In this paper, a new existence theorem of anti-periodic solutions for a class of strongly nonlinear evolution equations in Banach spaces is presented. The equations contain nonlinear monotone operators and a nonmonotone perturbation. Moreover, through an appropriate transformation, the existence of anti-periodic solutions for a class of second-order nonlinear evolution equations is verified. Our abstract results are illustrated by an example from quasi-linear partial differential equations with time anti-periodic conditions and an example from quasi-linear anti-periodic hyperbolic differential equations.

In this paper, a new existence theorem of anti-periodic solutions for a class of strongly nonlinear evolution equations in Banach spaces is presented. The equations contain nonlinear monotone operators and a nonmonotone perturbation. Moreover, through an appropriate transformation, the existence of anti-periodic solutions for a class of second-order nonlinear evolution equations is verified. Our abstract results are illustrated by an example from quasi-linear partial differential equations with time anti-periodic conditions and an example from quasi-linear anti-periodic hyperbolic differential equations.