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Nonlinear operators with partial integrals. (English) Zbl 0895.47049

The author gives sufficient conditions for the boundedness, continuity, Lipschitz continuity, or uniform continuity of nonlinear partial integral operators of Uryson type between so-called ideal function spaces.

MSC:

47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47G10 Integral operators
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References:

[1] Appell, J.; Zabrejko, P. P., (Nonlinear Superposition Operators (1990), Cambridge Univ. Press: Cambridge Univ. Press Berlin)
[2] Aleksandrov, V. M.; Kovalenko, E. V., (Problems of Continuum Mechanics with Mixed Boundary Conditions (1986), Nauka: Nauka Cambridge), (Russian)
[3] Kalitvin, A. S.; Zabrejko, P. P., On the theory of partial integral operators, J. Integral Equ. Appl., 3, 351-382 (1991) · Zbl 0747.45005
[4] Van Der Mee, C. V.M., Transport theory in \(L^p\) spaces, Integral Equ. Oper. Theory, 6, 405-443 (1983) · Zbl 0506.45011
[5] Agoshkov, V. I., (Generalized Solutions of Transport Equations and Their Smoothness Properties (1988), Nauka: Nauka Moscow), (Russian)
[6] Minin, I. N., (Theory of Radiation Transfer in the Atmosphere of Planets (1988), Nauka: Nauka Moscow), (Russian)
[7] Vekua, I. N., (New Methods for the Solution of Elliptic Equations (1948), OGIZ: OGIZ Moscow), (Russian)
[8] Appell, J.; Kalitvin, A. S.; Zabrejko, P. P., Boundary value problems for integro-differential equations of Barbashin type, J. Integral Equ. Appl., 6, 1-30 (1994) · Zbl 0808.45012
[9] APPELL J., KALITVIN A.S. & ZABREJKO P.P., Fixed point theorems in K-normed spaces and boundary value problems for nonlinear integro-differential equations of Barbashin type, Rivista Mat. Pura Appl.; APPELL J., KALITVIN A.S. & ZABREJKO P.P., Fixed point theorems in K-normed spaces and boundary value problems for nonlinear integro-differential equations of Barbashin type, Rivista Mat. Pura Appl. · Zbl 0901.45004
[10] Appell, J.; De Pascale, E.; Kalitvin, A. S.; Zabrejko, P. P., On the application of the Newton-Kantorovich method to nonlinear partial integral equations, Zeitschr. Anal. Anw., 15, 397-418 (1996) · Zbl 0852.47033
[11] Povolotskij, A. I.; Kalitvin, A. S., (Nonlinear Operators with Partial Integrals (1991), Izd. Ross. Gos. Pedag. Univ.: Izd. Ross. Gos. Pedag. Univ. Moscow), (Russian)
[12] Zabrejko, P. P., Nonlinear integral operators (Russian), Trudy Sem. Funk. Anal. (Voronezh), 8, 1-148 (1966)
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