Fractal Oscillations of Chirp Functions and Applications to Second-Order Linear Differential Equations

Abstract

We derive some simple sufficient conditions on the amplitude 𝑎(𝑥), the phase 𝜑(𝑥), and the instantaneous frequency 𝜔(𝑥) such that the so-called chirp function 𝑦(𝑥)=𝑎(𝑥)𝑆(𝜑(𝑥)) is fractal oscillatory near a point 𝑥=𝑥₀, where 𝜑^′(𝑥)=𝜔(𝑥) and 𝑆=𝑆(𝑡) is a periodic function on ℝ. It means that 𝑦(𝑥) oscillates near 𝑥=𝑥₀, and its graph Γ(𝑦) is a fractal curve in ℝ² such that its box-counting dimension equals a prescribed real number 𝑠∈[1,2) and the 𝑠-dimensional upper and lower Minkowski contents of Γ(𝑦) are strictly positive and finite. It numerically determines the order of concentration of oscillations of 𝑦(𝑥) near 𝑥=𝑥₀. Next, we give some applications of the main results to the fractal oscillations of solutions of linear differential equations which are generated by the chirp functions taken as the fundamental system of all solutions.

1. Introduction

The brilliant heuristic approach of Tricot [1] to the fractal curves such as the graph of functions 𝑦(𝑥)=𝑥^𝛼sin𝑥^−𝛽 and 𝑦(𝑥)=𝑥^𝛼cos𝑥^−𝛽 gave the main motivation for studying the fractal properties near 𝑥=0 of graph of oscillatory solutions of various types of differential equations: linear Euler-type equation 𝑦^′′+𝜆𝑥^−𝜎𝑦=0 (see [2]), general second-order linear equation 𝑦^′′+𝑓(𝑥)𝑦=0 (see [3]) where 𝑓(𝑥) satisfies the Hartman-Wintner asymptotic condition near 𝑥=0, half-linear equation (|𝑦^′|^𝑝−2𝑦^′)^′+𝑓(𝑥)|𝑦|^𝑝−2𝑦=0 (see [4]), linear self-adjoint equation (𝑝(𝑥)𝑦^′)^′+𝑞(𝑥)𝑦=0 (see [5]), and 𝑝-Laplace differential equations in an annular domain (see [6]).

A function 𝑦(𝑥) is said to be a chirp function if it possesses the form 𝑦(𝑥)=𝑎(𝑥)𝑆(𝜑(𝑥)), where 𝑎(𝑥) and 𝜑(𝑥) denote, respectively, the amplitude and phase of 𝑦(𝑥), and 𝑆(𝑡) is a periodic function on ℝ. In all previously mentioned papers [2–5], authors are dealing with the fractal oscillations of second-order differential equations and are deriving some sufficient conditions on the coefficients of considered equations such that all their solutions 𝑦(𝑥) together with the first derivative 𝑦^′(𝑥) admit asymptotic behaviour near 𝑥=0. It is formally written in the form of a chirp function, that is, 𝑦(𝑥)=𝑎(𝑥)sin(𝜑(𝑥)) and 𝑦^′(𝑥)=𝑏(𝑥)cos(𝜑(𝑥)) near 𝑥=0. According to it, one can say that the asymptotic formula for solutions of considered equations satisfies the chirp-like behaviour near 𝑥=0 (on the asymptotic formula for solutions near 𝑥=∞, see [7, 8]). Then, in the dependence of a prescribed real number 𝑠∈[1,2), authors give some asymptotic conditions on 𝑎(𝑥), 𝑏(𝑥), and 𝜑(𝑥) such that all solutions 𝑦(𝑥) are fractal oscillatory near 𝑥=0 with the fractal dimension 𝑠.

In this paper, independently of the asymptotic theory of differential equations, we firstly study the fractal oscillations of a chirp function; see Theorems 8 and 11. Second, taking two linearly independent chirp functions 𝑦₁(𝑥) and 𝑦₂(𝑥), we generate some new classes of fractal oscillatory linear differential equations which are not considered in [2–5] and have the general solution in the form of 𝑦(𝑥)=𝑐₁𝑦₁(𝑥)+𝑐₂𝑦₂(𝑥); see Theorems 16 and 17 (on some detailed description of the solution space of the second-order linear differential equations and on their constructions, we refer the reader to [9, 10]). Finally, we suggest that the reader considers the fractal oscillations near an arbitrary real point 𝑥=𝑥₀ instead of 𝑥=0 and studies the fractal oscillations near 𝑥=𝑥₀ from the left side and from both sides; see Theorem 24. Many examples are considered to show the originality of obtained results.

The chirp functions are also appearing in the time-frequency analysis; see for instance, [11–15] as well as in several applications of the time-frequency analysis; see for instance, [16–20].

2. Statement of the Main Results

We study some local asymptotic behaviours of fractal types for the so-called chirp function as follows: 𝑦(𝑥)=𝑎(𝑥)𝑆𝜑(𝑥),𝑥∈0,𝑡₀, (1) where 𝑡₀>0, the amplitude 𝑎=𝑎(𝑥) is a nontrivial function on (0,𝑡₀], the phase 𝜑=𝜑(𝑥) is singular at 𝑥=0 and 𝑆=𝑆(𝑡) is a periodic function on ℝ such as cos(𝑡) and sin(𝑡).

The chirp function (1) is sometimes written in the following form in which the so-called instantaneous frequency 𝜔(𝑥) appears instead of phase 𝜑(𝑥): 𝑦(𝑥)=𝑎(𝑥)𝑆^𝑡0_𝑥𝜔(𝜉)𝑑𝜉,𝑥∈0,𝑡₀. (2)

Let for some 𝑡₀>0, the functions 𝑎∈𝐶((0,𝑡₀]), 𝜑∈𝐶²((0,𝑡₀]), and 𝑆∈𝐶¹(ℝ) satisfy the following basic structural conditions: 𝑎(𝑥)>0,𝑎(𝑥)isboundedon0,𝑡₀,lim_𝑥→+0𝜑(𝑥)=∞,𝜑(𝑥)>0,𝜑^′(𝑥)<0,𝑥∈0,𝑡₀,forsome𝜏>0,|𝑆(𝑡+𝜏)|=|𝑆(𝑡)|∀𝑡∈ℝ,forsome𝜏₀∈ℝ,𝑆𝜏₀(𝑡)𝜏=0,𝑆≠0∀𝑡∈₀,𝜏₀+𝜏. (3)(4)(5)

Definition 1. A function 𝑦∈𝐶((0,𝑡₀]) is oscillatory near 𝑥=0, if there is a decreasing sequence 𝑥_𝑛∈(0,𝑡₀] such that 𝑦(𝑥_𝑛)=0 for 𝑛∈ℕ, and 𝑥_𝑛↘0 as 𝑛→∞; see Figure 1.
It is easy to prove the next proposition.

Figure 1 

𝑦is continuous, bounded, and oscillatory near 𝑥=0.

Proposition 2. Let 𝑎∈𝐶((0,𝑡₀]) satisfy 𝑎(𝑥)>0 on (0,𝑡₀], let 𝜑∈𝐶((0,𝑡₀]) be strictly decreasing on (0,𝑡₀], and let 𝑆∈𝐶(ℝ) satisfy (5). Then, the following two conditions are equivalent:(i) the chirp function 𝑦(𝑥)=𝑎(𝑥)𝑆(𝜑(𝑥)) is bounded and oscillatory near 𝑥=0;(ii) the amplitude 𝑎(𝑥) is bounded and lim_𝑥→+0𝜑(𝑥)=∞.

