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Impulsive problems for semilinear differential equations with nonlocal conditions. (English) Zbl 1188.34073

This paper deals with the impulsive differential equation with nonlocal conditions

u ' (t)=Au(t)+f(t,u(t)),0tb,tt i ,u(0)=u 0 -g(u),Δu(t i )=I i (u(t i )),i=1,2,,p,0<t 1 <t 2 <<t p <b,

where A:D(A)XX is the infinitesimal generator of a strongly continuous semigroup T(t),t0, X is a real Banach space, Δu(t i )=u(t i + )-u(t i - ), u(t i + ),u(t i - ) denotes the right and the left limit of u at t i , respectively and f,g,I i are appropriate continuous functions.

By using the Hausdorff measure of noncompactness and fixed point techniques, the author proves the existence of a mild solution without the Lipschitz continuity of the mapping f, in the cases when (i) g and I i are Lipschitz and the semigroup T(t),t>0, generated by the linear operator A is compact, and (ii) g is not Lipschitz and not compact.

34G10Linear ODE in abstract spaces
47D06One-parameter semigroups and linear evolution equations
47N20Applications of operator theory to differential and integral equations
34A37Differential equations with impulses