Periodic solutions of Abel equations with jumps. (English) Zbl 1448.34039
The authors study the existence of \(T\)-periodic solutions of Abel-type equations with jumps.
More precisely, they consider the equation \[ \theta' = P(\theta) + I(\theta) \tag{1} \] where \(P(\theta) = f_0 + \sum_{j=1}^mf_j\theta^j\) is a polynomial term in \(\theta\) with \(T\)-periodic coefficients of bounded variation and \(I(\theta)=\sum_{k=1}^n a_k(\theta)J_{\tau_k}(\theta)\) is a step function with jumps of state-dependent amplitude at state-dependent instants, which satisfies the condition \[ \sum_{k=1}^n a_k(\theta)=0. \tag{2} \]
Under suitable assumption, the existence of an absolutely continuous \(T\)-periodic solution \(\theta\) of (1) is proved with derivative having preassigned state-dependent discontinuities that satisfy (2).
The proof of these existence results is based on fixed point techniques, Schauder’s theorem and Banach-Picard contraction principle. The operators involved are obtained by means of a convolution.
The paper contains also some numerical examples, one of them concerning an Abel equation encountered in cosmology.
More precisely, they consider the equation \[ \theta' = P(\theta) + I(\theta) \tag{1} \] where \(P(\theta) = f_0 + \sum_{j=1}^mf_j\theta^j\) is a polynomial term in \(\theta\) with \(T\)-periodic coefficients of bounded variation and \(I(\theta)=\sum_{k=1}^n a_k(\theta)J_{\tau_k}(\theta)\) is a step function with jumps of state-dependent amplitude at state-dependent instants, which satisfies the condition \[ \sum_{k=1}^n a_k(\theta)=0. \tag{2} \]
Under suitable assumption, the existence of an absolutely continuous \(T\)-periodic solution \(\theta\) of (1) is proved with derivative having preassigned state-dependent discontinuities that satisfy (2).
The proof of these existence results is based on fixed point techniques, Schauder’s theorem and Banach-Picard contraction principle. The operators involved are obtained by means of a convolution.
The paper contains also some numerical examples, one of them concerning an Abel equation encountered in cosmology.
Reviewer: Alessandro Calamai (Ancona)
MSC:
| 34A36 | Discontinuous ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
Keywords:
Abel equation; state-dependent jumps; periodic solutions; iterated convolutions algorithm; Riccati equation; Einstein-Friedmann universeReferences:
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