On the qualitative and oscillatory behaviours of solutions of differential equations of several types, we refer the reader to [7, 8].

Example 3. The following three main types of chirp functions satisfy the conditions (3), (4), and (5): (i)the so-called (𝛼,𝛽)-chirps 𝑦₁(𝑥)=𝑥^𝛼cos𝑥^−𝛽,𝑦₂(𝑥)=𝑥^𝛼sin𝑥^−𝛽, (6) where 𝛼≥0 and 𝛽>0; the first studies on the (𝛼,𝛽)-chirps appeared in [1, 13, 14] from different point of views; (ii)the so-called logarithmic chirps 𝑦₁(𝑥)=𝑥^𝛾cos𝜌log𝑥,𝑦₂(𝑥)=𝑥^𝛾sin𝜌log𝑥, (7) where 𝛾>0 and 𝜌∈ℝ; this type of chirps appears in definition of the Lamperti transform (see [11]) and in the fundamental system of solutions of the famous Euler equation 𝑦^′′+𝜆𝑥⁻²𝑦=0 for 𝛾=1/2 and 𝜌=𝜆−1/4 (see [21]); (iii)the chirp function of exponential type 𝑦₁(𝑥)=𝑥𝑒^−1/(2𝑥)cos𝑒^1/𝑥,𝑦₂(𝑥)=𝑥𝑒^−1/(2𝑥)sin𝑒^1/𝑥. (8)

Example 4. Let 𝑦₁(𝑥)=𝑎(𝑥)cos(𝜑(𝑥)) and 𝑦₂(𝑥)=𝑎(𝑥)sin(𝜑(𝑥)), 𝑥∈(0,𝑡₀], where the amplitude 𝑎 and phase 𝜑 satisfy (3) and (4), respectively. Then, 𝑦(𝑥)=𝑐₁𝑦₁(𝑥)+𝑐₂𝑦₂(𝑥), 𝑐²₁+𝑐²₂>0, is also a chirp function which is bounded and oscillatory near 𝑥=0. Indeed, if 𝑐₁=0 or 𝑐₂=0, then 𝑦(𝑥)=𝑐₂𝑦₂(𝑥) or 𝑦(𝑥)=𝑐₁𝑦₁(𝑥); since 𝑆(𝑡)=cos𝑡 and 𝑆(𝑡)=sin𝑡 satisfy (5), both cases 𝑦(𝑥) are chirp functions; otherwise, we have 𝑦(𝑥)=𝐴𝑎(𝑥)sin(𝜑(𝑥)+𝐵), where 𝐴=(𝑐²₁+𝑐²₂)^1/2 and 𝐵=arctan(𝑐₁/𝑐₂); obviously, the amplitude 𝐴𝑎(𝑥) and the phase 𝜑(𝑥)+𝐵 satisfy the required conditions (3) and (4), respectively.

Next, let Γ𝑦=𝑡₁,𝑡₂∈ℝ²∶𝑡₁∈0,𝑡₀,𝑡₂=𝑦𝑡₁ (9) denotes the graph of a function 𝑦∈𝐶((0,𝑡₀]). In this paper, we show that the graph Γ(𝑦) of chirp function (1) is a fractal curve in ℝ² in respect to the Minkowski-Bouligand dimension dim_𝑀Γ(𝑦)=𝑠 (box-counting dimension) and the 𝑠-dimensional upper and lower Minkowski contents 𝑀^∗𝑠(Γ(𝑦)) and 𝑀^𝑠_∗(Γ(𝑦)) defined by dim_𝑀Γ𝑦=lim_𝜀→+0Γ2−log_𝜀𝑦,𝑀log𝜀_∗𝑠Γ𝑦=limsup_𝜀→+0(2𝜀)^𝑠−2Γ_𝜀𝑦,𝑀^𝑠_∗Γ𝑦=liminf_𝜀→+0(2𝜀)^𝑠−2Γ_𝜀𝑦. (10) Here, Γ_𝜀(𝑦) denotes the 𝜀-neighbourhood of graph Γ(𝑦) defined by Γ_𝜀𝑦=𝑡₁,𝑡₂∈ℝ²∶𝑑𝑡₁,𝑡₂,Γ𝑦≤𝜀,𝜀>0, (11) and 𝑑((𝑡₁,𝑡₂),Γ(𝑦)) denotes the distance from (𝑡₁,𝑡₂) to Γ(𝑦), and |Γ_𝜀(𝑦)| denotes the Lebesgue measure of Γ_𝜀(𝑦). On the box-counting dimension and the 𝑠-dimensional Minkowski content, we refer the reader to [22–27].

The main fractal properties considered in the paper are given in the next definitions.

Definition 5. For a given real number 𝑠∈[1,2) and a function 𝑦∈𝐶((0,𝑡₀]), which is bounded on (0,𝑡₀] and oscillatory near 𝑥=0, it is said that 𝑦(𝑥) is fractal oscillatory near 𝑥=0 with the fractal dimension 𝑠, if dim_𝑀Γ𝑦=𝑠,0<𝑀^𝑠_∗Γ𝑦≤𝑀^∗𝑠Γ𝑦<∞. (12) On the contrary, if there is no any 𝑠∈[1,2) such that Γ(𝑦) satisfies (12), then 𝑦(𝑥) is not fractal oscillatory near 𝑥=0.

Fractal oscillations can be understood also as a refinement of rectifiable and nonrectifiable oscillations. They are recently studied in [3, 4, 21, 28–30].

Example 6. The chirp functions 𝑦(𝑥)1=𝑥cos𝑥(𝑥)1,𝑦=𝑥sin𝑥 (13) are not fractal oscillatory near 𝑥=0. In fact, 𝑀^∗1(Γ(𝑦))=∞ and dim_𝑀Γ(𝑦)=1 (see [2]). It also implies that 𝑀^∗𝑠(Γ(𝑦))=0 for all 𝑠∈(1,2) (see [1, 23]), and thus the statement (12) is not satisfied for any 𝑠∈[1,2). Hence, 𝑦₁ and 𝑦₂ are not fractal oscillatory near 𝑥=0.

Example 7. Let 𝜌>0. It is clear that the chirp functions 𝑦(𝑥)=√𝑥cos𝜌log𝑥,𝑦(𝑥)=√𝑥sin𝜌log𝑥 (14) are oscillatory near 𝑥=0. Moreover, the length of Γ(𝑦) is finite (see [21]), and therefore we observe that 0<𝑀¹_∗(Γ(𝑦))≤𝑀^∗1(Γ(𝑦))<∞ and dim_𝑀Γ(𝑦)=1 (see [22]). Thus, such chirp functions are fractal oscillatory near 𝑥=0 with the fractal dimension 1.

In order to show that the chirp function (1) is fractal oscillatory near 𝑥=0, we need to impose on amplitude 𝑎(𝑥) the following additional structural condition: 𝑎∈𝐶¹0,𝑡₀,𝑎^′(𝑥)≥0for𝑥∈0,𝑡₀. (15)

Now we are able to state the first main result of the paper.

Theorem 8. Let the functions 𝑎(𝑥), 𝜑(𝑥), and 𝑆(𝑡) satisfy the structural conditions (3), (4), (5), (15), and 𝜑^′′(𝑥)>0on0,𝑡₀,limsup_𝑥→01−𝜑^′(𝑥)^′<∞. (16) Let the amplitude 𝑎(𝑥) and the phase 𝜑(𝑥) satisfy the following asymptotic conditions near 𝑥=0: liminf_𝑥→0−𝜑^′(𝑥)^2−𝑠𝑥₀𝑎(𝜉)𝑑𝜉>0,limsup_𝑥→0[𝑥𝑎(𝑥)]^{(𝑠−1)/(2−𝑠)𝑡0}_𝑥𝑎(𝜉)−𝜑^′(𝜉)𝑑𝜉<∞, (17)(18) where 𝑠∈(1,2) is a given real number. Then, the chirp function (1) is fractal oscillatory near 𝑥=0 with the fractal dimension 𝑠.

