In this paper we prove the of solutions, of some kinds of differential equations, and system of first order differential equations are bounded. Key words: Boundedness of solutions, first order ordinary  Differential equations, systems of differential equations and limit cycles

Abdulla Abid, Na'mh

AL –Azzawi, Saad N.

Joudha, Zainab Mohmmad

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.9, 2014  151 Bounded Solutions of Certain (Differential and first order systems of Differential Equations) Saad N. AL –Azzawi,             Na'mh Abdulla Abid          and       Zainab Mohmmad Joudha* Dept. of Math., College of Science for Women, Univ. of Baghdad Saad_naji2007@yahoo.com, namh_abed@yahoo.com, *zaina600@yahoo.com Abstract : In this paper we prove the of solutions, of some kinds of differential equations, and system of first order differential equations are bounded. Key words: Boundedness of solutions, first order ordinary  Differential equations, systems of differential equations and limit cycles ..  1- Introduction  A  solution x(t) is called bounded on [0, ∞  if there exist a positive constant  M such that   x(t)    for all t [0, ∞      . Boundedness is one of the mathematical properties that is needed in applications such as population growth, trophic   function, capacity, voltage, differential and integral equations, chemical reactions etc. [1], [2]. It can be determined by finding the functions that make the solution bounded, or by using the stability of limit cycles, or determining the bounded region in which if the solution inter the region then it stay their for all t . Differential equations of first order describe many real life applications such as population growth, nuclear decoy, Newton law of cooling, so the concept of boundedness of the solutions, is important in this sense. Mawhin [3] studied the boundedness of solutions of the differential equation.            and that of its corresponding difference equation               ,  m   Jose L. and Bravo and his colleagues, [4] studied the first order differential equation            and they defined a subset D of all continuous functions f: ℝxℝ ℝ such that f satisfies four conditions and defining  subsets of D called           of all points f such that            has bounded solutions.  Tineo [7] studied           +  Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.9, 2014  152 and showed that there exists      ℝ such that the equation has at least one or two separated bounded solutions . Also he showed that when f satisfies the concavity condition with respect to x, then the equation has exactly one or two separated bounded solutions. Rao [6] studied the bounded of solutions of differential equations        ,     (      )                                     and                  Where A is nxn constant matrix and B(t) is nxn variable matrix. 2- Preliminaries We can use the following concepts and theorems for studying boundedness. Theorem (1) [5]:- Let                 then  (1)  ∫           is convergent if p   and A is finite. (2)  ∫           is divergent if p   and A        A may be infinite. Theorem (2) [5]:-  Let                     then  (1) ∫           is convergent if p   and A is finite. (2) ∫           is divergent if p   and A    , A may be infinite. Remark [6]:- (1) Every solution inside the limit cycles is bounded. (2) If there exist one stable limit cycle then every solution is bounded. (3) If there exist one semi-stable limit cycle enclosing sink point then every solution is bounded. (4) If there exist one semi-stable limit cycle enclosing source point then the solution outside is unbounded. 3- Main Results In this section we find the conditions that ensure the boundedness of solutions. 3.1  Boundedness of solutions of certain differential  equations. The following lemmas concern the boundedness of solutions of specific third and first order ordinary differential equations. Lemma (1):- Every solution of                                                                                (1)                                  , is bounded if ∫ |    |      . Proof:- The general solution of the corresponding homogeneous part is                                                                                        It is clear that the  solution (2) is bounded. Now to find the particular of solution of equation (1), we apply the variation of parameters method.  For this purpose, let Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.9, 2014  153                            , where v1, v2 and v3 are functions should be determined. To do so, are can solve system of equations,                                             -p                                                                                                           -p2                                     It is a system of three parameters                  to find them we use Grammar method, are  can get the following :                                                     To find v1, v2     v3 , we integrate                  from 0 to           v1      ∫          v2      ∫               v3       ∫                Thus, the particular solution is, up=   ∫                    ∫                        ∫                               ∫                                    |  |     ∫ |    |     is bounded if ∫ |    |       Thus, the solution of equation (1) is bounded. Lemma (2):- Every solution of the initial value problem                                                                           (4) approaches  to zero as t     when f(t)       Proof:- The solution of this Bernoulli equation is                                                                                                                                     Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.9, 2014  154 Let z=   ,                        -2z = f(t)          +2z =- f(t) I.F.  =              [z    ] = -          z      ∫               z(t)= -    ∫              z(t) approaches zero as t                  Lemma (3):- Every solution of                                                                                         approach to zero as |    |                     . Proof:- The Equation (6) is first order linear equation, its general solution is         ∫                   |    |      ∫ |    |      | |              |    |          , then        |    |                                   .  