On a new principle in real analysis and its application to some well-known equations
Institute of Mathematics, NAS RA
Thursday, October 20, 2022
Conference room, Institute of Mathematics
We start with a result of a general nature (principle) related to arbitrary smooth real function of one variable. The result presents an evaluation method for the number of zeros of real functions. Then we apply this principle to studying the number of zeros of solutions of 19 well-known equations arising in physics, chemistry, biology, geology, and ecology. We consider a second-order ordinary differential equation which widely generalizes the Schrödinger equation, Emden-Fowler equation, Fisher equation, Kolmogorov-Petrovskii-Piskunov equation, Newell-Whitehead-Segel equation, Zeldovich equation, Van der Pol equation, Chandrasekar equation, generalized Emden-Fowler equation, Sturm-Liouville equation, Bessel equation, Legander equation, Laguerre equation. Also, we consider a generalization of the third-order Korteweg-De Vries equation. A part of them are partial differential equations which we considered in time-independent versions; others are ordinary differential equations. For all these equations, we give upper bounds for the number of zeros of their solutions. As far as we know the number of zeros was studied only for Sturm-Liouville type equations (including the Schrödinger equations), and for other listed above equations, the results are brand new.