# Linear and Nonlinear Perturbations of the Operator by V. G. Osmolovskii

By V. G. Osmolovskii

The perturbation concept for the operator div is of specific curiosity within the learn of boundary-value difficulties for the final nonlinear equation $F(\dot y,y,x)=0$. Taking as linearization the 1st order operator $Lu=C_{ij}u_{x_j}^i+C_iu^i$, it is easy to, less than convinced stipulations, regard the operator $L$ as a compact perturbation of the operator div. This publication offers effects on boundary-value difficulties for $L$ and the idea of nonlinear perturbations of $L$. in particular, worthy and adequate solvability stipulations in specific shape are chanced on for numerous boundary-value difficulties for the operator $L$. An analog of the Weyl decomposition is proved. The publication additionally incorporates a neighborhood description of the set of all ideas (located in a small local of a identified answer) to the boundary-value difficulties for the nonlinear equation $F(\dot y, y, x) = zero$ for which $L$ is a linearization. A category of units of all strategies to numerous boundary-value difficulties for the nonlinear equation $F(\dot y, y, x) = zero$ is given. the consequences are illustrated by way of a variety of purposes in geometry, the calculus of adaptations, physics, and continuum mechanics.

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