequences and of local maxima and local minima of attained at and , respectively. Now, put , and write (1.1) as a system

(3.36)

Consider of (2.18) and define a Lyapunov function by

(3.37)

where . By Remark 2.3, we see that and on . By (3.6) and (3.37), we have

(3.38)

Therefore

(3.39)

By (3.30), for . We see that is positive and decreasing (for the damped solution ) and hence

(3.40)

So, sequences and are decreasing:

(3.41)

for and

(3.42)

Further, due to Remark 2.3, the sequence is decreasing and the sequence is increasing. Consequently,

(3.43)

Remark 3.5.

There are two cases for the number from the proof of Theorem 3.4: and . Denote

(3.44)

If , then and hence , that is, .

Let . Consider an arbitrary sequence such that . By (3.40) we have . By (3.30) and (3.6), the sequence is bounded and so there exists a subsequence

(3.45)

such that , where is a point of the level curve:

(3.46)

Note that

(3.47)

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.

Proof.

The assertion follows from Theorems 2.4 and 3.4.

Remark 3.7.

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.