# Existence of Oscillatory Solutions of Singular Nonlinear Differential Equations

Authors: Irena Rachůnková; Lukáš Rachůnek; Jan Tomeček
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

Asymptotic properties of solutions of the singular differential equation(p(t)u′(t))′=p(t)f(u(t)) are described. Here, f is Lipschitz continuous on ℝ and has at least two zeros 0 and L>0. The function p is continuous on [0, ∞) and has a positive continuous derivative on (0, ∞) and p(0)=0. Further conditions for f and p under which the equation has oscillatory solutions converging to 0 are given.


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## Body

## 1. Introduction

Fork∈ℕ, k>1, and L∈(0,∞), consider the equation(1.1)u′′+k-1tu′=f(u),t∈(0,∞),
where(1.2)f∈Liploc⁡(R),f(0)=f(L)=0,f(x)<0,x∈(0,L),(1.3)∃B¯∈(-∞,0):f(x)>0,x∈[B¯,0).
Let us put(1.4)F(x)=-∫0xf(z)dzforx∈R.
Moreover, we assume that f fulfils(1.5)F(B¯)=F(L),
and denote (1.6)L0=inf⁡⁡{x<B¯:f(x)>0}≥-∞.
Due to (1.2)–(1.4), we see that F∈C1(ℝ) is decreasing and positive on (L0,0) and increasing and positive on (0,L].Equation (1.1) arises in many areas. For example, in the study of phase transitions of Van der Waals fluids [1–3], in population genetics, where it serves as a model for the spatial distribution of the genetic composition of a population [4, 5], in the homogenous nucleation theory [6], and in relativistic cosmology for description of particles which can be treated as domains in the universe [7], in the nonlinear field theory, in particular, when describing bubbles generated by scalar fields of the Higgs type in the Minkowski spaces [8]. Numerical simulations of solutions of (1.1), where f is a polynomial with three zeros, have been presented in [9–11]. Close problems about the existence of positive solutions can be found in [12–14].In this paper, we investigate a generalization of (1.1) of the form(1.7)(p(t)u′)′=p(t)f(u),t∈(0,∞),
where f satisfies (1.2)–(1.5) and p fulfils(1.8)p∈C[0,∞)∩C1(0,∞),p(0)=0,(1.9)p′(t)>0,t∈(0,∞),lim⁡t→∞p′(t)p(t)=0.
Equation (1.7) is singular in the sense that p(0)=0. If p(t)=tk-1, with k>1, then p satisfies (1.8), (1.9), and (1.7) is equal to (1.1).Definition 1.1.
A functionu∈C1[0,∞)∩C2(0,∞) which satisfies (1.7) for all t∈(0,∞) is called a  solution of (1.7).Consider a solutionu of (1.7). Since u∈C1[0,∞), we have u(0),u′(0)∈ℝ and the assumption,  p(0)=0 yields p(0)u′(0)=0. We can find M>0 and δ>0 such that |f(u(t))|≤M for t∈(0,δ). Integrating (1.7), we get (1.10)|u′(t)|=|1p(t)∫0tp(s)f(u(s))ds|≤Mp(t)∫0tp(s)ds≤Mt,t∈(0,δ).
Consequently, the condition(1.11)u′(0)=0
is necessary for each solution of  (1.7). Denote (1.12)usup⁡=sup⁡{u(t):t∈[0,∞)}.Definition 1.2.
Letu be a solution of (1.7). If usup<L, then u is called a damped solution.If a solutionu of (1.7) satisfies usup=L or usup>L, then we call u a bounding homoclinic solution or an escape solution. These three types of solutions have been investigated in [15–18]. Here, we continue the investigation of the existence and asymptotic properties of damped solutions. Due to (1.11) and Definition 1.2, it is reasonable to study solutions of (1.7) satisfying the initial conditions(1.13)u(0)=u0∈(L0,L],u′(0)=0.