Remark 9. Let 𝑠∈(1,2). Assume that (15) and (17) hold. Then, limsup_𝑥→0𝑥𝑎(𝑥)−𝜑^′(𝑥)^2−𝑠<∞ (19) implies limsup_𝑥→01−𝜑^′(𝑥)^{𝑠−1𝑡0}_𝑥𝑎(𝜉)−𝜑^′(𝜉)𝑑𝜉<∞ (20) and (18). Therefore, condition (18) is better than (19) in Theorem 8.

The proof of Remark 9 is presented in the Appendix.

Theorem 8 can be rewritten in the term of 𝜔(𝑥) instead of 𝜑(𝑥) as follows.

Corollary 10. Let the functions 𝑎(𝑥), 𝜔(𝑥), and 𝑆(𝑡) satisfy the structural conditions (3), (5), (15), and 𝜔∈𝐶¹0,𝑡₀,𝜔(𝑥)>0,𝜔^′(𝑥)<0𝑜𝑛0,𝑡₀,𝜔∉𝐿¹0,𝑡₀,limsup_𝑥→01𝜔(𝑥)^′<∞. (21) Let the amplitude 𝑎(𝑥) and the instantaneous frequency 𝜔(𝑥) satisfy the following asymptotic conditions near 𝑥=0: liminf_𝑥→0[𝜔(𝑥)]^2−𝑠𝑥₀𝑎(𝜉)𝑑𝜉>0,limsup_𝑥→0[𝑥𝑎(𝑥)]^{(𝑠−1)/(2−𝑠)𝑡0}_𝑥𝑎(𝜉)𝜔(𝜉)𝑑𝜉<∞, (22) where 𝑠∈(1,2) is a given real number. Then, the chirp function (2) is fractal oscillatory near 𝑥=0 with the fractal dimension 𝑠.

Now, we state analogous result to Theorem 8 in the case of 𝑠=1, which will be proved in Section 3.

Theorem 11. Assume that the function 𝑆(𝑡) satisfies (5), 𝑎∈𝐶([0,𝑡₀])∩𝐶¹((0,𝑡₀]), 𝜑∈𝐶²((0,𝑡₀]) and lim_𝑥→+0𝜑(𝑥)=∞. If 𝑎^′∈𝐿¹((0,𝑡₀]), and 𝜑^′𝑎∈𝐿¹((0,𝑡₀]), then the chirp function (1) is fractal oscillatory near 𝑥=0 with the fractal dimension 1.

In respect to some existing results on the fractal dimension of graph of chirp functions, previous theorems are the most simple and general. It is because in [31, 32] authors require some extra conditions on the chirp function 𝑦(𝑥) which are not easy to be satisfied in the application, for instance, the rapid convex-concave properties of 𝑦(𝑥) as in [31] and a condition on the curvature of 𝑦(𝑥) as in [32].

According to previous theorems, we can show the fractal oscillations of the so-called (𝛼,𝛽)-chirp as well as logarithmic chirp functions.

Example 12. We consider the (𝛼,𝛽)-chirp 𝑦(𝑥)=𝑥^𝛼𝑆(𝑥^−𝛽), where 𝑆(𝑡)=cos𝑡 or 𝑆(𝑡)=sin𝑡 and 𝛽>𝛼≥0. It is fractal oscillatory near 𝑥=0 with the fractal dimension 𝑠=2−(1+𝛼)/(1+𝛽). In fact, it is easy to see that 𝑎(𝑥)=𝑥^𝛼, 𝑆(𝑡), and 𝜑(𝑥)=𝑥^−𝛽 satisfy (3), (4), (5), (15), and (16). When 𝑠=2−(1+𝛼)/(1+𝛽), we see that −𝜑^′(𝑥)^2−𝑠𝑥₀𝑎(𝜉)𝛽𝑑𝜉=^{(1+𝛼)/(1+𝛽)}[(𝑥1+𝛼>0,𝑥𝑎)]^{(𝑠−1)/(2−𝑠)𝑡0}_𝑥𝑎(𝜉)−𝜑^′(𝜉=𝛽)𝑑𝜉𝛽−𝛼1−𝑥^𝛽−𝛼𝑡₀^𝛼−𝛽≤𝛽.𝛽−𝛼 (23) From Theorem 8, it follows that 𝑦(𝑥)=𝑥^𝛼𝑆(𝑥^−𝛽), 𝛽>𝛼≥0, is fractal oscillatory near 𝑥=0 with the fractal dimension 𝑠=2−(1+𝛼)/(1+𝛽).

Example 13. We consider the (𝛼,𝛽)-chirp 𝑦(𝑥)=𝑥^𝛼𝑆(𝑥^−𝛽) again, where 𝑆(𝑡)=cos𝑡 or 𝑆(𝑡)=sin𝑡. Now, we assume that 0<𝛽<𝛼. Applying Theorem 8, we easily see that it is fractal oscillatory near 𝑥=0 with the fractal dimension 1.

Example 14. We consider the logarithmic chirp functions 𝑦₁(𝑥)=𝑥^𝛾cos𝜌log𝑥,𝑦₂(𝑥)=𝑥^𝛾sin𝜌log𝑥, (24) where 𝛾>0 and 𝜌>0. Put 𝑆(𝑡)=cos(−𝑡) or 𝑆(𝑡)=sin(−𝑡), 𝑎(𝑥)=𝑥^𝛾, and 𝜑(𝑥)=−𝜌log𝑥. Then, we easily see that 𝑆(𝑡) satisfies (5) and that 𝑎^′(𝑥)=𝛾𝑥^𝛾−1∈𝐿¹((0,𝑡₀]) and 𝑎(𝑥)𝜑^′(𝑥)=−𝜌𝑥^𝛾−1∈𝐿¹((0,𝑡₀]). Theorem 11 implies that 𝑦₁(𝑥) and 𝑦₂(𝑥) are fractal oscillatory near 𝑥=0 with the fractal dimension 1.

Question 1. Is it possible to apply Theorem 8 on the exponential chirp given in Example 3(iii)?

Example 15. Let 𝑦₁(𝑥)=𝑎(𝑥)cos(𝜑(𝑥)) and 𝑦₂(𝑥)=𝑎(𝑥)sin(𝜑(𝑥)), 𝑥∈(0,𝑡₀], where the amplitude 𝑎(𝑥) and the phase 𝜑(𝑥) satisfy all assumptions of Theorem 8 for an arbitrary given 𝑠∈(1,2). Then, the chirp function 𝑦(𝑥)=𝑐₁𝑦₁(𝑥)+𝑐₂𝑦₂(𝑥), 𝑐²₁+𝑐²₂>0, is also fractal oscillatory near 𝑥=0 with the fractal dimension 𝑠. Indeed, similarly as in Example 4, 𝑦(𝑥) can be rewritten in the form 𝑦(𝑥)=𝑑₁sin(𝜑(𝑥)+𝑑₂), where 𝑑₁≠0 and 𝑑₂∈ℝ. It is clear that the function 𝑆(𝑡)=𝑑₁sin(𝑡+𝑑₂) satisfies the required condition (5) and hence Theorem 8 proves that 𝑦(𝑥) is fractal oscillatory near 𝑥=0 with the fractal dimension 𝑠.

