Investigation for periodic and almost periodic solutions of first-order ordinary differential equations

. Approaches to estimating for the number of periodic and almost periodic solutions of ordinary differential equations are considered. Conditions that allow determinating both the upper and lower bounds for these solutions are found. The existence of periodic and almost periodic solutions in problems are considered. Approaches to estimating for the number of periodic solutions of ordinary differential equations are considered. Conditions that allow determinating both the upper and lower bounds for these solutions are found. The existence and stability of almost periodic problems are considered


Introduction
The method of investigation of periodic solutions of first-order ordinary differential equations of the first-order is considered.It is developed based on the ideas of functional analysis.A large number of papers have been devoted to the study of periodic solutions.
The theoretical foundations of periodic solutions of differential equations were developed by H. Poincaré for the three-body problem [1] and A. Lyapunov for the problem of any mechanical system motion [2].Periodic solutions take an important part in the qualitative theory of differential equations and in applied problems [3,4].The analysis problems for periodic solutions of differential equations arise in classical mechanics, celestial mechanics, space robotics and modeling of economic processes [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22].However, there is no general approach to study periodic solutions of differential equations.There are several methods to solve this problem.Thus, the basic method to prove the existence of periodic solutions of differential equations is the Poincare-Andronov point mapping method, the method of directing functions, variational methods, the topological method, the Krylov-Bogolyubov averaging and so on.Note that these methods are difficult enough to be applied.
In this paper, the problem of estimating the number of periodic solutions of first-order differential equations is solved on the basis of the results of studies and with the use of derived numbers theory [18][19][20][21][22][23][24].
There are two classes of oscillatory processes -periodic and non-periodic.In theory and practice, an intermediate class of almost periodic oscillations is of great importance.
Almost periodic oscillations are oscillations close to periodic oscillations, which are composed of harmonics with incommensurable periods.The process, which consists of the sum of two periodic oscillations with incommensurate frequencies, is also an almost periodic oscillation.Fundamental results in the theory of periodic and almost periodic oscillations are obtained in the works [1][2][3][4][5][6][7].In many problems of classical and celestial mechanics, robotics and mechatronics [8][9][10][11][12][13], there are processes in which the time dependence is not periodic, but can be expressed through trigonometric sums.In this connection, the interest has arisen in the study of almost periodic solutions of differential equations and differential equations with almost periodic coefficients [3][4][5].Over the last years, the question of studying periodic and almost periodic functions in robotics, dynamic systems, stability theory, in particular, in control systems for space objects arose significantly [13][14][15][16][17].
Theorems of the method of derivatives for estimating the number of periodic solutions of first-order ordinary differential equations are formulated and proved.The use of the apparatus of derivative numbers makes it possible to weaken the restrictions imposed on the right-hand sides of differential equations.This will increase the degree of generality of the results.Upper and lower estimates for the number of periodic and almost periodic solutions of first-order ordinary differential equations are given.Conditions for the existence of periodic and stability of almost periodic solutions are established [25][26][27][28][29][30][31].

Periodic solutions of the first-order differential equations 2.1 The upper bound for a number of periodic solutions
Let the right side of equation ̇= (, ) is a continuous function with respect to the set of arguments and is function  -periodic with respect to t .Theorem 1.If the right-hand side of equation (1)  x x (2) Assume the contrary, namely, that the solutions 1 x and 2 x intersect.This means that there exists The subtraction From (3) it follows that (5) but due to (2) the following is obtained ), ( < ) ( x do not intersect.These solutions do not have common points for 0 < t , which follows from their  -periodicity.So, if equation ( 1) has two  -periodic solutions and inequality (2) is realized, then for all t ), ( < ) ( Taking into account the last inequality, the following contradictory chain of relations is obtained (8): This contradiction approves the assumption that equation ( 1 Proof.First of all, let us proove that the equation (1) has the property of existence and uniqueness for the solution under the conditions of the theorem.Assume for that purpose that if f is convex and continuous, then it is a Lipschitz function.