Note that if u0>L, then a solution u of the problem (1.7), (1.13) satisfies usup>L, and consequently u is not a damped solution. Assume that L0>-∞, then f(L0)=0, and if we put u0=L0, a solution u of (1.7), (1.13) is a constant function equal to L0 on [0,∞). Since we impose no sign assumption on f(x) for x<L0, we do not consider the case u0<L0. In fact, the choice of u0 between two zeros L0 and 0 of f has been motivated by some hydrodynamical model in [11].A lot of papers are devoted to oscillatory solutions of nonlinear differential equations. Wong [19] published an account on a nonlinear oscillation problem originated from earlier works of Atkinson and Nehari. Wong's paper is concerned with the study of oscillatory behaviour of second-order Emden-Fowler equations(1.14)y′′(x)+a(x)|y(x)|γ-1y(x)=0,γ>0,
where a is nonnegative and absolutely continuous on (0,∞). Both superlinear case (γ>1) and sublinear case (γ∈(0,1)) are discussed, and conditions for the function a giving oscillatory or nonoscillatory solutions of (1.14) are presented; see also [20]. Further extensions of these results have been proved for more general differential equations. For example, Wong and Agarwal [21] or Li [22] worked with the equation(1.15)(a(t)(y′(t))σ)′+q(t)f(y(t))=0,
where σ>0 is a positive quotient of odd integers, a∈C1(ℝ) is positive, q∈C(ℝ), f∈C1(ℝ), xf(x)>0, f′(x)≥0 for all x≠0. Kulenović and Ljubović [23] investigated an equation(1.16)(r(t)g(y′(t)))′+p(t)f(y(t))=0,
where g(u)/u≤m, f(u)/u≥k>0, or f′(u)≥k for all u≠0. The investigation of oscillatory and nonoscillatory solutions has been also realized in the class of quasilinear equations. We refer to the paper [24] by Ho, dealing with the equation(1.17)(tn-1Φp(u′))′+tn-1∑i=1NαitβiΦqi(u)=0,
where 1<p<n, αi>0, βi≥-p, qi>p-1, i=1,…,N, Φp(y)=|y|p-2y.Oscillation results for the equation(1.18)(a(t)Φp(x′))′+b(t)Φq(x)=0,
where a,b∈C([0,∞)) are positive, can be found in [25]. We can see that the nonlinearity f(y)=|y|γ-1y in (1.14) is an increasing function on ℝ having a unique zero at y=0.Nonlinearities in all the other (1.15)–(1.18) have similar globally monotonous behaviour. We want to emphasize that, in contrast to the above papers, the nonlinearity f in our (1.7) needs  not be globally monotonous. Moreover, we deal with solutions of (1.7) starting at a singular point t=0, and we provide an interval for starting values u0 giving oscillatory solutions (see Theorems 2.3, 2.10, and 2.16). We specify a behaviour of oscillatory solutions in more details (decreasing amplitudes—see Theorems 2.10 and 2.16), and we show conditions which guarantee that oscillatory solutions converge to 0 (Theorem 3.1).The paper is organized in this manner: Section2 contains results about existence, uniqueness, and other basic properties of solutions of the problem (1.7), (1.13). These results which mainly concern damped solutions are taken from [18] and extended or modified a little. We also provide here new conditions for the existence of oscillatory solutions in Theorem 2.16. Section 3 is devoted to asymptotic properties of oscillatory solutions, and the main result is contained in Theorem 3.1.


## 2. Solutions of the Initial Problem(1.7), (1.13)

Let us give an account of this section in more details. The main objective of this paper is to characterize asymptotic properties of oscillatory solutions of the problem (1.7), (1.13). In order to present more complete results about the solutions, we start this section with the unique solvability of the problem (1.7), (1.13) on [0,∞) (Theorem 2.1). Having such global solutions, we have proved (see papers [15–18]) that oscillatory solutions of the problem (1.7), (1.13) can be found just in the class of damped solutions of this problem. Therefore, we give here one result about the existence of damped solutions (Theorem 2.3). Example 2.5 shows that there are damped solutions which are not oscillatory. Consequently, we bring results about the existence of oscillatory solutions in the class of damped solutions. This can be found in Theorem 2.10, which is an extension of Theorem 3.4 of [18] and in Theorem 2.16, which are new. Theorems 2.10  and 2.16 cover different classes of equations which is illustrated by examples.Theorem 2.1 (existence and uniqueness).
Assume that (1.2)–(1.5), (1.8), (1.9) hold and that there exists CL∈(0,∞) such that
(2.1)0≤f(x)≤CLforx≥L
then the initial problem (1.7), (1.13) has a unique solution u. The solution u satisfies
(2.2)u(t)≥u0ifu0<0,u(t)>B¯ifu0≥0,fort∈[0,∞).Proof.
Letu0<0, then the assertion is contained in Theorem  2.1 of [18]. Now, assume that u0∈[0,L], then the proof of Theorem  2.1 in [18] can be slightly modified.For close existence results, see also Chapters 13 and 14 of [26], where this kind of equations is studied.Remark 2.2.
Clearly, foru0=0 and u0=L, the problem (1.7), (1.13) has a unique solution u≡0 and u≡L, respectively. Since f∈Liploc(ℝ), no solution of the problem (1.7), (1.13) with u0<0 or u0∈(0,L) can touch the constant solutions u≡0 and u≡L.
In particular, assume thatC∈{0,L}, a>0, u is a solution of the problem (1.7), (1.13) with u0<L, u0≠0, and (1.2), (1.8), and (1.9) hold. If u(a)=C, then u′(a)≠0, and if u′(a)=0, then u(a)≠C.The next theorem provides an extension of Theorem 2.4 in [18].Theorem 2.3 (existence of damped solutions).
Assume that (1.2)–(1.5), (1.8), and (1.9) hold, then for each u0∈[B¯,L), the problem (1.7), (1.13) has a unique solution. This solution is damped.Proof.
First, assume that there existsCL>0 such that f satisfies (2.1), then, by Theorem 2.1, the problem (1.7), (1.13) has a unique solution u satisfying (2.2). Assume that u is not damped, that is,
(2.3)sup⁡{u(t):t∈[0,∞)}≥L.
