Topics in metric fixed point teory and stability of dynamical systems

Abstract

In this thesis we introduce iterative methods approximating fixed points for nonlinear operators defined in infinite-dimensional spaces. The starting points are the Implicit and Explicit Midpoint Rules generating polygonal functions approximating a solution for an ordinary differential equation in finite-dimensional spaces. Our study has the purpose of determining suitable conditions on the mapping, the underlying space, the coefficients defining the method, in order to get strong convergence of the generated sequence to a common solution of a fixed point problem and a variational inequality. The contributions to this topic appear in the papers: G. Marino, R. Zaccone, On strong convergence of some midpoint type methods for nonexpansive mappings, J. Nonlinear Var. Anal., vol. 1 (2017), n. 2, 159-174; G. Marino, B. Scardamaglia, R. Zaccone, A general viscosity explicit midpoint rule for quasi-nonexpansive mappings, J. Nonlinear and Convex Anal., vol. 18 (2017), n. 1, 137-148; J. Garcia-Falset, G. Marino, R. Zaccone, An explicit midpoint algorithm in Banach spaces, to appear in J. Nonlinear and Convex Anal. (2017). Not rarely a fixed point iteration scheme is used to find a stationary state for a dynamical system. However the fixed points may not be stable. In view of this, we study some conditions under which the asymptotic stability for the critical points of a certain dynamical system is ensured. Our contribution to this topic appears in the paper: R. P. Agarwal, G. Marino, H. K. Xu, R. Zaccone, On the dynamics of love: a model including synergism, J. Linear and Nonlinear Anal., vol. 2, n. 1 (2016), 1-16.