Nash-Moser techniques for nonlinear boundary-value problems.
Near viability for fully nonlinear differential inclusions
We consider the nonlinear differential inclusion x′(t) ∈ Ax(t) + F(x(t)), where A is an m-dissipative operator on a separable Banach space X and F is a multi-function. We establish a viability result under Lipschitz hypothesis on F, that consists in proving the existence of solutions of the differential inclusion above, starting from a given set, which remain arbitrarily close to that set, if a tangency condition holds. To this end, we establish a kind of set-valued Gronwall’s lemma and a compactness...
Necessary And Sufficient Conditions For Existence Of Solutions To Equations With Noninvertible Linear Part.
Neumann boundary value problems across resonance
We obtain an existence-uniqueness result for a second order Neumann boundary value problem including cases where the nonlinearity possibly crosses several points of resonance. Optimal and Schauder fixed points methods are used to prove this kind of results.
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 solutions of equations on
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....
New variational principle and duality for an abstract semilinear Dirichlet problem
A new variational principle and duality for the problem Lu = ∇G(u) are provided, where L is a positive definite and selfadjoint operator and ∇G is a continuous gradient mapping such that G satisfies superquadratic growth conditions. The results obtained may be applied to Dirichlet problems for both ordinary and partial differential equations.
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.
Newton-type iterative methods for nonlinear ill-posed Hammerstein-type equations
We use a combination of modified Newton method and Tikhonov regularization to obtain a stable approximate solution for nonlinear ill-posed Hammerstein-type operator equations KF(x) = y. It is assumed that the available data is with , K: Z → Y is a bounded linear operator and F: X → Z is a nonlinear operator where X,Y,Z are Hilbert spaces. Two cases of F are considered: where exists (F’(x₀) is the Fréchet derivative of F at an initial guess x₀) and where F is a monotone operator. The parameter...