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

• 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: Point mapping method; first return function; phase-plane method; periodic solution; stability; variational method; semiregular solution
• Online Access: http://ejde.math.txstate.edu/Volumes/2016/124/abstr.html
• ISSN: 1072-6691

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.