Next, we pay attention to the fractal oscillations of solutions of linear differential equations generated by the system of functions as follows: 𝑦(𝑥)=𝑐₁𝑦₁(𝑥)+𝑐₂𝑦₂(𝑥),𝑐²₁+𝑐²₂𝑦>0,where₁(𝑥)=𝑎(𝑥)cos𝜑(𝑥),𝑦₂(𝑥)=𝑎(𝑥)sin𝜑(𝑥),𝑥∈0,𝑡₀. (25) It is not difficult to check that (25) is the fundamental system of all solutions of the following linear differential equation: 𝑦^′′+−2𝑅_𝑎−𝑅_𝜑′𝑦^′+−𝑅^′_𝑎+𝑅²_𝑎+𝑅_𝑎𝑅_𝜑′+𝜑^′2𝑦=0,𝑥∈0,𝑡₀, (26) where 𝑅_𝑓=𝑅_𝑓(𝑥)=𝑓^′(𝑥)/𝑓(𝑥) for some 𝐶¹-function 𝑓=𝑓(𝑥).

Theorem 16. Let the functions 𝑎, 𝜑∈𝐶²((0,𝑡₀]) satisfy structural conditions (3), (4), and (15) as well as the conditions (16), (17), and (18) in respect to a given real number 𝑠∈(1,2). Then, every nontrivial solution 𝑦∈𝐶²((0,𝑡₀]) of (26) is fractal oscillatory near 𝑥=0 with the fractal dimension 𝑠.

With the help of Theorem 11, we can state analogous result to Theorem 16 in the case of 𝑠=1.

Theorem 17. Assume that 𝑎∈𝐶([0,𝑡₀])∩𝐶²((0,𝑡₀]), 𝑎(𝑥)≠0 for 𝑥∈(0,𝑡₀], 𝜑∈𝐶²((0,𝑡₀]), lim_𝑥→+0𝜑(𝑥)=∞, 𝑎^′∈𝐿¹((0,𝑡₀]), and 𝜑^′𝑎∈𝐿¹((0,𝑡₀]). Then, every nontrivial solution 𝑦∈𝐶²((0,𝑡₀]) of (26) is fractal oscillatory near 𝑥=0 with the fractal dimension 1.

The previous two theorems will be proved in Section 4. Assumptions on the coefficients of (26) in general are different to those considered in [2–5].

When 𝑎^′/𝑎=−(1/2)𝜑^′′/𝜑^′, then (26) becomes the undamped equation 𝑦^′′+12𝑆_der𝜑^′+𝜑^′2𝑦=0,𝑥∈0,𝑡₀, (27) where 𝑆_der(𝑓) denotes the Schwarzian derivative of 𝑓 defined by 𝑆_der𝑓=𝑓^′′𝑓−32𝑓^′𝑓². (28) Hence, from Theorem 16, we obtain the following consequence.

Corollary 18. Let the functions 𝑎∈𝐶²((0,𝑡₀]) and 𝜑∈𝐶³((0,𝑡₀]) satisfy structural conditions (3), (4), and (15) as well as the conditions (16), (17), and (18) in respect to a given real number 𝑠∈(1,2). Let 𝑎^′/𝑎=−(1/2)𝜑^′′/𝜑^′. Then, every nontrivial solution 𝑦∈𝐶²((0,𝑡₀]) of (27) is fractal oscillatory near 𝑥=0 with the fractal dimension 𝑠.

As a consequence of Theorem 16, and Corollary 18 we derive the following examples for linear differential equations of second order having all the solutions to be fractal oscillatory near 𝑥=0.

Example 19. The so-called damped chirp equation 𝑦^′′+𝛽−2𝛼+1𝑥𝑦^′+𝛽²𝑥^2𝛽+2−𝛼𝛽−𝛼𝑥²𝑦=0,𝑥∈0,𝑡₀, (29) is fractal oscillatory near 𝑥=0 with the fractal dimension 2−(1+𝛼)/(1+𝛽), where 𝛽>𝛼≥0. When 𝑎(𝑥)=𝑥^𝛼 and 𝜑(𝑥)=𝑥^−𝛽, (26) becomes (29). It is easy to see that (3), (4), (15), and (16) are satisfied. In the same as in Example 12, we see that (17) and (18) hold for 𝑠=2−(1+𝛼)/(1+𝛽). Hence, Theorem 16 proves that every nontrivial solution of (29) is fractal oscillatory near 𝑥=0 with the fractal dimension 2−(1+𝛼)/(1+𝛽).
Now we assume that 0<𝛽<𝛼. Then, Theorem 17 implies that (29) is fractal oscillatory near 𝑥=0 with the fractal dimension 1.

Example 20. The following equation 𝑦^′′+1−2𝛾𝑥𝑦^′+𝛾²+𝜌²𝑥²𝑦=0,𝑥∈0,𝑡₀, (30) is fractal oscillatory near 𝑥=0 with the fractal dimension 1, where 𝛾>0 and 𝜌>0. In the case where 𝑎(𝑥)=𝑥^𝛾 and 𝜑(𝑥)=−𝜌log𝑥, (26) becomes (30). We see that 𝑎∈𝐶([0,𝑡₀])∩𝐶²((0,𝑡₀]), 𝜑∈𝐶²((0,𝑡₀]), lim_𝑥→+0𝜑(𝑥)=∞, 𝑎^′∈𝐿¹((0,𝑡₀]), and 𝜑^′𝑎∈𝐿¹((0,𝑡₀]). Therefore Theorem 17 implies that every nontrivial solution of (30) is fractal oscillatory near 𝑥=0 with the fractal dimension 1.

Question 2. What can we say about the application of Theorem 16 on the case of 𝜑(𝑥) given in Example 3(iii)?

At the end of this section, we suggest that the reader studies some invariant properties of fractal oscillations of the chirp function (1) in respect to the translation and reflexion. Analogously to Definitions 1 and 5, one can define the fractal oscillations near an arbitrary real point 𝑥=𝑥₀ as follows.

Definition 21. Let 𝑥₀∈ℝ and 𝛿>0. A function 𝑦∈𝐶((𝑥₀,𝑥₀+𝛿]) is oscillatory near 𝑥=𝑥₀, if there is a decreasing sequence 𝑏_𝑛∈(𝑥₀,𝑥₀+𝛿] such that 𝑦(𝑏_𝑛)=0, for 𝑛∈ℕ, and 𝑏_𝑛↘𝑥₀ as 𝑛→∞. Moreover, if the graph Γ(𝑦) satisfies the condition (12) for some 𝑠∈[1,2), then 𝑦(𝑥) is said to be fractal oscillatory near 𝑥=𝑥₀ with the fractal dimension 𝑠.

Definition 22. Let 𝑥₀∈ℝ and 𝛿>0. It is said that a function 𝑦∈𝐶([𝑥₀−𝛿,𝑥₀)) is fractal oscillatory near 𝑥=0 from the left side with the fractal dimension 𝑠∈[1,2), if there is an increasing sequence 𝑎_𝑛∈[𝑥₀−𝛿,𝑥₀) such that 𝑦(𝑎_𝑛)=0 for 𝑛∈ℕ, 𝑎_𝑛↗𝑥₀ as 𝑛→∞, and the graph Γ(𝑦) satisfies the condition (12), see Figure 2.

Figure 2 

𝑦 is oscillatory near 𝑥=0 from the left side.

