Integrable solutions of a nonlinear functional integral equation on an unbounded interval. (English) Zbl 1203.45004

Taoudi, Mohamed Aziz

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 9, 4131-4136 (2009).

Summary: We prove the existence of integrable solutions of a generalized functional-integral equation, which includes many key integral and functional equations that arise in nonlinear analysis and its applications. This is achieved by means of an improvement of a Krasnosel’skii type fixed point theorem recently proved by K. Latrach and the author [Nonlinear Anal., Theory Methods Appl. 66, No. 10, A, 2325–2333 (2007; Zbl 1128.45006)]. The result presented in this paper extends the corresponding result of [J. Banas and A. Chlebowicz, ibid. 70, No. 9, A, 3172–3179 (2009; Zbl 1168.45005)]. An example which shows the importance and the applicability of our result is also included.

MSC:

45G10	Other nonlinear integral equations
47H08	Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
47H10	Fixed-point theorems
47H30	Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)

Keywords:

fixed point theorem; measure of weak noncompactness; functional integral equation; superposition operator

Citations:

Zbl 1128.45006; Zbl 1168.45005

References:

[1]	Corduneanu, C., Integral equations and applications, (1991), Cambridge Univ. Press Cambridge · Zbl 0714.45002
[2]	Deimling, K., Nonlinear functional analysis, (1985), Springer Verlag Berlin · Zbl 0559.47040
[3]	Banas, J.; Chlebowicz, A., On existence of integrable solutions of a functional integral equation under Carathéodory condition, Nonlinear anal., (2008) · Zbl 1191.47087
[4]	Latrach, K.; Taoudi, M.A., Existence results for a generalized nonlinear Hammerstein equation on \(L_1\) spaces, Nonlinear anal., 66, 2325-2333, (2007) · Zbl 1128.45006
[5]	Appell, J.; Zabrejko, ()
[6]	Krasnosel’skii, M.A., On the continuity of the operator \(F u(x) = f(x, u(x))\), Dokl. akad. nauk SSSR, 77, 185-188, (1951), (in Russian)
[7]	Dieudonné, J., Sur LES espaces de Köthe, J. anal. math., 1, 81-115, (1951) · Zbl 0044.11703
[8]	Dunford, N.; Schwartz, J.T., Linear operators, part I: general theory, (1958), Interscience Publishers New York
[9]	Banas, J.; Rivero, J., On measures of weak noncompactness, Ann. math. pure appl., 151, 213-224, (1988) · Zbl 0653.47035
[10]	De Blasi, F.S., On a property of the unit sphere in Banach spaces, Bull. math. soc. sci. math. roumanie, 21, 259-262, (1977) · Zbl 0365.46015
[11]	Appell, J.; De Pascale, E., Su alcuni parametri connessi con la misura di non compattezza di Hausdorff in spazi di funzioni misurabili, Boll. unicone mat. ital. B(6), 3, 497-515, (1984) · Zbl 0507.46025
[12]	Banas, J.; Knap, Z., Measure of weak noncompactness and nonlinear integral equations of convolution type, J. math. anal. appl., 146, 353-362, (1990) · Zbl 0699.45002
[13]	Burton, T.A., Integral equations, implicit functions and fixed points, Proc. amer. math. soc., 124, 2383-2390, (1996) · Zbl 0873.45003
[14]	Liu, Y.C.; Li, Z.X., Schafer type theorem and periodic solutions of evolution equations, J. math. anal. appl., 316, 1, 237-255, (2006)
[15]	Liu, Y.C.; Li, Z.X., Krasnoselskii type fixed point theorems and applications, Proc. amer. math. soc., 136, 4, 1213-1220, (2008) · Zbl 1134.47040
[16]	Jachymski, J., On isac’s fixed point theorem for selfmaps of a Galerkin cone, Ann. sci. math. Québec, 18, 2, 169-171, (1994) · Zbl 0823.47057
[17]	Latrach, K.; Taoudi, M.A.; Zeghal, A., Some fixed point theorems of the Schauder and krasnosel’skii type and application to nonlinear transport equations, J. differential equations, 221, 1, 256-271, (2006) · Zbl 1091.47046
[18]	Burton, T.A., A fixed point theorem of Krasnoselskii, Appl. math. lett., 11, 85-88, (1998) · Zbl 1127.47318
[19]	Zabrejko, P.P.; Koshelev, A.I.; Krasnosel’skii, M.A.; Mikhlin, S.G.; Rakovshchik, L.S.; Stecenko, V.J., Integral equations, (1975), Noordhoff Leyden