Nonoscillatory half-linear differential equations and generalized Karamata functions

Nonlinear Analysis 64 (2006) 762 – 787 www.elsevier.com/locate/na Nonoscillatory half-linear differential equations and generalized Karamata function...

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Nonlinear Analysis 64 (2006) 762 – 787 www.elsevier.com/locate/na

Nonoscillatory half-linear differential equations and generalized Karamata functions Jaroslav Jaroša , Kusano Takaˆsib,∗ , Tomoyuki Tanigawac a Department of Mathematical Analysis, Faculty of Mathematics, Physics and Informatics, Comenius University,

824 48 Bratislava, Slovak Republic b Department of Applied Mathematics, Faculty of Science, Fukuoka University, 8-19-1 Nanakuma, Jonan-ku,

Fukuoka 814-0180, Japan c Department of Mathematics, Joetsu University of Education, Niigata 943-8512, Japan

Received 1 April 2005; accepted 27 May 2005 Dedicated to Professor Masuo Hukuhara on the occasion of his centennial birthday

Abstract We introduce a natural generalization of the concept of regularly varying functions in the sense of Karamata, and show that the class of generalized Karamata functions is a well-suited framework for the study of the asymptotic behavior of nonoscillatory solutions of the half-linear differential equation of the type (p(t)|y |−1 y ) + q(t)|y|−1 y = 0.

(A)

䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Half-linear differential equation; Nonoscillation; Slowly varying function; Regular variation

1. Introduction In our previous paper [6] it is shown how useful the regularly varying functions in the sense of Karamata are for the study of nonoscillation and asymptotic behavior of the ∗ Corresponding author.

E-mail addresses: [email protected] (J. Jaroš), [email protected] (K. Takaˆsi), [email protected] (T. Tanigawa). 0362-546X/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2005.05.045

J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

763

half-linear differential equation (|y |−1 y ) + q(t)|y|−1 y = 0,

t 0,

where > 0 is a constant and q : [0, ∞) → R is a continuous function. By deﬁnition, a measurable function f : [0, ∞) → (0, ∞) is regularly varying of index ∈ R if it satisﬁes lim

t→∞

f (t) = f (t)

for any > 0.

(1.1)

It is known [1] that such a function is characterized by the fact that it has the representation t (s) ds , t t0 (1.2) f (t) = c(t) exp s t0 for some t0 > 0 and some measurable functions c(t) and (t) such that lim c(t) = c ∈ (0, ∞) and

t→∞

lim (t) = .

t→∞

(1.3)

If c(t) ≡ c (constant) in (1.2), then f (t) is said to be a normalized varying function of index . The totality of regularly varying functions [or normalized regularly varying functions] of index is denoted by RV() [or n-RV()]. In the special case when = 0, we use the notation SV = RV(0) or n-SV = n-RV(0), referring to f ∈ SV [or f ∈ n-SV] as a slowly varying function [or a normalized slowly varying function]. A question naturally arises as to whether it is possible to investigate the more general half-linear equation of the type (p(t)|y |−1 y ) + q(t)|y|−1 y = 0,

t 0,

(A)

p : [0, ∞) → (0, ∞) and q : [0, ∞) → R being continuous, in the same spirit as in [6]. It is immediately obvious that the class of classical Karamata functions is not sufﬁcient to properly describe the possible asymptotic behavior of nonoscillatory solutions of (A) which are essentially affected by the function p(t), more precisely, by the integral ∞ t (p(s))−1/ ds in case (p(t))−1/ dt = ∞ (1.4) P (t) = a

or

(t) =

t

∞

a

(p(s))

−1/

ds

in case

∞ a

(p(t))−1/ dt < ∞.

(1.5)

Of paramount importance for us, therefore, is to set up the framework in which the dependence upon P (t) or (t) of solutions of (A) is accurately reﬂected. It is conjectured that the generalized Karamata functions which were introduced in [5] to analyze the linear equation (p(t)y ) + q(t)y = 0 will be effectively applicable to the half-linear equation (A) as well. The objective of this paper is to verify the truth of this conjecture. Let : [0, ∞) → (0, ∞) be a continuously differentiable function such that (t) > 0 for t 0

and

lim (t) = ∞.

t→∞

(1.6)

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J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

A measurable function f : [0, ∞) → (0, ∞) is said to be a regularly varying function of index with respect to if f ◦ −1 is deﬁned for all large t and is a regularly varying function of index in the sense of Karamata, where −1 denotes the inverse function of . The symbol RV () is used to denote the totality of regularly varying functions of index with respect to . From the deﬁnition it follows that f ∈ RV () if and only if it is written in the form f (t) = g((t)) for some g ∈ RV(). It is clear that RV() is the special case of RV () with (t) = t: RVt () = RV(). As a direct consequence of (1.2) one can derive the canonical representation for such generalized regularly varying functions, which implies that f ∈ RV () if and only if it is expressed in the form t (s) f (t) = c(t) exp (s) ds , t t0 , (1.7) t0 (s) for some t0 > 0 and some measurable functions c(t) and (t) satisfying (1.3). In case c(t) ≡ c (constant) in (1.7) f is called a normalized regularly varying function of index with respect to , and this fact is denoted by f ∈ n-RV (). Use is made of the notation SV = RV (0) or n-SV = n-RV (0) to justify the use of the terminology “slowly varying”instead of “regularly varying of index 0”. As indicated above we distinguish the two cases (1.4) and (1.5) and show that the set of generalized Karamata functions {RVP () : ∈ R} or {RV1/ () : ∈ R} formed with the choice of (t) = P (t) or (t) = 1/(t) provides a well-suited framework for the asymptotic analysis of Eq. (A) with p(t) satisfying (1.4) or (1.5), respectively. More speciﬁcally, we establish sharp conditions for (A) to have a pair of nonoscillatory solutions {y1 (t), y2 (t)} belonging to {RVP (1 ), RVP (2 )} or to {RV1/ (1 ), RV1/ (2 )} for some speciﬁed values of 1 and 2 . A duality existing between the results for the distinguished two cases would be worthy of attention. An essential part of our method, which is an extended adaptation of the one used in [6], is the application of the Banach contraction mapping principle to the construction of solutions of (A) of class RVP () or RV1/ (). It should be noticed that an attempt to apply the ﬁxed point techniques in the study of differential equations by means of regularly varying functions was ﬁrst made in the paper [4]. 2. Existence of generalized regularly varying solutions (the ﬁrst case) The problem of existence of generalized regularly solutions of half-linear differential equations of the form (A) is the subject of our investigations in this and the next sections. It is assumed that is a positive constant, and p : [0, ∞) → (0, ∞) and q : [0, ∞) → R are continuous functions. Note that no sign condition is imposed on q(t), so that it may take on positive and negative values in any neighborhood of inﬁnity. The characteristic feature of (A) is the fact that it has many fundamental qualitative properties in common with the corresponding linear differential equation (p(t)y ) + q(t)y = 0.

