We compute explicitly the oscillation constant for Euler type half-linear second-order differential equation having multi-different periodic coefficients.

1. Introduction

In literature, half-linear second-order differential equations are given by(1)rtΦx′′+ctΦx=0,Φs=sp-2s,p>1,where r,c are continuous functions and rt>0. It is well known that oscillation theory of (1) is very similar to that of the linear Sturm-Liouville differential equation, which is the special case of p=2 in (1); see [1].

In particular, (1) with λct instead of ct is said to be conditionally oscillatory if there exists a constant λ0 such that this equation is oscillatory for λ>λ0 and nonoscillatory for λ<λ0. λ0 is called the critical oscillation constant of this equation; see [2].

The half-linear Euler differential equation(2)Φx′′+γptpΦx=0,with the so-called critical oscillation constant γp=(p-1)/pp, plays an important role in the conditionally oscillatory half-linear differential equation.

Equation (2) can be regarded as a good comparative equation in the sense that (2) with γ instead of γp is oscillatory if and only if γ>γp (see [3]) and if rt=1 in (1), then this equation is oscillatory provided(3)limt→∞inftpct>γpand nonoscillatory if(4)limt→∞suptpct<γp;see [4].

In [5], perturbations of (2) being of the form(5)Φx′′+1tpγp+∑l=1nβjLogj2tΦx=0are investigated when limt→∞tpct=γp for constant βj(j=1,2,…,n). Here the notation(6)Logkt=∏j=1klogjt,Logkt=logk-1logt,Log1t=logtis used. It is shown that the constant μp≔1/2(p-1)/pp-1 plays a crucial role in (5). In particular, if n=1 in (5) this equation reduces to the so-called Riemann-Weber half-linear differential equation, and this equation is oscillatory if β1>μp and nonoscillatory otherwise. In general, if βj=μp for j=1,2,…,n-1, then (5) is oscillatory if and only if βn>μp.

One of the typical problems in the qualitative theory of various differential equations is to study what happens when constants in an equation are replaced by periodic functions which have same periods and different periods. Our investigation follows this line and it is mainly motivated by the paper [6].

In [7], the half-linear differential equation being of the form(7)1+∑j=1nαjLogj2t1-pΦx′′+1tpγp+∑j=1nβjLogj2tΦx=0is investigated for αj and βj are constants and the following result is obtained.

Theorem 1.

Suppose that there exists k∈2,…,n such that (8)βj+p-1γpαj=μp,j=1,…,k-1and βk+p-1γpαk≠μp. Then (7) is oscillatory if βk+p-1γpαk>μp and nonoscillatory if βk+p-1γpαk<μp.

In [8], the half-linear differential equation being of the form(9)rtΦx′′+γcttpΦx=0is considered for α-periodic positive functions r and c and it is shown that (9) is oscillatory if γ>K and nonoscillatory if γ<K, where K is given by (10)K=q-p1α∫0αdτrq-11-p1α∫0αcτdτ-1for p and q are conjugate numbers; that is, 1/p+1/q=1.

In [9], (9) and the half-linear differential equation being of the form(11)rtΦx′′+1tpγct+μdtlog2tΦx=0are considered for r,c, and d are α-periodic, positive functions defined on 0,∞ and it is shown that (9) is nonoscillatory if and only if γ≤γrc, where γrc is given by (12)γrc≔αpγp∫0αr1-qtdtp-1∫0αctdt.In the limiting case γ=γrc (11) is nonoscillatory if μ<μrd and it is oscillatory if μ>μrd, where μrd is given by (13)μrd=αpμp∫0αr1-qtdtp-1∫0αdtdt.

In [10], the half-linear differential equation being of the form (14)rtΦx′′+cttpΦx=0is considered for r:a,∞→R, (a>0), is a continuous function for which mean value Mr1-q exists and for which(15)0<inft∈a,∞rt≤supt∈a,∞rt<∞holds and c:a,∞→R, (a>0), is a continuous function having mean value Mc and it was shown that (14) is oscillatory if Mc>Γ and nonoscillatory if Mc<Γ, where Γ is given by (16)Γ=q-pMr1-q1-p.

