| Title | ON SOLVABILITY OF OPERATOR INCLUSIONS x ∈ Ax Bx + Cx IN BANACH ALGEBRAS AND DIFFERENTIAL INCLUSIONS |
| Publication Type | Journal Article |
| Year of Publication | 2010 |
| Authors | DHAGE, BAPURAOC |
| Secondary Title | Communications in Applied Analysis |
| Volume | 14 |
| Issue | 4 |
| Start Page | 567 |
| Pagination | 596 |
| Date Published | 12/2010 |
| Type of Work | scientific: mathematics |
| ISSN | 1083–2564 |
| AMS | 34A60, 47H10 |
| Abstract | In this paper, some hybrid fixed point theorems concerning the operator inclusions x ∈ Ax Bx + Cx in a Banach algebra are proved. They are applied to some first order ordinary differential inclusions of initial and boundary value problems for proving the existence theorems under mixed Lipschitz and compactness type conditions. Our results includes the multi-valued hybrid fixed point theorems of Dhage [1, 2, 3] as special cases.
|
| URL | http://www.acadsol.eu/en/articles/14/4/11.pdf |
| Short Title | OPERATOR INCLUSIONS IN BANACH ALGEBRAS AND DIFFERENTIAL INCLUSIONS |
| Refereed Designation | Refereed |
| Full Text | REFERENCES[1] B. C. Dhage, Multi-valued operators and fixed point theorems in Banach algebras, Disc. Math. Differential Inclusions, Control & Optimization 24 (2004), 97–122.
[2] B. C. Dhage, Multi-valued operators and fixed point theorems in Banach algebras II, Computers & Mathematics with Applications 48 (2004), 1461–1478. [3] B. C. Dhage, Multi-valued mappings and fixed points I, Nonlinear Functional Anal. & Appl. 10 (2005), 359–378. [4] M. A. Krasnoselskii, Topological Methods in The Theory of Nonlinear Integral Equations, Pergamon Press 1964. [5] E.Zeidler, Nonlinear Functional Analysis and Its Applications : Part I, Springer Verlag, 1985. [6] T. A. Burton, A fixed point theorem of Krasnoselskii, Appl. Math. Lett. 11 (1998), 85–88. [7] D. O’Regan, New fixed point results for 1-set contractive set-valued maps, Computers Math. Appl. 35 (1998), 27–34.
[8] B. C. Dhage, On a fixed point theorem in Banach algebras with applications, Appl. Math. Lett. 18 (2005), 273–280. [9] B. C. Dhage, A fixed point theorem in Banach algebras with applications to functional integral equations, Kyungpook Math. Journal 44 (2004), 145–155. [10] B. C. Dhage, Some nonlinear alternatives in Banach algebras with applications I, Nonlinear Studies 12 (2005), 271–283. [11] B. C. Dhage, Some nonlinear alternatives in Banach algebras with applications II, Kyungpook Math. Journal 45 (2005), 281–292. [12] B. C. Dhage, Local fixed point theory involving three operators in Banach algebras, Topological Methods in Nonlinear Anal. 24 (2004), 3773–86. [13] A. Granas and J, Dugundji, Fixed Point Theory, Springer Verlag, 2003. [14] A. Petru ̧sel, Multivalued operators and fixed points, P. U. M. A. 9 (1998), 165–170. [15] B. C. Dhage, Periodic boundary value problems of first order Carath ́eodory and discontinuous differential equations, Nonlinear Funct. Anal. & Appl. 13 (2008), 323–352. [16] R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodhina and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, Birkhauser Verlag 1992. [17] K. Deimling, Multivalued Differential Equations, W. de Gruyter, 1992. [18] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis , Vol. I: Theory, Kluwer Academic Publishers Dordrechet, 1997. [19] J. Banas and M. Lecho, Fixed points of the product of operators in Banach algebras, PanAmerican Math. Journal 12 (2002), 101–109. [20] B. C. Dhage, Multi-valued operators and fixed point theorems in Banach algebras I, Taiwanese J. Math. 10 (2006), 1025–1045. [21] H. Covitz, S. B. Nadler jr., Multivalued contraction mappings in generalized metric spaces, Israel J. Math. 8 (1970), 5–11. [22] J. Aubin and A. Cellina, Differential Inclusions, Springer Verlag 1984. [23] M. Martelli, A Rothe’s type theore for noncompact acyclic-valued maps, Bull. Un. Mat. Ital. 4 (1975), 70–76. [24] L. Rybinski, An application of the continuous selection theorem to the study of the fixed points of multivalued mappings, J. Math. Anal. Appl. 153 (1990), 391–396. [25] T. C. Lim, On fixed point stability for set-valued contraction mappings with applications to generalized differential equations, J. Math. Anal. Appl. 110 (1985), 436–441. [26] A. Lasota, and Z. Opial, An application of the Kakutani-Ky Fan theorem in the theory of ordinary differential equations , Bull. Acad. Pol. Sci. Ser. Sci. Math. Astronom. Phys. 13 (1965), 781–786. [27] J. J. Nieto, Periodic boundary value problems for first order impulsive ODEs, Nonlinear Anal. 51 (2002), 1223–1232. |