(A0 )

For example, the classical Sturm comparison and oscillation theorems for (A0 ) can be carried over almost literatim and verbatim to (A); see [2]. Thus, Eq. (A) cannot possess an oscillatory solution and a nonoscillatory solution simultaneously, so that all of its solutions

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are either oscillatory (in which case (A) is called oscillatory) or nonoscillatory (in which case (A) is called nonoscillatory). In what follows our exclusive attention will be given to the situation in which Eq. (A) is nonoscillatory. Basic to our subsequent discussions is the following lemma relating (A) to the ﬁrst-order nonlinear differential equation, called the generalized Riccati equation associated with (A), u + (p(t))−1/ |u|1+1/ + q(t) = 0.

(B)

Lemma 2.1. Eq. (A) is nonoscillatory if and only if Eq. (B) has a solution deﬁned in some neighborhood of inﬁnity. This lemma shows that the well-known Riccati equation technique for the linear equation (A0 ) allows natural generalization to the half-linear equation (A). It is convenient to use the asterisk notation deﬁned by ∗ = ||−1 = || sgn ,

∈ R, > 0,

(2.1)

in terms of which (A) can be rewritten as (p(t)(y )∗ ) + q(t)y ∗ = 0. The proof of Lemma 2.1 is straightforward. In fact, if (A) has a nonoscillatory solution y(t), then the function u(t) = p(t)(y (t)/y(t))∗ satisﬁes (B) for all sufﬁciently large t. Conversely, if u(t) is a solution of (B) on [t0 , ∞), then the function y(t) = ∗ t exp{ t0 (u(s)/p(s))1/ ds} satisﬁes (A) on [t0 , ∞). ∞ We start with the case of Eq. (A) in which the function p(t) satisﬁes 0 (p(t))−1/ dt =∞. With regard to q(t) we assume that it is conditionally integrable in the sense that ∞ T q(t) dt = lim q(t) dt exists and is ﬁnite. 0

T →∞

0

It will be shown that the set of regularly varying functions with respect to the function P (t) deﬁned by (1.4), {RVP () : ∈ R}, is the best framework for the asymptotic analysis of Eq. (A) under consideration, meaning that sharp criteria can be obtained for the existence of solutions belonging to RVP () for some speciﬁed values of . From formula (1.7) it follows that f ∈ RVP () for some if and only if it is expressed in the form t (s) ds , t t0 , (2.2) f (t) = c(t) exp 1/ P (s) t0 p(s) for some t0 > 0 and some measurable functions c(t) and (t) such that limt→∞ c(t) = c ∈ (0, ∞) and limt→∞ (t) = . Use is made of the symbol E() to denote the “generalized Euler constant” E() =

( + 1)+1

.

The following theorem is the main result of the present section.

(2.3)

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Theorem 2.1. Let c ∈ (−∞, E()) be ﬁxed arbitrarily and let 1 , 2 , 1 < 2 , denote the real roots of the equation ||1+1/ − + c = 0.

(2.4) 1/∗

Eq. (A) possesses a pair of solutions yi (t) ∈ n-RVP (i latory, if and only if ∞ q(s) ds = c. lim P (t) t→∞

), i = 1, 2, thus being nonoscil-

(2.5)

t

Proof (The “only if”part). Suppose that (A) has a nonoscillatory solution yi (t) ∈ n-RVP 1/∗ (i ), i = 1 or 2. By the representation formula (2.2) there exist positive constans ci , ti 1/∗

and a continuous function i (t) such that limt→∞ i (t) = i yi (t) = ci exp

t

ti

i (s) p(s)1/ P (s)

ds ,

and

t ti .

(2.6)

The function ui (t) = p(t)(yi (t)/yi (t))∗ satisﬁes the generalized Riccati equation (B) on [ti , ∞). Integrating (B) on [t, ∞), t ti , and noting that ui (t) = (i (t)/P (t))∗ → 0 as t → ∞, we have ∞ ∞ |p(s) ui (s)|1+1/ P (t) ui (t) = P (t) ds + P (t) q(s) ds. (2.7) p(s)1/ P (s)+1 t t Let t → ∞ in (2.7). Since P (t) ui (t) → i as t → ∞, we conclude that ∞ q(s) ds = i − |i |1+1/ = c, i = 1 or 2. lim P (t) t→∞

t

(The “if” part). Suppose that (2.5) holds. Put ∞ (t) = P (t) q(s) ds − c

(2.8)

t

and consider the function t + (s) + v (s) 1/∗ i i yi (t) = exp ds , p(s)P (s) ti

i = 1, 2.

(2.9)

By Lemma 2.1 yi (t) is a solution of Eq. (A) if vi (t) is chosen in such a way that the function ui (t) = (i + (t) + vi (t))/P (t) satisﬁes the generalized Riccati equation (B) on [ti , ∞) for some ti > 0. The differential equation for vi (t) then reads vi −

1/

p(t) P (t) i = 1, 2.

vi +

p(t)

1/

P (t)

[|i + (t) + vi |1+1/ − |i |1+1/ ] = 0, (2.10)

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767

It is convenient to rewrite (2.10) as ∗

( + 1)(i + (t))1/ −

vi + p(t)1/ P (t)

1 ∗ × |i + (t) + vi |1+1/ − 1 + (i + (t))1/ vi − |i |1+1/ = 0

vi +

p(t)1/ P (t)

(2.11) and transform it into (ri (t)vi ) + where

p(t)1/ P (t)

t

ri (t) = exp

(2.12)

∗

( + 1)(i + (s))1/ − p(s)1/ P (s)

1

and

ri (t)Fi (t, vi ) = 0,

ds

(2.13)

1 Fi (t, v) = |i + (t) + v|1+1/ − 1 + ∗

× (i + (t))1/ v − |i |1+1/ ,

i = 1, 2.

(2.14)

Let us express Fi (t, v) as Fi (t, v) = Gi (t, v) + hi (t), where

(2.15)

1 ∗ Gi (t, v) = |i + (t) + v|1+1/ − 1 + (i + (t))1/ v − |i + (t)|1+1/ , (2.16) hi (t) = |i + (t)|1+1/ − |i |1+1/ ,

i = 1, 2.

(2.17)

We ﬁrst assume that c = 0 in (2.4), in which case both i = 0, i = 1, 2. Since (t) → 0 as t → ∞ by (2.4), there exists t0 > 0 such that | (t)|

|i | , 4

so that

3 5 |i ||i + (t)| |i |, 4 4

t t0 , i = 1, 2.

We note that there exist positive constants Ki (), Li () and Mi () such that |Gi (t, v)| Ki ()v 2 , jGi (t, v) Li ()|v|, jv

(2.18)

|hi (t)|Mi ()| (t)|

(2.20)

(2.19)

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J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

for t t0 and |v| |i |/4, i = 1, 2. In fact, (2.19) and (2.20) follow readily from the mean value theorem, while (2.18) is a consequence of the L’Hospital rule: lim

v→0

1 1 Gi (t, v) 1 j2 Gi (t, v) = lim 1 + |i + (t)|1/−1 . = 2 v→0 jv 2 2 v2

Let us ﬁrst examine Eq. (2.12) with i = 1. The following properties of r1 (t) are needed: 1/∗

r1 (t) ∈ n-RVP (( + 1)1

lim t→∞ r1 (t)

∞

r1 (s) p(s)

t

− ),

1/

P (s)

lim r1 (t) = 0,

(2.21)

t→∞

ds =

1/∗

− ( + 1)1

∞ r1 (s) lim h(s) ds = 0 t→∞ r1 (t) t p(s)1/ P (s) lim h(t) = 0.