By (1.3)–(1.5), the inequality F(u0)≤F(L) holds. Since u fulfils (1.7), we have
(2.4)u′′(t)+p′(t)p(t)u′(t)=f(u(t))fort∈(0,∞).
Multiplying (2.4) by u′ and integrating between 0 and t>0, we get
(2.5)0<u′2(t)2+∫0tp′(s)p(s)u′2(s)ds=F(u0)-F(u(t)),t∈(0,∞),
and consequently
(2.6)0<∫0tp′(s)p(s)u′2(s)ds≤F(u0)-F(u(t)),t∈(0,∞).
By (2.3), we can find that b∈(0,∞] such that u(b)≥L, (u(∞)=limsupt→∞u(t)), and hence, according to (1.5),
(2.7)0<∫0bp′(s)p(s)u′2(s)ds≤F(u0)-F(u(b))≤F(B)-F(L)≤0,
which is a contradiction. We have proved that sup{u(t):t∈[0,∞)}<L, that is, u is damped. Consequently, assumption (2.1) can be omitted.Example 2.4.
Consider the equation(2.8)u′′+2tu′=u(u-1)(u+2),
which is relevant to applications in [9–11]. Here, p(t)=t2, f(x)=x(x-1)(x+2), L0=-2, and L=1. Hence f(x)<0 for x∈(0,1), f(x)>0 for x∈(-2,0), and
(2.9)F(x)=-∫0xf(z)dz=-x44-x33+x2.
Consequently, F is decreasing and positive on [-2,0) and increasing and positive on (0,1]. Since F(1)=5/12 and F(-1)=13/12, there exists a unique B¯∈(-1,0) such that F(B¯)=5/12=F(1). We can see that all assumptions of Theorem 2.3 are fulfilled and so, for each u0∈[B¯,1), the problem (2.8), (1.13) has a unique solution which is damped. We will show later (see Example 2.11), that each damped solution of the problem (2.8), (1.13) is oscillatory.In the next example, we will show that damped solutions can be nonzero and monotonous on[0,∞) with a limit equal to zero at ∞. Clearly, such solutions are not oscillatory.Example 2.5.
Consider the equation(2.10)u′′+3tu′=f(u),
where
(2.11)f(x)={-x3forx≤1,x-2forx∈(1,3),1forx≥3.
We see that p(t)=t3 in (2.10) and the functions f and p satisfy conditions (1.2)–(1.5), (1.8), and (1.9) with L=2. Clearly, L0=-∞. Further,
(2.12)F(x)=-∫0xf(z)dz={x44forx≤1,-x22+2x-54forx∈(1,3),-x+134forx≥3.
Since F(L)=F(2)=3/4, assumption (1.5) yields F(B¯)=B¯4/4=3/4 and B¯=-31/4. By Theorem 2.3, for each u0∈[-31/4,2), the problem (2.10), (1.13) has a unique solution u which is damped. On the other hand, we can check by a direct computation that for each u0≤1 the function
(2.13)u(t)=8u08+u02t2,t∈[0,∞)
is a solution of equation (2.10) and satifies conditions (1.13). If u0<0, then u<0,u′>0 on (0,∞), and if u0∈(0,1], then u>0, u′<0 on (0,∞). In both cases, limt→∞u(t)=0.In Example2.5, we also demonstrate that there are equations fulfilling Theorem 2.3 for which all solutions with u0<L, not only those with u0∈[B¯,L), are damped. Some additional conditions giving, moreover, bounding homoclinic solutions and escape solutions are presented in [15–17].In our further investigation of asymptotic properties of damped solutions the following lemmas are useful.Lemma 2.6.
Assume (1.2), (1.8), and (1.9). Let u be a damped solution of the problem (1.7), (1.13) with u0∈(L0,L) which is eventually positive or eventually negative, then
(2.14)lim⁡t→∞u(t)=0,lim⁡t→∞u′(t)=0.Proof.
Letu be eventually positive, that is, there exists t0≥0 such that
(2.15)u(t)>0fort∈[t0,∞).
Denote θ=inf{t0≥0:u(t)>0,t∈[t0,∞)}.
Letθ>0, then u(θ)=0 and, by Remark 2.2, u′(θ)>0. Assume that u′>0 on (θ,∞),  then u is increasing on (θ,∞), and there exists limt→∞u(t)=ℓ∈(0,L). Multiplying (2.4) by u′, integrating between θ and t, and using notation (1.4), we obtain
(2.16)u′2(t)2+∫θtp′(s)p(s)u′2(s)ds=F(u0)-F(u(t)),t∈(θ,∞).
Letting t→∞, we get
(2.17)lim⁡t→∞u′2(t)2=-lim⁡t→∞∫θtp′(s)p(s)u′2(s)ds+F(u0)-F(l).
