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 (phi(x'))' + lambda phi(x) = 0 satisfying x(0) = x(0) and x'(0) = x(1) where lambda is a positive parameter and phi : (-rho, rho) -> (-sigma, sigma) with 0 < rho <= infinity and 0 < sigma <= infinity is strictly increasing odd bijective and continuous on (-rho, rho). 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. (C) 2013 Elsevier Inc. All rights reserved.