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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 12, 2012

Measures of noncompactness in the study of solutions of nonlinear differential and integral equations

  • Józef Banaś EMAIL logo
From the journal Open Mathematics

Abstract

The aim of this paper is to make an overview of some existence results for nonlinear differential and integral equations. Those results were obtained by the author and his co-workers during last years with some help of the technique of measures of noncompactness and a fixed point theorem of Darbo type.

MSC: 47H08

[1] Akhmerov R.R., Kamenskii M.I., Potapov A.S., Rodkina A.E., Sadovskii B.N., Measures of Noncompactness and Condensing Operators, Oper. Theory Adv. Appl., 55, Birkhäuser, Basel, 1992 10.1007/978-3-0348-5727-7Search in Google Scholar

[2] Appell J., Banas J., Merentes N., Measures of noncompactness in the study of asymptotically stable and ultimately nondecreasing solutions of integral equations, Z. Anal. Anwend., 2010, 29(3), 251–273 http://dx.doi.org/10.4171/ZAA/140810.4171/ZAA/1408Search in Google Scholar

[3] Ayerbe Toledano J.M., Domınguez Benavides T., López Acedo G., Measures of Noncompactness in Metric Fixed Point Theory, Oper. Theory Adv. Appl., 99, Birkhäuser, Basel, 1997 http://dx.doi.org/10.1007/978-3-0348-8920-910.1007/978-3-0348-8920-9Search in Google Scholar

[4] Banaś J., Goebel K., Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math., 60, Marcel Dekker, New York, 1980 Search in Google Scholar

[5] Banaś J., Lecko M., Solvability of infinite systems of differential equations in Banach sequence spaces, J. Comput. Appl. Math., 2001, 137(2), 363–375 http://dx.doi.org/10.1016/S0377-0427(00)00708-110.1016/S0377-0427(00)00708-1Search in Google Scholar

[6] Banaś J., Olszowy L., Measures of noncompactness related to monotonicity, Comment. Math. Prace Mat., 2001, 41, 13–23 Search in Google Scholar

[7] Banaś J., Rzepka B., On existence and asymptotic behavior of solutions of infinite systems of differential equations, Panamer. Math. J., 2004, 14(1), 105–115 Search in Google Scholar

[8] Banas J., Rzepka B., Monotonic solutions of a quadratic integral equation of fractional order, J. Math. Anal. Appl., 2007, 332(2), 1371–1379 http://dx.doi.org/10.1016/j.jmaa.2006.11.00810.1016/j.jmaa.2006.11.008Search in Google Scholar

[9] Banas J., Zajac T., Solvability of a functional integral equation of fractional order in the class of functions having limits at infinity, Nonlinear Anal., 2009, 71(11), 5491–5500 http://dx.doi.org/10.1016/j.na.2009.04.03710.1016/j.na.2009.04.037Search in Google Scholar

[10] Darbo G., Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Math. Univ. Padova, 1955, 24, 84–92 Search in Google Scholar

[11] Deimling K., Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math., 596, Springer, Berlin-New York, 1977 10.1007/BFb0091636Search in Google Scholar

[12] Dunford N., Schwartz J.T., Linear Operators I, Pure Appl. Math., 7, Interscience, New York-London, 1958 Search in Google Scholar

[13] Granas A., Dugundji J., Fixed Point Theory, Springer Monogr. Math., Springer, New York, 2003 10.1007/978-0-387-21593-8Search in Google Scholar

[14] Kilbas A.A., Srivastava H.M., Trujillo J.J., Theory and Applications of Fractional Differential Equations, North-Holland Math. Stud., 204, Elsevier, Amsterdam, 2006 http://dx.doi.org/10.1016/S0304-0208(06)80001-010.1016/S0304-0208(06)80001-0Search in Google Scholar

[15] Maurin K., Analysis I, Biblioteka Matematyczna, 69, PWN, Warsaw, 1991 (in Polish) Search in Google Scholar

[16] Mazur S., Über die kleinste konvexe Menge, die eine gegebene kompakte Menge enthält, Studia Math., 1930, 2, 7–9 10.4064/sm-2-1-7-9Search in Google Scholar

[17] Mönch H., von Harten G.-F., On the Cauchy problem for ordinary differential equations in Banach spaces, Arch. Math. (Basel), 1982, 39(2), 153–160 http://dx.doi.org/10.1007/BF0189919610.1007/BF01899196Search in Google Scholar

[18] Mursaleen M., Mohiuddine S.A., Applications of measures of noncompactness to the infinite system of differential equations in l p spaces, Nonlinear Anal., 2012, 75(4), 2111–2115 http://dx.doi.org/10.1016/j.na.2011.10.01110.1016/j.na.2011.10.011Search in Google Scholar

[19] Persidskiĭ K.P., Countable systems of differential equations and stability of their solutions, III, Izv. Akad. Nauk Kazah. SSR Ser. Fiz.-Mat. Nauk, 1961, 9, 11–34 (in Russian) Search in Google Scholar

[20] Podlubny I., Fractional Differential Equations, Math. Sci. Engrg., 198, Academic Press, San Diego, 1999 Search in Google Scholar

[21] Valeev K.G., Zhautykov O.A., Infinite Systems of Differential Equations, Nauka Kazah. SSR, Alma-Ata, 1974 (in Russian) Search in Google Scholar

Published Online: 2012-10-12
Published in Print: 2012-12-1

© 2012 Versita Warsaw

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