Since the function ∫θt(p′(s)/p(s))u′2(s)ds is positive and increasing, it follows that it has a limit at ∞, and hence there exists also limt→∞u′(t)≥0. If limt→∞u′(t)>0, then L>l=limt→∞u(t)=∞, which is a contradiction. Consequently
(2.18)lim⁡t→∞u′(t)=0.
Letting t→∞ in (2.4) and using (1.2), (1.9) and ℓ∈(0,L), we get limt→∞u′′(t)=f(ℓ)<0, and so limt→∞u′(t)=-∞, which is contrary to (2.18). This contradiction implies that the inequality u′>0 on (θ,∞) cannot be satisfied and that there exists a>θ such that u′(a)=0. Since u>0 on (a,∞), we get by (1.2), (1.7), and (1.13) that (pu′)′<0 on (a,∞). Due to p(a)u′(a)=0, we see that u′<0 on (a,∞). Therefore, u is decreasing on (a,∞) and limt→∞u(t)=ℓ0∈[0,L). Using (2.16) with a in place of θ, we deduce as above that (2.18) holds and that limt→∞u′′(t)=f(ℓ0)=0. Consequently, ℓ0=0. We have proved that (2.14) holds provided θ>0.
Ifθ=0, then we take a=0 and use the above arguments. If u is eventually negative, we argue similarly.Lemma 2.7.
Assume (1.2)–(1.5), (1.8), (1.9), and
(2.19)p∈C2(0,∞),limsup⁡t→∞|p′′(t)p′(t)|<∞,(2.20)lim⁡x→0+f(x)x<0.
Let u be a solution of the problem (1.7), (1.13) with u0∈(0,L), then there exists δ1>0 such that
(2.21)u(δ1)=0,u′(t)<0fort∈(0,δ1].Proof.
Assume that suchδ1 does not exist, then u is positive on [0,∞) and, by Lemma 2.6, u satisfies (2.14). We define a function
(2.22)v(t)=p(t)u(t),t∈[0,∞).
By (2.19), we have v∈C2(0,∞) and
(2.23)v′(t)=p′(t)u(t)2p(t)+p(t)u′(t),(2.24)v′′(t)=v(t)[12p′′(t)p(t)-14(p′(t)p(t))2+f(u(t))u(t)],t∈(0,∞).
By (1.9) and (2.19), we get
(2.25)lim⁡t→∞[12p′′(t)p(t)-14(p′(t)p(t))2]=12lim⁡t→∞p′′(t)p′(t)⋅p′(t)p(t)=0.
Since u is positive on (0,∞), conditions (2.14) and (2.20) yield
(2.26)lim⁡t→∞f(u(t))u(t)=lim⁡x→0+f(x)x<0.
Consequently, there exist ω>0 and R>0 such that
(2.27)12p′′(t)p(t)-14(p′(t)p(t))2+f(u(t))u(t)<-ωfort≥R.
By (2.22), v is positive on (0,∞) and, due to (2.24) and (2.27), we get
(2.28)v′′(t)<-ωv(t)<0fort≥R.
Thus, v′ is decreasing on [R,∞) and limt→∞v′(t)=V. If V<0, then limt→∞v(t)=-∞, contrary to the positivity of v. If V≥0, then v′>0 on [R,∞) and v(t)≥v(R)>0 for t∈[R,∞). Then (2.28) yields 0>-ωv(R)≥-ωv(t)>v′′(t) for t∈[R,∞). We get limt→∞v′(t)=-∞ which contradicts V≥0. The obtained contradictions imply that u has at least one zero in (0,∞). Let δ1>0 be the first zero of u. Then u>0 on [0,δ1) and, by (1.2) and (1.7), u′<0 on (0,δ1). Due to Remark 2.2, we have also u′(δ1)<0.For negative starting value, we can prove a dual lemma by similar arguments.Lemma 2.8.
Assume (1.2)–(1.5), (1.8), (1.9), (2.19) and
(2.29)lim⁡x→0-f(x)x<0.
Let u be a solution of the problem (1.7), (1.13) with u0∈(L0,0), then there exists θ1>0 such that
(2.30)u(θ1)=0,u′(t)>0fort∈(0,θ1].The arguments of the proof of Lemma2.8 can be also found in the proof of Lemma 3.1 in [18], where both (2.20) and (2.29) were assumed. If one argues as in the proofs of Lemmas 2.7 and 2.8 working with a1, A1 and b1, B1 in place of 0, and u0, one gets the next corollary.Corollary 2.9.
Assume (1.2)–(1.5), (1.8), (1.9), (2.19), (2.20), and (2.29). Let u be a solution of the problem (1.7), (1.13) with u0∈(L0,0)∪(0,L).
(I) Assume that there existb1>0 and B1∈(L0,0) such that
(2.31)u(b1)=B1,u′(b1)=0,
then there exists θ>b1 such that
(2.32)u(θ)=0,u′(t)>0fort∈(b1,θ].
