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Global solutions of nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces. (English) Zbl 1121.45004

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)].

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations

Citations:

Zbl 1081.34077
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Full Text: DOI

References:

[1] Guo, F.; Liu, L. S.; Wu, Y. H.; Siew, P., Global solutions of initial value problems for nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces, Nonlinear Anal., 61, 1363-1382 (2005) · Zbl 1081.34077
[2] Guo, D. J., Solutions of nonlinear integro-differential equations of mixed type in Banach spaces, J. Appl. Math. Simulation, l2, 1-11 (1989)
[3] Guo, D. J., 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
[4] Guo, D. J.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press New York · Zbl 0661.47045
[5] Guo, D. J.; Lakshmikantham, V.; Liu, X. Z., Nonlinear Integral Equations in Abstract Spaces (1996), Kluwer: Kluwer Dordrecht
[6] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S., Theory of Impulsive Differential Equations (1989), World Scientific: World Scientific Singapore · Zbl 0719.34002
[7] Lakshmikantham, V.; Leela, S., Nonlinear Differential Equations in Abstract Space (1981), Pergamon Press: Pergamon Press New York · Zbl 0456.34002
[8] Lakshmikantham, V., Some problems in integro-differential equations of Volterra type, J. Integral Equations, 10, Suppl., 137-146 (1985) · Zbl 0598.45015
[9] Liu, L. S., Iterative method for solutions and coupled quasi-solutions of nonlinear Fredholm integral equations in ordered Banach spaces, Indian J. Pure Appl. Math., 27, 959-972 (1996) · Zbl 0862.45015
[10] Liu, L. S., Existence of global solutions of initial value problem for nonlinear integro-differential equations of mixed type in Banach space, J. Systems Sci. Math. Sci., 20, 112-116 (2000), (in Chinese) · Zbl 0962.45008
[11] Liu, L. S.; Wu, C. X.; Guo, F., Existence theorems of global solutions of initial value problem for nonlinear integro-differential equations of mixed type in Banach spaces and applications, Comput. Math. Appl., 47, 13-22 (2004)
[12] Liu, L. S.; Wu, C. X.; Guo, F., A unique solution of initial value problems for first order impulsive integro-differential equations of mixed type in Banach Spaces, J. Math. Anal. Appl., 275, 369-385 (2002) · Zbl 1014.45007
[13] Sun, J. L.; Ma, Y. H., Initial value problems for the second order mixed monotone type of impulsive differential equations in Banach Spaces, J. Math. Anal. Appl., 247, 506-516 (2000) · Zbl 0964.34008
[14] Zhang, J. Q., Solutions of second order impulsive integro-differential equations in Banach Spaces, Acta Math. Sci., 19, 565-572 (1999), (in Chinese) · Zbl 0954.45003
[15] Zhang, X. Y.; Sun, J. X., Solutions of nonlinear second order impulsive integro-differential equations of mixed type in Banach Spaces, J. Systems Sci. Math. Sci., 22, 428-438 (2002), (in Chinese) · Zbl 1052.45010
[16] Liu, L. S.; Guo, F.; Wu, C. X.; Wu, Y. H., Existence theorems of global solutions for nonlinear Volterra type integral equations in Banach spaces, J. Math. Anal. Appl., 309, 638-649 (2005) · Zbl 1080.45005
[17] Lou, B. D., Fixed point for operators in a space of continuous functions and applications, Proc. Amer. Math. Soc., 127, 2259-2264 (1999) · Zbl 0918.47046
[18] Deimling, K., Nonlinear Functional Analysis (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0559.47040
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