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

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.)
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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
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