The paper is concerned with the convergence problem of Newton type methods for finding zeros of nonlinear operators in Banach spaces. Some families of nonlinear operators are defined by different Lipschitz conditions and an " universal constant " is introduced so that a unified convergence determination of these methods is established for the defined… (More)
AIM To explore the clinical value of combining pyramidal tract mapping, microscopic-based neuronavigation, and intraoperative magnetic resonance imaging (iMRI) in the surgical treatment of epileptic foci involving sensorimotor cortex. MATERIAL AND METHODS We retrospectively analyzed 69 patients with focal epilepsy involving motor and sensory cortex. The… (More)
The maximal monotonicity notion in Banach spaces is extended to Riemannian manifolds of nonpositive sectional curvature, Hadamard manifolds, and proved to be equivalent to the upper semicontinuity. We consider the problem of finding a singularity of a multivalued vector field in a Hadamard manifold and present a general proximal point method to solve that… (More)
The problem is considered of best approximation of finite number of functions simultaneously. For a very general class of norms, characterization results are derived. The main part of the paper is concerned with proving uniqueness and strong uniqueness theorems. For a particular subclass, which includes the important special case of the Chebyshev norm, a… (More)
The γ-conditions for vector fields on Riemannian manifolds are introduced. The γ-theory and the α-theory for Newton's method on Riemannian manifolds are established under the γ-conditions. Applications to analytic vector fields are provided and the results due to Dedieu et al.
The convergence properties of Newton's method for systems of equations with constant rank derivatives are studied under the hypothesis that the derivatives satisfy some weak Lipschitz conditions. The unified convergence results, which include Kantorovich type theorems and Smale's point estimate theorems as special cases, are obtained.
Let F be a field and n ≥ 3. Suppose S 1 , S 2 ⊆ Mn(F) contain all rank-one idempotents. The structure of surjections φ : S 1 → S 2 satisfying ABA = 0 ⇐⇒ φ(A)φ(B)φ(A) = 0 is determined. Similar results are also obtained for (a) subsets of bounded operators acting on a complex or real Banach space X, (b) the space of Hermitian matrices acting on n-dimensional… (More)