ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 1, pp. 18–23. © Pleiades Publishing, Ltd., 2018.
Original Russian Text © S. N. Askhabov, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 1, pp. 20–26.
Nonlinear Singular Integro-Diﬀerential Equations
with an Arbitrary Parameter
S. N. Askhabov
Chechen State University, Groznyi, Russia
Chechen State Pedagogical University, Groznyi, Russia
Received July 14, 2016
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.
Keywords: maximally monotone operator, nonlinear singular integro-diﬀerential equations.
In the present paper, we study various classes of nonlinear equations containing singular integral and
integro-diﬀerential operators of the form
s − x
s − x
[b(s) · u(s)]
s − x
where the integral is regarded in the sense of the Cauchy–Lebesgue principal value. Using methods from
the theory of maximally monotone operators , in the real Lebesgue space L
() with power weight
, under suﬃciently clear constraints on the nonlinearity, we prove existence and
uniqueness theorems for the solution for three diﬀerent classes of nonlinear singular integro-diﬀerential
equations with an arbitrary positive parameter. In contrast to the papers  and , in which other
classes of nonlinear singular integro-diﬀerential equations were studied, we do not use inversion
formulas for singular integral operators. It should be noted that the application to nonlinear singular
integro-diﬀerential equations of other methods based on the Banach or Schauder principles  or the
implicit-function theorem  leads (see, for example, ), respectively, either to rigid constraints on the
parameter or to degeneracy of the nonlinearity (in the case of Lebesgue spaces) or to unclear constraints
on the nonlinearity (in the case of H
The results obtained in the present paper for p =2include, in particular, also the case of linear
singular integro-diﬀerential equations. In this connection, we note that the interest in singular
integro-diﬀerential equations (linear and nonlinear) is due, in particular, to their numerous and varied
applications to hydrodynamics and aerodynamics (the “plane-wing equation,” also known as the
“Prandtl equation”), to elasticity and automatic control theory, to stable processes with independent
increments and others (for details, see ,  and the bibliography given there).