Green ’ s Function and Periodic Solutions of a Spring-Mass System in which the Forces are Functionally Dependent on Piecewise Constant Argument

In this paper, damped spring-mass systems with generalized piecewise constant argument and with functional dependence on generalized piecewise constant argument are considered. These spring-mass systems have piecewise constant forces of the forms $Ax(\gamma(t))$ and $Ax(\gamma(t))+h(t,x_{t},x_{\gamma(t)})$, respectively. These spring-mass systems are examined without reducing them into discrete equations. While doing this examination, we make use of the results which have been obtained for differential equations with functional dependence on generalized piecewise constant argument in \cite{2}. Sufficient conditions for the existence and uniqueness of solutions of the spring-mass system with functional dependence on generalized piecewise constant argument are given. The periodic solution of the spring-mass system which has functional force is created with the help of the Green's function, and its uniqueness is proved. The obtained theoretical results are illustrated by an example. This illustration shows that the damped spring-mass systems with functional dependence on generalized piecewise constant argument with proper parameters has a unique periodic solution which can be expressed by Green's function.


Introduction and Preliminaries
Delay differential equations play an important role to model real world problems, and scientists [2][3][4][5] have done a lot of work on these types of equations.Differential equations with piecewise constant argument which are in the class of delay differential equations, have deviating arguments of retarded and/or advanced type [1,[6][7][8][9].Recently, numerous problems mainly related to the existence of periodic and almost periodic solutions [8,[10][11][12][13], oscillatory behavior of solutions [10,[14][15][16][17][18], global attractivity of the trivial solution [19] have been investigated for differential equations with piecewise of periodic solutions using Green's function, existence and uniqueness of almost periodic solutions, exponential stability of solutions and periodic solutions, boundedness of the solutions were obtained under certain assumptions.Besides, modeling by functional differential equations provide more detailed analysis for real life problems since they represent dependency on system's past and future situations of its current time.So, functional differential equations have a great importance, and much investigation has been carried out on the solutions and on the existence of periodic solutions for various types of them (see, e.g., [30][31][32][33] and references therein).
The general form of a damped spring-mass equation whose main expression come from Newton's second law and Hooke's law for a spring is which is called a damped harmonic oscillator, as well.
Here, m > 0 is the mass, c > 0 is the damping coefficient, k > 0 is the spring constant and x(t) is the displacement of the mass.The cases ∆ > 0, ∆ = 0, ∆ < 0 for the discriminant of the spring-mass system (1) show that the system (1) exhibits motion with over damped, critical damped and under damped, respectively.The spring-mass system (1) can include any external force H, in other words, the system (1) can typically be written as nonhomogeneous differential equation which is referred as a forced harmonic oscillator.Springmass systems have been widely used by many scientists in the fields such as physics [34][35][36][37], mathematics [14,38], biomechanics [39,40], electrical and computer engineering [41], biology [42].Dai and Singh [14] studied the oscillation problem for the damped spring-mass equation ( 2) taking the piecewise constant force Ax([t]) instead of the external force H.In the piecewise constant force Ax([t]), A specifies the magnitude of the force.
Let Z and R be the sets of all integers and real numbers, respectively.|| • || signs to the Euclidean norm in In this paper, we are interested in the following damped spring-mass systems with generalized piecewise constant argument and with functional dependence on generalized piecewise constant argument where x ∈ R, t ∈ R, and , k and A remark the mass, the coefficient of damping, the spring constant and the magnitude of the force, respectively.Let D be a subset of the product R × C × C and h : D → R denote a continuous functional force in system (4).Let C s = {φ ∈ C | φ 0 ≤ s} where 0 < s ∈ R, and C 0 (W ) be the set of all bounded and continuous functions on W . Here, In system (4), x t and x γ(t) mean that With z 1 = x, z 2 = x , the damped spring-mass sytems ( 3) and ( 4) can be reduced to the first-order differential equations as follows: and where the matrices depend on the parameters of the spring-mass systems (3) and ( 4).Here, continuous functional force . The spring-mass system (5) is a linear homogeneous system with argumentfunction γ(t), and the system ( 6) is a quasilinear system with functional dependence on argument-function γ(t).
The aim of the present paper is to examine damped springmass systems with generalized piecewise constant argument and with functional dependence on generalized piecewise constant argument without transforming them into discrete equations, assuming the systems exhibit harmonic motion with under damped.The fundamental matrix of the homogeneous spring-mass system (5) is constructed in several intervals for illustration.Sufficient conditions for the existence and uniqueness of solutions of (6) are found.Existence of periodic solutions of the system ( 6) is investigated by using Green's function which have been obtained for differential equations with functional dependence on piecewise constant argument of generalized type [1].Then, we prove the uniqueness of the periodic solution.

