Liu, Xinzhi; Guo, Dajun Method of upper and lower solutions for second-order impulsive integro-differential equations in a Banach space. (English) Zbl 0939.45004 Comput. Math. Appl. 38, No. 3-4, 213-223 (1999). The initial value problem is investigated for the second-order nonlinear impulsive integro-differential equations of Volterra type \[ x''=f(t,x,Tx), \quad\forall t\in J,\;t\neq t_i, \]\[ \Delta x|_{t= t_i}=- \sum^i_{j=1} \alpha_{ij} x(t_j)+ \sum^i_{j=1} \beta_{ij} x'(t_j), \]\[ \Delta x'|_{t=t_i}= -\sum^i_{j=1} \gamma_{ij} x(t_j), \]\[ x(0)=x_0,\;x'(0) =x_1, \] where \(f\in C[J\times E\times E,E]\), \(J=[0,a](a>0)\), \(0<t_1< \cdots< t_i < \cdots< t_m<a\), \(\alpha_{ij}\), \(\beta_{ij}\), \(\gamma_{ij}(i\geq j,\;i=1,2, \dots, m)\) are nonnegative constants \(x_0,x_1\in E\), and \[ (Tx)(t)= \int^t_0k (t,s) x(s)ds,\quad forall t\in J, \] \(k\in C[D,R_+]\), \(D=\{(t,s)\in J\times J: t\geq s\}\), \(R_+\) is the set of all nonnegative numbers, in a real Banach space by means of upper and lower solutions. Conditions for the existence of maximal and minimal solutions are established. Reviewer: I.Foltyńska (Poznań) Cited in 22 Documents MSC: 45J05 Integro-ordinary differential equations 45N05 Abstract integral equations, integral equations in abstract spaces 45G10 Other nonlinear integral equations 45L05 Theoretical approximation of solutions to integral equations Keywords:initial value problem; second-order nonlinear impulsive integro-differential equations of Volterra type; Banach space; upper and lower solutions; maximal and minimal solutions PDFBibTeX XMLCite \textit{X. Liu} and \textit{D. Guo}, Comput. Math. Appl. 38, No. 3--4, 213--223 (1999; Zbl 0939.45004) Full Text: DOI References: [1] Liu, X.; Guo, D., Initial value problems for first order impulsive integro-differential equations in Banach spaces, Communications on Appl. Nonlinear Anal., 2, 65-83 (1995) · Zbl 0858.34068 [2] Guo, D., Initial value problems for nonlinear second order impulsive integro-differential equations in Banach spaces, J. Math. Anal. Appl., 200, 1-13 (1996) · Zbl 0851.45012 [3] Guo, D.; Lakshmikantham, V.; Liu, X., Nonlinear Integral Equations in Abstract Spaces (1996), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0866.45004 [4] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press New York · Zbl 0661.47045 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.