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THE SET OF SOLUTIONS OF INTEGRODIFFERENTIAL EQUATIONS IN BANACH SPACES

Part of: Stability theory

Published online by Cambridge University Press:  01 December 2008

RAVI P. AGARWAL ,
DONAL O’REGAN  and
ANETA SIKORSKA-NOWAK
RAVI P. AGARWAL
Affiliation:
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA (email: agarwal@fit.edu)
DONAL O’REGAN
Affiliation:
Department of Mathematics, National University of Ireland, Galway, Ireland (email: donal.oregan@nuigalway.ie)
ANETA SIKORSKA-NOWAK
Affiliation:
Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland (email: anetas@amu.edu.pl)
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Abstract

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In this paper, we first prove an existence theorem for the integrodifferential equation (*)where f,k,x are functions with values in a Banach space E and the integral is taken in the sense of Henstock–Kurzweil–Pettis. In the second part of the paper we show that the set S of all solutions of the problem (*) is compact and connected in (C(Id,E),ω), where .

Keywords

integral equationsexistence theorempseudo-solutionset of solutionsmeasure of noncompactnessHenstock–Kurzweil–Pettis integral

MSC classification

Secondary: 34D09: Dichotomy, trichotomy 34D99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 78 , Issue 3 , December 2008 , pp. 507 - 522
Copyright
Copyright © 2009 Australian Mathematical Society

References

[1]Agarwal, R. P., Meehan, M. and O’Regan, D., ‘Positive solutions of singular integral equations—a survey’, Dynam. Systems Appl. 14(1) (2005), 1–37.Google Scholar
[2]Agarwal, R. P., Meehan, M. and O’Regan, D., Nonlinear Integral Equations and Inclusions (Nova Science Publishers, Hauppauge, NY, 2001).Google Scholar
[3]Agarwal, R. P. and O’Regan, D., ‘Existence results for singular integral equations of Fredholm type’, Appl. Math. Lett. 13(2) (2000), 27–34.Google Scholar
[4]Artstein, Z., ‘Topological dynamics of ordinary differential equations and Kurzweil equations’, J. Differential Equations 23 (1977), 224–243.CrossRefGoogle Scholar
[5]Banaś, J. and Rivero, J., ‘On measures of weak noncompactness’, Ann. Mat. Pura Appl. 125 (1987), 213–224.Google Scholar
[6]DeBlasi, F. S., ‘On a property of the unit sphere in a Banach space’, Bull. Math. Soc. Sci. Math. R. S. Roumanie 21 (1977), 259–262.Google Scholar
[7]Cao, S. S., ‘The Henstock integral for Banach valued functions’, SEA Bull. Math. 16 (1992), 36–40.Google Scholar
[8]Chew, T. S., ‘On Kurzweil generalized ordinary differential equations’, J. Differential Equations 76 (1988), 286–293.CrossRefGoogle Scholar
[9]Chew, T. S. and Flordeliza, F., ‘On x′=f(t,x) and Henstock–Kurzweil integrals’, Differential Integral Equations 4 (1991), 861–868.CrossRefGoogle Scholar
[10]Cichoń, M., ‘Convergence theorems for the Henstock–Kurzweil–Pettis integral’, Acta Math. Hungarica 92 (2001), 75–82.CrossRefGoogle Scholar
[11]Cichoń, M. and Kubiaczyk, I., ‘On the set of solutions of the Cauchy problem in Banach spaces’, Arch. Math. 63 (1994), 251–257.CrossRefGoogle Scholar
[12]Cichoń, M., Kubiaczyk, I. and Sikorska, A., ‘Henstock–Kurzweil and Henstock–Kurzweil–Pettis integrals and some existence theorems’, in: Proc. ISCM Herlany 1999. 2000, pp. 53–56.Google Scholar
[13]Cichoń, M., Kubiaczyk, I. and Sikorska, A., ‘The Henstock–Kurzweil–Pettis integrals and existence theorems for the Cauchy problem’, Czechoslovak Math. J. 54 (2004), 279–289.CrossRefGoogle Scholar
[14]Cramer, F., Lakshmikantham, V. and Mitchell, A. R., ‘On the existence of weak solution of differential equations in nonreflexive Banach spaces’, Nonlinear. Anal., TMA 2 (1978), 169–177.CrossRefGoogle Scholar
[15]Geitz, R. F., ‘Pettis integration’, Proc. Amer. Math. Soc. 82 (1991), 81–86.CrossRefGoogle Scholar
[16]Gordon, R. A., ‘The McShane integral of Banach-valued functions’, Illinois J. Math. 34 (1990), 557–567.CrossRefGoogle Scholar
[17]Gordon, R. A., The Integrals of Lebesgue, Denjoy, Perron and Henstock (American Mathematical Society, Providence, RI, 1994).CrossRefGoogle Scholar
[18]Gordon, R. A., ‘Riemann integration in Banach spaces’, Rocky Mountain J. Math. 21 (1991), 923–949.CrossRefGoogle Scholar
[19]Henstock, R., The General Theory of Integration, Oxford Math. Monographs (Clarendon, Oxford, 1991).CrossRefGoogle Scholar
[20]Januszewski, J., ‘On the existence of continuous solutions of nonlinear integral equations in Banach spaces’, Comment. Math. 30 (1990), 85–92.Google Scholar
[21]Krzyśka, S., ‘On the existence of continuous solutions of Urysohn and Volterra integral equations in Banach spaces’, Demonstratio Math. 2 (1995), 353–359.CrossRefGoogle Scholar
[22]Kubiaczyk, I., ‘Kneser type theorems for ordinary differential equations in Banach spaces’, J. Differential Equations 45 (1982), 139–146.CrossRefGoogle Scholar
[23]Kubiaczyk, I. and Sikorska, A., ‘Differential equations in Banach spaces and Henstock–Kurzweil integrals’, Discuss. Math. Differ. Incl. 19 (1999), 35–43.Google Scholar
[24]Lee, P. Y., Lanzhou Lectures on Henstock Integration, Ser. Real. Anal., 2 (World Scientific, Singapore, 1989).Google Scholar
[25]Meehan, M. and O’Regan, D., ‘Existence theory for nonlinear Volterra integrodifferential and integral equations’, Nonlinear Anal. TMA 31(3/4) (1998), 317–341.CrossRefGoogle Scholar
[26]Mitchell, A. R. and Smith, Ch., ‘An existence theorem for weak solutions of differential equations in Banach spaces’, in: Nonlinear Equations in Abstract Spaces (Academic Press, New York, 1978), pp. 387–404.CrossRefGoogle Scholar
[27]O’Regan, D., ‘Existence results for nonlinear integral equations’, J. Math. Anal. Appl. 192(3) (1995), 705–726.CrossRefGoogle Scholar
[28]O’Regan, D. and Meehan, M., Existence Theory for Integral and Integrodifferential Equations, Mathematics and Its Applications, 445 (Kluwer Academic, Dordrecht, 1998).CrossRefGoogle Scholar
[29]Pettis, B. J., ‘On integration in vector spaces’, Trans. Amer. Math. Soc. 44 (1938), 277–304.CrossRefGoogle Scholar
[30]Szufla, S., ‘Kneser’s theorem for weak solutions for ordinary differential equations in reflexive Banach spaces’, Bull. Polish Acad. Sci. Math. 26 (1978), 407–413.Google Scholar