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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.

Smooth bifurcation for a Signorini problem on a rectangle

Jan Eisner, Milan Kučera, Lutz Recke (2012)

Mathematica Bohemica

We study a parameter depending semilinear elliptic PDE on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. We describe smooth branches of smooth nontrivial solutions bifurcating from the trivial solution branch in eigenvalues of the linearized problem. In particular, the contact sets of these nontrivial solutions are intervals which change smoothly along the branch. The main tools of the proof...

Weakly coercive mappings sharing a value

J. M. Soriano (2011)

Czechoslovak Mathematical Journal

Some sufficient conditions are provided that guarantee that the difference of a compact mapping and a proper mapping defined between any two Banach spaces over 𝕂 has at least one zero. When conditions are strengthened, this difference has at most a finite number of zeros throughout the entire space. The proof of the result is constructive and is based upon a continuation method.

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