Material and Method
In this section, we obtain the fundamental matrix of the linear homogeneous equation without piecewise constant argument, i.e. the system x = Bx.Then, we create the matrix-function and the fundamental matrix of the linear homogeneous equation ( 5) with piecewise constant argument in several intervals using the construction of the fundamental matrix of the differential equations with generalized piecewise constant argument [1].Additionally, we give the assumptions needed for our study.We state the initial conditions depending on two cases of the initial value t 0 .
2.1.The fundamental matrix of the linear homogeneous equation without piecewise constant argument Consider the system Equation given by ( 7) is the linear homogenous part of ( 5) and ( 6) without piecewise constant argument.Let X(t, s) denote the fundamental matrix of solutions of (7) satisfying X(s, s) =I, s ∈ R, where I is the 2×2 identity matrix.Since it is assumed that systems ( 5) and ( 6) exhibit motion with under damped, X(t, s) for ( 7) is in the following form = e −α(t−s) X 11 (t, s) X 12 (t, s) X 21 (t, s) X 22 (t, s) , where its indices are given by We see that the fundamental matrix X(t, s) has elements depending on the model we consider.
2.2.The matrix-function and fundamental matrix of the homogeneous spring-mass system (5) In [1], a matrix-function M i (t), i ∈ Z, is introduced as follows for the systems and The matrix-function is important for the investigation of existence and uniqueness of periodic solutions.We find the matrix-function M i (t) for the linear homogeneous system (5) with piecewise constant argument as and its indices K i , L i , M i , N i are in the following form: and Let us fix t 0 ∈ R and assume without loss of generality that θ i < t 0 < ζ i , i ∈ Z. Z(t) = Z(t,t 0 ) with Z(t 0 ) = Z(t 0 ,t 0 ) =I is called a fundamental matrix of the system (5).Let θ i ≤ t 0 < θ i+1 for a fixed i ∈ Z.For interval t ∈ [t 0 , θ i+1 ], the fundamental matrix is given by The fundamental matrix of ( 5) is defined for increasing t and decreasing t as expressed in [1,6].In other words, if In this context, we create the fundamental matrix of (5) in three intervals t ∈ [θ i , θ i+1 ], t ∈ [θ i−1 , θ i ] for decreasing t and t ∈ [θ i+1 , θ i+2 ] for increasing t for illustration.It is possible to obtain the fundamental matrix for more intervals.
First, we obtain the fundamental matrix of ( 5) in the following form This fundamental matrix has indices as follows 5) is in the form where indices are listed below; . (5) for increasing value of t is

(cos(β
where indices are given by . sin(β (t 0 − θ i+1 )) and The fundamental matrix can be obtained as shown above for any other intervals.

Assumptions
For the damped spring-mass systems ( 5) and ( 6), we shall need the following assumptions throughout the paper: D. Arugaslan, N. Cengiz / Green's Function and Periodic Solutions of a Spring-Mass System in which the Forces are Functionally Dependent on Piecewise Constant Argument (S1) h satisfies the Lipschitz condition for some constant L > 0: where (t, µ 1 , η 1 ) and (t, µ 2 , η 2 ) ∈ D; Moreover, assume that system ( 6) is ω−periodic with the following conditions: (S5) there are two numbers ω ∈ R and p ∈ Z such that We can infer from (S1) and (S6) that the functional force ) also satisfies the Lipschitz condition and periodicity condition.
In addition to the above conditions, we can define the initial conditions for the damped spring-mass system (6) with the functional force for the cases corresponding to t 0 ≤ ζ i or For a fixed number t 0 ∈ R, the functions µ, η ∈ K and some i ∈ Z, if θ i ≤ t 0 < θ i+1 : (K1) there exists a solution z (t) = z (t,t 0 , µ) satisfying the initial condition (K2) there exists a solution z (t) = z (t,t 0 , µ, η) satisfying the initial conditions z t 0 (s) = µ (s) and is a solution of ( 6) with (K1) or (K2) on an interval [t 0 ,t 0 + a) , a > 0, if: (i) it satisfies the initial condition, (iii) the derivative z (t) exists for t ≥ t 0 with the possible exception of the points θ i , where one-sided derivatives exists, (iv) equation ( 6) is satisfied by z (t) for all t > t 0 , except, possibly, the points of θ and it holds for the right derivative of z (t) at points θ i .
[1] A function z (t) is a solution of ( 6)(( 5)) on R if: (i) z (t) is continuous, (ii) the derivative z (t) exists for all t ∈ R with the possible exception of the points θ i , i ∈ Z, where one-sided derivatives exists, (iii) equation ( 6)(( 5)) is satisfied by z (t) for all t ∈ R, except, points of θ and it holds for the right derivative of z (t) at points θ i .

Results
In this section, we give the sufficient conditions for the existence and uniqueness of solutions and periodic solutions of the damped spring-mass system (6).We create the periodic solution using Green's function with the initial condition corresponding to the Poincare criterion for differential equations with generalized piecewise constant argument.

