ISSN 1066-369X, Russian Mathematics, 2018, Vol. 62, No. 6, pp. 43–55.
Allerton Press, Inc., 2018.
Original Russian Text
Kh.A. Khachatryan H.S. Petrosyan, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 6, pp. 48–62.
One Initial Boundary-Value Problem for Integro-Diﬀerential
Equation of the Second Order With Power Nonlinearity
and H. S. Petrosyan
Institute of Mathematics, National Academy of Sciences of Armeniya
pr. Marshala Bagramyana 24/5, Erevan, 0019 Armenia
Armenian National Agrarian University
ul. Teryana 72, Erevan, 0009, Armenia
Received March 28, 2017
Abstract—In the Sobolev space W
) we investigate one initial boundary-value problem for
integro-diﬀerential equation of the second order with power nonlinearity on a semi-axis. Assuming
that summary-diﬀerence even kernel serves for the considered kernel as minorant in the sense of
M. A. Krasnosel’skii, we prove the existence of a nonnegative (nontrivial) solution in the Sobolev
). We also calculate the limits of constructed solution at inﬁnity.
Keywords: nonnegative solution, iteration, limit of solution, Sobolev space, monotonicity.
The present paper is devoted to the investigation of an initial-boundary-value problem for an integro-
diﬀerential equation of the second order
(t)dt, x ∈ R
y(0) = 0,y∈ W
≡ [0, +∞), (2)
with respect to a desired function y(x),whereμ>0, α ∈ (0, 1).ThekernelK(x, t) isadeﬁned on the
continuous function, which takes nonnegative values, and
K(x, t)dt ≤ μ, x ∈ R
Problem (1)–(2) is applied in the theory of nonlocal interaction, in p-adic mathematical physics ([1–3]).
In the case, when the kernel K(x, t) depends on the diﬀerence of its arguments: K(x, t)=K(x− t),
K(τ)dτ = μ and
τK(τ)dτ < 0, problem (1)–(2) and certain its generalizations were
investigated in papers [4–7]. –.
In the present paper, with essentially diﬀerent constraints on the kernel K(x, t) we prove the existence
of a nonnegative (nontrivial) solution in the Sobolev space W
). We also calculate the limit of the
constructed solution at inﬁnity.
Section 1 is devoted to reducing problem (1)–(2) to the solution to a nonlinear integral equation on
the positive semi-axis. In addition, we adduce certain auxiliary facts from the theory of conservative
integral equations of convolution type.
In Section 2, we introduce special sequential approximations for the obtained nonlinear integral
equation. We prove that the indicated iterations have a point-wise limit as n →∞.InSection3we
calculate the limit of the constructed solution at inﬁnity.