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Existence of positive solutions for \(n\)th-order nonlinear impulsive singular integro-differential equations in Banach spaces. (English) Zbl 1140.45017

The author considers a boundary value problem for an \(n\)-th order nonlinear impulsive integro-differential equation in an ordered Banach space on an infinite interval. He establishes the existence of positive solutions by virtue of the well-known theorems on fixed points for completely continuous operators.

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45M20 Positive solutions of integral equations
45J05 Integro-ordinary differential equations
45G05 Singular nonlinear integral equations
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H05 Monotone operators and generalizations
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References:

[1] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
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[3] Guo, D., A class of \(n\) th-order impulsive integro-differential equations in Banach spaces, Computers Math. Appl., 44, 1339-1356 (2002) · Zbl 1035.45009
[4] Guo, D., Multiple positive solutions of a boundary value problem for \(n\) th-order impulsive integro-differential equations in a Banach space, Nonlinear Anal., 56, 985-1006 (2004) · Zbl 1054.45007
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[6] Guo, D., Extremal solutions for \(n\) th-order impulsive integro-differential equations on the half-line in Banach spaces, Nonlinear Anal., 65, 677-696 (2006) · Zbl 1098.45013
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[8] Guo, D.; Lakshmikantham, V.; Liu, X. Z., Nonlinear Integral Equations in Abstract Spaces (1996), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0866.45004
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