Lemma (4):- Every solution of ;    (      )                                        Converges to zero if f(t) is continuous on [0,       ∫             . Proof:- Rewrite equation (7) as         (      )   Now by integrating both sides  from zero to t, we get u(t)=      ∫            Now  |    |  |     ∫           |=      ∫            Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.9, 2014  155 Which is bounded and approaching zero as t→ .  3.2  Boundedness of solution of systems of first order differential equation. Lemma (1):- Let                                   such that ‖    ‖      t    and          ∫                                                                                   If B(t) is a real nxn continuous matrix for t   with ∫ ‖         ‖                                                                                  then every solution of yꞋ=B(t)y, is bounded on [0,    Proof:- The solution 0f yꞋ=B(t)y  can be written as: y(t) = x(t)- ∫                                                            Where      is the fundemental matrix of the system x'=A(t)x, such that       . It should be noted that x(t)=      , since‖    ‖   , t                                  , then        is bounded in view of the condition(8)           ‖      ‖    ‖    ‖ . By taking the norm to equation (10), we get                   ‖    ‖       ∫ ‖         ‖‖    ‖      Now by using Gronwall-Reid-Bellman inequality, then  ‖    ‖           ∫ ‖         ‖      the result follows, by using the inequality (9). Lemma (2):- If all the solutions of x'=A(t)x                                                                                            (11)  are bounded, then all the solutions of (11) x'=A(t)x+f(t,x)                                                                                 (12) are also bounded provided that the following conditions are satisfied: (1)  ‖      ‖  |    |‖ ‖ (2)  ∫            (3)        ∫                    (4)    Tr A(t) = 0. Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.9, 2014  156 Proof:- Expressing the solution z(t) of  x'=A(t)x+f(t,x) in terms of the solution x(t) of x'=A(t)x, we have    z(t)= x(t)+ ∫        (      )                                                                                        Where      is the fundamental matrix of (11) with      = I. It should be noted that x(t) =       . Since all the solutions of (11) are bounded, then ‖    ‖  is bounded, and since                                     ∫             , then ‖      ‖ is bounded also, that ‖      ‖ is bounded. Now, let           ‖      ‖    ‖    ‖  then, ‖    ‖       ∫     ‖    ‖     and by using Gronwall-Reid-Bellman inequality, and from condition (2) we have the result.  Lemma (3):- If all the solutions of x'=A(t)x ,                                                                       approach zero as t    then the same is true for the solutions of system x'=A(t)x+f(t,x) provided  that conditions (1) and (2) in lemma(2) together with condition (3) or (4), hold. Proof:- Expressing the solution z(t) of  x'=A(t)x+f(t,x) in terms of the solution x(t) of x'=A(t)x, we have    z(t)= x(t)+ ∫        (      )                                                                                       Where      is the fundamental matrix of x'=A(t)x with      = I. It should be noted that z(0) =           , since all the solutions of (11) are approaching zero as t   , we have  ‖    ‖ is approaching 0 as      Thus, in the view of  condition(3) and the fact that ‖    ‖  is approaching 0 as    , we have ‖      ‖ is approaching 0 as    . Since both ‖    ‖  and ‖      ‖ are approaching 0 as     and are continuous , then they are bounded for all t                     ‖      ‖    ‖    ‖  Therefore, from equation (14) we obtain: ‖    ‖  ‖    ‖‖  ‖  ∫ ‖      ‖‖        ‖     and from the condition (1) of lemma (2), we have  ‖    ‖       ∫ |    |‖    ‖     Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.4, No.9, 2014  157 Now by applying Gronwall-Reid-Bellman inequality we obtain, for all t     ‖    ‖           ∫ |    |                  ‖    ‖      , since all the solutions of (11) approach zero as t   then, the same is true for (12) .   REFERNCES   [1].  Arturas Stikonas ," Differential Equations    , Course of Lectures. Vilnius University , Lithuania 2012. [2]. James D. Neiss." Differential Dynamical Systems". Philadelphia : SIAM , Mathematical Modeling and Computation 2007.  [3]. Jean Mawhin ," Bounded solutions  Differential Vs difference equations", Electronic Journal of Differential equations Conf. 17 (2009) 11.159-170 .  [4]. Jose L.Bravo, Manuel Fernandez and Antonio Tineo," The Set of first order Differential Equations with periodic or bounded solutions", Extracta Mathematica Vol. 16, 2001, No.2, 293 – 295. [5]. Murray Spiegel," Theory and Applications of Advanced Calculus", McGraw-Hill Company, London 1974.  [6]. Rao M," Ordinary Differential Equations". Theory and Applications, Edward Arnold U.K. 1981.  [7]. Tineo, A. first Order Ordinary Differential Equations .with Several Bounded Separated solutions , J. Math. Apral. April 225, (1998), 359-372. The IISTE is a pioneer in the Open-Access hosting service and academic event management.  The aim of the firm is Accelerating Global Knowledge Sharing.  More information about the firm can be found on the homepage:  http://www.iiste.org  CALL FOR JOURNAL PAPERS There are more than 30 peer-reviewed academic journals hosted under the hosting platform.   Prospective authors of journals can find the submission instruction on the following page: http://www.iiste.org/journals/  All the journals articles are available online to the readers all over the world without financial, legal, or technical barriers other than those inseparable from gaining access to the internet itself.  Paper version of the journals is also available upon request of readers and authors.   MORE RESOURCES Book publication information: http://www.iiste.org/book/  IISTE Knowledge Sharing Partners EBSCO, Index Copernicus, Ulrich's Periodicals Directory, JournalTOCS, PKP Open Archives Harvester, Bielefeld Academic Search Engine, Elektronische Zeitschriftenbibliothek EZB, Open J-Gate, OCLC WorldCat, Universe Digtial Library , NewJour, Google Scholar   

Bounded Solutions of Certain (Differential and first order systems of Differential Equations)

https://www.iiste.org/Journals/index.php/MTM/article/download/15084/15209

Bounded Solutions of Certain (Differential and first order systems of Differential Equations)

Abstract

Similar works

Full text

Available Versions

International Institute for Science, Technology and Education (IISTE): E-Journals