> 0,

(2.22)

if h(t) ∈ C[0, ∞) and (2.23)

t→∞

The ﬁrst of (2.21) is a direct consequence of the deﬁnition (2.13) of r1 (t). To prove the 1/∗ second of (2.21) it sufﬁces to note that ( + 1)1 − < 0 since 1 is the smaller root of (2.4) and apply the general fact that any f ∈ RVP () satisﬁes limt→∞ f (t) = 0 if < 0. The truth of (2.22) and (2.23) is ensured by application of L’Hospital’s rule. Let 1 > 0 be a constant such that 1 < min{1, |1 |/4} and 2 1/∗

− ( + 1)1

[K1 () + L1 () + M1 ()] 1 1

(2.24)

and choose t1 > 0 so that |(t)| 21 ,

t t1

(2.25)

and r1 (t)

t

∞

r1 (s) p(s)

1/

P (s)

ds

2 1/∗

− ( + 1)1

,

t t1 .

(2.26)

Let C0 [t1 , ∞) denote the set of all continuous functions on [t1 , ∞) tending to zero as t → ∞; C0 [t1 , ∞) is a Banach space with the sup-norm v = sup{|v(t)| : t t1 }. Deﬁne the set V1 ⊂ C0 [t1 , ∞) and the integral operator F1 by V1 = {v ∈ C0 [t1 , ∞) : |v(t)| 1 , t t1 }, F1 v(t) = r1 (t)

t

∞

r1 (s) p(s)1/ P (s)

F1 (s, v(s)) ds,

(2.27) t t1 ,

(2.28)

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where F1 (t, v) is given by (2.14) (i = 1). It can be shown that F1 is a contraction mapping on V1 . In fact, if v ∈ V1 , then, using (2.15), (2.18), (2.20) and (2.24)–(2.26), we obtain ∞ r1 (s) [|G1 (s, v(s))| + |h1 (s)|] ds r1 (t) t p(s)1/ P (s) ∞ r1 (s) [K1 ()v(s)2 + M1 ()|(s)|] ds r1 (t) t p(s)1/ P (s) ∞ r1 (s) [K1 () + M1 ()] 21 ds r1 (t) t p(s)1/ P (s) 2 [K1 () + M1 ()] 21 1 , t t1 . 1/∗ − ( + 1)1

|F1 v(t)|

Since F1 (t, v(t)) → 0 as t → ∞, we have limt→∞ F1 v(t) = 0 by (2.23). It follows that F1 v ∈ V , so that F1 maps V1 into itself. Furthermore, if v, w ∈ V1 , then, using (2.19) and (2.22), we ﬁnd that |F1 v(t) − F1 w(t)|

r1 (t)

∞

r1 (s) p(s)1/ P (s)

t

2 1/∗

− ( + 1)1

|G1 (s, v(s)) − G1 (s, w(s))| ds

L1 () 1 v − w ,

t t1 ,

which implies that F1 v − F1 w

2L1 ()

v 1/∗ 1

− ( + 1)1

− w .

In view of (2.24) this shows that F1 is a contraction mapping on V1 . Therefore, there exists a unique ﬁxed element v1 ∈ V1 such that v1 = F1 v1 , which is equivalent to the integral equation v1 (t) = r1 (t)

t

∞

r1 (s) p(s)1/ P (s)

F1 (s, v1 (s)) ds,

t t1 .

(2.29)

Differentiation of (2.29) shows that v1 (t) satisﬁes the differential equation (2.12) with i = 1 on [t1 , ∞), and so the function y1 (t) deﬁned by (2.9) (i = 1) with this v1 (t) becomes a solution of the half-linear differential equation (A) on [t1 , ∞). Since limt→∞ v1 (t) = 0, we 1/∗ conclude that y1 (t) ∈ n-RVP (1 ). Our next step is to solve Eq. (2.12) with i = 2 to construct a larger solution y2 (t) ∈ 1/∗ n-RVP (2 ) by means of formula (2.9) (i = 2). It is easy to verify that 1/∗

r2 (t) ∈ n-RVP (( + 1)2

− ),

lim r2 (t) = ∞

t→∞

(2.30)

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J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

and that, for any ﬁxed t2 > 0, t r2 (s) ds = lim > 0, 1/∗ t→∞ r2 (t) t p(s)1/ P (s) ( + 1)2 − 2 t r2 (s) h(s) ds if h(t) ∈ C[0, ∞) and lim t→∞ r2 (t) t p(s)1/ P (s) 2 lim h(t) = 0.

(2.31)

(2.32)

t→∞

1/∗

The veriﬁcation of (2.30)–(2.32) is based on the inequality ( + 1)2 − > 0, which is a consequence of the fact that 2 is the larger root of (2.4). Let 2 > 0 be a constant small enough so that 2 min{1, |2 |/4} and 2 1/∗

( + 1)2

−

[K2 () + L2 () + M2 ()] 2 1.

(2.33)

Choose t2 > 0 so large that |(t)| 22 , and r2 (t)

t

t2

t t2 r2 (s)

p(s)

1/

P (s)

(2.34)

ds

2 1/∗

( + 1)2

−

,

t t2

(2.35)

and deﬁne V2 = {v ∈ C0 [t2 , ∞) : |v(t)| 2 , t t2 }, t r2 (s) F2 v(t) = − F2 (s, v(s)) ds, r2 (t) t2 p(s)1/ P (s)

(2.36) t t2 ,

(2.37)

where F2 (t, v) is given by (2.14) (i = 2). An easy caculation shows that F2 is a contraction mapping on V2 , so that there exists a unique function v2 ∈ V such that v2 = F2 v2 , that is, t r2 (s) v2 (t) = − F2 (s, v2 (s)) ds, t t2 . (2.38) r2 (t) t2 p(s)1/ P (s) It is clear that v2 (t) satisﬁes the differential equation (2.12) (i = 2), and so the function y2 (t) deﬁned by (2.9) (i = 2) provides a solution of (A) on [t2 , ∞). Since limt→∞ v2 (t) = 0, we 1/∗ see that y2 (t) ∈ n-RVP (2 ) as desired. It remains to examine the case c = 0 in (2.5). Then, 1 = 0 and 2 = 1 are the real roots of (2.4) and Theorem 2.1 asserts that (A) has two solutions y1 (t) ∈ n-RVP (0) = n-SVP and y2 (t) ∈ n-RVP (1). The existence of the larger solution y2 (t) is guaranteed by the aforementioned arguments, which, however, do not apply to the construction of the smaller solution y1 (t) because of the fact that 1 = 0. The solution y1 (t) will now be sought in the form t (s) + w (s) 1/∗ 1 0 y1 (t) = exp ds (2.39) p(s)P (s) t1

J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

for some constant t1 > 0, where 0 (t) = P (t) w1 (t) then becomes w1 |0 (t) + w1 |1+1/ + = 0. P (t) p(t)1/ P (t)+1

∞ t

771

q(s) ds. The differential equation for

(2.40)

We want to solve the following integrated version of (2.40): ∞ |0 (s) + w1 (s)|1+1/ w1 (t) = P (t) ds. p(s)1/ P (s)+1 t Let > 0 be a constant satisfying 1 1/ (2 ) max 2, 1 + < 1.

