Sturm ’ s Theorems for Canonical Equation of the Second Order with Non-monotonous Coefficients

For canonical equation 0 ) ( = ⋅ + ′ ′ y x a y , if a(x) is strictly monotonous function, we have determined number of zeros and exact locations of zero solutions which are main weak points of classic Sturm’s theorems. However, if in equation, y x a y ⋅ − = ′ ′ ) ( , a(x) is non-monotonous function, then function y x a y x F ⋅ − = ) ( ) , ( is not monotonous, as well and its inverse function is the multi-valued one. This naturally influences both oscillarity and zero oscillations. Therefore, this case has been considered in this work. When the coefficient a(x) is non-monotonous function, we have first determined all intervals of monotony of coefficient. Then in each interval, we have determined number of zeros and very precise locations of zeros, linearly particular solutions y1 and y2 and thus improved Sturm’s theorems, which are almost 180 years old.

Then on the semi-axis [ ) , 0 +∞ both particular solutions of the equation are oscillating and have infinite number of zeros.This is in accordance with the physical meaning of the coefficient a(x) which represent difference between forces which trigger oscillations and resistance.Since 0 ) ( ≠ x a , i.e. a(x)>0, from the equation if and only if y(x)=0, which means that zero solutions correspond to curving points of the solution.The curving points are simple curves with steep tangent and they cannot be contacts, i.e. curving points with horizontal tangent, which derives from the derivative of the given equation: If ξ is zero solution of y(x), i.e. , which follows based on a classic theorem according to which the solutions of the linear differential equation of the second order cannot have double zeros.Based on this, it follows that for the quality of the curve, including quality of zero solutions, derivative ) (x a′ is important, as well as its fore-sign.Furthermore, it is very important whether a(x) is monotonous or not, which Sturm could not see due to his algebraic approach to the problem.If coefficient a(x) is non-monotonous function, then the number of zero solutions of the equation 0 is determined by firstly determining intervals of monotony.Then in each interval separately the number of zeros and locations of zeros are determined.Sturm also sensed this to some point when he defined his famous theorem on minimum number of zeros (case when a(x) is strictly ), but he did not have better mathematical apparatus to establish and introduce a definite proof for it.
However, despite of this, each basic literature on oscillating solutions of differential equation contains also Sturm's theorems on minimum number of zero oscillations in canonical interval which goes as follows: Theorem 1.1 Let us have the following equation: with two more equations with constant coefficients:

Sturm's theorems for canonical equation
In this case the following is valid: 1. each non-trivial solution of the equation each non-trivial solution of the equation π , where m is a whole number and 1 ≥ m .Sturm proved Theorem 1.1 by method of comparing coefficients of differential equation (1.3) by framing them between two constants k and K, which are coefficients of equations of harmonic oscillations.In this way he applied avant-garde approach in using method of differential inequalities and nonequations for evaluation of squarely insolvable equations.However, we cannot find this theorem satisfying, because it does not determine better and more precise zero locations.Furthermore, it does not determine the precise number of zeros either, but it only specifies the minimum number of zeros.
Today, there are works on zero solutions of differential oscillating equation, but they are based on theory of groups (Boruvka, Gregus, Biernacki, Suyama), while the thorough sources on these results can be found in famous monographs, those of Kamke [1] and Lamberto Cesari [2].These works have not found their places in large books and manuals on differential equations (Eins, Hartman, Coddington-Levnison, Matveev, Pontryagin, etc).We also do not have impression that they are simple and adequate for engineering practice.According to our results, the right method for determination of number of zeros and very precise locations of zero oscillations is the method of iteration sequences, which is the most direct, but it has not been used, because it has not been worked out.
By applying this method we have solved not the equation (1.3), but its canonical equation:

Milena Lekic, Ljubica Lalovic and Miloje Rajovic
By series of works since 1993, we have shown that the equation (1.6), when the coefficient a(x) fulfills the conditions 1) and 2), is solved most easily if integral form is submitted to iterations, regardless of general nature of the coefficient a(x).Thus, the solution in the form of iteration sequences is easily obtained: Even at first glance, it can be seen that for a(x)>0, this solution has the structure of sine and cosine, and these are exactly obtained for a=const.If a(x) is limited (due to continuity) on canonical interval , then the upper sequences made of successive integrals (iterations) are actually some nonelementary functions similar to simple Euclid's sine and cosine which depend on a(x).Nevertheless, this a(x) does not change drastically their usual main properties -schedule of zeros, change of sign, limitation, curving points in zeros, changes of curves, oscillating nature.Testing this, we have obtained the base for statement which says that solutions of the equation (1.6) given with (1.9) have the following form: and that general solution is: By using the theorem on mean value of integral and method of geometric means in some definite interval [ ] x , 0 , but which can also be big, we easily obtained very precise approximate formulae for the above generalized (general) sine and cosine.However this is now complex argument which depends on a(x).Namely, we obtain: the same arrangement is obtained for zeros of derivatives.
3. Theorem (1.1) on number of zeros, which is not less than m, can now be formulated more precisely.If we put 1 = ϕ in (1.3), then the condition K ≤ < ϕ 0 is satisfied for K=1.In this case the second condition: with horizontal lines depends considerably on behavior of the coefficient a(x).If a(x) also has oscillations on its side, then the number of sections can be considerably higher, which leads to higher number of zeros and consequently to higher frequencies.Then the following theorem is valid: Theorem 1.3.If a(x) is monotonously rising function on [0,X] and if the value F(X) in the end point x is maximum of the function F(X) and π π , then the solution y 1 has precisely m+1 zero on a closed interval [0,X], while the other solution y 2 has exactly m zeros on the same interval.
The reason for solution y 1 to have one zero more than solution y 2 lies in the fact that sine always has one zero more than cosine in a closed interval which contains the whole number of full periods of these functions.This is obvious both from analytic sequences for sinx and cosx and from iteration sequences for the same functions.

