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Nonlocal impulsive Cauchy problems for evolution equations. (English) Zbl 1214.47083

The authors consider the following nonlocal impulsive Cauchy problem for evolution equations

d dt[u(t)+F(t,u(t))]+Au(t)=f(t,u(t)),0tK,tt i ,u(0)+g(u)=u 0 ,Δu(t i )=I i (u(t i )),i=1,2,,p,0<t 1 <t 2 <<t p <K,

where -A:D(A)XX is the infinitesimal generator of an analytic semigroup {T(t);t0}, X is a real Banach space, Δu(t i )=u(t i + )-u(t i - ), and F,f,g,I i are appropriate continuous functions. Existence results are obtained by combining operator semigroups, the techniques of approximate solutions, the Hausdorff measure of noncompactness, and fixed point theory. These results generalize and improve existing results in the literature, since neither the Lipschitz continuity nor the compactness assumption on the nonlocal item and impulsive functions is required. An example to illustrate the abstract results is also presented.

47N20Applications of operator theory to differential and integral equations
47J35Nonlinear evolution equations
34A37Differential equations with impulses
34K05General theory of functional-differential equations
34K30Functional-differential equations in abstract spaces
34K40Neutral functional-differential equations
47H08Measures of noncompactness and condensing mappings, K-set contractions, etc.