(2.41)

(2.42)

Choose t1 > 0 so that |0 (t)| ,

t t1

(2.43)

and consider the set W1 ⊂ C0 [t1 , ∞) and the mapping G1 : W1 → C0 [t1 , ∞) deﬁned by W1 = {w ∈ C0 [t1 , ∞) : |w(t)| , t t1 } and Gw(t) = P (t)

∞

|0 (s) + w(s)|1+1/ p(s)1/ P (s)+1

t

(2.44)

ds,

t t1 .

(2.45)

It is a matter of easy calculation to verify that G is a contraction mapping on W1 . The ﬁxed element w1 ∈ W1 of G satisﬁes (2.41), and hence is a solution of (2.40) on [t1 , ∞). Then, the function y1 (t) given by (2.39) with this w1 (t) is a solution of the half-linear equation (A) on [t1 , ∞). Since 0 (t) + w1 (t) → 0 as t → ∞, y1 (t) belongs to n-SVP by the representation theorem. This completes the proof of Theorem 2.1. Let us now consider the case where ∞ lim P (t) q(s) ds = E(), t→∞

(2.46)

t

where E() is given by (2.3). Eq. (A) with q(t) satisfying (2.46) can be regarded as a perturbation of the generalized Euler equation (p(t)|y |−1 y ) +

E() p(t)

1/

P (t)+1

|y|−1 y = 0,

(2.47)

which is nonoscillatory and has the solution y(t) = P (t)/(+1) ∈ n-RVP (/( + 1)). We will present a nonoscillation theorem for (A) showing that (A) has a solution y(t) ∈ n-RVP (/(+1)) provided the perturbation q(t)−E()/p(t)1/ P (t)+1 is small in some sense.

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J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

Theorem 2.2. Assume that (2.46) holds. Put ∞ (t) = P (t) q(s) ds − E().

(2.48)

t

Suppose that ∞

|(t)|

p(t)1/ P (t)

∞

p(t)

(2.49)

(t) 1/

dt < ∞,

P (t)

dt < ∞,

(t) =

∞

|(s)| p(s)1/ P (s)

t

ds.

(2.50)

Then (A) has a solution y(t) ∈ n-RVP (/( + 1)) such that y(t) = P (t)/(+1) f (t) with f (t) ∈ n-SVP satisfying limt→∞ f (t) = f (∞) ∈ (0, ∞). Proof. We look for a solution of (A) expressed in the form t A + (s) + v(s) 1/∗ ds y(t) = exp p(s)P (s) t0

(2.51)

for some t0 > 0 and v : [t0 , ∞) → R, where A = (/ + 1) . The following differential equation for v(t) is obtained via the generalized Riccati equation (B): v −

p(t)

1/

P (t)

v+

p(t)

1/

1+1/

P (t)

[|A + (t) + v|1+1/ − A

] = 0,

(2.52)

which can be rewritten as r(t)

(r(t)v ) + where

p(t)1/ P (t)

t

r(t) = exp

F (t, v) = 0,

∗

( + 1)(A + (s))1/ − p(s)1/ P (s)

1

and

(2.53)

ds

(2.54)

1 ∗ 1+1/ . F (t, v) = |A + (t) + v|1+1/ − 1 + (A + (t))1/ v − A

(2.55)

Let t0 > 0 be such that |(t)|

A 4

for t t0 .

(2.56)

Noting that ∗

1/

|( + 1)(A + (t))1/ − | = ( + 1)|(A + (t))1/ − A | m()|(t)|, t t0

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for some constant m() > 0, we see in view of (2.54) and (2.49) that limt→∞ r(t) = r(∞) ∈ (0, ∞), from which it follows that r(s) 2 r(t)

for s t t1 ,

(2.57)

provided t1 > t0 is sufﬁciently large. We express F (t, v) as F (t, v) = G(t, v) + h(t), where

(2.58)

1 G(t, v) = |A + (t) + v|1+1/ − 1 + ∗

× (A + (t))1/ v − |A + (t)|1+1/ , 1+1/

h(t) = |A + (t)|1+1/ − A

.

There exist positive constants K(), L() and M() such that jG 2 |G(t, v)|K()v , (t, v) L()|v|, |h(t)| M()|(t)| jv

(2.59) (2.60)

(2.61)

for t t1 and |v| A /4. Choose T > t1 so large that 4M() (t)

A 4

and

162 K()M()

(2.62)

∞ T

(s) p(s)1/ P (s)

ds 1.

(2.63)

Let C [T , ∞) be deﬁned by

|v(t)| C [T , ∞) = v ∈ C0 [T , ∞) : sup <∞ , t T (t)

(2.64)

which is a Banach space equipped with the norm v = supt T |v(t)|/ (t). Deﬁne V = {v ∈ C [T , ∞) : |v(t)|4M() (t), t T }, ∞ r(s) Fv(t) = F (s, v(s)) ds, t T . r(t) t p(s)1/ P (s)

(2.65) (2.66)

Using (2.63) and (2.64), we can show that F is a contraction mapping on V. In fact, that v ∈ V implies Fv ∈ V is a consequence of the following calculations: ∞ ∞ r(s) M()|(s)| |h(s)| ds 2 ds = 2M() (t), t T , r(t) t p(s)1/ P (s) p(s)1/ P (s) t

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J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

r(t)

t

∞

r(s) p(s)1/ P (s)

|G(s, v(s))| ds 2

∞ t

2

∞

K()v(s)2

ds p(s)1/ P (s) K()[4M() (s)]2

ds p(s)1/ P (s) ∞ (s) 323 K()M()2 (t) ds p(s)1/ P (s) ∞ t (s) 323 K()M()2 ds (t) p(s)1/ P (s) T 2M() (t), t T . t

Furthermore, if v, w ∈ V , then we have r(t) 2

∞

r(s)

|G(s, v(s)) − G(s, w(s))| ds p(s)1/ P (s) 4L()M() (s) |v(s) − w(s)| ds p(s)1/ P (s) t ∞ |v(s) − w(s)| (s)2 82 L()M() ds 1/ (s) p(s) P (s) t ∞ (s) ds v − w , t T , 82 L()M() (t) p(s)1/ P (s) T

|Fv(t) − Fw(t)|

t ∞

which implies that Fv − Fw 21 v − w . Thus, F is a contraction mapping, and so it has a ﬁxed element v ∈ V , which solves the integral equation v(t) =

r(t)

∞

r(s) p(s)1/ P (s)

t

F (s, v(s)) ds,

t T

(2.67)

and hence is a solution of the differential equation (2.53) on [T , ∞). This function v(t) gives rise to a solution y(t) of (A) via formula (2.51). Since (t) + v(t) → 0 as t → ∞, it follows that y(t) ∈ n-RVP (/( + 1)) has the form y(t) = P (t)/(+1) f (t), where f (t) admits the expression

t

f (t) = exp 1

∗

1/

(A + (s) + v(s))1/ − A p(s)1/ P (s)

ds

∈ n-SVP ,

t T .

