| The aim of this work is to develop an existence theory for various classes of nonlinear abstract evolution equations.; The investigation was carried out using monotonicity and compactness methods, elements of operator theory, nonlinear analysis and semigroup theory, as well as fixed point techniques and integral inequalities.; The main contributions of our study are the following. First, we establish an existence theory for a class of nonlinear nonlocal Cauchy problems in Banach spaces for both the autonomous and nonautonomous cases under various conditions. Secondly, we prove the existence of anti-periodic solutions to a class of first and second order nonmonotone evolution equations with discontinuous nonlinearities. Finally, we investigate the well-posedness of a phase field model governed by two coupled hyperbolic integro-differential equations with memory in which the classical Fourier law is replaced by the Gurtin-Pipkin constitutive assumption. |
