Functional differential equations of delay type and nonlinear evolution in \(L_ p\)-spaces
Nonlinear nonautonomous functional differential equations with \(L_ p\)- initial functions are studied by means of the theory of evolution operators in \(L_ p\)-spaces. The equations have the form \[ \dot x(t)=L(t,x_ t)+f(t,x_ t),\quad t\in [a,b], \] with initial conditions \[ x_ a=\phi \in L_ p([-r,0];R^ n),\quad x(a)=\eta \in R^ n, \] \[ L(t,\phi)=\sum^{m}_{j=1}A_ j(t)\phi (-r_ j)+\int^{0}_{- r}D(t,\theta)\phi (\theta)du. \] The function \(f: [a,b]\times L_ p\to R^ n\) is assumed to be Lipschitz in the second variable and to satisfy another condition with respect to the first variable. Evolution operators for the linear and nonlinear equations are constructed and it is shown that the evolution operator in \(L_ p\times R^ n\) determines the solution of the initial value problem in the exact sense. A variation of parameters formula is developed for the nonlinear equation.
Nonlinear nonautonomous functional differential equations
scientific article; zbMATH DE number 4038293
scientific article