In [6], the half-linear differential equation being of the form(17)rt+∑j=1nαjtLogj2t1-pΦx′′+1tpct+∑j=1nβjtLogj2tΦx=0is considered for T-periodic functions r,c,αj, and βj,j=1,2,…,n, and rt>0 and the following result was obtained.

Theorem 2.

Let r,c,αj, and βj(j=1,2,…,n) be T-periodic continuous functions, rt>0, and their mean values over the period T are denoted by r~,c~,α~j, and β~j(j=1,2,…,n).

If c~r~p-1>γp, then (17) is oscillatory and if c~r~p-1<γp, then it is nonoscillatory.

Let c~r~p-1=γp. If there exists k∈1,…,n such that (18)β~jr~p-1+p-1γpα~jr~-1=μp,j=1,2,…,k-1

(if k≠1), and β~kr~p-1+p-1γpαkr~-1≠μp, then (17) is oscillatory if(19)β~kr~p-1+p-1γpα~kr~-1>μp

and nonoscillatory if(20)β~kr~p-1+p-1γpα~kr~-1<μp.

Our research is motivated by the paper [6], where the oscillation constant is computed for (17) with the periodic coefficients having same T-period. However, if these periodic functions have different periods what would be the oscillation constant is not investigated. Thus, in this paper we investigate the oscillation constant for (17) with periodic coefficients having different periods. In this paper we consider two types of periodic coefficients which have different periods for (17). In the first type we consider these periodic coefficient functions having the least common multiple and in the second type, we consider these periodic coefficient functions which do not have least common multiple. We give some corollaries which illustrate the first type’s cases that our results compile the known results in [6] but in the second type only our results can be applied.

In Section 2, we recall the concept of half-linear-trigonometric functions and their properties. In Section 3 we compute the oscillation constant for (17) with periodic coefficients which have different periods. Additionally we show that if the different periods coincide, then our results compile with the known results in [6]. Thus, our results extend and improve the results of [6].

2. Preliminaries

We start this section with recalling the concept of half-linear-trigonometric functions; see [1] or [4]. Consider the following special half-linear equation being of the form (21)Φx′′+p-1Φx=0and denote its solution by x=xt given by the initial conditions x0=0, x′0=1. We see that the behavior of this solution is very similar to that of the classical sine function. We denote this solution by sinpt and its derivative by sinpt′=cospt. These functions are 2πp-periodic, where πp≔2π/psinπ/p, and satisfy the half-linear Pythagorean identity(22)sinptp+cosptp=1,t∈R.Every solution of (21) is of the form xt=Csinpt+φ, where C and φ are real constants; that is, it is bounded together with its derivative and periodic with the period 2πp. The function u=Φcospt is a solution to the reciprocal equation of (21);(23)Φ-1u′′+p-1q-1Φ-1u=0,Φ-1u=uq-2u,q=pp-1,which is an equation of the form as in (21), so the functions u and u′ are also bounded.

Let xt be a nontrivial solution of (1) and we consider the half-linear Prüfer transformation which is introduced using the half-linear-trigonometric functions(24)xt=ρtsinpφt,x′t=r1-qtρttcospφt,where ρt=xtp+rqtx′tp and Prüfer angle φt is a continuous function defined at all points where xt≠0.

Then φt satisfies the following differential equation:(25)φ′t=1tr1-qtcospφtp-Φcospφtsinpφt+tpctp-1sinpφtp;see [9].

3. Main Results

We need the following lemma in order to prove our main Theorem 4.

Lemma 3.