Existence and uniqueness of solutions
The following lemmas give necessary conditions for existence and uniqueness of solutions of the damped springmass system (6).
Proof.Consider the initial condition (K1) and so a solution of the form z (t) = (z 1 (t), z 2 (t)) T = z (t,t 0 , µ) with θ i ≤ t 0 ≤ ζ i < θ i+1 for fixed i ∈ Z.The proof for the initial condition (K2) can be performed as in the case of functional differential equations [43].

Existence and uniquness of periodic solutions
In addition to the assumptions, let ζ 0 = 0 without loss of generality, and Besides, the matrix Q is the monodromy matrix defined by where p ∈ Z such that θ k+p = θ k + ω and Eigenvalues of the matrix Q or Z(ω), ρ j , j = 1, 2, are called multipliers [1].For the spring-mass system (6), using the formula (8) we find the matrix G k as follows where and The matrix Q can be obtained in terms of the matrices G k with the value p ∈ Z corresponding the sequences θ = (θ i ) and ζ = (ζ i ), i ∈ Z. So, the multipliers can be found, and periodicities of the solutions of the spring-mass system ( 6) can be researched.As a result, the existence of periodic solutions is certain if there exists a unit multiplier.However, periodic solutions can also be found in the absence of unit multipliers i.e. in the non-critical case.In this study, a periodic solution of the damped spring-mass system ( 6) is created for the non-critical case with the help of Green's function.In the interval t ∈ θ j , θ j+1 , with Z(t) = Z(t, 0), t ∈ R, the solution z(t) = z(t, 0, z 0 ) satisfies the following integral equation ζ k e B(θ k+1 −s) f (s, z s , z γ(s) )ds( 9) The solution ( 9) is a periodic solution of the system (6) if the initial condition is taken according to Poincare criterion [24], [25], in the following form where det[I − Z(ω)] −1 = 0. Substuting the initial condition (10) in the equation ( 9), we obtain the integral equation This solution z(t) is a continuous function.Thus, Green's function G p (t, s), t, s ∈ [0, ω] for the periodic solution can be constructed in t ∈ θ j , θ j+1 , j = 0, 1, ..., p − 1 as follows So, the periodic solution of the system ( 6) is expressed in the form In the next theorem, the sufficient conditions for the springmass system (6) to have a unique ω-periodic solution are given.
Proof.Let the complete metric space C λ (R) denote the sets of all continuous and ω−periodic functions on R. Define on C λ (R) an operator such that where t ∈ θ j , θ j+1 , j = 0, 1, 2, ..., p − 1 and It can be seen that ∏ : So, the condition 2 RLω < 1 shows the uniqueness of the periodic solution (11).The proof is completed.

An Example
i, consider the linear nonhomogeneous spring-mass system with piecewise constant argument of generalized type and delayed argument Let τ = 1 and the initial condition z t 0 (s) = µ(s) = (0.02029980811, −0.01018243128) T , s ∈ [−1, 0].Taking z 1 = x and z 2 = x , spring-mass system (12) can be reduced to the following nonhomogeneous differential equation .

Discussion and Conclusion
Differential equations are important to model real world problems in many areas.Nevertheless, the modeling the problems with differential equations may not reflect reality if we ignore the effects of delays and discontinuities.For this reason, differential equations with deviating argument that produce more realistic models have great importance.The differential equations with deviating argument include delay differential equations, functional differential equations, differential equations with piecewise constant argument and differential equations with generalized piecewise constant argument.Many scientists have worked on the theory and applications of these equations.Moreover, Akhmet contributed to these studies by introducing differential equations with functional dependence on generalized piecewise constant argument.This contribution increases the realism of the models.In applications, the spring-mass system has an importance in many areas such as physic, mathematics, biomechanics, biology.In our study, we modeled the spring-mass systems using differential equations with generalized piecewise constant argument and with functional dependence on generalized piecewise constant argument.So, we obtain more realistic and detailed analysis.Later, we analyze the qualitative behaviors of these spring-mass systems, and give the sufficient conditions for the existence and uniqueness of periodic solutions.Periodicity provides information about behavior of solution for the other intervals, with knowledge of the qualitative behavior of the system in a particular interval.Therefore, periodicity in a system is a desired feature, and a lot of study about existence of periodic solutions are available in the literature.Our examination is considerable since it is obvious that periodicity is significiant in both theory and practice.In the literature, differential equations with deviating argument are generally studied by reducing into discrete equations.We examine our models without reducing them into discrete equations.It shows our work's novelty from the other studies in the literature.

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. Arugaslan, N. Cengiz / Green's Function and Periodic Solutions of a Spring-Mass System in which the Forces are Functionally Dependent on Piecewise Constant Argument then we have
. Arugaslan, N. Cengiz / Green's Function and Periodic Solutions of a Spring-Mass System in which the Forces are Functionally Dependent on Piecewise Constant Argument D