On solvability of the BVPS for the fourth‐order Emden‐Fowler type equations

Abstract

Solvability of the boundary value problems (BVPs) for the fourth‐order Emden‐Fowler type equations x ⁽⁴⁾ = q(t)|x| ^p sgn x is investigated by using the quasilinearization process. We modify the equation to a quasi‐linear form x^( 4) – k ⁴ x = F_k (t,x) for various values of k. Our considerations are based on a fact that the modified quasi‐linear problem has a solution of the same oscillatory type as the linear part x ⁽⁴⁾ – k ⁴ x has. We show that original problem in some cases also has a solution of definite type and establish sufficient conditions for multiple solutions of the given BVP.

First Published Online: 14 Oct 2010