Singular Integro-Differential Equations with Hilbert Kernel and Monotone Nonlinearity

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2017.3.7108 Singular Integro-Differential Equations with Hilbert Kernel and Monotone Nonlinearity Askhabov, S. N. Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 3. Abstract: In this paper applying methods of trigonometric series we establish that the singular integro-differential operator with the Hilbert kernel \((Gu)(x)=-\frac{1}{2\pi}\int\nolimits_{-\pi}^{\pi} u'(s) {\rm ctg}\frac{s-x}{2}\,ds\) with the domain \(D(G)=\{u(x):\, u(x)\ \text{absolutely continuous with}\ u'(x)\in L_{p'}(-\pi,\pi)\) and \(u(-\pi)=u(\pi)=0\}\), where \(p'=p/(p-1)\), \({1<p<\infty}\), is a strictly positive, symmetric and potential. Using this result and the method of maximal monotone operators, we investigate three different classes of nonlinear singular integro-differential equations with the Hilbert kernel, containing an arbitrary parameter, in the class of \(2\pi\)-periodic real functions. The solvability and uniqueness theorems, covering also the linear case, are established under transparent restrictions. In contrast to previous papers devoted to other classes of nonlinear singular integro-differential equations with the Cauchy kernel, this one is based on inverting of the superposition operator generating the nonlinearity in the equations considered, and on the proof of the coercivity of this inverse operator. The corollaries are given that illustrate the obtained results. Keywords: nonlinear singular integro-differential equations, Hilbert kernel, method of maximal monotone operators Language: Russian Download the full text For citation: Askhabov S. N. Singular integro-differential equations with hilbert kernel and monotone nonlinearity // Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp. 11-20. DOI 10.23671/VNC.2017.3.7108 + References 1. Askhabov S. N. Nelineinye singuljarnye integralnye uravnenija v prostranstvakh Lebega [Nonlinear Singular Integral Equations in Lebesgue Spaces]. Grozny, Chechenskii Gos. Univ., 2013. 136 p. (in Russian). 2. Askhabov S. N. Application of the Method of Maximal Monotone Operators to Nonlinear Singular Integro-Differential Equations. Vestnik Chechenskogo gosudarstvennogo universiteta [Bulletin of the Chechen State University], 2015, no. 1, pp. 7-12 (in Russian). 3. Bari N. K. Trigonometricheskie rjady [Trigonometric Series]. Moscow, Fizmatgiz, 1961. 672 p. (in Russian). 4. Vainberg M.M. Variacionnyi metod i metod monotonnykh operatorov v teorii nelineinykh uravnenii [Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations]. Moscow, Nauka, 1972. 416 p. (in Russian). 5. Gajewski H., Groger K., Zacharias K. Nelineinye operatornye uravnenija i operatornye differentsialnye uravnenija [Nonlinear Operator Equations and Operator Differential Equations]. Moscow, Mir, 1978. 336 p. (in Russian). 6. Zhikov V. V. Monotonnyi operator [Monotone Operator]. Moscow, Sovetskaja Encyclopedia, vol. 3, 1982. 592 p. (in Russian). 7. Kogan Kh. M. On a Singular Integro-Differential Equation. Uspekhi matematicheskikh nauk [Russian Mathematical Surveys], 1965, vol. 20, no. 3(123), pp. 243-244. (in Russian). 8. Kogan Kh. M. On a singular integro-differential equation. Differentsialnye uravneniya [Differntial Equations], 1967, vol. 3, no. 2, pp. 278-293 (in Russian). 9. Magomedov G. M. The Method of Monotonicity in the Theory of Nonlinear Singular Integral and Integro-Differential equations. Differentsialnye uravneniya [Differntial Equations], 1977, vol. 13, no. 6, pp. 1106-1112 (in Russian). 10. Wolfersdorf L. v. Monotonicity Methods for Nonlinear Singular Integral and Integro-Differential Equations. ZAMM, 1983, vol. 63, no. 6, pp. 249-259. 11. Schleiff M. Untersuchungen Einer Linearen Singularen Integrodifferentialgleichung der Tragflugeltheorie. Wiss. Z. Univ. Halle. Math.-nat. Reihe, 1968, vol. 17, pp. 981-1000. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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