(2.68)

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Since |(t) + v(t)|A /2, t T , by the mean value theorem, we have ∗

1/

|(A + (t) + v(t))1/ − A |N ()[|(t)| + (t)],

t T

for some constant N () > 0. Using this inequality and conditions (2.49) and (2.50), we conclude form (2.68) that limt→∞ f (t) = f (∞) ∈ (0, ∞). This completes the proof of Theorem 2.2. Remark 2.1. If in particular p(t) ≡ 1 on [a, ∞), then the function P (t) deﬁned by (1.1) can be taken to be P (t)=t, so that, as remarked before, a regularly varying function of index with respect to P is a regularly varying function of index in the sense of Karamata. This observation shows that Theorems 2.1 and 2.2 generalize, respectively, Theorems 2.1 and 2.2 in our previous paper [6], which is designed to generalize the pioneering nonoscillation theory for linear differential equations of the form (A0 ) developed by Howard and Mari´c in [3] (see also [9]). ∞ Remark 2.2. It would be of interest to notice that the limit limt→∞ P (t) t q(s) ds appearing in (2.5) has played a central role in formulating the Hille-type oscillation criteria for Eq. (A) with p(t) satisfying (1.4); see [7]. 3. Existence of generalized regularly varying solutions (the second case) ∞ We now turn to the case of Eq. (A) in which p(t) satisﬁes 0 (p(t))−1/ dt < ∞ and show that in this case the asymptotic behavior of nonoscillatory solutions of (A) is best understood in the framework {RV1/ () : ∈ R} of regularly varying functions with respect to 1/(t), where (t) is deﬁned by (1.5). Thus, this section is concerned exclusively with solutions of (A) belonging to RV1/ () for some , that is, those solutions y(t) which are expressed in the form t (s) y(t) = c(t) exp ds , t t0 1/ (s) t0 p(s) for some t0 > 0 and some functions c(t) and (t) such that limt→∞ c(t) = c ∈ (0, ∞) and limt→∞ (t) = . The main results of this section are given by the following theorem which corresponds to Theorem 2.1 in the preceding section. Theorem 3.1. Let c ∈ (−∞, E()) be ﬁxed arbitrarily and let 1 , 2 , 1 < 2 , denote the two real roots of the equation | |1+1/ + + c = 0.

(3.1) 1/∗

Eq. (A) possesses a pair of solutions yi (t) ∈ n-RV1/ ( i nonoscillatory, if and only if ∞ 1 lim (s)+1 q(s) ds = c. t→∞ (t) t

), i = 1, 2, thus being

(3.2)

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J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787 1/∗

Proof (The “only if” part). Let yi (t) be a solution of class n-RV1/ ( i which, by the representation theorem, is expressed as t i (s) yi (t) = ci exp ds , t ti , i = 1 or 2, 1/ (s) ti p(s)

), i = 1 or 2,

(3.3) 1/∗

where ci , ti are positive constants and i (t) is a function satisfying limt→∞ i (t) = i . ∗ Put ui = p(t)(yi (t)/yi (t))∗ and vi = (t)ui (t)1/ , i = 1 or 2. Rewriting the generalized Riccati equation (B) for ui (t) in terms of vi (t), we have ((t)vi∗ ) +

( + 1)vi∗ + |vi |+1 p(t)1/

+ (t)+1 q(t) = 0,

t ti .

Integration of (3.4) over [t, ∞) yields ∞ ( + 1)vi (s)∗ + |vi (s)|+1 1 ds vi (t)∗ = (t) t p(s)1/ ∞ 1 + (s)+1 q(s) ds, (t) t

(3.4)

(3.5)

where we have used the fact limt→∞ (t)vi (t)∗ = 0, which is an immediate consequence 1/∗ of (3.3). Letting t → ∞ in (3.5) and noting that (3.3) implies limt→∞ vi (t) = i , we conclude that ∞ 1

i = ( + 1) i + | i |1+1/ + lim (s)+1 q(s) ds t→∞ (t) t or ∞ 1 (s)+1 q(s) ds = −(| i |1+1/ + i ) = c. lim t→∞ (t) t (The “if” part). Suppose that (3.2) holds. Deﬁne the function by t + (s) + v (s) 1/∗ i i ds , i = 1, 2, yi (t) = exp p(s)(s) ti where (t) is given by ∞ 1 (s)+1 q(s) ds − c. (t) = (t) t

(3.6)

(3.7)

In order for yi (t) to be a solution of (A) it sufﬁces to determine a constant ti > 0 and a function vi : [ti , ∞) → R in such a way that ui (t) = ( i + (t) + vi (t))/(t) satisﬁes the generalized Riccati equation (B) for t ti . We then have the differential equation for vi (t): vi + | i + (t) + vi |1+1/ vi + p(t)1/ (t) p(t)1/ (t)

1 1+1/ (t) − | i | = 0, (3.8) + 1+

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which is rewritten as (ri (t)vi ) + where

ri (t)Fi (t, vi ) = 0,

p(t)1/ (t)

t

ri (t) = exp

(3.9)

∗

( + 1)( i + (s))1/ + p(s)1/ (s)

1

and 1+1/

ds

(3.10)

1 ∗ − 1+ ( i + (t))1/ v

Fi (t, v) = | i + (t) + v| 1 + 1+ i (t) − | i |1+1/ .

(3.11)

Suppose that c = 0 in (3.2). Then, the roots 1 , 2 of (3.1) are different from zero. Let t0 > 0 be such that |(t)|

| i | , 4

so that

5 3 | i || i + (t)| | i |, 4 4

t t0 , i = 1, 2.

(3.12)

We express Fi (t, v) as Fi (t, v) = Gi (t, v) + hi (t),

(3.13)

where Gi (t, v) = | i + (t) + v|

1+1/ ∗

1 − 1+

× ( i + (t))1/ v − | i + (t)|1+1/ , 1 1+1/ 1+1/ hi (t) = | i + (t)| (t). − | i | + 1+

(3.14) (3.15)

As easily veriﬁed, there exist positive constants Ki (), Li () and Mi () such that |Gi (t, v)| Ki ()v 2 , jGi jv (t, v) Li ()|v|,

(3.16)

|hi (t)|Mi ()|(t)|

(3.18)

(3.17)

for t t0 and |v| | i |/4, i = 1, 2. We ﬁrst solve Eq. (3.9) with i = 1. For this purpose we observe that 1/∗

r1 (t) ∈ n-RV1/ (( + 1) 1 lim

t→∞

r1 (t)

t

∞

r1 (s) p(s)

1/

(s)

+ ),

ds =

lim r1 (t) = 0,

t→∞

−

1/∗

( + 1) 1

+

> 0,

(3.19) (3.20)

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J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

∞ r1 (s)h(s) ds = 0 t→∞ r1 (t) t p(s)1/ (s) lim h(t) = 0.

if h(t) ∈ C[0, ∞) and

lim

(3.21)

t→∞

The ﬁrst of (3.19) follows from (3.10) combined with the fact that limt→∞ (t) = 0, and 1/∗ the second is implied by the inequality ( + 1) 1 + < 0 which holds because 1 is the smaller root of (3.1). Both (3.20) and (3.21) are veriﬁed by means of the L’Hospital rule. Let 1 > 0 be a constant such that 1 < min{1, | 1 |/4} and −2

1/∗

( + 1) 1

+

[K1 () + L1 () + M1 ()] 1 1

(3.22)

and take t1 t0 so that |(t)| 21 , and r1 (t)

t

∞

t t1 r1 (s)

p(s)

1/

(s)

(3.23)

ds

−2

1/∗

( + 1) 1

+

,

t t1 .