(II) Assume that there exista1>0 and A1∈(0,L) such that
(2.33)u(a1)=A1,u′(a1)=0,
then there exists δ>a1 such that
(2.34)u(δ)=0,u′(t)<0fort∈(a1,δ].Note that if all conditions of Lemmas2.7 and 2.8 are satisfied, then each solution of the problem (1.7), (1.13) with u0∈(L0,0)∪(0,L) has at least one simple zero in (0,∞). Corollary 2.9 makes possible to construct an unbounded sequence of all zeros of any damped solution u. In addition, these zeros are simple (see the proof of Theorem 2.10). In such a case, u has either a positive maximum or a negative minimum between each two neighbouring zeros. If we denote sequences of these maxima and minima by {An}n=1∞ and {Bn}n=1∞, respectively, then we call the numbers |An-Bn|,n∈ℕ amplitudes of u.In [18], we give conditions implying that each damped solution of the problem (1.7), (1.13) with u0<0 has an unbounded set of zeros and decreasing sequence of amplitudes. Here, there is an extension of this result for u0∈(0,L).Theorem 2.10 (existence of oscillatory solutions I).
Assume that (1.2)–(1.5), (1.8), (1.9), (2.19), (2.20), and (2.29) hold, Then each damped solution of the problem (1.7), (1.13) with u0∈(L0,0)∪(0,L) is oscillatory and its amplitudes are decreasing.Proof.
Foru0<0, the assertion is contained in Theorem  3.4 of [18]. Let u be a damped solution of the problem (1.7), (1.13) with u0∈(0,L). By (2.2) and Definition 1.2, we can find L1∈(0,L) such that
(2.35)B¯<u(t)≤L1fort∈[0,∞).Step 1.
Lemma2.7 yields δ1>0 satisfying (2.21). Hence, there exists a maximal interval (δ1,b1) such that u′<0 on (δ1,b1). If b1=∞, then u is eventually negative and decreasing. On the other hand, by Lemma 2.6, u satisfies (2.14). But this is not possible. Therefore, b1<∞ and there exists B1∈(B¯,0) such that (2.31) holds. Corollary 2.9 yields θ1>b1 satisfying (2.32) with θ=θ1. Therefore, u has just one negative local minimum B1=u(b1) between its first zero δ1 and second zero θ1.Step 2.
By (2.32) there exists a maximal interval (θ1,a1), where u′>0. If a1=∞, then u is eventually positive and increasing. On the other hand, by Lemma 2.6, u satisfies (2.14). We get a contradiction. Therefore a1<∞ and there exists A1∈(0,L) such that (2.33) holds. Corollary 2.9 yields δ2>a1 satisfying (2.34) with δ=δ2. Therefore u has just one positive maximum A1=u(a1) between its second zero θ1 and third zero δ2.Step 3.
We can continue as in   Steps1 and 2 and get the sequences {An}n=1∞⊂(0,L) and {Bn}n=1∞⊂[u0,0) of positive local maxima and negative local minima of u, respectively. Therefore u is oscillatory. Using arguments of the proof of Theorem  3.4 of [18], we get that the sequence {An}n=1∞ is decreasing and the sequence {Bn}n=1∞ is increasing. In particular, we use (2.5) and define a Lyapunov function Vu by
(2.36)Vu(t)=u′2(t)2+F(u(t))=F(u0)-∫0tp′(s)p(s)u′2(s)ds,t∈(0,∞),
then
(2.37)Vu(t)>0,Vu′(t)=-p′(t)p(t)u′2(t)≤0fort∈(0,∞),(2.38)Vu′(t)<0fort∈(0,∞),t≠an,bn,n∈N.
Consequently,
(2.39)cu∶=lim⁡t→∞Vu(t)≥0.
So, sequences {Vu(an)}n=1∞={F(An)}n=1∞ and {Vu(bn)}n=1∞={F(Bn)}n=1∞ are decreasing and
(2.40)lim⁡n→∞F(An)=lim⁡n→∞F(Bn)=cu.
Finally, due to (1.4), the sequence {An}n=1∞ is decreasing and the sequence {Bn}n=1∞ is increasing. Hence, the sequence of amplitudes {An-Bn}n=1∞ is decreasing, as well.Example 2.11.
Consider the problem (1.7), (1.13), where p(t)=t2 and f(x)=x(x-1)(x+2). In Example 2.4, we have shown that (1.2)–(1.5), (1.8), and (1.9) with L0=-2, L=1 are valid. Since
(2.41)lim⁡t→∞p′′(t)p′(t)=lim⁡t→∞1t=0,lim⁡x→0f(x)x=lim⁡x→0(x-1)(x+2)=-2<0,
we see that (2.19), (2.20), and (2.29) are satisfied. Therefore, by Theorem 2.10, each damped solution of (2.8), (1.13) with u0∈(-2,0)∪(0,1) is oscillatory and its amplitudes are decreasing.Example 2.12.
