Ji, Shaochun; Wen, Shu Nonlocal Cauchy problem for impulsive differential equations in Banach spaces. (English) Zbl 1225.34082 Int. J. Nonlinear Sci. 10, No. 1, 88-95 (2010). 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. Cited in 25 Documents MSC: 34K30 Functional-differential equations in abstract spaces 34K45 Functional-differential equations with impulses 47N20 Applications of operator theory to differential and integral equations Keywords:impulsive differential equations; nonlocal conditions; Hausdorff measure of noncompactness; fixed point; mild solution PDFBibTeX XMLCite \textit{S. Ji} and \textit{S. Wen}, Int. J. Nonlinear Sci. 10, No. 1, 88--95 (2010; Zbl 1225.34082)