We present, under a weak majorant condition, a local convergence analysis for the Gauss-Newton method for injective-overdetermined systems of equations in a Hilbert space setting. Our results provide under the same information a larger radius of convergence and tighter error estimates on the distances involved than in earlier studies such us [10, 11, 13, 14, 18]. Special cases and numerical exa...

We represent new estimates of errors of quadrature formula, formula of numerical differentiation and approximation using Taylor polynomial. To measure the errors we apply representation of the remainder in Taylor formula by least concave majorant of the modulus of continuity of the n−th derivative of an n−times differentiable function. Our quantitative estimates are special applications of a mo...

We show that any set containing a positive proportion of the primes contains a 3-term arithmetic progression. An important ingredient is a proof that the primes enjoy the socalled Hardy-Littlewood majorant property. We derive this from a rather more general result which, because of a close analogy with a classical argument of Tomas and Stein from Euclidean harmonic analysis, might be called a r...

We present new semilocal convergence theorems for Newton methods in a Banach space. Using earlier general conditions we find more precise error estimates on the distances involved using the majorant principle. Moreover we provide a better information on the location of the solution. In the special case of Newton’s method under Lipschitz conditions we show that the famous Newton–Kantorovich hypo...

We provide a local convergence analysis of inexact Newton–like methods in a Banach space setting under flexible majorant conditions. By introducing center–Lipschitz–type condition, we provide (under the same computational cost) a convergence analysis with the following advantages over earlier work [9]: finer error bounds on the distances involved, and a larger radius of convergence. Special cas...

In the article, existence of solutions for Van der Pol differential equation is proved, and approximate structure such in analyticity domain obtained. proof, majorant method was applied not to right side equation, as per usual, but solution nonlinear under consideration. Results numerical study are presented.