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Nonlocal Cauchy problem for impulsive differential equations in Banach spaces. (English) Zbl 1225.34082

From the introduction: We discuss the impulsive Cauchy problem with nonlocal conditions
\[ u'(t)=Au(t)+f(t,u(t)),\quad t\in [0,b],\;t\neq t_i,\;i=1,2,\dots,p,\tag{1} \]
\[ \Delta u(t_i)=u(t^+_i)-u(t^-_i)=I_i(u(t_i)),\quad i=1,2,\dots,p, \tag{2} \]
\[ u(0)=g(u)+u_0, \tag{3} \]
where \(A\) is the infinitesimal generator of a strongly continuous semigroup of bounded linear operators \(T(t)\) on a Banach space \(X\); \(f:[0,b]\times X\to X\); \(0<t_1<t_2< \cdots < t_p < t_{p+1}=b\); \(I_i: X\to X\), \(i=1,2,\dots,p\), are impulsive functions and \(g:PC([0,b];X)\to X\).
We derive some sufficient conditions for the solution of differential equation (1)–(3), combining impulsive conditions and nonlocal conditions. Our results are achieved by applying the Hausdorff measure of noncompactness and a fixed point theorem. Neither the semigroup \(T(t)\) nor the function \(f\) is needed to be compact in our results.

MSC:

34K30 Functional-differential equations in abstract spaces
34K45 Functional-differential equations with impulses
47N20 Applications of operator theory to differential and integral equations
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