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Measures of weak noncompactness and nonlinear integral equations of convolution type. (English) Zbl 0699.45002

The authors prove that the equation \(x(t)=f[t,\int^{\infty}_{0}k(t- s)x(\phi (s))ds]\) has a monotone solution \(x\in L^ 1(0,\infty)\) if suitable conditions are imposed on the functions f, k, and \(\phi\). The proof builds on measures of weak noncompactness [F. S. De Blasi, Bull. Math. Soc. Sci. Math. R. S. R. n. Ser. 21(69), 259-262 (1977; Zbl 0365.46015)], a fixed point principle of G. Emmanuele [Bull. Math. Soc. Sci. Math. Repub. Soc. Roum. Nouv. Ser. 25, 353-358 (1981; Zbl 0482.47027)], and certain properties of nonlinear superposition operators [P. P. Zdrejko and the referee, J. Aust. Math. Soc. Ser. A 47, 186- 210 (1989; Zbl 0683.47045)].
Reviewer: J.Appell

MSC:

45G05 Singular nonlinear integral equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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[1] Appell, J., Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator, J. Math. Anal. Appl., 83, 251-263 (1981) · Zbl 0495.45007
[2] Appell, A.; De Pascale, E., Su alcuni parametri connesi con la misura di non compattezza di Hausdorff in spazi di functioni misurabili, Boll. Un. Mat. Ital. B (6), 3, 497-515 (1984) · Zbl 0507.46025
[3] Appell, J.; Zabrejko, P. P., Continuity properties of the superposition operator, Preprint Univ. Augsburg, No. 131 (1987) · Zbl 0627.47033
[4] Askhabov, S. N.; Mukhtarov, K. S., On a class of nonlinear integral equations of convolution type, Differentsial’nye Uravneniya, 23, 512-514 (1987), [Russian] · Zbl 0627.45005
[5] Babayan, A. O., Wiener-Hopf integral equations with degenerating symbol, Dokl. Akad. Nauk Armian. SSR, 73, 24-28 (1981), [Russian] · Zbl 0497.45006
[6] Banaś, J., Integrable solutions of Hammerstein and Urysohn integral equations, J. Austral. Math. Soc., 46, 61-68 (1989) · Zbl 0666.45008
[7] Banaś, J.; Rivero, J., On measures of weak noncompactness, Ann. Mat. Pura Appl., 151, 213-224 (1988) · Zbl 0653.47035
[8] De Blasi, F. S., On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. R.S. Roumanie (N.S.), 21, 259-262 (1977) · Zbl 0365.46015
[9] Dieudonné, J., Sur les espaces de Köthe, J. Analyse Math., 1, 81-115 (1951) · Zbl 0044.11703
[10] Dunford, N.; Pettis, B. J., Linear operators on summable functions, Trans. Amer. Math. Soc., 47, 323-392 (1940) · Zbl 0023.32902
[11] Dunford, N.; Schwartz, J., Linear operators, I, Intersci. Publ. (1963) · Zbl 0128.34803
[12] Emmanuele, G., Measure of weak noncompactness and fixed point theorems, Bull. Math. Soc. Sci. Math. R.S. Roumanie (N.S.), 25, 253-258 (1981) · Zbl 0482.47027
[13] Jingqi, Y., On non-negative solutions of a kind of non-linear convolution equation, Acta Math. Sinica (N.S.), 1, 201-205 (1985) · Zbl 0614.45012
[14] Krasnosel’skii, M. A., On the continuity of the operator \(Fu (x) = f(x, u (x))\), Dokl. Akad. Nauk SSSR, 77, 185-188 (1951), [Russian] · Zbl 0042.12101
[15] Krasnosel’skii, M. A.; Zabrejko, P. P.; Pustyl’nik, J. I.; Sobolevskii, P. J., Integral Operators in Spaces of Summable Functions (1966), Nauka: Nauka Moscow, (English translation: Noordhoff, Leyden, 1976) · Zbl 0145.39703
[16] Krzyz, J., On monotonity-preserving transformations, Ann. Univ. Mariae Curie-Sklodowska Sect. A, 6, 91-111 (1952) · Zbl 0053.03602
[17] Rudin, W., Real and Complex Analysis (1966), McGrax-Hill: McGrax-Hill New York · Zbl 0148.02904
[18] Zabrejko, P. P.; Koshelev, A. I.; Krasnosel’skii, M. A.; Mikhlin, S. G.; Rakovshchik, L. S.; Stetsenko, V. J., Integral Equations (1968), Nauka: Nauka Moscow, (English translation: Noordhoff, Leyden, 1975) · Zbl 0159.41001
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