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Abstract. Given T > 0 , the Abel-like equation θ = f 0 + ∑ j∈N f j θ 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.Mathematics subject classification (2010): 34A37, 34B15, 34B37, 34C25.