Existence and global asymptotic behavior of positive solutions for combined second-order differential equations on the half-line

Abstract

AbstractWe are concerned with the existence, uniqueness and global asymptotic behavior of positive continuous solutions to the second-order boundary value problem1A⁢(A⁢u′)′+a1⁢(t)⁢uσ1+a2⁢(t)⁢uσ2=0,t∈(0,∞),$\frac{1}{A}(Au^{\prime})^{\prime}+a_{1}(t)u^{\sigma_{1}}+a_{2}(t)u^{\sigma_{2}% }=0,\quad t\in(0,\infty),$subject to the boundary conditionslimt→0+⁡u⁢(t)=0${\lim_{t\rightarrow 0^{+}}u(t)=0}$,limt→∞⁡u⁢(t)/ρ⁢(t)=0${\lim_{t\rightarrow\infty}{u(t)}/{\rho(t)}=0}$, whereσ1,σ2<1${\sigma_{1},\sigma_{2}<1}$andAis a continuous function on[0,∞)${[0,\infty)}$which is positive and differentiable on(0,∞)${(0,\infty)}$such that∫011/A⁢(t)⁢𝑑t<∞${\int_{0}^{1}{1}/{A(t)}\,dt<\infty}$and∫0∞1/A⁢(t)⁢𝑑t=∞${\int_{0}^{\infty}{1}/{A(t)}\,dt=\infty}$. Here,ρ⁢(t)=∫0t1/A⁢(s)⁢𝑑s${\rho(t)=\int_{0}^{t}{1}/{A(s)}\,ds}$fort>0${t>0}$anda1,a2${a_{1},a_{2}}$are nonnegative continuous functions on(0,∞)${(0,\infty)}$that may be singular att=0${t=0}$and satisfying some appropriate assumptions related to the Karamata regular variation theory. Our approach is based on the sub-supersolution method.