Definition 23. Let 𝑥₀∈ℝ, and let 𝛿>0. It is said that a function 𝑦∈𝐶([𝑥₀−𝛿,𝑥₀)∪(𝑥₀,𝑥₀+𝛿]) is two-sided fractal oscillatory near 𝑥=0 with the fractal dimension 𝑠∈[1,2), if there is an increasing sequence 𝑎_𝑛∈[𝑥₀−𝛿,𝑥₀) and a decreasing sequence 𝑏_𝑛∈(𝑥₀,𝑥₀+𝛿] such that 𝑦(𝑎_𝑛)=𝑦(𝑏_𝑛)=0 for 𝑛∈ℕ, 𝑎_𝑛↗𝑥₀ and 𝑏_𝑛↘𝑥₀ as 𝑛→∞, and the graph Γ(𝑦) satisfies the condition (12); see Figure 3.

Figure 3 

𝑦 is two-sided oscillatory near 𝑥=0.

With the help of Theorem 8, we state the following result.

Theorem 24. Let 𝑥₀∈ℝ, and let 𝛿>0. Let the functions 𝑎,𝜔∈𝐶¹((0,𝛿]) satisfy structural conditions (3), (4), (5), (15), and (16) as well as conditions (17) and (18) in respect to a given real number 𝑠∈(1,2). Then, one has(i)the chirp function 𝑦(𝑥)=𝑎(𝑥−𝑥₀)𝑆(𝜑(𝑥−𝑥₀)), 𝑥∈(𝑥₀,𝑥₀+𝛿] is fractal oscillatory near 𝑥=𝑥₀ with the fractal dimension 𝑠; (ii)the chirp function 𝑦(𝑥)=𝑎(𝑥₀−𝑥)𝑆(𝜑(𝑥₀−𝑥)), 𝑥∈[𝑥₀−𝛿,𝑥₀) is fractal oscillatory near 𝑥=𝑥₀ from the left side with the fractal dimension 𝑠; (iii)the chirp function 𝑦(𝑥)=𝑎(|𝑥−𝑥₀|)𝑆(𝜑(|𝑥−𝑥₀|)), 𝑥∈[𝑥₀−𝛿,𝑥₀)∪(𝑥₀,𝑥₀+𝛿] is two-sided fractal oscillatory near 𝑥=𝑥₀ with the fractal dimension 𝑠.

3. Proof for the Fractal Oscillations of Chirp Functions

In this section, we give the proofs of the main results dealing with the fractal oscillations of chirp functions.

By Definition 5, it follows that if for a prescribed real number 𝑠∈(1,2) there are two positive constants 𝑐₁ and 𝑐₂ such that 𝑐₁𝜀^2−𝑠≤Γ_𝜀𝑦≤𝑐₂𝜀^2−𝑠,𝜀∈0,𝜀₀, (31) for some 𝜀₀>0, then a function 𝑦∈𝐶((0,𝑡₀]) is fractal oscillatory near 𝑥=0 with the fractal dimension 𝑠. The following lemma plays the essential role in the proof of (31).

Lemma 25. Let 𝑦∈𝐶((0,𝑡₀]) be a bounded function on (0,𝑡₀], and let 𝑎_𝑛∈(0,𝑡₀] be a decreasing sequence of consecutive zeros of 𝑦(𝑥) such that 𝑎_𝑛→0. Let 𝜀₀>0, and let 𝑘=𝑘(𝜀) be an index function satisfying 𝑎_𝑛−𝑎_𝑛+1≤𝜀∀𝑛≥𝑘(𝜀),𝜀∈0,𝜀₀. (32) Then Γ_𝜀𝑦≥_{(𝜀)𝑛≥𝑘}max_{𝑎𝑥∈𝑛+1,𝑎𝑛}𝑦(𝑥)𝑎_𝑛−𝑎_𝑛+1∀𝜀∈0,𝜀₀. (33)

Proof. It is exactly the same as [3, pp. 2350].

Let us remark that for some 𝜀₀>0, we say that a function 𝑘=𝑘(𝜀) is an index function on (0,𝜀₀) if  𝑘∶(0,𝜀₀)→ℕ  and lim_𝜀→0𝑘(𝜀)=∞.

Lemma 26. Let 𝑦∈𝐶[𝑎,𝑏]. Then, Γ_𝜀𝑦≤4𝜋𝜀length𝑦+4𝜋𝜀²,forevery𝜀>0. (34)

Proof. Let 𝜀>0. Set 𝑥₁=𝑎, and set 𝑥_𝑖+1=max𝑥∈𝑥_𝑖,𝑏∶𝑑𝑡,𝑦(𝑡),𝑥_𝑖,𝑦𝑥_𝑖≤𝜀for𝑡∈𝑥_𝑖,𝑥, (35) for 𝑖=1,2,…. Then, there exists 𝑛≥2 such that 𝑥_𝑛=𝑏. Set 𝑁=max{𝑛∈𝐍∶𝑥_𝑛<𝑏}. We see that 𝑁≥1, 𝑎=𝑥₁<𝑥₂<⋯<𝑥_𝑖<𝑥_𝑖+1<⋯<𝑥_𝑁<𝑥_𝑁+1=𝑏, (36) and if 𝑁≥2, then 𝑑𝑥_𝑖,𝑦𝑥_𝑖,𝑥_𝑖+1,𝑦𝑥_𝑖+1=𝜀,𝑖=1,2,…,𝑁−1. (37) We will show that Γ_𝜀𝑦⊂^𝑁_𝑖=1𝐵_2𝜀𝑥_𝑖,𝑦𝑥_𝑖, (38) where 𝐵_2𝜀𝑡₁,𝑡₂=𝜏₁,𝜏₂∈𝐑²∶𝑑𝑡₁,𝑡₂,𝜏₁,𝜏₂≤2𝜀. (39) Let (𝜏₁,𝜏₂)∈Γ_𝜀(𝑦). Then there exists 𝜉∈[𝑎,𝑏] such that 𝑑((𝜏₁,𝜏₂),(𝜉,𝑦(𝜉)))≤𝜀. Because of the definition of 𝑥_𝑖+1, we find that 𝜉∈[𝑥_𝑖,𝑥_𝑖+1] for some 𝑖∈{1,2,…,𝑁}, so that 𝑑𝜉,𝑦(𝜉),𝑥_𝑖,𝑦𝑥_𝑖≤𝜀. (40) Hence, it follows that 𝑑𝜏₁,𝜏₂,𝑥_𝑖,𝑦𝑥_𝑖≤𝑑𝜏₁,𝜏₂,(𝜉)(𝜉),𝑥𝜉,𝑦+𝑑𝜉,𝑦_𝑖,𝑦𝑥_𝑖≤2𝜀, (41) which means that (𝜏₁,𝜏₂)∈𝐵_2𝜀(𝑥_𝑖,𝑦(𝑥_𝑖)). Therefore, we obtain (38). By (38), we conclude that Γ_𝜀𝑦≤^𝑁_𝑖=1𝐵_2𝜀𝑥_𝑖,𝑦𝑥_𝑖=4𝑁𝜋𝜀². (42) When 𝑁=1, from (42), it follows that Γ_𝜀𝑦≤4𝜋𝜀²≤4𝜋𝜀length𝑦+4𝜋𝜀². (43) Now, we assume that 𝑁≥2. We observe that length𝑦=^𝑁_𝑖=1length𝑦|_{[𝑥𝑖,𝑥𝑖+1]}≥^𝑁_𝑖=1𝑑𝑥_𝑖,𝑦𝑥_𝑖,𝑥_𝑖+1,𝑦𝑥_𝑖+1≥^𝑁−1_𝑖=1𝑑𝑥_𝑖,𝑦𝑥_𝑖,𝑥_𝑖+1,𝑦𝑥_𝑖+1=(𝑁−1)𝜀, (44) that is, 𝑁𝜀≤length𝑦+𝜀. (45) Combining (42) with (45), we obtain Γ_𝜀𝑦≤4𝜋𝜀length𝑦+4𝜋𝜀². (46)

