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On existence and uniqueness of solutions for ordinary differential equations with nonlinear boundary conditions

Alessandro Calamai (2004)

Bollettino dell'Unione Matematica Italiana

We prove an existence and uniqueness theorem for a nonlinear functional boundary value problem, that is, an ordinary differential equation with a nonlinear boundary condition. The proof is based on a Global Inversion Theorem of Ambrosetti and Prodi, which is applied to the boundary operator restricted to the manifold of the global solutions to the equation. Our result is a generalization of an analogous existence and uniqueness theorem of G. Vidossich, as it is shown with some examples.

On existence of solutions to degenerate nonlinear optimization problems

Agnieszka Prusińska, Alexey Tret'yakov (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We investigate the existence of the solution to the following problem min φ(x) subject to G(x)=0, where φ: X → ℝ, G: X → Y and X,Y are Banach spaces. The question of existence is considered in a neighborhood of such point x₀ that the Hessian of the Lagrange function is degenerate. There was obtained an approximation for the distance of solution x* to the initial point x₀.

On forward and inverse uncertainty quantification for a model for a magneto mechanical device involving a hysteresis operator

Olaf Klein (2023)

Applications of Mathematics

Modeling real world objects and processes one may have to deal with hysteresis effects but also with uncertainties. Following D. Davino, P. Krejčí, and C. Visone (2013), a model for a magnetostrictive material involving a generalized Prandtl-Ishlinski-operator is considered here. Using results of measurements, some parameters in the model are determined and inverse Uncertainty Quantification (UQ) is used to determine random densities to describe the remaining parameters and their uncertainties....

On mesh independence and Newton-type methods

Owe Axelsson (1993)

Applications of Mathematics

Mesh-independent convergence of Newton-type methods for the solution of nonlinear partial differential equations is discussed. First, under certain local smoothness assumptions, it is shown that by properly relating the mesh parameters H and h for a coarse and a fine discretization mesh, it suffices to compute the solution of the nonlinear equation on the coarse mesh and subsequently correct it once using the linearized (Newton) equation on the fine mesh. In this way the iteration error will be...

On monotone-like mappings in Orlicz-Sobolev spaces

Vesa Mustonen, Matti Tienari (1999)

Mathematica Bohemica

We study the mappings of monotone type in Orlicz-Sobolev spaces. We introduce a new class ( S m ) as a generalization of ( S + ) and extend the definition of quasimonotone map. We also prove existence results for equations involving monotone-like mappings.

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