Viability for delay evolution equations with nonlocal initial conditions. (English) Zbl 1326.34103
The authors consider the problem
\[
u'(t)\in Au(t)+f(t,u_t),\,\, t\in R_+,
\]
\[ u(t)=g(u)(t), \,\, t\in [-\sigma, 0], \] where \(X\) is a real Banach space; \(A : D(A) \subset X \rightsquigarrow X\) is the infinitesimal generator of a nonlinear semigroup; \(K : R_+ \rightsquigarrow \overline{D(A)};\) \(L = \{(t, \phi) \in R_+ \times C_\sigma \,\, :\,\, \phi(0) \in K(t)\};\) \(f : L \to X\) be a continuous function which is Lipschitz with respect to its second argument; \(g : C_b([-\sigma ,+\infty ); \overline{D(A)}) \to C([-\sigma , 0]; \overline{D(A)})\). They prove a necessary and a sufficient condition for \(L\) to be globally viable with respect to \((A, f , g).\) An application to a comparison problem is also included.
\[ u(t)=g(u)(t), \,\, t\in [-\sigma, 0], \] where \(X\) is a real Banach space; \(A : D(A) \subset X \rightsquigarrow X\) is the infinitesimal generator of a nonlinear semigroup; \(K : R_+ \rightsquigarrow \overline{D(A)};\) \(L = \{(t, \phi) \in R_+ \times C_\sigma \,\, :\,\, \phi(0) \in K(t)\};\) \(f : L \to X\) be a continuous function which is Lipschitz with respect to its second argument; \(g : C_b([-\sigma ,+\infty ); \overline{D(A)}) \to C([-\sigma , 0]; \overline{D(A)})\). They prove a necessary and a sufficient condition for \(L\) to be globally viable with respect to \((A, f , g).\) An application to a comparison problem is also included.
Reviewer: Andrej V. Plotnikov (Odessa)
MSC:
| 34K09 | Functional-differential inclusions |
| 34K30 | Functional-differential equations in abstract spaces |
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