Main results
If non-continuous function a(x) is not monotonous and if it oscillates infinite number of times remaining positive, as, for example, in equation where M is high enough so that a(x) can remain positive for each x>0, then the number of zeros must be searched separately in every interval of monotony either of rising or falling.
for those values of x which satisfies the equation:  Therefore, for the arrangements of abscissas of the functions a(x) and F(x) the following inequality is valid: This intricacy, originally, was characteristic for Sturm's theorems.Theorem 2.1 In one interval of monotony of the function a(x), the abscissas of extremes of functions a(x) and are interlaced successively.
If we now find solutions of the equation (1.15) and (1.16) in each interval of the monotony of F(x), then the number of zeros is searched separately in every interval.First we take the first interval of monotony on which, depending on a(x), the function F(x) reaches its minimum and maximum.On graph 3, it is the , i.e. up to the height of maximum of the function F(x), let there be p 1 number of sections, while on interval [ ] δ η, , i.e. up to the height of minimum of the function F(x), let there be another p 2 number of sections.Then the total number of sections of the horizontal lines where the zero in the point x=0 is also obligatorily is included.Since π π Completely analogically the minimum of the function F(x) is between two whole multiples of the number π , i.e. π π , i.e.In this way we obtained that on the interval [ ] δ , 0 , the total number of zeros is of the sine solution: The procedure is continued and in the next interval of monotony the number of zeros will be M 3 -m 2 in the rising part of the interval, while in the falling part of the interval there will be M 3 -m 4 , where M 3 is the maximum and m 4 is the minimum of the function F(x) and so on.If we now sum up the number of these sections, then it is the number of zero solutions of y 1 .
However, the coefficient a(x) is considered on canonical interval on which it oscillates the canonical number of times, so this procedure should be finished after many steps in some point until which, according to our construction, there are 2n-1 maximums M 1 , M 3 ,, M 5 ,…, M 2n-1 and 2n-2 minimums m 2 , m 4 , m 6 ,,…,m 2n- 2 .Furthermore, in the rest of canonical interval, the solution y 1 can have M zeros more, although the number M can be possibly equal to zero.

Theorem 2.2
The number of zeros of the oscillating solutions of the equation 0 on canonical interval on which a(x) is positive continuous non-monotonous function, which has oscillations from its side, should be as follows: where M i is the number of multiples of number π up to the maximum of the function F(x) on i th interval, while m i is number of multiples of number π up to the minimum of the function F(x) on the same interval and M is number of multiples of number π on the last interval within [0,X] where 0 ) ( > ′ x a and from which a(x) starts monotonously to rise.
, so the total number of zeros of oscillations is ( ) Example 2 As for differential equation 0 ) sin ( 2 , which also . Therefore, the number of zeros, i.e. the number of frequencies is ( )

Conclusion
The linear canonical differential equation ( 1 We have also reduced the famous Sturm's equation (1.3) to the equation (1.1).Sturm determined the conditions for this equation in the sense that, when its solutions are oscillations, they have infinite number of zeros, while when they are monotonous, they have maximum one Sturm's zero.According to classical qualitative Sturm's theorems, some ten of them, the zeros of oscillations 1 y and 2 y mutually intertwine, between every two zeros, there is one zero of the derivative of the other zero and the number of zeros cannot be less than some number m.However, the main deficiency of the Sturm's theorems is the fact that they determine neither the exact number of zeros not their locations.Furthermore, as the greatest algebraist of his time, Sturm noticed the importance of fore-sign of the coefficient of the equation for the nature of solution, but he did not realize the importance of monotonous i.e. non-monotonous nature of the initiator of oscillation.In this work, we have actually considered the case when the coefficient of the equation is non-monotonous function.We have split that interval into smaller intervals of monotony.For each of these, we have determined number of zeros and very precise locations of zeros as cross section of the function of frequency

3 )
Since a(x)>0 and x>0, then the equation (2.3) can have solutions only in area e. in which a(x) is a falling function.Therefore, if η is an abscissa of maximum of the function F(x) and ξ is an abscissa of the maximum of the function a(x), then η ξ < , which is very important for these Sturm's theorems.

Fig. 2
Fig.2 intersect non-monotonous graph of the function F(x) Fig.4 this way, we have improved classical Sturm's theories which are almost 150 years old.
1.14)which are some complete functions of the complex argument.Now, based on aforesaid, it has not been difficult to determine approximately much better zero locations, which Sturm completely missed.The following theorem is valid: