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We study local well-posedness of the Cauchy problem for the generalized Camassa-Holm equation ${\partial }_{t}u-\partial {³}_{txx}u+2\kappa {\partial }_{x}u+{\partial }_x[g\left(u\right)/2]=\gamma (2{\partial }_{x}u\partial {²}_{xx}u+u\partial {³}_{xxx}u)$ for the initial data u₀(x) in the Besov space $B_{p,r}^s\left(ℝ\right)$ with max(3/2,1 + 1/p) < s ≤ m and (p,r) ∈ [1,∞]², where g:ℝ → ℝ is a given $C^m$-function (m ≥ 4) with g(0)=g’(0)=0, and κ ≥ 0 and γ ∈ ℝ are fixed constants. Using estimates for the transport equation in the framework of Besov spaces, compactness arguments and Littlewood-Paley theory, we get a local well-posedness result.