NONLINEAR EXTENSION OF THE PERRON-FROBENIUS THEORY VIA THE EXTENDED

The Perron–Frobenius theorem (1907, 1912) states that every irreducible non-negative matrix has a prime largest eigenvalue. This theory was later extended by M. Krein and M. Rutman (1948) to operators in Banach spaces, and by G. Birkhoff and R. Varga (1958) to essentially positive matrices, including those that do not preserve vector cones. The talk will cover a number of my recent studies, including those conducted jointly with N. Valeev, which led to several new and interesting discoveries in the theory of finding bifurcations of solutions to parametrized equations. In particular, it turns out that the method of the extended Rayleigh quotient allows one to obtain a new proof of the Perron–Frobenius theorem, extending it to new classes of mappings. In addition, this method allows one to generalize the Perron-Frobenius theory to nonlinear equations, including nonlinear systems of partial differential equations, where bifurcation values ​​are considered instead of eigenvalues.The report will also consider the generalization of the Weyl inequality for eigenvalues ​​of operators to bifurcation values ​​of nonlinear problems.