International Journal of Mathematics and Mathematical Sciences

Volume 2007 (2007), Article ID 65947, 14 pages

http://dx.doi.org/10.1155/2007/65947

Research Article

## Nonlinear Integrodifferential Equations of Mixed Type in Banach Spaces

^{1}Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań 61-614, Poland^{2}University of Marketing and Management, Ostroroga 9a, Leszno 64-100, Poland

Received 27 February 2007; Revised 28 April 2007; Accepted 19 June 2007

Academic Editor: Marco Squassina

Copyright © 2007 Aneta Sikorska-Nowak and Grzegorz Nowak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Linked References

- 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. View at Zentralblatt MATH · View at MathSciNet - R. Henstock,
*The General Theory of Integration*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, NY, USA, 1991. View at Zentralblatt MATH · View at MathSciNet - P. Y. Lee,
*Lanzhou Lectures on Henstock Integration*, vol. 2 of*Series in Real Analysis*, World Scientific, Teaneck, NJ, USA, 1989. View at Zentralblatt MATH · View at MathSciNet - Z. Artstein, “Topological dynamics of ordinary differential equations and Kurzweil equations,”
*Journal of Differential Equations*, vol. 23, no. 2, pp. 224–243, 1977. View at Zentralblatt MATH · View at MathSciNet - T. S. Chew and F. Flordeliza, “On ${x}^{\prime}=f\left(t,x\right)$ and Henstock-Kurzweil integrals,”
*Differential and Integral Equations*, vol. 4, no. 4, pp. 861–868, 1991. View at Zentralblatt MATH · View at MathSciNet - 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. View at Zentralblatt MATH · View at MathSciNet - J. Kurzweil, “Generalized ordinary differential equations and continuous dependence on a parameter,”
*Czechoslovak Mathematical Journal*, vol. 7(82), pp. 418–449, 1957. View at Zentralblatt MATH · View at MathSciNet - S. S. Cao, “The Henstock integral for Banach-valued functions,”
*Southeast Asian Bulletin of Mathematics*, vol. 16, no. 1, pp. 35–40, 1992. View at Zentralblatt MATH · View at MathSciNet - B. J. Pettis, “On integration in vector spaces,”
*Transactions of the American Mathematical Society*, vol. 44, no. 2, pp. 277–304, 1938. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Cichoń, I. Kubiaczyk, and A. Sikorska-Nowak, “Henstock-Kurzweil and Henstock-Kurzweil-Pettis integrals and some existence theorems,” in
*International Scientific Conference on Mathematics (Herl'any, 1999)*, pp. 53–56, University of Technology, Košice, Slovakia, 2000. View at Zentralblatt MATH · View at MathSciNet - 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. View at Zentralblatt MATH · View at MathSciNet - R. P. Agarwal, M. Meehan, and D. O'Regan,
*Nonlinear Integral Equations and Inlusions*, Nova Science Publishers, Hauppauge, NY, USA, 2001. - 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. L. Karakostas and P. Ch. Tsamatos, “Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundary-value problems,”
*Electronic Journal of Differential Equations*, vol. 2002, no. 30, pp. 1–17, 2002. View at Zentralblatt MATH · View at MathSciNet - A. Karoui, “Existence and approximate solutions of nonlinear integral equations,”
*Journal of Inequalities and Applications*, vol. 2005, no. 5, pp. 569–581, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - R. K. Miller, J. A. Nohel, and J. S. W. Wong, “A stability theorem for nonlinear mixed integral equations,”
*Journal of Mathematical Analysis and Applications*, vol. 25, pp. 446–449, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. O'Regan, “Existence results for nonlinear integral equations,”
*Journal of Mathematical Analysis and Applications*, vol. 192, no. 3, pp. 705–726, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. O'Regan and M. Meehan,
*Existence Theory for Nonlinear Integral and Integrodifferential Equations*, vol. 445 of*Mathematics and Its Applications*, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1998. View at Zentralblatt MATH · View at MathSciNet - 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. View at Zentralblatt MATH · View at MathSciNet - F. S. De Blasi, “On a property of the unit sphere in a Banach space,”
*Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie*, vol. 21(69), no. 3-4, pp. 259–262, 1977. View at Zentralblatt MATH · View at MathSciNet - J. Banaś and J. Rivero, “On measures of weak noncompactness,”
*Annali di Matematica Pura ed Applicata*, vol. 151, no. 1, pp. 213–224, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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. View at MathSciNet - I. Kubiaczyk, “On a fixed point theorem for weakly sequentially continuous mappings,”
*Discussiones Mathematicae. Differential Inclusions*, vol. 15, no. 1, pp. 15–20, 1995. View at Zentralblatt MATH · View at MathSciNet - M. Cichoń, I. Kubiaczyk, and A. Sikorska-Nowak, “The Henstock-Kurzweil-Pettis integrals and existence theorems for the Cauchy problem,”
*Czechoslovak Mathematical Journal*, vol. 54(129), no. 2, pp. 279–289, 2004. View at Zentralblatt MATH · View at MathSciNet - R. A. Gordon, “Riemann integration in Banach spaces,”
*The Rocky Mountain Journal of Mathematics*, vol. 21, no. 3, pp. 923–949, 1991. View at Zentralblatt MATH · View at MathSciNet - M. Cichoń, “Convergence theorems for the Henstock-Kurzweil-Pettis integral,”
*Acta Mathematica Hungarica*, vol. 92, no. 1-2, pp. 75–82, 2001. View at Zentralblatt MATH · View at MathSciNet - R. H. Martin Jr., “Nonlinear Operators and Differential Equations in Banach Spaces,” Robert E. Krieger, Melbourne, Fla, USA, 1987. View at Zentralblatt MATH · View at MathSciNet
- 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. View at Zentralblatt MATH · View at MathSciNet - A. P. Solodov, “On conditions for the differentiability almost everywhere of absolutely continuous Banach-valued functions,”
*Moscow University Mechanics Bulletin*, vol. 54, no. 4, pp. 29–32, 1999. View at Zentralblatt MATH · View at MathSciNet - 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. View at Zentralblatt MATH · View at MathSciNet - M. Cichoń, “On solutions of differential equations in Banach spaces,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 60, no. 4, pp. 651–667, 2005. View at Zentralblatt MATH · View at MathSciNet - A. R. Mitchell and C. Smith, “An existence theorem for weak solutions of differential equations in Banach spaces,” in
*Nonlinear Equations in Abstract Spaces (Proc. Internat. Sympos., Univ. Texas, Arlington, Tex, 1977)*, V. Lakshmikantham, Ed., pp. 387–403, Academic Press, New York, NY, USA, 1978. View at Zentralblatt MATH · View at MathSciNet