We will discuss Kellogg's iterations in eigenvalue problems for normal operators. A certain generalisation of the convergence theorem is shown.

The weak convergence of the iterative generated by $J(u_{n+1}-u_n)=\tau (F{u}_n-J{u}_n)$, $n\ge 0$, $(0<\tau =min\{1,\frac{1}{\lambda }\})$ to a coincidence point of the mappings $F,J:V\to V^{☆}$ is investigated, where $V$ is a real reflexive Banach space and $V^{☆}$ its dual (assuming that $V^{☆}$ is strictly convex). The basic assumptions are that $J$ is the duality mapping, $J-F$ is demiclosed at $0$, coercive, potential and bounded and that there exists a non-negative real valued function $r(u,\eta )$ such that $$\underset{u,\eta \in V}{sup}\left\{r(u,\eta )\right\}=\lambda <\infty$$$$...$$

The purpose of this paper is to study global existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations. By constructing a special Banach space and employing fixed-point theorems, some sufficient conditions are obtained for the global existence and uniqueness of solutions of this kind of equations involving Caputo fractional derivatives and multiple base points. We apply the results to solve the forced logistic model with multi-term fractional...