Indeed, let
) , ( 0 0 x t be an arbitrary point of the plane, and let  and  be some positive numbers.Let us show that for the domain In the proof of Theorem 14 [19] it is noted that for each fixed t function f has a bounded right derivative, which is an increasing function with respect to x .Let us show that there exists K , such that for all Since for each fixed increases monotonically, it can take its extreme values only at the ends of the interval.Therefore, in order to prove the required inequality, it suffices to show that the functions . Then, taking into account the fact that for any fixed , (16) which means that f is a Lipschitzian function.
Let the conditions of the theorem be satisfied.Let us show that in this case the equation (1) can not have more than two  -periodic solutions.Assume, bu contradiction, that the equation ( 1) has three  -periodic solutions 1 y , 2 y and 3 y .As shown above, the function f satisfies the Lipschitz condition, and consequently, the equation (1) has the property of existence and uniqueness for solutions, and therefore, if for 0 = t (0), < (0) < (0) The function ) (t g is continuous, since the functions f and  are continuous and since it follows from inequality (18) and the definition of  that there exists a number Thus, the function g is continuous, non-negative and there is an interval which the function g takes only positive values in.
Considering this, let the identity (5) be integrated in the range from 0 to  .Since the functions 1 y , 2 y and 3 y are  -periodic, the following is obtained: This contradiction shows that if the conditions of the theorem are satisfied, the equation  is also defined and continuous, and from the equation in variations it follows that for any


is strictly convex, then the theorem is very simple to prove.Indeed, let the equation ( 1) have four  -periodic solutions ) , ( i x t takes the same values at the ends of the interval, since due to  -periodicity, and finally, it has a derivative at each point x .Thus, for the function all the conditions of Rolle's theorem are satisfied, and, consequently, there exist points ) , ( ), , ( ), , ( '   from which it is finally obtained that So, if the equation ( 1) has four  -periodic solutions, then the function ) (x  takes the same values at three different points, which contradicts the assumption of its convexity.Therefore, in order to prove the theorem, it is sufficient to show that the function x and continuous in the set of arguments.Thus, all the functions entering the right-hand side of (32) are bounded, and therefore, the function x x that has a bounded right derivative on this interval.Then, according to Theorem 14 [19], in order to prove the strict convexity of be an arbitrary right-hand derived number of the function n h be a sequence which this number is realized on, and 0 x t '  and using the equation in variations, the following is obtaind:  .First of all, note that from the estimation obtained above As mentioned above, the second term is 0 , therefore, the first term denoted by I(x) is also equal to zero.Estimating ) (x I from below, the following is realized:   the arguments similar with ones for the right derivative of this function, it is obvious that the left derivative of the function is also bounded above.Thus, on the basis of Theorem 1 [19] Hence, in view of the continuity of the functions ' f and  in the set of arguments, it follows (49

Lower bound of  -periodic solutions number
Pliss V.A. in [3] constructed an example demonstrating that this series of theorems can not be continued.Here we present another approach, which let obtain the information on a number of  -periodic solutions of the equation (1).Theorem 5. Let the right-hand side of the equation ( 1 then the equation ( 1) has no less than  of  -periodic solutions.Here it is assumed that .
By the theorem condition But then, as shown by H. W. Knobloch [8] and J. Mawhin [9], in the band there exists at least one  -periodic solution of the equation (1).Since there are n such bands, then equation (1) has no less than n  -periodic solutions.
Combining Theorem 5 with the statements in [19], a number of statements can be proved giving more exact information on the number of  -periodic solutions of the equation (1).As and example, let us prove one such statement.
x for each fixed t .Let us show that in the selected band there is only one  -periodic solution of the equation (1).
Let, on the contrary, in the band there are two  -periodic solutions of the equation ( 1 Thus, the obtained contradiction shows that in the band there exists no more than one  -periodic solution of equation ( 1).But, as follows from the Theorem 5, in this band there necessarily exists at least one  -periodic solution of the equation (1).
Hence it can be finally concluded that in each band , , there is only one  -periodic solution of equation ( 1).Since there are n such bands, then the equation ( 1) has exactly n  -periodic solutions.
In particular, it follows from Theorem 5 that if the right-hand side of the equation ( 1) is representable in the form ), , ( = ) , ( , then the equation (1) satisfy the requirements of Theorem 5.

