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Implicit functions, nonlinear integral equations, and the measure of noncompactness of the superposition operator. (English) Zbl 0495.45007


MSC:

45G10 Other nonlinear integral equations
45P05 Integral operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
58C15 Implicit function theorems; global Newton methods on manifolds
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References:

[1] {\scM. Z. Berkolajko}, personal communication.
[2] Berkolajko, M.Z; Rutickij, Ja.B, On operators in Hölder spaces, Dokl. akad. nauk SSSR, 192, No, 6, 1199-1201, (1970), (Russian)
[3] Bondarenko, V.A; Zabrejko, P.P, The superposition operator in Hölder function spaces, Dokl. akad. nauk SSSR, 222, No. 6, 739-743, (1975), (Russian) · Zbl 0328.47040
[4] Daneš, J, Fixed point theorems, Nemytskij and Urysohn operators, and continuity of nonlinear mappings, Comm. math.k univ. carol., 11, No. 3, 481-500, (1970) · Zbl 0202.14802
[5] Gol’denstein, L.S; Markus, A.S, On a measure of noncompactness of bounded sets and linear operators, Stud. alg. math. anal., kishinjov, 45-54, (1965), (Russian)
[6] Krasnosel’skij, M.A.k, On the continuity of the operator fu(x) = f(x, u(x)), Dokl. akad. nauk, SSSR, 77, No. 2, 185-188, (1951), (Russian) · Zbl 0042.12101
[7] Krasnosel’skij, M.A, Topological methods in the theory of nonlinear integral equations, (1956), Tech. Teor. Lit Moscow, [English translation: Pergamon, Oxford, 1964] · Zbl 0072.09702
[8] Krasnosel’skij, M.A; Zabrejko, P.P; Pustyl’nik, Je.I; Sobolevskij, P.Je, Integral operators in spaces of summable functions, (1966), Nauka Moscow, [English translation: Noordhoff, Leyden, 1976]
[9] Ladyzhenskij, L.A, Conditions for the complete continuity of the Urysohn integral operator in the space of continuous functions, Dokl. akad. naukk SSSR, 96, No. 5, 1105-1108, (1951), (Russian)
[10] Nussbaum, R.D.k, The radius of the essential spectrum, Duke math. J., 38, No. 3, 473-478, (1970) · Zbl 0216.41602
[11] Pachale, H, Über den urysohnschen integraloperator, Arch. math., 10, No. 2/3, 134-136, (1959) · Zbl 0088.08301
[12] Petryshyn, W.V, A new fixed point theorem and its applications, Bull. amer. math. soc., 78, No. 2, 225-229, (1972) · Zbl 0231.47030
[13] Ruthickij, Ja.B, On a nonlinear operator in Orlicz spaces, Dopov. akad. naukk ukrssr, 3, 161-166, (1952), (Ukranian)
[14] Sadovskij, B.N; Sadovskij, B.N.k, Limit-compact and condesing operators, Usp. mat. nauk., Russ. math. surverys, 27, 85-155, (1972), (Russian) [English translation: · Zbl 0243.47033
[15] Shen-Van, Van, Complete continuity and strong continuity of the Urysohn integral operators, Dokl. akad. nauk SSSR, 151, No. 5, 1011-1013, (1963), (Russian)
[16] Zabrejko, P.P, On the continuity and complete continuity of Urysohn operators, Dokl. akad. naukk SSSR, 161, No. 5, 1007-1010, (1965), (Russian) · Zbl 0125.35701
[17] Zabrekjo, P.P, Nonlinear integral operators, Trudyk sem. funck, anal., 8, (1966), (Russian)
[18] Zabrekjo, P.P; Kolesov, Ju.S.k; Krasnosel’skij, M.A, Implicit functions and the bogoljubov-Krylov averaging principle, Dokl. akad. nauk SSSR, 184, No. 3, 111-114, (1969), (Russian) · Zbl 0181.42001
[19] Zabrejko, P.P; Appell, J, Condensing operators and implicit function theorems, Qual. approx. meth. stud. operator equations, 5, 3-14, (1980), (Russian)
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