Zhang, Xinguang; Liu, Lishan; Wu, Yonghong Global solutions of nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces. (English) Zbl 1121.45004 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 8, 2335-2349 (2007). The authors study the existence of global solutions of the following initial value problem for second order impulsive integro-differential equations of mixed type in Banach space \(E\), \[ x''(t)=f(t,x,x',Tx,Sx), \;t\in J:=[0,a], \;\;t\not=t_{k}, \;k=1,\dots,m \]\[ \Delta x| _{t=t_{k}}=I_k(x(t_{k}),x'(t_{k})), \;k=1,\dots,m \]\[ \Delta x'| _{t=t_{k}}=\overline I_k(x(t_{k}),x'(t_{k})), \;k=1,\dots,m \]\[ x(0)=x_{0}, \;\;x'(0)=x_{1}, \]where \(T\) and \(S\) are linear operators defined by \[ (Tx)(t)=\int_{0}^{t}k(t,s)x(s)ds, \;\;(Sx)(t)=\int_{0}^{a}h(t,s)x(s)ds, \;t\in J, \] and \(k\in C(D,R), \;D=\{(t,s)\in J\times J: t\geq s\}, \;h\in C(J\times J,R), \;\Delta x| _{t=t_{k}}\) denotes the jump of \(x(t)\) at \(t=t_{k}\), i.e., \(\Delta x| _{t=t_{k}}=x(t_{k}^{+})-x(t_{k}^{-})\), where \(x(t_{k}^{+})\) and \(x(t_{k}^{-})\) represent the right and left limits of \(x(t)\) at \(t=t_{k}\) respectively. The proofs are based upon a generalization of Darbo’s fixed point theorem. The results of the present paper improve those obtained for the same problem by F. Guo, L. Liu, Y. Wu, and P.-F. Siew [Nonlinear Anal., Theory Methods Appl. 61, No. 8(A), 1363–1382 (2005; Zbl 1081.34077)]. Reviewer: Mouffak Benchohra (Sidi Bel Abbes) Cited in 14 Documents MSC: 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations 45G10 Other nonlinear integral equations Keywords:global solution; impulsive integro-differential equation; initial value problem, ordered Banach space Citations:Zbl 1081.34077 PDFBibTeX XMLCite \textit{X. Zhang} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 67, No. 8, 2335--2349 (2007; Zbl 1121.45004) Full Text: DOI References: [1] Guo, F.; Liu, L. S.; Wu, Y. 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