Estimation of almost periodic solutions number of first-order ordinary differential equations
Approaches to estimating for the number of periodic and almost periodic solutions of ordinary differential equations are considered.Conditions that allow determinating both the upper and lower bounds for these solutions are found.The existence and stability of almost periodic solutions in problems are considered.
There are two classes of oscillatory processes -periodic and non-periodic.In theory and practice, an intermediate class of almost periodic oscillations is of great importance.
Almost periodic oscillations are oscillations close to periodic oscillations, which are composed of harmonics with incommensurable periods.The process, which consists of the sum of two periodic oscillations with incommensurate frequencies, is also an almost periodic oscillation.
The theory of almost periodic oscillations is developed in the works of mathematicians P. Bolh and H. Bohr [6].They laid the foundations of the theory of almost periodic functions and quasi-periodic functions, proved the theorem on the expansion of functions into a Fourier series and the theorem on a quasi-periodic function.Fundamental results in the theory of periodic and almost periodic oscillations were obtained in the works of N.M.Krylov, N.N.Bogolyubov, Yu.A. Mitropolsky, A.M. Samoylenko, V.A. Pliss and others .
In many problems of classical mechanics, celestial mechanics, robotics and mechatronics, there are processes in which the time dependence is not periodic, but is expressed in terms of trigonometric sums.In this connection, interest has arisen in the study of almost periodic solutions of differential equations and equations with almost periodic coefficients [7][8][9][10][11][12][13][14].
Over the last years, the question of studying almost periodic functions in robotics, dynamic systems, stability theory, in particular, in control systems for space objects arose significantly [9][10][11][12].

Upper bound for number of almost periodic solutions
Let the right-hand side Proof.Let the conditions of the assertion be satisfied, and let, on the contrary, equation x respectively.In exactly the same way as was done in the proof of Theorem 1 [19], only taking into account the almost periodicity of the solutions, we can show that the solutions Then, on the one hand, from (1), taking into account (56) and (58), we have The resulting contradiction proves that Eq. ( 9 Repeating the arguments given in the proof of Theorem 2 [19], it is easy to check that 0, > = ) (

 T G
and from the definition of the function G and from the convexity with respect x to the function f it follows that for any 0 > ' T there will be ), ( The resulting contradiction proves the validity of the assertion.Note.All of equation ( 1  is a function that is continuous in a set of arguments.Theorem 9.If for some  = 1,2,3 function  −2 (, ) is continuous in the set of arguments and convex with respect to  for each fixed , and there exists an instant  * such that  −2 ( * , ) is strictly convex, then equation ( 9) can have at most  almost periodic solutions.

A lower bound for the number of almost periodic solutions
On the basis of the previous theorems, the authors obtain the conditions to determine the maximum possible number of almost periodic solutions in first-order differential equation.Now the problem of the existence of almost periodic solutions for the equation is under consideration, since this allows for the determination of the minimum possible number of almost periodic solutions for the differential equation considered.
So, consider the first-order differential equation ( 1) where f is a function continuous on  2 that is almost periodic in t uniformly in x in every compact set and such that equation (11) has the property of existence and uniqueness of its solutions.
To prove the existence of almost periodic solution for equation ( 1), the result obtained should be used.Let it be formulated in the form of the following theorem.
This study allows to determine the minimum possible number of almost periodic solutions for the considered differential equation.Consider the first-order differential equation ( 1), where  is a function continuous on  2 almost periodic in  uniformly in  on each compact set and such that equation (1) has the property of existence and uniqueness of solutions.In proving the existence of an almost periodic solution of equation ( 1), the results obtained in [9]    Let, for definiteness, (0) < (0)

