Quasilinearization and multiple solutions of the Emden‐Fowler type equation

Abstract

Existence and multiplicity of solutions of the problem x” = ‐q(t) |x|^p sign x (i), x(0) = x(1) = 0 (ii) are investigated by reducing equation (i) to a quasi‐linear one so that both equations are equivalent in some domain O. If a solution of corresponding quasi‐linear problem is located in the domain of equivalence O, then this solution solves the original problem also. If this process of quasilinearization is possible for multiple essentially different linear parts, then multiple solutions to the problem (i), (ii) exist.

Darbe nagrinejamas taip vadinamas Emdeno‐Faulerio kvazitiesines diferencialines lygties homogeninio kraštinio uždavinio sprendiniu egzistavimas ir daugialypumas. Parodyta, kad šio uždavinio sprendinio daugialypumas priklauso nuo tam tikru būdu gautos kvazilinearizuotos lygties tiesines dalies savybiu.

Key words:

• quasi‐linear equation
• i‐nonresonant linear part
• i‐type solution
• quasi‐linearization