Measures of noncompactness in the study of solutions of nonlinear differential and integral equations
Józef Banaś
Open Mathematics, Tome 10 (2012), p. 2003-2011 / Harvested from The Polish Digital Mathematics Library

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.

Publié le : 2012-01-01
EUDML-ID : urn:eudml:doc:269227
@article{bwmeta1.element.doi-10_2478_s11533-012-0120-9,
     author = {J\'ozef Bana\'s},
     title = {Measures of noncompactness in the study of solutions of nonlinear differential and integral equations},
     journal = {Open Mathematics},
     volume = {10},
     year = {2012},
     pages = {2003-2011},
     zbl = {1277.47067},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0120-9}
}
Józef Banaś. Measures of noncompactness in the study of solutions of nonlinear differential and integral equations. Open Mathematics, Tome 10 (2012) pp. 2003-2011. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0120-9/

[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

[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/1408[Crossref][WoS] | Zbl 1214.47044

[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/10.1007/978-3-0348-8920-9[Crossref] | Zbl 0885.47021

[4] Banaś J., Goebel K., Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math., 60, Marcel Dekker, New York, 1980 | Zbl 0441.47056

[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)00708-1[Crossref] | Zbl 0997.34048

[6] Banaś J., Olszowy L., Measures of noncompactness related to monotonicity, Comment. Math. Prace Mat., 2001, 41, 13–23 | Zbl 0999.47041

[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 | Zbl 1053.34056

[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.2006.11.008[Crossref] | Zbl 1123.45001

[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.2009.04.037[Crossref] | Zbl 1181.45010

[10] Darbo G., Punti uniti in trasformazioni a codominio non compatto, Rend. Sem. Math. Univ. Padova, 1955, 24, 84–92 | Zbl 0064.35704

[11] Deimling K., Ordinary Differential Equations in Banach Spaces, Lecture Notes in Math., 596, Springer, Berlin-New York, 1977 | Zbl 0361.34050

[12] Dunford N., Schwartz J.T., Linear Operators I, Pure Appl. Math., 7, Interscience, New York-London, 1958 | Zbl 0084.10402

[13] Granas A., Dugundji J., Fixed Point Theory, Springer Monogr. Math., Springer, New York, 2003 | Zbl 1025.47002

[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)80001-0[Crossref]

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

[16] Mazur S., Über die kleinste konvexe Menge, die eine gegebene kompakte Menge enthält, Studia Math., 1930, 2, 7–9 | Zbl 56.0091.01

[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/BF01899196[Crossref]

[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.2011.10.011[Crossref]

[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)

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

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