On some nonlocal evolution equations in Banach spaces (Q1885329)

Let \(E\) be a reflexive Banach space. Denote its dual by \(E'\). The paper is concerned with the double nonlinear evolution inclusion \(f\in (Au)'+Bu, \;\;0<t<T\), with the initial condition \(v^0\in (Au)(0)\), where \(A\) and \(B\) are two (possible multivalued) maximal monotone operators, \(A\subset E\times E'\), \(B\subset L^p(0,T;E)\times L^q(0,T;E')\), with \(p,q\in (1,\infty)\) such that \(1/p + 1/q =1\). The main result says that there is at least a solution whenever the two operators fulfill some boundedness condition, \(A\) is a compact subdifferential, and \(B\) is causal and coercive. The result is an extension of a previous existence result in Hilbert framework by the same author [J. Differ. Integral Equ. 15, No. 8, 897--922 (2002; Zbl 1014.35051)]. The proof is also new. It is based on a variable time-step discretization of a regularized version of the evolution inclusion. An application to nonlinear diffusion phenomena with memory is discussed.