Existence of solutions for second-order differential equations with discontinuous right-hand side

We consider an ordinary differential equation of second order with constant coefficients and a discontinuous right-hand side. First we use the point mapping method defining first return functions, then we use the phase-plane method. We establish both the existence and non-existence of periodic...

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Main Authors: Alexander M. Kamachkin, Dmitriy K. Potapov, Victoria V. Yevstafyeva
Format: Article
Language:English
Published: Texas State University 2016-05-01
Series:Electronic Journal of Differential Equations
Subjects:
Online Access:http://ejde.math.txstate.edu/Volumes/2016/124/abstr.html
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author Alexander M. Kamachkin
Dmitriy K. Potapov
Victoria V. Yevstafyeva
author_facet Alexander M. Kamachkin
Dmitriy K. Potapov
Victoria V. Yevstafyeva
author_sort Alexander M. Kamachkin
collection DOAJ
description We consider an ordinary differential equation of second order with constant coefficients and a discontinuous right-hand side. First we use the point mapping method defining first return functions, then we use the phase-plane method. We establish both the existence and non-existence of periodic solutions (including stable ones) and oscillatory solutions depending on the coefficients of the equation. By the variational method, we prove the existence of nonzero semiregular solutions for a boundary-value problem.
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spelling doaj.art-3247c7c5324741e7af39a4721f470bfa2022-12-21T19:41:07ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912016-05-012016124,19Existence of solutions for second-order differential equations with discontinuous right-hand sideAlexander M. Kamachkin0Dmitriy K. Potapov1Victoria V. Yevstafyeva2 Saint Petersburg State Univ., Russia Saint Petersburg State Univ., Russia Saint Petersburg State Univ., Russia We consider an ordinary differential equation of second order with constant coefficients and a discontinuous right-hand side. First we use the point mapping method defining first return functions, then we use the phase-plane method. We establish both the existence and non-existence of periodic solutions (including stable ones) and oscillatory solutions depending on the coefficients of the equation. By the variational method, we prove the existence of nonzero semiregular solutions for a boundary-value problem.http://ejde.math.txstate.edu/Volumes/2016/124/abstr.htmlPoint mapping methodfirst return functionphase-plane methodperiodic solutionstabilityvariational methodsemiregular solution
spellingShingle Alexander M. Kamachkin
Dmitriy K. Potapov
Victoria V. Yevstafyeva
Existence of solutions for second-order differential equations with discontinuous right-hand side
Electronic Journal of Differential Equations
Point mapping method
first return function
phase-plane method
periodic solution
stability
variational method
semiregular solution
title Existence of solutions for second-order differential equations with discontinuous right-hand side
title_full Existence of solutions for second-order differential equations with discontinuous right-hand side
title_fullStr Existence of solutions for second-order differential equations with discontinuous right-hand side
title_full_unstemmed Existence of solutions for second-order differential equations with discontinuous right-hand side
title_short Existence of solutions for second-order differential equations with discontinuous right-hand side
title_sort existence of solutions for second order differential equations with discontinuous right hand side
topic Point mapping method
first return function
phase-plane method
periodic solution
stability
variational method
semiregular solution
url http://ejde.math.txstate.edu/Volumes/2016/124/abstr.html
work_keys_str_mv AT alexandermkamachkin existenceofsolutionsforsecondorderdifferentialequationswithdiscontinuousrighthandside
AT dmitriykpotapov existenceofsolutionsforsecondorderdifferentialequationswithdiscontinuousrighthandside
AT victoriavyevstafyeva existenceofsolutionsforsecondorderdifferentialequationswithdiscontinuousrighthandside