Let φt=φ1t+∑j=1nφ2jt+φ3t+φ4t+∑j=1nφ5jt+M (M is a suitable constant) be a solution of the equation(26)φ′t=φ1′t+∑j=1nφ2j′t+φ3′t+φ4′t+∑j=1nφ5j′t,where(27)φ1′t=1trtcospφtp,φ2j′t=αjttLogj2tcospφtpj=1,…,n,φ3′t=-1tΦcospφtsinpφt,φ4′t=ctp-1tsinpφtp,φ5j′t=βjtp-1tLogj2tsinpφtpj=1,…,n,with r,c,αj, and βj(j=1,2,…,n) are periodic functions having different T1,T2,Pj, and Qj(j=1,2,…,n) periods, respectively, and rt>0 and(28)θt=1T1∫tt+T1φ1sds+∑j=1n1Pj∫tt+Pjφ2jsds+1ξ∫tt+ξφ3sds+1T2∫tt+T2φ4sds+∑j=1n1Qj∫tt+Qjφ5jsds,where ξ is one of the following T1,T2,Pj, and Qj(j=1,2,…,n) periods. Then θt is a solution of(29)θ′t=1tr∗+∑j=1nαj∗Logj2t+o1Logn2tcospθtp+1p-1tc∗+∑j=1nβj∗Logj2t+o1Logn2tsinpθtp-1tΦcospθtsinpθt,where(30)r∗=1T1∫0T1rsds,c∗=1T2∫0T2csds,αj∗=1Pj∫0Pjαjsds,βj∗=1Qj∫0Qjβjsdsfor j=1,2,…,nand φτ-θt=∘1 as t→∞.

Proof.

Taking derivative of θt, we have(31)θ′t=1T1∫tt+T1φ1′sds+∑j=1n1Pj∫tt+Pjφ2j′sds+1ξ∫tt+ξφ3′sds+1T2∫tt+T2φ4′sds+∑j=1n1Qj∫tt+Qjφ5j′sds=1T1∫tt+T11srscospφspds+∑j=1n1Pj∫tt+PjαjssLogj2scospφspds-1ξ∫tt+ξ1sΦcospφssinpφsds+1T2∫tt+T2csp-1ssinpφspds+∑j=1n1Qj∫tt+Qjβjssp-1Logj2ssinpφspds.

Using integration by parts, we get(32)θ′t=1T1t∫tt+T1rτcospφτpdτ+1t∑j=1n1Pj∫tt+PjαjτLogj2τcospφτpdτ-1ξt∫tt+ξΦcospφτsinpφτdτ+1T2t∫tt+T2cτp-1sinpφτpdτ+1t∑j=1n1Qj∫tt+Qjβjτp-1Logj2τsinpφτpdτ-1T1∫tt+T11s2∫st+T1rτcospφτpdτds-∑j=1n1Pj∫tt+Pj1s2∫tt+PjαjτLogj2τcospφτpdτds+1ξ∫tt+ξ1s2∫st+ξΦcospφτsinpφτdτds-1T2∫tt+T21s2∫st+T2cτp-1sinpφτpdτds-∑j=1n1Qj∫tt+Qj1s2∫tt+Qjβjτp-1Logj2τsinpφτpdτds.

Let f be a continuous T-periodic function and f∗=1/T∫0Tfsds; then integration by parts yields (33)1T∫tt+Tfslogj2sds=f∗logj2t1+O1tlogt.By using (33) and ∫tt+Tfsds=∫0Tfsds for any T-periodic function and Pythagorean identity, the expressions(34)r1-qtcospφtp,-Φcospφtsinpφt,ctp-1sinpφtpare bounded. Thus we get(35)θ′t=1T1t∫tt+Trτcospφτpdτ+1t∑j=1n1Pj∫tt+PjαjτLogj2τcospφτpdτ-1ξt∫tt+ξΦcospφτsinpφτdτ+1T2t∫tt+T2cτp-1sinpφτpdτ+1t∑j=1n1Qj∫tt+Qjβjτp-1Logj2τsinpφτpdτ+O1t.

