The article deals with Robinson’s generalized Newton method for nonlinear inclusion
\[ f(p^*) \in K, \]
where \(f: \;M \to {\mathbb R}^n\) is a \(C^1\) map from a Riemannian manifold \(M\) in \({\mathbb R^n}\), \(K\) is a nonempty closed cone in \({\mathbb R}^n\). The generalized Newton iteration \(\{p_k\}\) is defined by means of the following algorithm: for \(k = 0,1,\dots\), having \(p_0,p_1,\ldots,p_k\), determine \(p_{k+1}\) by the formula \(p_{k+1} = \exp_{p_k} v_k\), where \(v_k \in \Lambda(p_k)\) and such that \(\|v_k\|: = \min \{\|v\|: \;v \in \Lambda(p_k)\}\); the set \(\Lambda(p)\), for each \(p \in M\) is defined by
\[ \Lambda(p): = \{v \in T_pM: \;f(p) + Df(p)v \in K\}. \]
The main results are Kantorovich’s type theorem about the convergence of the iteration \(\{p_k\}\) under the assumption that \(Df\) satisfies the center \(L\)-average Lipschitz condition on a ball \(B(p_0,r)\) (for any point \(p \in B(p_0,r)\) and any geodesic \(c\) connecting \(p_0, p\) with \(l(c) < r\) the inequality
\[ \|W_{p_0}^{-1}\| \cdot \|Df(p)P_{c,p,p_0} - Df(p_0)\| \leq \int_0^{l(c)} L(u) \, du \]
holds) or the \(L\)-average Lipschitz condition on a ball \(B(p_0,r)\) (for any two points \(p,q \in B(p_0,r)\) and any geodesic \(c\) connecting \(p, q\) with \(d(p_0,p) + l(c) < r\) the inequality
\[ \|W_{p_0}^{-1}\| \cdot \|Df(q)P_{c,q,p} - Df(p_0)\| \leq \int_{d(p_0,p)}^{d(p,p) + l(c)} L(u) \, du \]
holds); here \(W_{p_0}\) is a convex process carrying \(T_{p_0}M\) onto \({\mathbb R}^n\). In particular, the special cases when \(Df\) is Lipschitz continuous, or \(f \in C^k\), or \(f\) is analytic map are considered.