We deal with the problem ⎧ -Δu = f(x,u) + λg(x,u), in Ω, ⎨ ($P_{\lambda }$) ⎩ $u_{\mid \partial \Omega }=0$ 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 ($P_{\lambda }$) admits a non-zero, non-negative strong solution $u_{\lambda }\in {\bigcap }_{p\ge 2}{W}^{2,p}\left(\Omega \right)$ such that $li{m}_{\lambda \to 0⁺}\left|\right|u_{\lambda }{\left|\right|}_{W^{2,p}\left(\Omega \right)}=0$ for all p ≥ 2. Moreover, the function $\lambda \mapsto I_{\lambda }\left(u_{\lambda }\right)$ is negative and decreasing in ]0,λ*[, where $I_{\lambda }$ is the energy functional related to ($P_{\lambda }$).

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...

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...

We study the convergence of the iterations in a Hilbert space $V,x_{k+1}=W\left(P\right)x_k,W\left(P\right)z=w=T(Pw+(I-P)z)$, 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 $W\left(P\right)$ is continuous and the Lipschitz constant $∥(I-P)W\left(P\right)∥<1$. If an operator $W\left(P_1\right)$ 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 $W\left(P_2\right)$ is defined and continuous in $V$ and satisfies $∥(I-P_2)W\left(P_2\right)∥\le ∥(I-P_1)W\left(P_1\right)∥$.

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.

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.

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.

A simple proof is given of a basic surjectivity result for monotone operators. The proof is based on the dynamical systems method (DSM).