Consider the problem (1.7), (1.13), where
(2.42)p(t)=tk1+tl,k>l≥0,f(x)={x(x-1)(x+3),forx≤0,x(x-1)(x+4),forx>0,
then L0=-3, L=1,
(2.43)lim⁡t→∞p′′(t)p′(t)=0,lim⁡x→0-f(x)x=-3,lim⁡x→0+f(x)x=-4.
We can check that also all remaining assumptions of Theorem 2.10 are satisfied, and this theorem is applicable here.Assume thatf does not fulfil (2.20) and (2.29). It occurs, for example, if f(x)=-|x|αsignx with α>1 for x in some neighbourhood of 0, then Theorem 2.10 cannot be applied. Now, we will give another sufficient conditions for the existence of oscillatory solutions. For this purpose, we introduce the following lemmas.Lemma 2.13.
Assume (1.2)–(1.5), (1.8), (1.9), and
(2.44)∫1∞1p(s)ds=∞,(2.45)∃ϵ>0:f∈C1(0,ϵ),f′≤0on⁡(0,ϵ).
Let u be a solution of the problem (1.7), (1.13) with u0∈(0,L), then there exists δ1>0 such that
(2.46)u(δ1)=0,u′(t)<0fort∈(0,δ1].Proof.
Assume that suchδ1 does not exist, then u is positive on [0,∞) and, by Lemma 2.6, u satisfies (2.14). In view of (1.7) and (1.2), we have u′<0 on (0,∞). From (2.45), it follows that there exists t0>0 such that
(2.47)0<u(t)<ϵ,fort∈[t0,∞).
Motivated by arguments of [27], we divide  (1.7) by f(u) and integrate it over interval [t0,t]. We get
(2.48)∫t0t(p(s)u′(s))′f(u(s))ds=∫t0tp(s)dsfort∈[t0,∞).
Using the per partes integration, we obtain
(2.49)p(t)u′(t)f(u(t))+∫t0tp(s)f′(u(s))u′2(s)f2(u(s))ds=p(t0)u′(t0)f(u(t0))+∫t0tp(s)ds,t∈[t0,∞).
From (1.8) and (1.9), it follows that there exists t1∈(t0,∞) such that
(2.50)p(t0)u′(t0)f(u(t0))+∫t0tp(s)ds≥1,t∈[t1,∞),
and therefore
(2.51)p(t)u′(t)f(u(t))+∫t0tp(s)f′(u(s))u′2(s)f2(u(s))ds≥1,t∈[t1,∞).
From the fact that f′(u(s))≤0 for s>t0 (see (2.45)), we have
(2.52)p(t)u′(t)f(u(t))+∫t1tp(s)f′(u(s))u′2(s)f2(u(s))ds≥1,t∈[t1,∞),
then
(2.53)p(t)u′(t)f(u(t))≥1-∫t1tp(s)f′(u(s))u′2(s)f2(u(s))ds>0,t∈[t1,∞),(2.54)p(t)u′(t)f(u(t))(1-∫t1tp(s)f′(u(s))u′2(s)f-2(u(s))ds)≥1,t∈[t1,∞).
Multiplying this inequality by -f′(u(t))u′(t)/f(u(t))≥0, we get
(2.55)(ln⁡(1-∫t1tp(s)f′(u(s))u′2(s)f2(u(s))ds))′≥-(ln⁡|f(u(t))|)′,t∈[t1,∞),
and integrating it over [t1,t], we obtain
(2.56)ln⁡(1-∫t1tp(s)f′(u(s))u′2(s)f2(u(s))ds)≥ln⁡(f(u(t1))f(u(t))),
and therefore,
(2.57)1-∫t1tp(s)f′(u(s))u′2(s)f2(u(s))ds≥f(u(t1))f(u(t)),t∈[t1,∞).
According to (2.53), we have
(2.58)p(t)u′(t)f(u(t))≥f(u(t1))f(u(t)),t∈[t1,∞),
and consequently,
(2.59)u′(t)≤f(u(t1))1p(t),t∈[t1,∞).
Integrating it over [t1,t], we get
(2.60)u(t)≤u(t1)+f(u(t1))∫t1t1p(s)ds,t∈[t1,∞).
From (2.44), it follows that
(2.61)lim⁡t→∞u(t)=-∞,
which is a contradiction.By similar arguments, we can prove a dual lemma.Lemma 2.14.
Assume (1.2)–(1.5), (1.8), (1.9), (2.44), and
(2.62)∃ϵ>0:f∈C1(-ϵ,0),f′≤0on⁡(-ϵ,0).
Let u be a solution of the problem (1.7), (1.13) with u0∈(L0,0), then, there exists θ1>0 such that
(2.63)u(θ1)=0,u′(t)>0fort∈(0,θ1].Following ideas before Corollary2.9, we get the next corollary.Corollary 2.15.