If we add and subtract the below terms in the right side of this equation(36)1T1t∫tt+T1rτcospθtpdτ+1t∑j=1n1Pj∫tt+PjαjτLogj2τcospθtpdτ-1ξt∫tt+ξΦcospθtsinpφτdτ+1T2t∫tt+T2cτp-1sinpθtpdτ+1t∑j=1n1Qj∫tt+Qjβjτp-1Logj2τsinpθtpdτwe can rewrite this equation as(37)θ′t=1T1t∫tt+T1rτcospθtpdτ+1t∑j=1n1Pj∫tt+PjαjτLogj2τcospθtpdτ-1ξt∫tt+ξΦcospθtsinpφτdτ+1T2t∫tt+Tcτp-1sinpθtpdτ+1t∑j=1n1Qj∫tt+Qjβjτp-1Logj2τsinpθtpdτ+1T1t∫tt+T1rτcospφτp-cospθtpdτ+1t∑j=1n1Pj∫tt+PjαjτLogj2τcospφτp-cospθtpdτ-1ξt∫tt+ξΦcospφτsinpφτ-Φcospθtsinθtdτ+1T2t∫tt+T2cτp-1sinpφtp-sinpθtpdτ+1t∑j=1n1Qj∫tt+Qjβjτp-1Logj2τsinpφtp-sinpθtpdτ+O1t.

And using the half-linear-trigonometric functions, we have(38)cospφτp-cospθtp≤p∫θtφτΦcospscosps′ds≤constφτ-θt,Φcospφτsinpφτ-Φcospθtsinpθt≤∫θtφτΦcospssinps′ds≤constφτ-θt,sinpφtp-sinpθtp≤constφτ-θt.

By the Mean Value Theorem we can write(39)θt=φ1t1+∑j=1nφ2jt2j+φ3t3+φ4t4+∑j=1nφ5jt5jfor t1∈t,t+T1, t2j∈t,t+Pj,j=1,2,…,n, t3∈t,t+ξ, t4∈t,t+T2, and t5j∈t,t+Qj,j=1,2,…,n; thus(40)φτ-θt≤φ1τ-φ1t1+∑j=1nφ2jτ-φ2jt2j+φ3τ-φ3t3+φ4τ-φ4t4+∑j=1nφ5jτ-φ5jt5j.This implies that (41)φτ-θt≤o1t,φτ-θt=o1as t⟶∞.And using r∗, c∗, αj∗, βj∗, and (33), we get(42)θ′t=1tr∗+1+O1tlogt∑j=1nαj∗Logj2tcospθtp+1p-1tc∗+1+O1tlogt∑j=1nβj∗Logj2tsinpθtp-1tΦcospθtsinpθt+O1t.The term O1/t can be written as cospθp+sinpθpO1/t; hence we get (43)θ′t=1tr∗+1+O1tlogt∑j=1nαj∗Logj2t+O1tcospθtp+1p-1tc∗+1+O1tlogt∑j=1nβj∗Logj2t+O1tsinpθtp-1tΦcospθtsinpθt.

Now since all the terms of O1/tlogt/logj2t are O1/t as t→∞ for j=1,2,…,n, then all these terms are asymptotically less than o1/logn2t. Hence we get(44)θ′t=1tr∗+∑j=1nαj∗Logj2t+o1Logn2tcospθtp+1p-1tc∗+∑j=1nβj∗Logj2t+o1Logn2tsinpθtp-1tΦcospθtsinpθt.

The main result of this paper is as follows.

Theorem 4.

Let r,c,αj, and βj,j=1,2,…,n, are periodic functions which have different T1,T2,Pj, and Qj,j=1,2,…,n, periods, respectively, and rt>0 in (17).

(17) is oscillatory if c∗r∗p-1>γp and nonoscillatory if c∗r∗p-1<γp, where r∗ and c∗ are defined in Lemma 3.

Let c∗r∗p-1=γp. If there exists k∈2,…,n such that (45)βj∗r∗p-1+p-1γpαj∗r∗-1=μp,j=1,…,k-1

and βk∗r∗p-1+p-1γpαk∗r∗-1≠μp, then (17) is oscillatory if(46)βk∗r∗p-1+p-1γpαk∗r∗-1>μp

and nonoscillatory if (47)βk∗r∗p-1+p-1γpαk∗r∗-1<μp,

where αj∗ and βj∗,j=1,2,…,n, are defined in Lemma 3.

Proof.

The statement (i) is proved in [10]. It remains to prove the statement (ii) in full generality.

