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Abstract
In this paper, we give existence and uniqueness results of nontrivial positive solution of the singular and non-autonomous kind of Duffing oscillator by using fixed point index theory.
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References
- Agarwal, R.P. and O’Regan, D., Infinite interval problems for differential, difference and integral equations, Kluwer Academic Publisher, Dordrecht, 2001.
- Avramescu, C. and Vladimirescu, C., ”Existence of solutions to second order ordinary differrential equations having finite limits at ±∞”, Electronic Journal of Differen-tial Equations, Vol. 2004(2004), No. 18, pp. 1.12.
- Bonheure, D. and Torres, P.J., ”Bounded and homoclinic-like solutions of a second-order singular differential equation”, Bull. Lond. Math. Soc. 44, 47–54 (2012).
- Corduneanu, C., Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973.
- Erbe, L.H. and Mathsen, R.M., ”Positive solutions for singular nonlinear boundary value problems”, Nonlin. Anal. 46 979–86, 2001.
- Eze, E.O., Ujumadu, R.N., Ezugorie, G.S., and Uchenna, E.O., ”Existence of Homoclinic Orbits and Conditions for the Onset of Chaotic Behavior in a Perturbed Double-Well Oscillator”, International Journal of Mathematical Analysis, Vol. 14, 2020, no. 7, 329 – 343.
- Eze, S.C., ”Analysis of fractional Duffing oscillator”, Revista Mexicana de Fısica 66 (2) 187–191 ( 2020).
- Minh´os, F. and Carrasco, H., ”Existence of homoclinic solutions for nonlinear second order discontinuous problems”, Mediterranean Journal of Mathematics 13(6)(2016).
- Alessio, F., Caldiroli, P., and Montecchiari, P., ”On The Existence of Homoclinic Orbits for the Asymptotically Periodic Duffing Equation”, Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center, Volume 12, 1998, 275–292.
- Guo, D. and Lakshmikantaham, V., Nonlinear Problems in Abstract Cones, Academic Press, San Diego, 1988.
- Jim´enez, S., Gonz´alez, J.A., and V´azquez, L., ”Fractional Duffing’s equation and geometrical resonance”, International Journal of Bifurcation and Chaos, 23 (2013), 1350089-13.
- Murray, J.D., Mathematical biology, An introduction, Springer-Verlag, 2001.
- Torres, P.J., ”Existence and Stability of Periodic Solutions of a Duffing Equation by Using a New Maximum Principle”, Mediterr. j. math. 1 (2004), 479–486.
- Torres, P.J., ”Guided waves in a multi-layered optical structure”, Nonlinearity 19 (2006) 2103–2113.
- Cheng, Z. and Yuan, Q., ”Damped superlinear Duffing equation with strong singularity of repulsive type”, Journal of Fixed Point Theory and Applications volume 22, Article number: 37 (2020).
- Zeidler, E., Nonlinear Functional Analysis and its applications , Vol. I, Fixed point theorems, Springer-Verlag, New-York 1986.
- Zima, M., ”On positive solutions of boundary value problems on the half-line”, J. Math. Anal. Appl. 259 (2001) 127–136
References
Agarwal, R.P. and O’Regan, D., Infinite interval problems for differential, difference and integral equations, Kluwer Academic Publisher, Dordrecht, 2001.
Avramescu, C. and Vladimirescu, C., ”Existence of solutions to second order ordinary differrential equations having finite limits at ±∞”, Electronic Journal of Differen-tial Equations, Vol. 2004(2004), No. 18, pp. 1.12.
Bonheure, D. and Torres, P.J., ”Bounded and homoclinic-like solutions of a second-order singular differential equation”, Bull. Lond. Math. Soc. 44, 47–54 (2012).
Corduneanu, C., Integral Equations and Stability of Feedback Systems, Academic Press, New York, 1973.
Erbe, L.H. and Mathsen, R.M., ”Positive solutions for singular nonlinear boundary value problems”, Nonlin. Anal. 46 979–86, 2001.
Eze, E.O., Ujumadu, R.N., Ezugorie, G.S., and Uchenna, E.O., ”Existence of Homoclinic Orbits and Conditions for the Onset of Chaotic Behavior in a Perturbed Double-Well Oscillator”, International Journal of Mathematical Analysis, Vol. 14, 2020, no. 7, 329 – 343.
Eze, S.C., ”Analysis of fractional Duffing oscillator”, Revista Mexicana de Fısica 66 (2) 187–191 ( 2020).
Minh´os, F. and Carrasco, H., ”Existence of homoclinic solutions for nonlinear second order discontinuous problems”, Mediterranean Journal of Mathematics 13(6)(2016).
Alessio, F., Caldiroli, P., and Montecchiari, P., ”On The Existence of Homoclinic Orbits for the Asymptotically Periodic Duffing Equation”, Topological Methods in Nonlinear Analysis Journal of the Juliusz Schauder Center, Volume 12, 1998, 275–292.
Guo, D. and Lakshmikantaham, V., Nonlinear Problems in Abstract Cones, Academic Press, San Diego, 1988.
Jim´enez, S., Gonz´alez, J.A., and V´azquez, L., ”Fractional Duffing’s equation and geometrical resonance”, International Journal of Bifurcation and Chaos, 23 (2013), 1350089-13.
Murray, J.D., Mathematical biology, An introduction, Springer-Verlag, 2001.
Torres, P.J., ”Existence and Stability of Periodic Solutions of a Duffing Equation by Using a New Maximum Principle”, Mediterr. j. math. 1 (2004), 479–486.
Torres, P.J., ”Guided waves in a multi-layered optical structure”, Nonlinearity 19 (2006) 2103–2113.
Cheng, Z. and Yuan, Q., ”Damped superlinear Duffing equation with strong singularity of repulsive type”, Journal of Fixed Point Theory and Applications volume 22, Article number: 37 (2020).
Zeidler, E., Nonlinear Functional Analysis and its applications , Vol. I, Fixed point theorems, Springer-Verlag, New-York 1986.
Zima, M., ”On positive solutions of boundary value problems on the half-line”, J. Math. Anal. Appl. 259 (2001) 127–136