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 ′ ) ) ′ + λ φ ( x ) = 0 satisfying x ( 0 ) = x 0 and x ′ ( 0 ) = x 1 , where λ is a positive parameter and φ : ( − ρ , ρ ) → ( − σ , σ ) with 0 < ρ ⩽ ∞ and 0 < σ ⩽ ∞ is strictly increasing odd bijective and continuous on ( − 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.