Assume (1.2)–(1.5), (1.8), (1.9), (2.44), (2.45), and (2.62). Let u be a solution of the problem (1.7), (1.13) with u0∈(L0,0)∪(0,L), then the assertions I and II of Corollary 2.9 are valid.Now, we are able to formulate another existence result for oscillatory solutions. Its proof is almost the same as the proof of Theorem2.10 for u0∈(L0,0) and the proof of Theorem 3.4 in [18] for u0∈(0,L). The only difference is that we use Lemmas 2.13, 2.14,  and Corollary 2.15, in place of Lemmas 2.7, 2.8, and Corollary 2.9, respectively.Theorem 2.16 (existence of oscillatory solutions II).
Assume that (1.2)–(1.5), (1.8), (1.9), (2.44), (2.45), and (2.62) hold, then each damped solution of the problem (1.7), (1.13) with u0∈(L0,0)∪(0,L) is oscillatory and its amplitudes are decreasing.Example 2.17.
Let us consider (1.7) with
(2.64)p(t)=tα,t∈[0,∞),f(x)={-|x|λsgn⁡x,x≤1,x-2,x∈(1,3),1,x≥3,
where λ and α are real parameters.Case 1.
Letλ∈(1,∞) and α∈(0,1], then all assumptions of Theorem 2.16 are satisfied. Note that f satisfies neither (2.20) nor (2.29) and hence Theorem 2.10 cannot be applied.Case 2.
Letλ=1 and α∈(0,∞), then all assumptions of Theorem 2.10 are satisfied. If α∈(0,1], then also all assumptions of Theorem 2.16 are fulfilled, but for α∈(1,∞), the function p does not satisfy (2.44), and hence Theorem 2.16 cannot be applied.


## 3. Asymptotic Properties of Oscillatory Solutions

In Lemma2.6 we show that if u is a damped solution of the problem (1.7), (1.13) which is not oscillatory then u converges to 0 for t→∞. In this section, we give conditions under which also oscillatory solutions converge to 0.Theorem 3.1.
Assume that (1.2)–(1.5), (1.8), and (1.9) hold and that there exists k0>0 such that
(3.1)lim⁡inf⁡t→∞p(t)tk0>0,
then each damped oscillatory solution u of the problem (1.7), (1.13) with u0∈(L0,0)∪(0,L) satisfies
(3.2)lim⁡t→∞u(t)=0,lim⁡t→∞u′(t)=0.Proof.
Consider an oscillatory solutionu of the problem (1.7), (1.13) with u0∈(0,L).Step 1.
Using the notation and some arguments of the proof of Theorem2.10, we have the unbounded sequences {an}n=1∞, {bn}n=1∞, {θn}n=1∞, and {δn}n=1∞, such that
(3.3)0<δ1<b1<θ1<a1<δ2<⋯<δn<bn<θn<an<δn+1<⋯,
where u(θn)=u(δn)=0, u(an)=An>0 is a unique local maximum of u in (θn,δn+1), u(bn)=Bn<0 is a unique local minimum of u in (δn,θn), n∈ℕ. Let Vu be given by (2.36) and  then (2.39) and (2.40) hold and, by (1.2)–(1.4), we see that
(3.4)lim⁡t→∞u(t)=0⟺cu=0.
Assume that (3.2) does not hold. Then cu>0. Motivated by arguments of [28], we derive a contradiction in the following steps.Step 2 (estimates ofu).
By (2.36) and (2.39), we have
(3.5)lim⁡n→∞u′2(δn)2=lim⁡n→∞u′2(θn)2=cu>0,
and the sequences {u′2(δn)}n=1∞ and {u′2(θn)}n=1∞ are decreasing. Consider n∈ℕ. Then u′2(δn)/2>cu and there are αn,βn satisfying an<αn<δn<βn<bn and such that
(3.6)u′2(αn)=u′2(βn)=cu,u′2(t)>cu,t∈(αn,βn).
Since Vu(t)>cu for t>0 (see (2.39)), we get by (2.36) and (3.6) the inequalities cu/2+F(u(αn))>cu and cu/2+F(u(βn))>cu, and consequently F(u(αn))>cu/2 and F(u(βn))>cu/2. Therefore, due to (1.4), there exists c̃>0 such that
(3.7)u(αn)>c̃,u(βn)<-c̃,n∈N.
Similarly, we deduce that there are α̃n, β̃n satisfying bn<α̃n<θn<β̃n<an+1 and such that
(3.8)u(α̃n)<-c̃,u(β̃n)>c̃,n∈N.
The behaviour of u and inequalities (3.7) and (3.8) yield
(3.9)|u(t)|>c̃,t∈[βn,α̃n]∪[β̃n,αn+1],n∈N.Step 3 (estimates ofβn-αn).
We prove that there existc0,c1∈(0,∞) such that
(3.10)c0<βn-αn<c1,n∈N.
Assume on the contrary that there exists a subsequence satisfying limℓ→∞(βℓ-αℓ)=0. By the mean value theorem and (3.7), there is ξℓ∈(αℓ,βℓ) such that 0<2c̃<u(αℓ)-u(βℓ)=|u′(ξℓ)|(βℓ-αℓ). Since F(u(t))≥0 for t∈[0,∞), we get by (2.16) the inequality
(3.11)|u′(t)|<2F(u0),t∈[0,∞),
and consequently
(3.12)0<2c̃≤2F(u0)lim⁡l→∞(βl-αl)=0,
which is a contradiction. So, c0 satisfying (3.10) exists. Using the mean value theorem again, we can find τn∈(αn,δn) such that u(δn)-u(αn)=u′(τn)(δn-αn) and, by (3.6),
(3.13)δn-αn=-u(αn)u′(τn)=u(αn)|u′(τn)|<A1cu.
