SEPARATE CONTRACTION AND EXISTENCE OF PERIODIC SOLUTIONS IN TOTALLY NONLINEAR DELAY DIFFERENTIAL EQUATIONS

Abstract

In this study, we employ the fixed point theorem of Krasnoselskii and
the concepts of separate and large contractions to show the existence
of a periodic solution of a highly nonlinear delay differential equation.
Also, we give a classification theorem providing sufficient conditions for
an operator to be a large contraction, and hence, a separate contraction.
Finally, under slightly different conditions, we obtain the existence of
a positive periodic solution.

Keywords

• Krasnoselskii’s fixed point theorem,Large contraction,Periodic solution,Positive periodic solution,Separate contraction,Totally nonlinear delay differential equations

SEPARATE CONTRACTION AND EXISTENCE OF PERIODIC SOLUTIONS IN TOTALLY NONLINEAR DELAY DIFFERENTIAL EQUATIONS

Abstract

References

1. Burton, T. A. Stability by Fixed Point Theory for Functional Differential Equations (Dover, New York, 2006).
2. Kaufmann, E. R. and Raffoul, Y. N. Periodicity and stability in neutral nonlinear dynamic equations with functional delay on a time scale, Electron. J. Differential Equations No. 27, 12pp, 2007.
3. Kaufmann, E. R. and Raffoul, Y. N. Periodic solutions for a neutral nonlinear dynamical equation on a time scale, J. Math. Anal. Appl. 319 (1), 315–325, 2006.
4. Liu, Y. and Li, Z. Schaefer type theorem and periodic solutions of evolution equations, J. Math. Anal. Appl. 316, 237–255, 2006.
5. Liu, Y. and Li, Z. Krasnoselskii type fixed point theorems and applications, Proc. Amer. Math. Soc. 136 (4), 1213–1220, 2008.
6. Raffoul, Y. N. Periodic solutions for neutral nonlinear differential equations with functional delay, Electron. J. Differential Equations 102, 1–7, 2003.
7. Raffoul, Y. N. Positive periodic solutions in neutral nonlinear differential equations, E. J. Qualitative Theory of Diff. Equ. 16, 1–10, 2007.
8. Smart, D. R. Fixed Points Theorems (Cambridge Univ. Press, Cambridge, 1980).