We consider (17); let xt be the nontrivial solution of (17) and φt is the Prüfer angle of (17) given in (24). Then (48)φt=φ1t+∑j=1nφ2jt+φ3t+φ4t+∑j=1nφ5jt+Mis a solution of(49)φ′t=φ1′t+∑j=1nφ2j′t+φ3′t+φ4′t+∑j=1nφ5j′t,where(50)φ1′t=1trtcospφtp,φ2j′t=αjttLogj2tcospφtpj=1,2,…,n,φ3′t=-1tΦcospφtsinpφt,φ4′t=ctp-1tsinpφtp,φ5j′t=βjtp-1tLogj2tsinpφtpj=1,2,…,n.

By the help of Lemma 3, θt is a solution of(51)θ′t=1tr∗+∑j=1nαj∗Logj2t+o1Logn2tcospθtp+1p-1tc∗+∑j=1nβj∗Logj2t+o1Logn2tsinpθtp-1tΦcospθtsinpθt,where r∗,c∗,αj∗, and βj∗,j=1,2,…,n, are given in Lemma 3.

This equation is a “Prüfer angle” equation for the following second-order half-linear differential equation (52)r∗+∑j=1nα∗jLogj2t+o1Logn2t1-pΦx′′+1tpc∗+j=1n∑j=1nβ∗jLogj2t+o1Logn2tΦx=0,which is the same as the following equation:(53)RtΦx′′+1tpc∗r∗p-1+∑j=1nβ∗jr∗p-1Logj2t+o1Logn2tΦx=0.Suppose that assumption (ii) of Theorem 4 is satisfied and that (46) holds for k∈1,2,…,n-1. Then (53) is oscillatory as a direct consequence of Theorem 1. If (46) holds for k=n, let ε>0 be so small that still (54)β∗nr∗p-1-ε+p-1γpr∗-1α∗j-ε>μpand consider the following equation:(55)R1tΦx′′+1tpc∗r∗p-1+∑j=1nβ∗jr∗p-1Logj2t+β∗nr∗p-1-εLogn2tΦx=0,where R1t=1+∑j=1nα∗j/r∗/logj2t+α∗n/r∗-ε/logn2t1-p. This equation is a Sturmian minorant for sufficiently large t in (53) and (54) and Theorem 1 implies that this minorant equation is oscillatory and hence (53) is oscillatory as well. This means that the Prüfer angle θt of the solution of (52) is unbounded and by Lemma 3 the Prüfer angle φt of the solution of (17) is unbounded as well. Thus, (17) is oscillatory. A slightly modified argument implies that (17) is nonoscillatory provided that (47) holds.

Corollary 5.

If the periods of the functions r,c,αj, and βj,j=1,2,…,n, in (17) coincide with T-period, which is given in [6], then our oscillation constants overlap to their oscillation constants and our main result compiles with the result given in [6].

Corollary 6.

If there exists a lcmT1,T2,Pj,Qj,j=1,2,…,n, and the period T which is given in [6] is chosen as lcmT1,T2,Pj,Qj,j=1,2,…,n, then our oscillation constants overlap to their oscillation constants and our main result compiles with the result given in [6].

Remark 7.

If for j=1,2,…,nlcmT1,T2,Pj,Qj is not defined, then only our result can be applied whereas the result given in [6] can not.

Example 8.

Consider the nonlinear equation (17) for p=3, rt=2+cosax+b,(a,b∈R), α1t=cos3t, α2t=sin8t, β1t=sin4t, β2t=sin2t, and ct=2+sin6t. In this case T1=2π/a, P1=2π/3, P2=π/4, Q1=π/2, Q2=π, and T2=π/3 are periods of these functions, respectively. Because of these functions being periodic functions and rt positive defined we can use Theorem 4 for all a≠0 and we obtain(56)c∗r∗p-1=a2π∫02π/a2+cosas+bds3π∫0π/32+sin6sds3-1=8,γ3=3-133=827.Thus we get c∗r∗p-1>γ3 for all a≠0 and considered equation is oscillatory. Here the important point to note is that while we cannot apply Theorem 2 which is given in [6] for this example if we choose a=5, then lcm2π/a,2π/3,π/4,π/2,π,π/3 is not defined, we can apply our Theorem 4.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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