Anti-periodic solutions of Abel differential equations with state dependent discontinuities

Abstract

Given T > 0 , the Abel-like equation θ ′ = f0 + ∑ j∈N f jθ j is generalized to the case where θ and θ ′ are real functions on [0,T ] subject to given state dependent discontinuities. Each f j is a real function of bounded variation for which f j(0) = (−1) j+1 f j(T ) . Under appropriate conditions, this equation is shown to admit a solution of bounded variation on [0,T ] which is T -anti-periodic in the sense that θ (0) = −θ (T) . The contraction principle yields a bound for the rate of uniform convergence to the solution of a sequence of iterates.