Nonlinear integral equations with potential-type kernels on a segment. (English. Russian original) Zbl 1471.45004

J. Math. Sci., New York 235, No. 4, 375-391 (2018); translation from Sovrem. Mat., Fundam. Napravl. 60, 5-22 (2016).
Summary: We study various classes of nonlinear equations containing operators of potential type (Riesz potential). By the method of monotone operators in the Lebesgue spaces of real-valued functions \(L_p(a, b)\) we prove global theorems on the existence, uniqueness, estimates, and methods of construction of their solutions. We present applications that illustrate the results obtained.


45G10 Other nonlinear integral equations
47H05 Monotone operators and generalizations
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[1] S. N. Askhabov, Singular Integral Equations and Equations of Convolution Type with Monotone Nonlinearity [in Russian], Maikop (2004).
[2] S. N. Askhabov, Nonlinear Equations of Convolution Type [in Russian], Fizmatlit, Moscow (2009).
[3] Askhabov, SN, Nonlinear equations with weight operators of potential type in Lebesgue spaces, Vestn. Samar. Tekh. Univ. Ser. Fiz.-Mat., 4, 160-164, (2011)
[4] Askhabov, SN, Approximate solution of nonlinear discrete equations of convolution type, Sovr. Mat. Fundam. Naprav., 45, 18-31, (2012)
[5] Askhabov, SN, Nonlinear integral equations with kernels of convolution type on a semiaxis, Vladikavkaz. Mat. Zh., 15, 3-11, (2013) · Zbl 1287.45002
[6] Brezis, H.; Browder, FE, Some new results about Hammerstein equations, Bull. Am. Math. Soc. (N.S.), 80, 567-572, (1974) · Zbl 0286.45007
[7] Brezis, H.; Browder, FE, Nonlinear integral equations and systems of Hammerstein type, Adv. Math., 18, 115-147, (1975) · Zbl 0318.45011
[8] R. E. Edwards, Fourier Series. A Modern Introduction, Vols. 1, 2, Grad. Texts Math., 64, Springer-Verlag, New York-Heidelberg-Berlin (1979).
[9] H. Gajewski, K. Gröger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin (1974). · Zbl 0289.47029
[10] Kachurovskiy, RI, Nonlinear monotone operators in Banach spaces, Usp. Mat. Nauk, 23, 121-168, (1968)
[11] V. B. Moroz, “Hammerstein equations with kernels of Riesz potential type,”Abstr. Int. Conf. “Boundary-Value Problems, Special Functions, and Fractional Calculus,” Minsk (1996), pp. 249-254.
[12] A. M. Nakhushev, Fractional Calculus and Its Applications [in Russian], Fizmatlit, Moscow (2003). · Zbl 1066.26005
[13] A. D. Polyanin and A. V. Manzhirov, Handbook on Integral Equations [in Russian], Nauka, Moscow (1978).
[14] D. Porter and D. Stirling, Integral Equations. A Practical Treatment, from Spectral Theory to Applications, Cambridge Univ. Press, Cambridge (1990). · Zbl 0714.45001
[15] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Integrals and Derivatives of Fractional Order and Their Applications [in Russian], Minsk (1987). · Zbl 0617.26004
[16] M. M. Vaynberg, Variational Methods of Investigation of Nonlinear Operators [in Russian], Moscow (1956).
[17] M. M. Vaynberg, Variational Method and Monotone Operators Method in Theory of Nonlinear Equations [in Russian], Nauka, Moscow (1972).
[18] P. P. Zabreyko, A. I. Koshelev et al., Integral Equations [in Russian], Nauka, Moscow (1968).
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