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A bifurcation theory for some nonlinear elliptic equations

Biagio Ricceri (2003)

Colloquium Mathematicae

We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ( P λ ) ⎩ u Ω = 0 where Ω ⊂ ℝⁿ is a bounded domain, λ ∈ ℝ, and f,g: Ω×ℝ → ℝ are two Carathéodory functions with f(x,0) = g(x,0) = 0. Under suitable assumptions, we prove that there exists λ* > 0 such that, for each λ ∈ (0,λ*), problem ( P λ ) admits a non-zero, non-negative strong solution u λ p 2 W 2 , p ( Ω ) such that l i m λ 0 | | u λ | | W 2 , p ( Ω ) = 0 for all p ≥ 2. Moreover, the function λ I λ ( u λ ) is negative and decreasing in ]0,λ*[, where I λ is the energy functional related to ( P λ ).

A convergence analysis of Newton-like methods for singular equations using outer or generalized inverses

Ioannis K. Argyros (2005)

Applicationes Mathematicae

The Newton-Kantorovich approach and the majorant principle are used to provide new local and semilocal convergence results for Newton-like methods using outer or generalized inverses in a Banach space setting. Using the same conditions as before, we provide more precise information on the location of the solution and on the error bounds on the distances involved. Moreover since our Newton-Kantorovich-type hypothesis is weaker than before, we can cover cases where the original Newton-Kantorovich...

A convergence result and numerical study for a nonlinear piezoelectric material in a frictional contact process with a conductive foundation

El-Hassan Benkhira, Rachid Fakhar, Youssef Mandyly (2021)

Applications of Mathematics

We consider two static problems which describe the contact between a piezoelectric body and an obstacle, the so-called foundation. The constitutive relation of the material is assumed to be electro-elastic and involves the nonlinear elastic constitutive Hencky's law. In the first problem, the contact is assumed to be frictionless, and the foundation is nonconductive, while in the second it is supposed to be frictional, and the foundation is electrically conductive. The contact is modeled with the...

A convergent nonlinear splitting via orthogonal projection

Jan Mandel (1984)

Aplikace matematiky

We study the convergence of the iterations in a Hilbert space V , x k + 1 = W ( P ) x k , W ( P ) z = w = T ( P w + ( I - P ) z ) , where T maps V into itself and P is a linear projection operator. The iterations converge to the unique fixed point of T , if the operator W ( P ) is continuous and the Lipschitz constant ( I - P ) W ( P ) < 1 . If an operator W ( P 1 ) satisfies these assumptions and P 2 is an orthogonal projection such that P 1 P 2 = P 2 P 1 = P 1 , then the operator W ( P 2 ) is defined and continuous in V and satisfies ( I - P 2 ) W ( P 2 ) ( I - P 1 ) W ( P 1 ) .

A converse to the Lions-Stampacchia theorem

Emil Ernst, Michel Théra (2009)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we show that a linear variational inequality over an infinite dimensional real Hilbert space admits solutions for every nonempty bounded closed and convex set, if and only if the linear operator involved in the variational inequality is pseudo-monotone in the sense of Brezis.

A converse to the Lions-Stampacchia Theorem

Emil Ernst, Michel Théra (2008)

ESAIM: Control, Optimisation and Calculus of Variations

In this paper we show that a linear variational inequality over an infinite dimensional real Hilbert space admits solutions for every nonempty bounded closed and convex set, if and only if the linear operator involved in the variational inequality is pseudo-monotone in the sense of Brezis.

A Dirichlet problem with asymptotically linear and changing sign nonlinearity.

Marcello Lucia, Paola Magrone, Huan-Song Zhou (2003)

Revista Matemática Complutense

This paper deals with the problem of finding positive solutions to the equation -∆[u] = g(x,u) on a bounded domain 'Omega' with Dirichlet boundary conditions. The function g can change sign and has asymptotically linear behaviour. The solutions are found using the Mountain Pass Theorem.

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