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 Advances in Pure Mathematics, 2011, 1, 243-244 doi:10.4236/apm.2011.14043 Published Online July 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM An Inequality for Second Order Differential Equation with Retarded Argument Erdoğan Şen Department of Mathematics, Namık Kemal University, Tekirdağ, Turkey E-mail: esen@nku.edu.tr Received April 23, 2011; revised May 10, 2011; accepted May 20, 2011 Abstract Applications of differential equations with retarded argument can be encountered in the theory of automatic control, in the theory of self-oscillatory systems, in the study of problems connected with combustion in rocket engines, in a number of problems in economics, biophysics. The problems in this areas can be solved reducing differential equations with retarded argument. In this work an important inequality for second order differential equation with retarded argument is obtained. Keywords: Differential Equation with Retarded Argument, Inequality 1. Introduction In this study we consider the equation  0Lwtwt Mtwtw t (1) on an interval I. Where  is a real parameter; Mtt and are continuous functions on I; 0tt 10t tIt and for each . 2. An Inequality for Second Order Differential Equation with Retarded Argument Theorem. Let us denote by every point with 0jwhich is satisfying the mean-value theorem for a continuous solu- tion jwt of (1) on ,jjtttI  jtIfor each  and , where jJJ is an index set. Also let us as- sume that 0supMtM where 0M is a real number. Then for all jt in the equation 000eejjjjjkt tkt twtwt wt 0jj (2) where 1/ 222003,1 2jjjwtwtw tkM  Proof. From the mean-value theorem we can write the followings: 0jjjjjwt wttwtt  0jjjj jwttwtw tt and 0jjj jwttwtw t   (3) 2jjut wtNow we let . Thus uwwww where jjwt wt. Then u wwwwwwww    From the definition of a derivative it follows that ww . Also jjwt wt. Therefore 0022jjjjjutwtwtwtw tw (4) 0LwSince satisfies we have  jjjjjwtwtM twtt  and hence applying (3)  000jjj jjjjwtwtM wttwt Mwt wt    (5) 244 E. ŞEN Using (5) in (4) we obtain     0000002221 2jjjjjjjjutwt wtwtwt Mwt wtMwtwt  200jjMwt Now applying the fact 2200jjw t2jjwtwtwt we get 2020020113321 2jjjjutM wtMwtMwt  20jwt or 2jjkut 2ut This is equivalent to 2jjjtku t20ukuku tu (6) And these inequalities lead directly to (2). Indeed con-sider the right inequality which can be written as 22e2e0jjkt ktuku u. It is equivalent to 0jjtt0t0jt If we integrate from to obtaining 0220ee 0jjjktktjut ut or 02()0ejjjkt tjut ut The left inequality in (6) similarly implies 0()00e,jjjjkt tjjwtwtt t and therefore 00000ee,jjjjjjjkt tkt tjjwtwtwtt t 0 jj. The case which is just (2) for tt0jjttmay be considered analogically. 3. References [1] E. A. Coddington and N. Levinson, “Theory of Ordinary Differential Equations,” McGraw-Hill, New York, 1955. [2] S. B. Norkin, “Differential Equations of the Second Order with Retarded Argument,” AMS, Providence, 1972. Copyright © 2011 SciRes. APM