A nonoscillation theorem for sublinear Emden–Fowler equations

Consider the sublinear Emden–Fowler equation (E)y″+a(x)|y|γ-1y=0(E)y″+a(x)|y|γ-1y=0, 0<γ<10<γ<1, and where a(x)a(x) is a positive continuous function on (0,∞)(0,∞). Define φ(x)=x(γ+3)/2a(x)φ(x)=x(γ+3)/2a(x). We prove in this paper that if φ(x)φ(x) is nonincreasing and limx→∞φ(x)=0, then equation (E) is nonoscillatory.