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Sikorska-Nowak, Aneta

Existence theory for integrodifferential equations and Henstock-Kurzweil integral in Banach spaces. (English) Zbl 1148.26010

J. Appl. Math. 2007, Article ID 31572, 12 p. (2007).
The author considers the initial value problem for a nonlinear Volterra type integro-differential equation \[ x'(t)=f\left(t, x(t), \int\limits_0^t k(t, s, x(s)) ds \right), \] where \(f\), \(k\) and \(x\) are functions with values in a Banach space, and the integral is taken in the sense of HL [see S. S. Cao, Southeast Asian Bull. Math. 16, No. 1, 35–40 (1992; Zbl 0749.28007)]. In the first part of the paper some well-known definitions and results from the literature are presented and discussed, in particular, the Kuratowski and Hausdorff measures of noncompactness as well as Henstock-Kurzweil integral in Banach space. Using these notions and a fixed point theorem due to H. Mönch [Nonlinear Anal., Theory Methods Appl. 4, 985–999 (1980; Zbl 0462.34041)], the author proves local solvability of the above initial value problem, provided the functions \(f\) and \(k\) belong to the \(L^1\)-Carathéodory class, and satisfy some additional conditions expressed in terms of the measure of noncompactness.
Reviewer: Oleh Omel’chenko (Berlin)

Cited in 5 Documents

MSC:

26A39 Denjoy and Perron integrals, other special integrals
45J05 Integro-ordinary differential equations
45D05 Volterra integral equations
26E20 Calculus of functions taking values in infinite-dimensional spaces

Keywords:

Volterra type integro-differential equation; Banach space valued functions; Henstock-Kurzweil integral

Citations:

Zbl 0749.28007; Zbl 0462.34041
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\textit{A. Sikorska-Nowak}, J. Appl. Math. 2007, Article ID 31572, 12 p. (2007; Zbl 1148.26010)
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References:

[1] S. S. Cao, “The Henstock integral for Banach-valued functions,” Southeast Asian Bulletin of Mathematics, vol. 16, no. 1, pp. 35-40, 1992. · Zbl 0749.28007
[2] V. G. \vCelidze and A. G. D\vzvar\vsvili, Theory of the Denjoy Integral and Some of Its Applications, Tbilis. Gos. Univ., Tbilisi, Russia, 1978. · Zbl 0471.26005
[3] R. A. Gordon, The Integrals of Lebesgue, Denjoy, Perron, and Henstock, vol. 4 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, USA, 1994. · Zbl 0807.26004
[4] R. Henstock, The General Theory of Integration, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, NY, USA, 1991. · Zbl 0745.26006
[5] P. Y. Lee, Lanzhou Lectures on Henstock Integration, vol. 2 of Real Analysis, World Scientific, Teaneck, NJ, USA, 1989. · Zbl 0699.26004
[6] T. S. Chew, “On Kurzweil generalized ordinary differential equations,” Journal of Differential Equations, vol. 76, no. 2, pp. 286-293, 1988. · Zbl 0666.34041
[7] T. S. Chew and F. Flordeliza, “On x\(^{\prime}\)=f(t,x) and Henstock-Kurzweil integrals,” Differential Integral Equations, vol. 4, no. 4, pp. 861-868, 1991. · Zbl 0733.34004
[8] I. Kubiaczyk and A. Sikorska-Nowak, “Differential equations in Banach space and Henstock-Kurzweil integrals,” Discussiones Mathematicae. Differential Inclusions, vol. 19, no. 1-2, pp. 35-43, 1999. · Zbl 0962.34043
[9] J. Kurzweil, “Generalized ordinary differential equations and continuous dependence on a parameter,” Czechoslovak Mathematical Journal, vol. 7, no. 82, pp. 418-449, 1957. · Zbl 0090.30002
[10] A. Sikorska-Nowak, “Retarded functional differential equations in Banach spaces and Henstock-Kurzweil integrals,” Demonstratio Mathematica, vol. 35, no. 1, pp. 49-60, 2002. · Zbl 1011.34066
[11] R. P. Agarwal, M. Meehan, and D. O/Regan, “Positive solutions of singular integral equations-a survey,” Dynamic Systems and Applications, vol. 14, no. 1, pp. 1-37, 2005. · Zbl 1076.45001
[12] R. P. Agarwal, M. Meehan, and D. O/Regan, Nonlinear Integral Equations and Inclusions, Nova Science Publishers, New York, NY, USA, 2001.
[13] R. P. Agarwal and D. O/Regan, “Existence results for singular integral equations of fredholm type,” Applied Mathematics Letters, vol. 13, no. 2, pp. 27-34, 2000. · Zbl 0973.45002
[14] S. Krzyśka, “On the existence of continuous solutions of Urysohn and Volterra integral equations in Banach spaces,” Demonstratio Mathematica, vol. 28, no. 2, pp. 353-360, 1995. · Zbl 0849.45007
[15] M. Meehan and D. O/Regan, “Existence theory for nonlinear Volterra integrodifferential and integral equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 31, no. 3-4, pp. 317-341, 1998. · Zbl 0891.45004
[16] D. O/Regan, “Existence results for nonlinear integral equations,” Journal of Mathematical Analysis and Applications, vol. 192, no. 3, pp. 705-726, 1995. · Zbl 0851.45003
[17] D. O/Regan and M. Meehan, Existence Theory for Nonlinear Integral and Integrodifferential Equations, vol. 445 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1998. · Zbl 0905.45003
[18] H. Mönch, “Boundary value problems for nonlinear ordinary differential equations of second order in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 4, no. 5, pp. 985-999, 1980. · Zbl 0462.34041
[19] J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, vol. 60 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980. · Zbl 0441.47056
[20] A. Ambrosetti, “Un teorema di esistenza per le equazioni differenziali negli spazi di Banach,” Rendiconti del Seminario Matematico della Università di Padova, vol. 39, pp. 349-361, 1967. · Zbl 0174.46001
[21] R. A. Gordon, “Riemann integration in Banach spaces,” The Rocky Mountain Journal of Mathematics, vol. 21, no. 3, pp. 923-949, 1991. · Zbl 0764.28008
[22] G. Ye, P. Y. Lee, and C. Wu, “Convergence theorems of the Denjoy-Bochner, Denjoy-Pettis and Denjoy-Dunford integrals,” Southeast Asian Bulletin of Mathematics, vol. 23, no. 1, pp. 135-143, 1999. · Zbl 0940.28012
[23] A. P. Solodov, “On conditions for the differentiability almost everywhere of absolutely continuous Banach-valued functions,” Moscow University Mathematics Bulletin, vol. 54, no. 4, pp. 29-32, 1999. · Zbl 0961.26012
[24] R. H. Martin Jr., “Nonlinear Operators and Differential Equations in Banach Spaces,” Robert E. Krieger, Melbourne, Fla, USA, 1987. · Zbl 0649.47039
[25] A. Alexiewicz, Functional Analysis, PWN, Warsaw, Poland, 1969. · Zbl 0181.13001
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