| Abstract: |
| In this work, we study the existence of solutions to the following generalized nonlinear two-parameter problem
\begin{equation}
a(u, v) = \lambda b(u, v) + \mu m(u, v) + \varepsilon F(u, v),
\end{equation}
for a triple $(a, b, m)$ of continuous, symmetric bilinear forms on a real separable Hilbert space $V$ and nonlinear form $F$.
This problem is a natural abstraction of nonlinear problems that occur for
a large class of differential operators, various elliptic pde`s with nonlinearities in either the differential equation and/or the boundary conditions being a special subclass.
We will start by discussing a Fredholm alternative for the associated linear two-parameter eigenvalue problem and then we will use this characterization to construct solutions to various nonlinear problems. |
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