Necessary And Sufficient Conditions For Existence Of Solutions To Equations With Noninvertible Linear Part.
New convergence criteria for the Newton-Kantorovich method and some applications to nonlinear integral equations
New hybrid iterative schemes for an infinite family of nonexpansive mappings in Hilbert spaces.
New iteration methods for asymptotically nonexpansive mappings in uniformly smooth real Banach spaces
New iteration methods for pseudocontractive and accretive operators in arbitrary Banach spaces
New iterative approximation methods for a countable family of nonexpansive mappings in Banach spaces.
New iterative scheme for finite families of equilibrium, variational inequality, and fixed point problems in Banach spaces.
New iterative schemes for asymptotically quasi-nonexpansive mappings.
New perturbed iterations for a generalized class of strongly nonlinear operator inclusion problems in Banach spaces.
New sufficient convergence conditions for the secant method
We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses “Lipschitz-type” and center-“Lipschitz-type” instead of just “Lipschitz-type” conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.
New unifying convergence criteria for Newton-like methods
We present a local and a semilocal analysis for Newton-like methods in a Banach space. Our hypotheses on the operators involved are very general. It turns out that by choosing special cases for the "majorizing" functions we obtain all previous results in the literature, but not vice versa. Since our results give a deeper insight into the structure of the functions involved, we can obtain semilocal convergence under weaker conditions and in the case of local convergence a larger convergence radius....
Newton-Kantorovich method and its global convergence.
Newton's methods for variational inclusions under conditioned Fréchet derivative
Estimates of the radius of convergence of Newton's methods for variational inclusions in Banach spaces are investigated under a weak Lipschitz condition on the first Fréchet derivative. We establish the linear convergence of Newton's and of a variant of Newton methods using the concepts of pseudo-Lipschitz set-valued map and ω-conditioned Fréchet derivative or the center-Lipschitz condition introduced by the first author.
Newton-type iterations as approximations of Chebyshev's method.
Nondiscrete induction and an inversion-free modification of Newton's method
Nondiscrete induction and double step secant method.
Nonexpansive mappings and iterative methods in uniformly convex Banach spaces.
Nonlinear separable equations in linear spaces and commutative Leibniz algebras
We consider nonlinear equations in linear spaces and algebras which can be solved by a "separation of variables" obtained due to Algebraic Analysis. It is shown that the structures of linear spaces and commutative algebras (even if they are Leibniz algebras) are not rich enough for our purposes. Therefore, in order to generalize the method used for separable ordinary differential equations, we have to assume that in algebras under consideration there exist logarithmic mappings. Section 1 contains...
Nonlinear symmetric positive systems
Non-monotoneous parallel iteration for solving convex feasibility problems
The method of projections onto convex sets to find a point in the intersection of a finite number of closed convex sets in an Euclidean space, sometimes leads to slow convergence of the constructed sequence. Such slow convergence depends both on the choice of the starting point and on the monotoneous behaviour of the usual algorithms. As there is normally no indication of how to choose the starting point in order to avoid slow convergence, we present in this paper a non-monotoneous parallel algorithm...