(3.24)

It can be shown with the help of (3.16)–(3.18) and (3.22)–(3.24) that the mapping F1 deﬁned by ∞ r1 (s) F1 (s, v(s)) ds, t t1 , (3.25) F1 v(t) = r1 (t) t p(s)1/ (s) is a contraction on the set V1 = {v ∈ C0 [t1 , ∞) : |v(t)| 1 , t t1 }. In fact, F1 maps V1 into itself, because v ∈ V1 implies ∞ r1 (s) [|G1 (s, v(s))| + |h1 (s)|] ds |F1 v(t)| r1 (t) t p(s)1/ (s) ∞ r1 (s) [K1 ()v(s)2 + M1 ()|(s)|] ds r1 (t) t p(s)1/ (s) ∞ r1 (s) [K1 () + M1 ()] 21 ds r1 (t) t p(s)1/ (s) −2 [K1 () + K2 ()] 21 1 , t t1 1/∗ ( + 1) 1 + and F1 is a contraction, because v, w ∈ V1 imply ∞ r1 (s) |F1 v(t) − F2 w(t)| |G1 (s, v(s)) − G1 (s, w(s))| ds r1 (t) t p(s)1/ (s) −2 L1 () 1 v − w , t t1 . 1/∗ ( + 1) 1 +

(3.26)

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779

Consequently, F1 has a ﬁxed element v1 ∈ V1 . Clearly v1 (t) is a solution if the integral equation ∞ r1 (s) F1 (s, v1 (s)) ds, t t1 , (3.27) v1 (t) = r1 (t) t p(s)1/ (s) which is an integrated version of (3.9). Therefore, by using v1 (t) in (3.6) (i = 1), we obtain a solution y1 (t) of (A) on [t1 , ∞). Since 1 + (t) + v1 (t) → 1 as t → ∞, y1 (t) is shown 1/∗ to belong to n-RV1/ ( 1 ) by the representation formula. This establishes the existence of a smaller solution y1 (t) of (A). To construct a larger solution y2 (t) of (A) we need to solve the differential equation (3.9) with i = 2. We begin by noting that r2 (t) given by (3.10) (i = 2) satisﬁes: 1/∗

r2 (t) ∈ n-RV1/ (( + 1) 2 lim

t→∞

r2 (t)

t

r2 (s) p(s)1/ (s)

t2

+ ),

ds =

lim r2 (t) = ∞,

t→∞

1/∗

( + 1) 2

t r2 (s)h(s) ds = 0 t→∞ r2 (t) t p(s)1/ (s) 2 lim h(t) = 0.

+

> 0,

(3.28) (3.29)

if h(t) ∈ C[0, ∞) and

lim

(3.30)

t→∞

Here t2 t0 is any ﬁxed constant. Let 2 > 0 be a constant such that 2 < min{1, | 2 |/4} and 2 1/∗

( + 1) 2

+

[K2 () + L2 () + M2 ()] 2 1

(3.31)

and choose t2 > t0 so large that |(t)| 22 , and r2 (t)

t

t2

t t2 r2 (s)

p(s)

1/

(s)

(3.32)

ds

2 1/∗

( + 1) 2

+

,

t t2 .

Then one can prove without difﬁculty that the mapping F2 deﬁned by t r2 (s) − F2 v(t) = F2 (s, v(s)) ds, t t2 r2 (t) t2 p(s)1/ (s)

(3.33)

(3.34)

is a contraction on the set V2 = {v ∈ C0 [t2 , ∞) : |v(t)| 2 , t t2 }. The ﬁxed element v2 ∈ V2 of F2 satisﬁes the integral equation t r2 (s) − v2 (t) = F2 (s, v2 (s)) ds, t t2 r2 (t) t2 p(s)1/ (s)

(3.35)

(3.36)

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J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

and a fortiori the differential equation (3.9) (i = 2) on [t2 , ∞). The desired larger solution 1/∗ y2 (t) is obtained by using this v2 (t) in the formula (3.6) (i =2). That y2 (t) ∈ n-RV1/ ( 2 ) follows from the fact that 2 + (t) + v2 (t) → 2 as t → ∞. We next suppose that c = 0 in (3.2), in which case the real roots of (3.1) are 1 = −1 and

2 = 0. Since 1 = 0, we need only to construct the larger solution y2 (t) ∈ n-RV1/ (0) = n-SV1/ corresponding to the root 2 = 0. The differential equation to be solved in this case becomes

1 1+1/ v2 + |(t) + v2 | v2 + (t) = 0, + 1+ p(t)1/ (t) p(t)1/ (t) (3.37) which can be written as

v2 1 1+1/ (t) = 0. |(t) + v2 | + + 1+ (t) p(t)1/ (t)+1

(3.38)

It sufﬁces to solve, instead of (3.38), the integral equation

t 1 1+1/ v2 (t) = −(t) |(s) + v (s) ds, (s)| + 1 + 2 1/ (s)+1 t2 p(s) t t2 (3.39) for some t2 > 0. Put (t) = sup |(s)|1/(+1) s t

(3.40)

and let t2 > t0 be such that 3 · 21+1/ (t2 ) 1.

(3.41)

Deﬁne V2 = {v ∈ C0 [t2 , ∞) : |v(t)| (t2 ) , t t2 }, (3.42)

t 1 F2 v(t) = −(t) (s) ds, |(s) + v2 (s)|1+1/ + 1 + 1/ +1 (s) t2 p(s) t t2 . (3.43) If v ∈ V2 , then F2 v ∈ V2 , since ∞ 21+1/ [|(s)|1+1/ |F2 v(t)| (t) 1/ (s)+1 t2 p(s) + |v2 (s)|1+1/ + |(s)|] ds ∞ 1+1/ (t) [(t2 )(+1)(1+1/) 2 1/ (s)+1 t2 p(s) + (t2 )+1 + (t2 )+1 ] ds 21+1/ (t2 )+1 [(t2 )1+1/ + 2]3 · 21+1/ (t2 )+1 (t2 )

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for t t2 . Thus, F2 maps V2 into itself. Furthermore, if v, w ∈ V2 , then |F2 v(t) − F2 w(t)| t (t) ||(s) + v(s)|1+1/ − |(s) + w(s)|1+1/ | ds 1/ (s)+1 t2 p(s) t 1 (t) 1 + ((t2 )+1 + (t2 ) )1/ |v(s) − w(s)| ds 1/ (s)+1 t2 p(s) 1 1/ 1+ 2 (t2 ) v − w 21+1/ (t2 ) v − w , t t2 , from which it follows that F2 v − F2 w 21+1/ (t2 ) v − w 13 v − w . This shows that F2 is a contraction mapping on V2 . Let v2 ∈ V2 be the ﬁxed element of F2 . Then, v2 (t) is a solution of (3.39), and so the function y2 (t) given by t

y2 (t) = exp

t2

(s) + v2 (s) p(s)(s)

1/∗

ds ,

t t2 ,

is the desired solution of (A) belonging to n-SV1/ . Thus the proof of Theorem 3.1 is complete. We are now in a position to deal with the critical case where lim

t→∞

1 (t)

∞

t

(s)+1 q(s) ds = E().

