An $L_2$-approach to second-order nonlinear functional evolutions involving $M$-accretive operators in Banach spaces

Abstract

The objective of this paper is to initiate the study of second-order nonlinear functional evolutions of the type $$ \begin{cases} & u''(t) \in A(t)u(t) + G(t,u_t),\quad t>0,\\ & u(0)=x, \ \ u_0 = \phi, \ \ \sup_{t \ge 0} \lbrace \|u(t)\| \rbrace < +\infty, \end{cases} \tag*{($P$)} $$ in a real, uniformly smooth Banach space $X$ with strongly monotone duality mapping. The operators $A(t)$ are $m$-accretive and the operators $G$ are Lipschitzian. The problem is lifted into the space $L_2([-r,\infty);X),$ in which it becomes an elliptic-type problem of the type $$\mathcal Au+\mathcal Bu+\mathcal G(\cdot,u_\cdot) \ni 0$$ with $\mathcal A$ and $\mathcal B$ $m$-accretive. Unperturbed results of Xue, Song and Ma are extended to the present case. The main difficulty in the solvability of these problems is due to the presence of a delay and the fact that certain monotonicity properties of some real-valued functions (defined via the duality mapping) which are present in the homogeneous case do not continue to hold in the perturbed case.