The existence and approximation of the periodic solutions for system of second order nonlinear differential equationsby using Lebesgue integrable

In this paper we study the existence and approximation of the periodicsolutions for system of second order nonlinear differential equations according to Lebesgue integrable concept and by assuming that each of measurable at t and bounded by the functions Lebesgue integrable functions numerical -- analytic method has been used to study the periodic The solutions of ordinary differential equations which were introduced by A.M. Samoilenko


ABSTRACT
In this paper we study the existence and approximation of the periodicsolutions for system of second order nonlinear differential equations according to Lebesgue integrable concept and by assuming that each of measurable at t and bounded by ) , , , the functions Lebesgue integrable functions numerical --analytic method has been used to study the periodic The solutions of ordinary differential equations which were introduced by A.M. Samoilenko.

INTRODUCTION
There are many subjects in physics and technology use mathematical methods that depends on the nonlinear differential equations, and it became clear that the existence of the periodic solutions and its algorithm structure from more important problems in the present time, because of the great possibility for employment the electronic computers the numerical analytic method [6] which suggested by Samoilenko to study the periodic solutions for the linear and nonlinear differential equations became the effective mean to find the periodic solutions and its algorithm structure and this method include uniformly sequences of the periodic functions and the result of that study is the using of the periodic solutions on a wide rang in the difference of the new processes in industry and technology as in the studies [1,2,4,5]. We study in this paper the system of the non linear differential equations with the form       3  2  1  2  2  1  1  2  1  2  1   ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  D  y  y  y  and  D  x  x  x  D  y  y  y  D  x  x are two chosen points so that

Definition 1 [6] :-
The system of nonlinear differential equations (1) where the righthand side defined and continuous and periodic in t of period T in the domain (2) is said to be system -T if 1-The two sets are not empty T  T  R  J  N  R  T  T  R  T  R  T   QC  QC  QC  QH  T  QC  H  N  Q  T   QC  T  QC  T  QC  QH  T  T  QC  H  N  Q  T  T is not greater than 1 that is 4  4  2  3  5  2  3  1  5  4   3  2  3  4  2  2  1  1 , , ,

Definition 2 [6]:-
The value of intermediary ) , ( if the intermediary is unique in that point .
Section One : The periodic approximate solution for the system (1) Assume that each of  )  ,  ,  ,  ,  (  )  ,  ,  ,  ,  (  y  x  y  x  t  g  and  y  x  y  x  t  f  &  &  &  & be vector function and continuous and defined in the interval [0,T] then the inequality

Lemma 1 :-
The existence and approximation …. 85 and also  (15) and (16) we conclude that the inequality (14) holds for be defined, Continuous and periodic in t on the interval [0,T] then each of )) , , are also continuous and periodic in t defined on the same interval By lemma 1 we obtain The existence and approximation ….

Theorem 1:-
Assume that the system (1) satisfy the inequalities (3),(4),(5) and the conditions (11), (12) and if each of are measurable functions at t in the system (1) and defined in the domain (2) and satisfy the inequalities above and if the inequalities (9), (10) satisfies  also  and  if  for  the  system  a  periodic  solution  ) , then the two sequences of the functions which are a unique solution for the system (1) and satisfies the following inequality  (25),(26) we find that each of the sequences of the functions are defined and continuous in the domain (2) and periodic in t of period T . By lemma 1 and from (25) when m=0 we obtain )) , , , also from the relation (19) and by lemma 1 when m=0 we obtain )) , , By using (21) we get , also from the relation (26) when m=0 we obtain )) , , , also from the relation (20) when m=0 we obtain )) , , and by using the mathematical induction we can prove the truth of the following inequalities for are uniformly convergent in the domain (27) and thereupon each of the two ends for the two sequences are periodicity and continuous in the same domain . from the previous proof we find Now when m=1 in (25) and by using (23) we find (19) and by using (21) we find also when m=1 in (26) and by using (24) we find also when m=1 in (22) and by using (20) we find And so by using the mathematical induction we can prove the truth of the following two inequalities : Now we take the maximum value for the two sides of the inequality (36)for And in addition for that the using of lemma 1 and the relation (40) the inequality (30) satisfies for m ≥ 0 now we prove that ) , , ( and by repetition (50) we obtain and then we have from the last inequality that are unique solution for the system (1)

Section Two : The existence of the periodic solution for the system (1)
The problem of the existence of the periodic solution for the system (1) is connected with unique form with existence zero for all of the two functions ) , ( ), , ( it is the end of the sequence ) , , (

Theorem 2:-
If were the assumptions and the conditions of the theorem 1 be given then the following inequalities : also by the equations (52), (54) and the conditions (30) we have By the helping of the theorem 2 we introduce the following theorem and take into consideration the truth of the two inequalities (55) and (56)for 0 ≥ m .

Theorem 3:-
Let the right hand side from the system (1) are defined in the on R 1 and assuming that the two sequences of the functions (53),(54) satisfies the following inequalities :

Remark 2:-
The theorem 4 confirm the stability of the solution for the system of non linear differential equations that is when a slight change happening in the point ) , ( [for this remark return to [3] ].