(3.44)

The prototype of Eq. (A) satisfying (3.44) is the generalized Euler equation of the second kind (p(t)|y |−1 y ) +

E() p(t)1/ (t)+1

|y|−1 y = 0,

(3.45)

which is nonoscillatory and has the solution y(t)=(t)/(+1) ∈ n-RV1/ (−/( + 1)). We will show that, under some additional conditions, Eq. (A) subject to (3.44) is nonoscillatory and has a solution y(t) ∈ n-RV1/ (−/( + 1)). Theorem 3.2. Assume that (3.44) holds. Put (t) =

1 (t)

t

∞

(s)+1 q(s) ds − E().

(3.46)

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J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

Suppose that ∞

|(t)|

p(t)1/ (t) ∞

(t) p(t)

1/

(t)

dt < ∞,

(3.47)

dt < ∞,

(t) =

∞

|(s)| p(s)1/ (s)

t

ds.

(3.48)

Then, Eq. (A) possesses a solution y(t) ∈ n-RV1/ (−/( + 1)) such that y(t) = (t)/(+1) f (t)

with f (t) ∈ n-SV1/ satisfying lim f (t) = f (∞) ∈ (0, ∞). t→∞

Proof. We seek a solution y(t) of (A) having the form t (s) − A + v(s) 1/∗ y(t) = exp ds , p(s)(s) t0

A =

. +1

(3.49)

The generalized Riccati equation (B) for u(t) = ((t) − A + v(t))/(t) yields the differential equation for v(t): v +

p(t)1/ (t)

v+

|(t) − A + v|1+1/ + (1 + 1 )(t) + E() − A p(t)1/ (t)

= 0. (3.50)

Let us transform (3.50) into (r(t)v) + where

r(t) p(t)1/ (t)

t

r(t) = exp

F (t, v) = 0,

∗

( + 1)((s) − A )1/ + p(s)1/ (s)

1

and

(3.51)

ds

1 ∗ ((t) − A )1/ v F (t, v) = |(t) − A + v|1+1/ − 1 + 1 1+1/ + 1+ . (t) − A

(3.52)

(3.53)

Let t0 > 0 be such that |(t)|

A , 4

t t0 .

(3.54)

Since ∗

1/

|( + 1)((t) − A )1/ + | = ( + 1)|A t t0

− (A − (t))1/ | m()|(t)|,

J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

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for some constant m() > 0, condition (3.46) implies that (t) ∈ n-SV1/ and limt→∞ (t)= (∞) ∈ (0, ∞). It follows that there exists t1 t0 such that r(s) 2 r(t)

for s t t1 .

(3.55)

We write F (t, v) = G(t, v) + h(t), where

(3.56)

1 G(t, v) = |(t) − A + v|1+1/ − 1 + ∗

× ((t) − A )(1/) v − |(t) − A |1+1/ , 1 1+1/ 1+1/ (t). − A + 1+ h(t) = |(t) − A |

(3.57) (3.58)

It is easy to see that there exist positive constants K(), L() and M() such that |G(t, v)|K()v 2 , jG L()|v|, (t, v) jv |h(t)| M()|(t)|

(3.59) (3.60) (3.61)

for t t1 and |v| A /4. Choose T > t1 so large that A , t T , 4 ∞ (t) dt 1 162 K()M() 1/ p(t) (t) T 4M() (t)

(3.62) (3.63)

and deﬁne V = {v ∈ C [T , ∞) : |v(t)|4M() (t), t T }, ∞ r(s) Fv(t) = F (s, v(s)) ds, t T . r(t) t p(s)1/ (s)

(3.64) (3.65)

If v ∈ V , then we have for t T ∞ ∞ r(s) M()|(s)| |h(s)| ds 2 ds = 2M() (t), 1/ r(t) t p(s) (s) p(s)1/ (s) t ∞ ∞ r(s) K()v(s)2 |G(s, v(s))| ds 2 ds 1/ r(t) t p(s) (s) p(s)1/ (s) t ∞ (s)2 3 2 32 K()M() ds p(s)1/ (s) t 323 K()M()2 (t)2 2M() (t),

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J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

which ensures that |Fv(t)|4M() (t), t T , that is, Fv ∈ V . If v, w ∈ V , then ∞ r(s) |G(s, v(s)) − G(s, w(s))| ds |Fv(t) − Fw(t)| r(t) t p(s)1/ (s) ∞ (s)|v(s) − w(s)| 82 L()M() ds p(s)1/ (s) t ∞ (s)2 |v(s) − w(s)| 2 = 8 L()M() ds (s) p(s)1/ (s) t ∞ (s) ds v − w , t T , 82 L()M() (t) p(s)1/ (s) T which implies that Fv − Fw 21 v − w , that is, F is a contraction mapping on V. Let v ∈ V be the ﬁxed element of F. It satisﬁes the integral equation ∞ r(s) v(t) = F (s, v(s)) ds (3.66) r(t) t p(s)1/ (s) and hence the differential equation (3.51) for t T , and so the function y(t) given by the formula (3.49) with this v(t) provides a solution of (A) on [T , ∞) belonging to nRV1/ (−/( + 1)). In view of (3.49) one can write y(t) = ((t)/(t0 ))/(+1) f (t), where t ((s) − A + v(s))1/∗ − (−A )1/∗ ds , t T . (3.67) f (t) = exp p(s)1/ (s) t0 Since there exists a constant N () > 0 such that ∗

∗

|((t) − A + v(t))1/ − (−A )1/ |N ()[|(t)| + (t)] for t T and |v|A /4, using (3.47) and (3.48), we conclude that f (t) ∈ n-SV1/ and limt→∞ f (t) = f (∞) ∈ (0, ∞). This completes the proof of Theorem 3.2. ∞ Remark 3.1. We note that the limit limt→∞ (1/(t)) t (s)+1 q(s) ds appearing in (3.2) has been employed in [8] to establish oscillation criteria of Hille type for Eq. (A) with p(t) satisfying (1.5). 4. Examples Examples illustrating our main results will be presented below. Consider the equation (t |y |−1 y ) + kt sin(t )|y|−1 y = 0,

t 1,

where > 0, , , > 0 and k = 0 are real constants.