Lemma 27. Let 𝑦∈𝐶(0,𝑡₀] be bounded on (0,𝑡₀], and let 𝜀>0. Then, there exists 𝐶>0 such that, for every 𝜀>0 and 𝑡∈(0,𝑡₀], Γ_𝜀𝑦≤𝐶𝑡sup_{(]𝑥∈0,𝑡}𝑦(𝑥)+𝜀length𝑦|_[𝑡,𝑡0]+𝜀+𝜀². (47)

Proof. Let 𝜀>0, and let 𝑡∈(0,𝑡₀]. From the geometry point of view, it is clear that Γ_𝜀𝑦|_(0,𝑡]⊂𝑅, (48) where 𝑅 is a rectangle in 𝐑² defined by 𝑅=𝑡₁,𝑡₂∶𝑡₁∈[],𝑡−𝜀,𝑡+𝜀₂∈−𝜀−sup_{(]𝑥∈0,𝑡}𝑦(𝑥),𝜀+sup_{(]𝑥∈0,𝑡}𝑦(𝑥). (49) Hence, we have Γ_𝜀𝑦_(0,𝑡]≤(2𝜀+𝑡)2𝜀+2sup_{𝑥∈(0,𝑡]}𝑦(𝑥)=4𝜀²+2𝑡+2sup_{(]𝑥∈0,𝑡}𝑦(𝑥)𝜀+2𝑡sup_{(]𝑥∈0,𝑡}𝑦(𝑥)≤4𝜀²+𝐶₁𝜀+2𝑡sup_{(]𝑥∈0,𝑡}𝑦(𝑥), (50) where 𝐶₁=2(𝑡₀+2sup_{𝑥∈(0,𝑡0]}|𝑦(𝑥)|). Lemma 26 implies that Γ_𝜀𝑦|_[𝑡,𝑡0]≤4𝜋𝜀length𝑦|_[𝑡,𝑡0]+4𝜋𝜀². (51) Consequently, we see that Γ_𝜀𝑦≤Γ_𝜀𝑦|_(0,𝑡]+Γ_𝜀𝑦|_[𝑡,𝑡0]≤4𝜀²+𝐶₁𝜀+2𝑡sup_{(]𝑥∈0,𝑡}𝑦(𝑥)+4𝜋𝜀length𝑦|_[𝑡,𝑡0]+4𝜋𝜀²≤𝐶𝑡sup_{(]𝑥∈0,𝑡}𝑦(𝑥)+𝜀length𝑦|_[𝑡,𝑡0]+𝜀+𝜀², (52) where 𝐶=max{𝐶₁,4(𝜋+1)}.

Lemma 28. Let 𝑦∈𝐶¹(0,𝑡₀] be bounded on (0,𝑡₀]. Assume that ^𝑡0₀𝑦^′(𝑥)𝑑𝑥=∞,limsup_𝑥→0𝑥sup_{(]𝜉∈0,𝑥}𝑦(𝜉)^{(𝑠−1)/(2−𝑠)𝑡0}_𝑥𝑦^′(𝜉)𝑑𝜉<∞, (53)(54) for some 𝑠∈(1,2). Then, there exists 𝑐₂>0 such that Γ_𝜀𝑦≤𝑐₂𝜀^2−𝑠,𝜀∈(0,1). (55)

Proof. Let 𝜀∈(0,1) and let 𝑠∈(1,2) be satisfy (54). By (53), there exists 𝑥₁∈(0,𝑡₀), which depends on 𝜀 and 𝑠, such that ^𝑡0_𝑥1𝑦^′(𝑡)𝑑𝑡=𝜀^1−𝑠. (56) Therefore, we have length𝑦|_{[𝑥1,𝑡0]}=^𝑡0_𝑥11+𝑦^′(𝑥)²≤𝑑𝑥_𝑡0𝑥11+𝑦^′(𝑥)𝑑𝑥=𝑡₀−𝑥₁+𝜀^1−𝑠≤𝑡₀+𝜀^1−𝑠. (57) By (54), there exists 𝐶₁>0 such that 𝑥sup_{(]𝜉∈0,𝑥}𝑦(𝜉)^{(𝑠−1)/(2−𝑠)𝑡0}_𝑥𝑦^′(𝜉)𝑑𝜉≤𝐶₁,𝑥∈0,𝑡₀, (58) which implies that 𝑥₁sup_{𝜉∈0,𝑥1}𝑦(𝜉)^{(𝑠−1)/(2−𝑠)𝑡0}_𝑥1𝑦^′(𝜉)𝑑𝜉≤𝐶₁, (59) that is, 𝑥₁sup_{𝜉∈(0,𝑥1]}𝑦(𝜉)≤𝐶₂𝜀^2−𝑠, (60) where 𝐶₂=𝐶₁^{(2−𝑠)/(𝑠−1)}. From Lemma 27, (57), and (60), it follows that Γ_𝜀𝑦≤𝐶𝐶₂𝜀^2−𝑠+𝜀𝑡₀+𝜀^1−𝑠+𝜀+𝜀²=𝐶𝐶₂+1+𝑡₀+1𝜀^𝑠−1+𝜀^𝑠𝜀^2−𝑠≤𝐶𝐶₂+𝑡₀+3𝜀^2−𝑠,𝜀∈(0,1). (61)

In order to show Theorems 8 and 11, we need the following two geometric lemmas.

Lemma 29 (see [1]). If Γ⊆ℝ² is a simple curve (i.e., its parameterization is a bijection) and  length(Γ)<∞, then length(Γ)=lim_𝜀→0Γ_𝜀,2𝜀 (62) where Γ_𝜀 denotes the 𝜀-neighborhood of the graph Γ.

Now, we are able to prove Theorem 8.

