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Nonlinear Singular Integro-Differential Equations with an Arbitrary Parameter

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Abstract

The maximally monotone operator method in real weighted Lebesgue spaces is used to study three different classes of nonlinear singular integro-differential equations with an arbitrary positive parameter. Under sufficiently clear constraints on the nonlinearity, we prove existence and uniqueness theorems for the solution covering in particular, the linear case as well. In contrast to the previous papers in which other classes of nonlinear singular integral and integro-differential equations were studied, our study is based on the inversion of the superposition operator generating the nonlinearities of the equations under consideration and the establishment of the coercitivity of the inverse operator, as well as a generalization of the well-known Schleiff inequality.

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References

  1. H. Gajewski, K. Groger, and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademie-Verlag, Berlin, 1974; Mir, Moscow, 1978).

    MATH  Google Scholar 

  2. G. M. Magomedov, “The monotonicity method in the theory of nonlinear singular integral and integrodifferential equations,” Differ. Uravn. 13 (6), 1106–1112 (1977).

    MATH  Google Scholar 

  3. L. v. Wolfersdorf, “Monotonicity methods for nonlinear singular integral and integro-differential equations,” Z. Angew. Math. Mech. 63 (6), 249–259 (1983).

    Article  MathSciNet  MATH  Google Scholar 

  4. A. I. Guseinov and Kh. Sh. Mukhtarov, Introduction to the Theory of Nonlinear Singular Integral Equations (Nauka, Moscow, 1980) [in Russian].

    Google Scholar 

  5. S. N. Askhabov, Nonlinear Singular Integral Equations in Lebesgue Spaces (Izd. Chechen. Gos. Univ., Groznyi, 2013) [in Russian].

    MATH  Google Scholar 

  6. Kh. M. Kogan, “On a singular integro-differential equation,” Differ. Uravn. 3 (2), 278–293 (1967).

    MathSciNet  Google Scholar 

  7. M. Schleiff, “Untersuchungen einer linearen singularen Integrodifferentialgleichung der Tragflügeltheorie,” Wiss. Z. Martin-Luther-Univ. Halle-WittenbergMath.-Natur. Reihe 17, 981–1000 (1968).

    MathSciNet  MATH  Google Scholar 

  8. S. N. Askhabov, Nonlinear Equations of Convolution Type (Fizmatlit, Moscow, 2009) [in Russian].

    MATH  Google Scholar 

  9. S. N. Askhabov, “Nonlinear convolution-type equations in Lebesgue spaces,” Mat. Zametki 97 (5), 643–654 (2015) [Math. Notes 97 (5–6), 659–668 (2015)].

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Chechen State University, Groznyi, Russia

    S. N. Askhabov

  2. Chechen State Pedagogical University, Groznyi, Russia

    S. N. Askhabov

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  1. S. N. Askhabov
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Correspondence to S. N. Askhabov.

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Original Russian Text © S. N. Abyzov, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 1, pp. 20–26.

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Askhabov, S.N. Nonlinear Singular Integro-Differential Equations with an Arbitrary Parameter. Math Notes 103, 18–23 (2018). https://doi.org/10.1134/S0001434618010029

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