Branching of solutions of the abstract kinetic equation (Q1810281)

This paper is devoted to the branching phenomenon for solutions of the nonlinear operator equation \(\widehat G(\alpha,f)= 0\), where \[ \widehat G(\alpha, f):= \widetilde A(\alpha)= {\partial f(x,y)\over\partial x}- \widehat F(\alpha,f) \] and (a) \(\widehat A(\alpha): X\to Y\) is a linear closed operator depending on \(\alpha\in \mathbb{R}^1\) such that \(\{0\}\in C\sigma(\widehat A(\alpha))\), and the closure \(\text{cl}(\text{Ran\,} \widehat A(\alpha))\) of the range \(\widehat A(\alpha)\) coincides with \(Y\); here \(\sigma(\widehat A(\alpha))\) denotes the continuous spectrum of \(\widehat A(\alpha)\). (b) \(\widehat F(\alpha, f): \mathbb{R}^1\times X\to Y\) is a nonlinear bounded operator analytic with respect to both arguments in the neighborhood of zero. In the case that the Fréchet derivative of the associated operator is a locally perturbed Fredholm operator, sufficient conditions for the branching of solutions are studied. As an example of the application of this theory, the Boltzmann equation is considered.