SOME PEROV’S AND KRASNOSEL’SKII TYPE FIXED POINT RESULTS AND APPLICATION

TitleSOME PEROV’S AND KRASNOSEL’SKII TYPE FIXED POINT RESULTS AND APPLICATION
Publication TypeJournal Article
Year of Publication2015
AuthorsOUAHAB, ABDELGHANI
Volume19
Issue4
Start Page623
Pagination19
Date Published2015
ISSN1083-2564
AMS47H10, 47H30, 54H25
Abstract

In this paper, we establish a single and multivalued version of a Perov type fixed point theorem. Also in generalized Banach spaces, we extend the Krasnosel’skii type fixed point theorem for the sum of B +A, where B is an expansive operator and A is a continuous map. Finally, our results are used to prove the existence of solutions for impulsive differential inclusions.

URLhttp://www.acadsol.eu/en/articles/19/4/12.pdf
Refereed DesignationRefereed
Full Text

REFERENCES
[1] Z. Agur, L. Cojocaru, G. Mazaur, R.M. Anderson and Y.L. Danon, Pulse mass measles vaccination
across age cohorts, Proc. Nat. Acad. Sci. USA. 90 (1993), 11698–11702.
[2] C. S. Barroso, Krasnosel’skii’s fixed point theorem for weakly continuous maps, Nonlinear Anal. 55 (2003), 25–31.
[3] D. D. Bainov and P. S. Simeonov, Systems with Impulse Effect, Ellis Horwood Ltd., Chichister, 1989.
[4] C. S. Barroso and E. V. Teixeira, A topological and geometric approach to fixed points results
for sum of operators and applications, Nonlinear Anal. 60 (2005), 625–650.
[5] M. Benchohra, J. Henderson, and S. K. Ntouyas, Impulsive Differential Equations and Enclusions
Contemporary Mathematics and its Applications, 2. Hindawi Publishing Corporation, New York, 2006.
[6] M. Boriceanu, Krasnosel’skii-type theorems for multivalued operators, Fixed Point Theory 9 (2008), 35–45.
[7] T. A. Burton and C. Kirk, A fixed point theorem of Krasnoselskii-Schaefer type, Math. Nachr. 189 (1998), 23–31.
[8] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes
in Mathematics, Springer-Verlag, Vol. 580, Berlin-Heidelberg-New York, 1977.
[9] R. Cristescu, Order Structures in Normed Vector Spaces, Editura S¸tiint¸ific˘a ¸si Enciclopedic˘a,
Bucure¸sti, 1983 (in Romanian).
[10] S. Djebali, L. Gorniewicz and A. Ouahab, Solutions Sets for Differential Equations and Inclusions,
De Gruyter Series in Nonlinear Analysis and Applications 18. Berlin: de Gruyter, 2013.
[11] J. Garcia-Falset, Existence of fixed points for the sum of two operators, Math. Nachr. 12
(2010), 1726–1757.
[12] J. Garcia-Falset, K. Latrach, E. Moreno-G´alvez and M. A Taoudi, Schaefer-Krasnosel’skii fixed
point theorems using a usual measure of weak noncompactness, J. Differential Equations 252
(2012), 3436–3452.
[13] J. Garcia-Falset and O. Mu˜niz-P´erez, Fixed point theory for 1-set weakly contractive and
pseudocontractive mappings, Appl. Math. Comput. 219 (2013), 6843–6855.
[14] J. R. Graef, J. Henderson and A. Ouahab, Impulsive Differential Inclusions. A Fixed Point
Approach, De Gruyter Series in Nonlinear Analysis and Applications 20, Berlin: de Gruyter,
2013.
[15] L. G´orniewicz, Topological Fixed Point Theory of Multivalued Mappings, Mathematics and its
Applications, Kluwer Academic Publishers, 495, Dordrecht, 1999.
[16] L. G´orniewicz and A. Ouahab, Some fixed point theorems of a Krasnosel’skii type and application
to differential inclusions, Fixed Point Theory, to appear.
[17] A. Halanay and D. Wexler, Teoria Calitativa a Systeme cu Impulduri, Editura Republicii
Socialiste Romania, Bucharest, 1968.
[18] J. Henderson and A. Ouahab, Some multivalued fixed point theorems in topological vector
spaces, Journal of Fixed Point Theory, to appear.
[19] Sh. Hu and N. Papageorgiou, Handbook of Multivalued Analysis, Volume I: Theory, Kluwer,
Dordrecht, 1997.
[20] M. A. Krasnosel’skii, Some problems of nonlinear analysis, Amer. Math. Soc. Transl. Ser. (2)
10 (1958), 345–409.
[21] E. Kruger-Thiemr, Formal theory of drug dosage regiments, J. Theoret. Biol. 13 (1966), 212–
235.
[22] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations,
World Scientific, Singapore, 1989.
[23] Y. Liu and Z. Li, Krasnosel’skii-type fixed point theorems, Proc. Amer. Math. Soc., 136
(2008), 1213–1220.
[24] V. D. Milman and A. A. Myshkis, On the stability of motion in the presence of impulses, Sib.
Math. J. (in Russian) 1 (1960) 233–237.
[25] A. I. Perov, On the Cauchy problem for a system of ordinary differential equations, Pviblizhen.
Met. Reshen. Differ. Uvavn., 2, (1964), 115–134 (in Russian).
[26] I. R. Petre, A multivalued version of Krasnosel’skii’s theorem in generalized Banach spaces,
An. S¸t. Univ. Ovidius Constant¸a, 22 (2014), 177–192.
[27] A. I. Perov, A. V. Kibenko, On a certain general method for investigation of boundary value
problems, Izv. Akad. Nauk SSSR, Ser. Mat., 30 1966, 249–264 (in Russian).
[28] I. R. Petre and A. Petru¸sel, Krasnosel’skii’s theorem in generalized Banach spaces and applications,
Electron. J. Qual. Theory Differ. Equ. (2012), no. 85, 20 pp.
[29] I. A. Rus, The theory of a metrical fixed point theorem: theoretical and applicative relevances,
Fixed Point Theory 9 (2008), 541–559.
[30] R. S. Varga, Matrix Iterative Analysis, Springer Series in Computational Mathematics, 27,
Springer-Verlag, Berlin, 2000.
[31] A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific,
Singapore, 1995.
[32] T. Xiang and R. Yuan, A class of expansive-type Krasnosel’skii fixed point theorems, Nonlinear
Anal. 71 (2009), 3229–3239.