Global existence and nonexistence of solutions for second-order nonlinear differential equations

This paper deals with the global existence and nonexistence of solutions of the second-order nonlinear differential equation (φ(x′))′+λφ(x)=0(φ(x′))′+λφ(x)=0 satisfying x(0)=x0x(0)=x0 and x′(0)=x1x′(0)=x1, where λ   is a positive parameter and φ:(−ρ,ρ)→(−σ,σ)φ:(−ρ,ρ)→(−σ,σ) with 0<ρ⩽∞0<ρ⩽∞ and 0<σ⩽∞0<σ⩽∞ is strictly increasing odd bijective and continuous on (−ρ,ρ)(−ρ,ρ). Necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic. Examples are given to illustrate our main result. Finally, a nonexistence result for the equation with a damping term is discussed as an application to our result.