Analytic Cauchy problem for the $\mu$-Camassa-Holm equation and its non-quasilinear version

We solve the Cauchy problems for the $\mu$-Camassa-Holm integro-partial differential equation of Khesin-Lenells-Misio\l{}ek and its non-quasilinear version introduced by Qu-Fu-Liu in the complex-analytic framework. These equations have nonlocal nature at two levels: they involve a pseudo-differential operator of negative order and this operator is defined in terms of an integral of the unknown function. We prove unique solvability of the Cauchy problems and provide an estimate of the lifespan of the solutions. Our method is the Ovsyannikov type argument by Batrostichi-Himonas-Petronilho about a scale of Banach spaces of analytic functions, but the non-quasilinearity is dealt with in a different way from theirs. Indeed, we use a simple reduction which is just a nonlocal version of a classical trick.