Proof of Theorem 8. Let 𝑠∈(1,2), and let 𝑦(𝑥) be a chirp function given by (1). We note here that it is enough to show that 𝑦(𝑥) satisfies (31).
At the first, let 𝑥_𝑛 be a sequence defined by 𝑥_𝑛=𝜑⁻¹(𝜏₀+𝑛𝜏) for all sufficiently large 𝑛∈ℕ. From (4), it follows that 𝜑⁻¹(𝑡) is decreasing. Hence, 𝑥_𝑛 is decreasing as well as 𝑥_𝑛→0 as 𝑛→∞ because of lim_𝑥→+0𝜑(𝑥)=∞ (see (4)). We note that 𝑦(𝑥_𝑛)=0 and 𝑦(𝑥)≠0 on (𝑥_𝑛+1,𝑥_𝑛) for all sufficiently large 𝑛∈ℕ. Also, −1/𝜑^′(𝑥) is an increasing function because of (16). The mean value theorem shows that 𝜏−𝜑^′𝑥_𝑛+1≤𝑥_𝑛−𝑥_𝑛+1≤𝜏−𝜑^′𝑥_𝑛. (63) Now, let 𝜀₀∈(0,1). Let 𝑘(𝜀) be the smallest natural number satisfying 𝜏−𝜑^′𝑥_𝑘(𝜀)≤𝜀∀𝜀∈0,𝜀₀. (64) Such 𝑘(𝜀) exists for every 𝜀∈(0,𝜀₀), since 𝑥_𝑛→0 as 𝑛→∞ and lim_𝑥→+0𝜑^′(𝑥)=−∞ (this equality is true because 𝜑^′∉𝐿¹(0,𝑡₀) since lim_𝑥→+0𝜑(𝑥)=∞). Moreover, since 𝑥_𝑛 is decreasing and −1/𝜑^′(𝑥) is increasing, we obtain −𝜑^′𝑥_𝑛≥𝜏𝜀⁻¹∀𝑛≥𝑘(𝜀). (65) Combining (63) and (65), it is easy to deduce that such defined 𝑘(𝜀) satisfies condition (32).
By (16), there exists 𝐿>0 such that (1/(−𝜑^′(𝑥)))^′≤𝐿 for 𝑥∈(0,𝑡₀], which means that −−𝜑^′′(𝑥)−𝜑^′(𝑥)≤−𝐿𝜑^′(𝑥),𝑥∈0,𝑡₀. (66) Integrating (66) on [𝜑⁻¹(𝑡+2𝜏),𝜑⁻¹(𝑡)], we have log−𝜑^′𝜑⁻¹(𝑡+2𝜏)−𝜑^′𝜑⁻¹(𝑡)≤2𝐿𝜏,𝑡∈𝜑𝑡₀,∞, (67) that is, −𝜑^′𝜑⁻¹(𝑡)≥−𝑒^−2𝐿𝜏𝜑^′𝜑⁻¹(𝑡+𝜏),𝑡∈𝜑𝑡₀,∞. (68) By the definition of 𝑘(𝜀) and (64), we see that 𝜏−𝜑^′𝑥_{𝑘(𝜀)−1}>𝜀. (69) Hence, from (68), it follows that 𝜏𝜀⁻¹>−𝜑^′𝑥_{𝑘(𝜀)−1}=−𝜑^′𝜑⁻¹𝜏₀+(𝑘(𝜀)−1)𝜏≥−𝑒^−2𝐿𝜏𝜑^′𝜑⁻¹𝜏₀+(𝑘(𝜀)−1)𝜏+2𝜏=−𝑒^−2𝐿𝜏𝜑^′𝜑⁻¹𝜏₀+(𝑘(𝜀)+1)𝜏=−𝑒^−2𝐿𝜏𝜑^′𝑥_𝑘(𝜀)+1, (70) which implies that −𝜑^′𝑥_𝑘(𝜀)+1≤𝑀𝜀⁻¹,𝜀>0, (71) where 𝑀=𝜏𝑒^2𝐿𝜏.
Next, it is clear that the assumption (5) ensures a real number 𝑀₀>0 such that 𝑀₀=max_𝑡∈ℝ|𝑆(𝑡)|=max_{[]𝑡∈0,𝜏}|𝑆(𝑡)|. (72) Hence, from (15), we have max_{𝑥𝑥∈𝑛+1,𝑥𝑛}𝑦(𝑥)=max_{𝑥𝑥∈𝑛+1,𝑥𝑛}𝑎(𝑥𝑆𝜑(𝑥)𝑥)≥𝑎_𝑛+1max_{𝑥𝑥∈𝑛+1,𝑥𝑛}𝑆𝜑(𝑥)=𝑀₀𝑎𝑥_𝑛+1. (73) Now, from (63) and (73), it follows that max_{𝑥𝑥∈𝑛+1,𝑥𝑛}𝑦(𝑥)𝑥_𝑛−𝑥_𝑛+1≥𝑀₀𝑎𝑥_𝑛+1𝑥_𝑛−𝑥_𝑛+1≥𝜏𝑀₀𝑎𝑥_𝑛+1−𝜑^′𝑥_𝑛+1=𝜏𝑀₀𝑎𝜑⁻¹𝜏₀+(𝑛+1)𝜏−𝜑^′𝜑⁻¹𝜏₀+()𝜏.𝑛+1 (74) Since 𝜑⁻¹(𝑡) is decreasing as well as −1/𝜑^′(𝑥) is increasing and 𝑎(𝑥) is nondecreasing and positive, we conclude that the function 𝑎𝜑⁻¹(𝑡)−𝜑^′𝜑⁻¹(𝑡)isnonincreasing. (75) Hence, we observe that _{(𝜀)𝑛≥𝑘}max_{𝑥𝑥∈𝑛+1,𝑥𝑛}𝑦(𝑥)𝑥_𝑛−𝑥_𝑛+1≥𝜏𝑀_{0(𝜀)𝑛≥𝑘}𝑎𝜑⁻¹𝜏₀+(𝑛+1)𝜏−𝜑^′𝜑⁻¹𝜏₀+(𝑛+1)𝜏≥𝜏𝑀₀^∞_𝑘(𝜀)𝑎𝜑⁻¹𝜏₀+(𝑡+1)𝜏−𝜑^′𝜑⁻¹𝜏₀+(𝑡+1)𝜏𝑑𝑡=𝑀₀^{𝜑−1(𝜏0}₀^{+(𝑘(𝜀)+1)𝜏)}𝑎(𝜉)𝑑𝜉=𝑀₀^{𝑥𝑘(𝜀)+1}₀𝑎(𝜉)𝑑𝜉. (76) Next, from assumption (17), we get some 𝑀₁>0 such that ^𝑥₀𝑎(𝜉)𝑑𝜉≥𝑀₁−𝜑^′(𝑥)^{−(2−𝑠)},𝑥∈0,𝑡₀. (77) Now, from (76), (77), and (71), we obtain _{(𝜀)𝑛≥𝑘}max_{𝑥𝑥∈𝑛+1,𝑥𝑛}𝑦(𝑥)𝑥_𝑛−𝑥_𝑛+1≥𝑀₀𝑀₁−𝜑^′𝑥_𝑘(𝜀)+1^{−(2−𝑠)}≥𝑀₀𝑀₁𝑀𝜀^{−1−(2−𝑠)}=𝑀₀𝑀₁𝑀^{−(2−𝑠)}𝜀^2−𝑠. (78) Thus, using Lemma 25, we have Γ_𝜀𝑦≥𝑐₁𝜀^2−𝑠,𝜀∈0,𝜀₀, (79) for some 𝑐₁>0.
Next we prove that 𝑦(𝑥) satisfies (53). Assume to the contrary that ^𝑡0₀𝑦^′(𝑥)𝑑𝑥<∞. (80) Then, we have length𝑦=^𝑡0₀1+𝑦^′(𝑥)²≤𝑑𝑥_𝑡001+𝑦^′(𝑥)𝑑𝑥<∞. (81) From Lemma 29, it follows that length𝑦=lim_𝜀→+0Γ_𝜀𝑦.2𝜀 (82) On the other hand, by (79), we see that lim_𝜀→+0Γ_𝜀𝑦2𝜀=∞. (83) This is a contaradiction, and hence 𝑦(𝑥) satisfies (53).
Finally, we show that 𝑦(𝑥) also satisfies inequality (54). Because of (5), we obtain a constant 𝑐>0 such that max_𝑡∈ℝ|𝑆(𝑡)|=max_{[]𝑡∈0,𝜏}|𝑆(𝑡)|≤𝑐,max_𝑡∈ℝ𝑆^′(𝑡)=max_{[]𝑡∈0,𝜏}𝑆^′(𝑡)≤𝑐. (84) By (15), we find that sup_{(]𝑡∈0,𝑥}𝑦(𝑡)=sup_{(]𝑡∈0,𝑥}|𝑎(𝑡)|𝑆𝜑(𝑡)≤𝑐𝑎(𝑥). (85) Hence, from (1) and (15), it follows that 𝑦^′(𝑥)≤𝑐𝑎^′(𝑥)+𝑎(𝑥)−𝜑^′(𝑥), (86) and therefore ^𝑡0_𝑥𝑦^′(𝜉)𝑑𝜉≤𝑐𝑎𝑡₀+𝑐^𝑡0_𝑥𝑎(𝜉)−𝜑^′(𝜉)𝑑𝜉. (87) Using (85) and (87), we see that 𝑥sup_{(]𝜉∈0,𝑥}𝑦(𝜉)^{(𝑠−1)/(2−𝑠)𝑡0}_𝑥𝑦^′(𝜉𝑡)𝑑𝜉≤𝑐𝑎₀[𝑐𝑥𝑎(𝑥)]^{(𝑠−1)/(2−𝑠)}+𝑐[𝑐𝑥𝑎(𝑥)]^{(𝑠−1)/(2−𝑠)𝑡0}_𝑥𝑎(𝜉)−𝜑^′(𝜉𝑡)𝑑𝜉≤𝑐𝑎₀𝑐𝑡₀𝑎𝑡₀^{(𝑠−1)/(2−𝑠)}+𝑐^1/(2−𝑠)[𝑥𝑎(𝑥)]^{(𝑠−1)/(2−𝑠)𝑡0}_𝑥𝑎(𝜉)−𝜑^′(𝜉)𝑑𝜉. (88) By (18) we conclude that (54) holds.
By Lemma 28, there exists 𝑐₂>0 such that Γ_𝜀𝑦≤𝑐₂𝜀^2−𝑠,𝜀∈0,𝜀₀. (89) Thus, we have proved that the chirp function 𝑦(𝑥) given by (1) satisfies the desired inequality (31). This completes the proof of Theorem 8.

