ISSN 0001-4346, Mathematical Notes, 2018, Vol. 103, No. 2, pp. 290–296. © Pleiades Publishing, Ltd., 2018.
Original Russian Text © B. I.
Efendiev, 2018, published in Matematicheskie Zametki, 2018, Vol. 103, No. 2, pp. 295–302.
The Dirichlet Problem for an Ordinary Continuous
Second-Order Diﬀerential Equation
Institute of Applied Mathematics and Automation, Nalchik, Russia
Received November 10, 2015
Abstract—The extremum principle for an ordinary continuous second-order diﬀerential equation
with variable coeﬃcients is proved and this principle is used to establish the uniqueness of the
solution of the Dirichlet problem. The problem under consideration is equivalently reduced to the
Fredholm integral equation of the second kind and the unique solvability of this integral equation is
Keywords: continuous diﬀerential equation, fractional integro-diﬀerential operator, Dirichlet
problem, extremum principle.
In the interval 0 <x<l, we consider the equation
Lu ≡ u
u + b(x)u
u + d(x)u = f(x), (1)
where u = u(x),
u(t) ds (2)
is the continuous integro-diﬀerential operator of order [α, β] , 
sign(η − ξ)
|η − t|
(η − ξ)
u(t),n− 1 <s≤ n, n ∈ N,
is the fractional integro-diﬀerential operator (in the sense of Riemann–Liouville) of order s , , Γ(s)
is the Euler gamma function, and
1 <α<β<2, 0 <γ<δ<1,a(x),b(x),c(x),d(x),f(x) ∈ C[0,l].
Equation (1) belongs to the class of continuous diﬀerential equations (see , ). The operator (2)
was introduced in  and its properties were studied in ; in particular, it was proved that the continuous
integro-diﬀerential operator is positive and a formula for continuous integration by parts was obtained.
In , the inversion operator for the operator (2) was constructed and analogs of the Newton–Leibniz
formula for the operator (2) were obtained. In the monographs [4, pp. 32, 90], [5, p. 135], boundary-value
problems for equations containing operators of the form (2) were studied; in particular, in [4, p. 145], for
Eq. (1) with a(x)=−λ, b(x)=c(x)=d(x)=0, the solution of the Dirichlet problem was constructed
in explicit form and a unique solvability condition was given.