Existence of solutions and Galerkin approximations for nonlinear functional evolution equations (Q802283)

One examines the abstract nonlinear functional evolution equation: \[ (*)\quad \dot x(t)+A(t,x_ t)x(t)=0,\quad x_ 0=\Phi \] in a Banach space X, where \(x_ t=x(t+\theta)\) for \(\theta\in [-r,0]\) and \(x_ 0\in C=C([-r,0],x)\). The existence of a local solution of (*), uniformly continuous, X-valued and differentiable almost everywhere is proved under the conditions: \(X^*\) is uniformly convex. The domain at \(A_ 1(\cdot,\cdot,\cdot)\) with \(A_ 1(t,\psi,v)=A(t,\psi)v\) is the set \([0,T)\times C_ 0\times D\), where D is a subset of X and \(C_ 0\) consists of all \(f\in C\) with f(t)\(\in \bar D\cup M\), \(t\in [-r,0]\). Here \(M=\{\Phi(t);\) \(t\in [-r,0]\}\). For every \((t,\psi)\in [0,T]\times C_ 0\), \(A(t,\psi)v\) is m-accretive in v. For every \(t,s\in [0,T)\), \(\psi_ 1,\psi_ 2\in C_ 0\), \(v\in D\), \[ \| A(t,\psi_ 1)v-A(s,\psi_ 2)v\| \leq \ell (\| \psi_ 1\|_ C,\| \psi_ 2\|_ C,\| v\|)[| t-s| (1+\| A(s,\psi_ 2)v\|)+\| \psi_ 1-\psi_ 2\|_ C]. \] Sufficient conditions for the convergence of the Galerkin method solving (*) in the simplified case of \(A(t,x_ t)x(t)=A(t)x(t)-G(t,x_ t)\) with \(G(t,\Phi)\) satisfying Lipschitz type conditions with respect to both variables \(t\in [0,T]\) and \(\Phi\in C\), are also formulated.