Similarly, we can find ηn∈(δn,βn) such that
(3.14)βn-δn=u(βn)u′(ηn)=|u(βn)||u′(ηn)|<|B1|cu.
If we put c1=(A1+|B1|)/cu, then (3.10) is fulfilled. Similarly, we can prove
(3.15)c0<β̃n-α̃n<c1,n∈N.Step 4 (estimates ofαn+1-αn).
We prove that there existc2∈(0,∞) such that
(3.16)αn+1-αn<c2,n∈N.
Put m1=min{f(x):B1≤x≤-c̃}>0. By (3.9), B1≤u(t)<-c̃ for t∈[βn,α̃n], n∈ℕ. Therefore,
(3.17)f(u(t))≥m1,t∈[βn,α̃n],n∈N.
Due to (1.9), we can find t1>0 such that
(3.18)p′(t)p(t)2F(u0)<m12,t∈[t1,∞).
Let n1∈ℕ fulfil αn1≥t1, then, according to (2.4), (3.11), (3.17), and (3.18), we have
(3.19)u′′(t)>-m12+m1=m12,t∈[βn,α̃n],n≥n1.
Integrating (3.19) from bn to βn and using (3.6), we get 2cu>m1(bn-βn) for n≥n1. Similarly we get 2cu>m1(α̃n-bn) for n≥n1. Therefore
(3.20)4m1cu>α̃n-βn,n≥n1.
By analogy, we put m2=min{-f(x):c̃≤x≤A1}>0 and prove that there exists n2∈ℕ such that
(3.21)4m2cu>αn+1-β̃n,n≥n2.
Inequalities (3.10), (3.15), (3.20), and (3.21) imply the existence of c2 fulfilling (3.16).Step 5 (construction of a contradiction).
Chooset0>c1 and integrate the equality in (2.37) from t0 to t>t0. We have
(3.22)Vu(t)=Vu(t0)-∫t0tp′(τ)p(τ)u′2(τ)dτ,t≥t0.
Choose n0∈ℕ such that αn0>t0. Further, choose n∈ℕ, n>n0 and assume that t>βn,  then, by (3.6),
(3.23)∫t0tp′(τ)p(τ)u′2(τ)dτ>∑j=n0n∫αjβjp′(τ)p(τ)u′2(τ)dτ>cu∑j=n0n∫αjβjp′(τ)p(τ)dτ=cu∑j=n0n[ln⁡p(τ)]αjβj.
By virtue of (3.1) there exists c3>0 such that p(t)/tk0>c3 for t∈[t0,∞). Thus, lnp(t)>lnc3+k0lnt and
(3.24)∫t0tp′(τ)p(τ)u′2(τ)dτ>cu∑j=n0n[ln⁡c3+k0ln⁡t)]αjβj=cuk0∑j=n0nln⁡βjαj.
Due to (3.10) and c1<αn0, we have
(3.25)1<βjαj<1+c1αj<2,j=n0,…,n,
and the mean value theorem yields ξj∈(1,2) such that
(3.26)ln⁡βjαj=(βjαj-1)1ξj>βj-αj2αj,j=n0,…,n.
By (3.10) and (3.16), we deduce
(3.27)βj-αjαj>c0αj,αj<jc2+α1,j=n0,…,n.
Thus,
(3.28)βj-αjαj>c0jc2+α1,j=n0,…,n.
Using (3.24)–(3.28) and letting t to ∞, we obtain
(3.29)∫t0∞p′(τ)p(τ)u′2(τ)dτ≥cuk0∑n=n0∞ln⁡βnαn≥12cuk0∑n=n0∞βn-αnαn≥12cuk0∑n=n0∞c0nc2+α1=∞.
Using it in (3.22), we get limt→∞Vu(t)=-∞, which is a contradiction. So, we have proved that cu=0.
Using (2.4) and (3.4), we have
(3.30)lim⁡t→∞(u′2(t)2+∫0tp′(s)p(s)u′2(s)ds)=F(u0)-F(0)=F(u0).
Since the function ∫0t(p′(s)/p(s))u′2(s)ds is increasing, there exists
(3.31)lim⁡t→∞∫0tp′(s)p(s)u′2(s)ds≤F(u0).
Therefore, there exists
(3.32)lim⁡t→∞u′2(t)=l2.
If ℓ>0, then limt→∞|u′(t)|=ℓ, which contradicts (3.4). Therefore, ℓ=0 and (3.2) is proved.
Ifu0∈(L0,0), we argue analogously.



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*Source: 408525-2011-05-19.xml*