2 .
) can have two different  - periodic solutions.TheoremIf the right-hand side of (1.1) is a convex function for each fixed can have no more than two different  -periodic solutions. 0

( 1 )Theorem 3 .
can not have three  -periodic solutions.If the right-hand side of the equation (1) set of arguments, which is a function convex with respect tox for each t , and there exists a moment ] strictly convex, then the equation (1) can not have more than three  -periodic solutions.of the equation(1), that starts at the point and continuous, then the function '

.
The limiting transition under the integral sign is admissible, since the right derivative  ' ' ) ( of the function '  exists and it is finite on ] E3S Web of Conferences 458, 09011 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345809011Repeating these arguments, for an arbitrary left derived number of the function  '  at a pointx , the followng estimation is obtained: its left derived numbers are uniformly bounded below.Therefore, according to Theorem 9


and if there are no roots of the function ' in a neighborhood of the point 0 x , then the function  '  strictly increases in this neighborhood.Suppose that '  increases, but not strictly.Then there exists a segment ] is constant on.But, as follows from the remark made above, this segment can be only zero, since if there is a neighborhood with It is easy to see that the left derivatives of the function r are nonnegative.Besides, repeating for the left derivative of the function  '

Theorem 6 .
Let the right-hand side of the equation (1) satisfy all the requirements of Theorem 5, and for any definiteness, assume that in the band function on  2 .Theorem 7. If the right-hand side of equation (1) for each fixed  is an increasing function with respect to , and there exists an instant ] then equation(1) can not have at most one almost periodic solution.


go back to the functions P and Q .It can be seen directly from their definition that they are almost periodic and that follows from the existence of y functions P and Q a common  -almost period.Then, for the function chosen '  from (2.9), taking into account the above properties of the function G , we have (70) On the other side, .

are used. Theorem 10 ..
Let the right-hand side of equation (1) be such that (, ) decreases with respect to  ∈ [, ] for each fixed .Then, if (1) has a bounded solution (), such that {(); 0 ≤  < ∞} ⊂ [, ], then it has an almost periodic solution () whose range of values is in interval [a, b].Note.If in equation (1) the variable is changed, setting that  = −, then the following equation is obtained:   = −(−, ).(75) It is clear that if equation (1) has an almost periodic solution, then equation (75) also has an almost periodic solution, and vice versa.Theorem 11.If the right-hand side of equation (1) is a function decreasing in  for each fixed , and lim →−∞ (, ) = +∞ and lim →+∞ (, ) = −∞(76)uniformly with respect to , then equation (1) has an almost periodic solution.Proof.By virtue of the assumption that Therefore, the solution of equation (1) starting at any point 0x , Thus, equation (1) has a bounded solution, which implies that equation (1) has at least one almost periodic solution taking into account Theorem.
and almost periodic with respect to t uniformly in x in this band, i.e., in the band 12 are satisfied, which implies that equation (80) has an almost periodic solution in the band exist two almost periodic solutions.Suppose the contrary.
Moreover, taking into account the convexity of the function f , i.e. that for all itself, as in some neighborhood of it, by continuity the following strict inequality is realized: * t the equation in variations it follows that , by virtue of the existence and uniqueness of the solutions of equation (1), strictly increases with respect to each fixed.It follows that the right derivative of the function, which we denote by, can be written in the form.
is bounded, since it increases with respect tox by virtue of the existence and uniqueness of the solutions of equation (1.1), and, consequently, It can assumed that the chosen  and  are also suitable for the .
we conclude that the function r is Then all the obtained results can be represented in a unified form. ' E3S Web of Conferences 458, 09011 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345809011proven that this solution also can not get into the domain which implies that there exists a bounded solution for equation (79) and for equation (1) as well, considering the assumption thst the functions  and  are bounded.Thus, all the requirements of the proved statement hold, which allows us to conclude that Equation (80) has a bounded solution in the band It is obvious that in the band: E3S Web of Conferences 458, 09011 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345809011 i 