(4.1)

J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

785

∞ (I) Suppose that . Then, the function p(t) = t satisﬁes 1 (p(t))−1/ dt = ∞ and the function P (t) given by (1.4) can be taken to be log t ( = ), (4.2) P (t) = ( − )/ t ( > ). − We observe by partial integration that, if > 1 + , then ∞ 1 1+− 1 + − ∞ − s sin(s ) ds = t cos(t ) + s cos(s ) ds t t 1 1 + − 1+−2 = t 1+− cos(t ) − t sin(t ) 2 (1 + − )(1 + − 2) ∞ −2 − s sin(s ) ds, (4.3) 2 t from which it follows that ∞ lim P (t) s sin(s ) ds = 0 (4.4) t→∞

t

if > 1 + − + . Theorem 2.1 (with c = 0) then implies that Eq. (4.1) has a pair of solutions y1 (t) ∈ n-SVlog t and y2 (t) ∈ n-RVlog t (1) if = , (4.5) y1 (t) ∈ n-SVt (−)/ and y2 (t) ∈ n-RVt (−)/ (1) if > . In view of the relation RVt () = RV(),

> 0, ∈ R,

(4.6)

which is veriﬁed without difﬁculty, we have in particular − SVt (−)/ = SV and RVt (−)/ (1) = RV if > . Using this fact in the second of (4.5), we see that (4.1) has two distinct regularly varying solutions in the sense of Karamata provided > and > 1 + − + . Let c ∈ (−∞, E()), where E() is deﬁned by (2.3), and deﬁne ⎧ ( = ), ⎪ ⎨ t (log t)+1 +1 q0 (t) = (4.7) 1 ⎪ ⎩ − ( > ). t −+1 Then, applying Theorem 2.1 again, we see that for any k = 0 the equation (t |y |−1 y ) + [cq 0 (t) + kt sin(t )]|y|−1 y = 0

(4.8)

has two solutions 1/∗

yi (t) ∈ n-RVlog t (i

),

i = 1, 2

if = , − 1/∗ 1/∗ i yi (t) ∈ n-RVt (−)/ (i ) = n-RV ,

i = 1, 2

if > ,

where i , i = 1, 2, are the real roots of the equation ||1+1/ − + c = 0.

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J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

Let c = E() in (4.8). Then the coefﬁcient q(t) = cq 0 (t) + kt sin(t ) satisﬁes (2.46) if (4.4) holds, in which case the function (t) appearing in (2.48) takes the form ∞ (t) = P (t) ks sin(s ) ds. (4.9) t

Using (4.3) it can be shown that this (t) satisﬁes (2.49) and (2.50) provided > 1+−+. The following statements then follow from Theorem 2.2. (i) If = and > 1+, then (4.8) with c=E() has a solution y(t) ∈ n-RVlog t (/(+1)) such that y(t) = (log t)/(+1) f (t) with f (t) ∈ n-SVlog t satisfying limt→∞ f (t) = f (∞) ∈ (0, ∞). (ii) If > and > 1 + − + , then (4.8) with c = E() has a solution y(t) ∈ n-RVt (−)/ (/( + 1)) = n-RV(( − )/( + 1)) such that y(t) = t (−)/(+1) f (t) with f (t) ∈ n-SVt (−)/ = n-SV satisfying limt→∞ f (t) = f (∞) ∈ (0, ∞). ∞ (II) Suppose that < . Then, p(t) = t satisﬁes 1 (p(t))−1/ dt < ∞ and the function (t) given by (1.5) reduces to (−)/ . (4.10) t (t) = − A straightforward calculation shows that ∞ 1 lim (s)+1 s sin(s ) ds = 0 t→∞ (t) t

(4.11)

if 1 (4.12) > 1 + + ( + 1)( − ). From Theorem 3.1 it follows that Eq. (4.1) has two solutions − , y2 (t) ∈ n-SVt (−)/ = n-SV y1 (t) ∈ n-RVt (−)/ (−1) = n-RV for any k = 0 if (4.12) holds. We now consider the equation ( − )+1 −−1 −1 t + kt sin(t ) |y|−1 y = 0, (t |y | y ) + c

(4.13)

where c > 0 is a constant. Note that condition (3.2) is satisﬁed for this equation. Therefore, it follows from Theorem 3.1 that if c < E() and (4.12) holds, then (4.13) possesses two solutions − 1/∗ 1/∗ , i = 1, 2,

i yi (t) ∈ n-RVt (−)/ ( i ) = n-RV where i , i = 1, 2, are the real roots of (3.1): | |1+1/ + + c = 0.

J. Jaroš et al. / Nonlinear Analysis 64 (2006) 762 – 787

Let c = E(). Then, the function (t) deﬁned by (3.46) reduces to ∞ k (s)+1 s sin(s ) ds (t) = (t) t

787

(4.14)

and, as easily checked, (t) satisﬁes (3.47) and (3.48) provided (4.12) is satisﬁed. Applying Theorem 3.2, we assert that under the condition (4.12) Eq. (4.13) with c=E() has a solution y(t) ∈ n-RVt (−)/ (−/( + 1)) = n-RV(( − )/( + 1)) such that y(t) = t (−)/(+1) f (t)

with f (t) ∈ n-SV satisfying lim f (t) = f (∞) ∈ (0, ∞). t→∞

References [1] N.H. Bingham, C.M. Goldie, J.L. Teugels, Regular Variation, Encyclopedia of Mathematics and its Applications, vol. 27, Cambridge University Press, Cambridge, 1987. [2] Á. Elbert, A half-linear second order differential equation, Colloquia Math. Soc. Janos Bolyai 30; Qualitative Theory of Differential Equations, Szeged, 1979, pp. 153–180. [3] H.C. Howard, V. Mari´c, Regularity and nonoscillation of solutions of second order linear differential equations, Bull. T. CXIV de Acad. Serbe Sci. et Arts, Classe Sci. Mat. Nat. Sci. Math. 22 (1997) 85–98. [4] J. Jaroš, T. Kusano, Remarks on the existence of regularly varying solutions for second order linear differential equations, Publ. Inst. Math. (Beograd) (N.S.) 72 (86) (2002) 113–118. [5] J. Jaroš, T. Kusano, Self-adjoint differential equations and generalized Karamata functions, Bull. T. CXXIX de Acad. Serbe Sci. et Arts, Classe Sci. Mat. Nat. Sci. Math. 29 (2004) 25–60. [6] J. Jaroš, T. Kusano, T. Tanigawa, Nonoscillation theory for second order half-linear differential equations in the framework of regular variation, Results Math. 43 (2003) 129–149. [7] T. Kusano,Y. Naito, Oscillation and nonoscillation theorems for second order quasilinear differential equations, Acta. Math. Hungar. 76 (1997) 81–99. [8] T. Kusano, Y. Naito, A. Ogata, Strong oscillation and nonoscillation of quasilinear differential equations of second order, Differential Equations and Dynamical Systems 2 (1994) 1–10. [9] V. Mari´c, Regular Variation and Differential Equations, Lecture Notes in Mathematics, vol. 1726, Springer, Berlin, Heidelberg, New York, 2000.

Nonoscillatory half-linear differential equations and generalized Karamata functions

Nonoscillatory half-linear differential equations and generalized Karamata functions

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