Proof of Theorem 11. Let 𝑦(𝑥) be the chirp function (1). It is easy to see that 𝑦(𝑥) is oscillatory near 𝑥=0. By (5), there exists 𝑀>0 such that |𝑆(𝑡)|≤𝑀,𝑆^′(𝑡)≤𝑀,𝑡∈ℝ. (90) Hence, we observe that 𝑦^′(𝑥)≤𝑎^′(𝑥)𝑆𝜑(𝑥)+𝑎(𝑥)𝑆𝜑(𝑥)𝜑^′(𝑥𝑎)≤𝑀^′(𝑥)+𝑀𝑎(𝑥)𝜑^′(𝑥), (91) for 𝑥∈(0,𝑡₀]. From (81), it follows that length(𝑦)<∞. Lemma 29 shows that length𝑦=lim_𝜀→0Γ_𝜀𝑦,2𝜀 (92) which implies that 𝜀length(Γ)≤|Γ_𝜀(𝑦)|≤4𝜀length(Γ), 𝜀∈(0,𝜀₀), for some 𝜀₀>0. Therefore, 𝑦(𝑥) is fractal oscillatory near 𝑥=0 with the fractal dimension 1.

4. Proof for the Fractal Oscillations of (26)

In this section, we give the proofs for the fractal oscillations of the linear second-order differential equation (26) considered as an application of the main results on the fractal oscillation of chirp functions.

Before we present the proofs of Theorems 16 and 17, we make the following observation. Since 𝑦₁(𝑥)=𝑎(𝑥)cos𝜑(𝑥),𝑦₂(𝑥)=𝑎(𝑥)sin𝜑(𝑥) (93) are solutions of (26), we see that 𝑦(𝑥)=𝑐₁𝑦₁(𝑥)+𝑐₂𝑦₂(𝑥) is a fundamental system of all solutions of (26). Assume that 𝑐²₁+𝑐²₂>0, and set 𝑆(𝑡)=𝑐₁sin𝑡+𝑐₂cos𝑡. Then, 𝑆(𝑡) clearly satisfies (5).

Proof of Theorem 16. Applying Theorem 8 on 𝑦(𝑥)=𝑐₁𝑦₁(𝑥)+𝑐₂𝑦₂(𝑥), we conclude that 𝑦(𝑥) is fractal oscillatory with the fractal dimension 𝑠.

Proof of Theorem 17. Theorem 11 implies that 𝑦(𝑥)=𝑐₁𝑦₁(𝑥)+𝑐₂𝑦₂(𝑥) is fractal oscillatory near 𝑥=0 with the fractal dimension 1.

Appendix

Proof of Remark 9. By (17), there exists 𝑐₁>0 such that 𝑐₁≤−𝜑^′(𝑥)^2−𝑠𝑥₀𝑎(𝜉)𝑑𝜉,𝑥∈0,𝑡₀. (A.1) From (15), it follows that 𝑐₁≤−𝜑^′(𝑥)^2−𝑠𝑎(𝑥)^𝑥₀𝑑𝜉=𝑥𝑎(𝑥)−𝜑^′(𝑥)^2−𝑠,𝑥∈0,𝑡₀, (A.2) which implies that 1−𝜑^′(𝑥)≤𝑐₁^{−1/(2−𝑠)}𝑥^1/(2−𝑠)[𝑎(𝑥)]^1/(2−𝑠),𝑥∈0,𝑡₀. (A.3) By (19), there exists 𝑐₂>0 such that 𝑥𝑎(𝑥)−𝜑^′(𝑥)^2−𝑠≤𝑐₂,𝑥∈0,𝑡₀, (A.4) and hence −𝜑^′(𝑥)≤𝑐₂^1/(2−𝑠)𝑥^{−1/(2−𝑠)}[𝑎(𝑥)]^{−1/(2−𝑠)},𝑥∈0,𝑡₀. (A.5) Therefore, we see that 1−𝜑^′(𝑥)^{𝑠−1𝑡0}_𝑥𝑎(𝜉)−𝜑^′(𝜉)𝑑𝜉≤𝑐₁^{−(𝑠−1)/(2−𝑠)}𝑥^{(𝑠−1)/(2−𝑠)}[𝑎(𝑥)]^{(𝑠−1)/(2−𝑠)}𝑐₂^1/(2−𝑠)×^𝑡0_𝑥𝜉^{−1/(2−𝑠)}𝑎(𝜉)^{−(𝑠−1)/(2−𝑠)}𝑑𝜉≤𝑐₃𝑥^{(𝑠−1)/(2−𝑠)}[𝑎(𝑥)]^{(𝑠−1)/(2−𝑠)}[𝑎(𝑥)]^{−(𝑠−1)/(2−𝑠)}×^𝑡0_𝑥𝜉^{−1/(2−𝑠)}𝑑𝜉=𝑐₃2−𝑠𝑠−11−𝑥^{(𝑠−1)/(2−𝑠)}𝑡₀^{−(𝑠−1)/(2−𝑠)}≤𝑐₃2−𝑠𝑠−1,𝑥∈0,𝑡₀, (A.6) so that (20) holds. We conclude that [𝑥𝑎(𝑥)]^{(𝑠−1)/(2−𝑠)𝑡0}_𝑥𝑎(𝜉)−𝜑^′(𝜉=(𝑥)𝑑𝜉𝑥𝑎)−𝜑^′(𝑥)^{2−𝑠(𝑠−1)/(2−𝑠)}×1−𝜑^′(𝑥)^{𝑠−1𝑡0}_𝑥𝑎(𝜉)−𝜑^′(𝜉)𝑑𝜉. (A.7) By (19) and (20), we conclude that (18) is satisfied. The proof of Remark 9 is complete.