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.

Multiple Positive Solutions via Index Theory for Singular Boundary Value Problems with Derivative Dependence

“…Some results on the existence of one solution can be found in [1][2][5][6][7][12][13]17,[22][23][24][29][30]. There are some results on the existence of multiple solutions for the equation x + Φ(t)f (t, x, x ) = 0 when f has no singularities at x = 0 and x = 0 (see [8,9,10,11,13,16,19,21,31]). If f (t, x, x ) = f (t, x) is singular at x = 0, there are many results on the existence of multiple solutions (see [3,4,14,15,18,20,24,27,28]).…”

Positive Solutions for Second Order Singular Boundary Value Problems with Derivative Dependence on Infinite Intervals

“…Moreover, motivated by many results on the existence of multiple positive solutions for singular second-order boundary value problems on finite intervals (see [1,3,14,19]) we establish the existence of multiple positive solutions for (1.1) when f is singular at x = 0 and z = 0.…”

Two‐point Boundary Value Problems at Resonance

“…We investigate solvability of the problem (1.1), (1.2) applying the quasilinearization method [8,9]. We try to reduce the equation (1.1) to a quasi-linear one of the form…”