### Paper Menu >>

### Journal Menu >>

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; M t 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 j wt of (1) on , jj tttI j tIfor each and , where jJ J is an index set. Also let us as- sume that 0 sup M tM where 0 M is a real number. Then for all j t in the equation 00 0 ee jj jj j kt tkt t wtwt wt 0 j j (2) where 1/ 2 2 2 00 3 ,1 2 j jj wtwtw tkM Proof. From the mean-value theorem we can write the followings: 0j jjj j wt wtt wt t 0j j jj j wttwtw tt and 0j jj j wttwtw t (3) 2 jj ut wt Now we let . Thus uwwww where j j wt wt. Then u wwwwwwww From the definition of a derivative it follows that ww . Also j j wt wt. Therefore 00 22 jj j jj utwtwtwtw t w (4) 0Lw Since satisfies we have jjjjj wtwtM twtt and hence applying (3) 0 00 j jj jj jj wtwtM wtt wt Mwt wt (5) 244 E. ŞEN Using (5) in (4) we obtain 00 00 00 22 21 2 jj j jj jj j utwt wtwt wt Mwt wt Mwtwt 2 00 jj Mwt Now applying the fact 2 2 00 jj w t 2jj wtwtwt we get 2 0 2 00 2 0 1 13 3 21 2 j jj j utM wt Mwt Mwt 2 0 j wt or 2 j j kut 2 ut This is equivalent to 2 j jj tku t 20uku ku tu (6) And these inequalities lead directly to (2). Indeed con- sider the right inequality which can be written as 22 e2e0 jj kt kt uku u . It is equivalent to 0j j tt0 t0j t If we integrate from to obtaining 0 2 2 0 ee 0 jj j kt kt j ut ut or 0 2() 0ejj j kt t j ut ut The left inequality in (6) similarly implies 0 () 00 e, jj jj kt t jj wtwtt t and therefore 00 000 ee, jj jj jjj kt tkt t jj wtwtwtt t 0 j j. The case which is just (2) for tt0 j j ttmay 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 |