CC BY
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Серия «Математика»
2021. Т. 36. С. 57—68
Онлайн-доступ к журналу: http://mathizv.isu.ru
УДК 517.968.4 MSC 45G05
DOI https://doi.org/10.26516/1997-7670.2021.36.57
On the Solvability of a Class of Nonlinear Urysohn Integral Equations on the Positive Half-line*
Kh. A. Khachatryan1'2'3, H. S. Petrosyan1'4
1 Lomonosov Moscow State University, Moscow, Russian Federation
2 Yerevan State University, Yerevan, Republic of Armenia
3 Institute of Mathematics of NAS of Armenia, Yerevan, Republic of Armenia
4 Armenian National Agrarian University,Yerevan, Republic of Armenia
Abstract. The paper investigates the Urysohn's nonlinear integral equation on the positive half-line. Some special cases of this equation have specific applications in different areas of modern natural science. In particular, such equations arise in the kinetic theory of gases, in the theory of p-adic open-closed strings, in mathematical theory of the spatiotemporal spread of the epidemic, and in theory of radiative transfer in spectral lines. The existence theorem for nonnegative nontrivial and bounded solutions is proved. Some qualitative properties of the constructed solution are studied. Specific applied examples of the Urysohn's kernel satisfying all the conditions of the approved theorem are provided.
Keywords: Urysohn equation, monotonicity, Caratheodory condition, iterations, bounded solution.
* This work was supported by the Russian Scientific Foundation (project no. 19-1100223)
1. Introduction
We consider the following class of Urysohn's nonlinear integral equations
oo
f (%) = J u(x,t,f (t))dt, x e R+ := [0, (1.1)
0
with respect to unknown essentially bounded real function f (x). In equation (1.1) Urysohn's kernel u(x, t, z) is defined on set R+ x R+ x R, accepts real values, satisfies condition of criticality
u(x, t, 0) = 0, (x, t) e R+ x R+ (1.2)
and some other conditions (see formulation of main result).
Note that Urysohn's type nonlinear integral equations have a numerous applications in the most diverse areas of modern natural science. In particular under certain representations of the Urysohn's kernel, the equation (1.1) arises in kinetic theory of gases (in the framework of modified model of nonlinear Boltzmann equation), in mathematical theory of spatial-temporal spread of epidemics, in dynamic theory of p-adic closed-open strings, in radiative transfer theory of spectral lines (see [2-5; 15] and references there in).
In papers [8; 11-13] the equation (1.1) has been investigated in sufficient details in the case of monotone Urysohn's operator, for which as a linear minorant in the sense of M.A. Krasnoselskii is Wiener-Hopf conservative integral operator (or Wiener-Hopf-Hankell substochastic operator). Moreover for some "q > 0 Urysohn's kernel satisfies the following integral inequality
oo
J u(x,t,rj)dt < rj, x e R+. (1.3)
0
In this work under essentially weak constraints on the Urysohn's kernel, the existence of a nonnegative nontrivial solution is proved, as well as some qualitative properties of constructed solution is investigated.
Before proceeding to the formulation of the main result of this work, we introduce some notations.
Let K0 — is defined on the set R+, measurable function, satisfying conditions
Ko(x) > 0, ж e R+, Ко e Li(R+) П M(R+), (1.4)
o
J Ko(x)dx = 1, (1.5)
0
where M(R+) — is the space of essentially bounded on R+ functions with norm
||/1| = esssup \f (x)\. (1.6)
Let A(^) — is defined on R+ continuous function possesses the following properties
0 < eo := inf X(x) < X(x) < 1, x e R+, X(x) t on R+, (1.7)
lim X(x) = 1, (1 - X(x))x e Li(R+). (1.8)
Further, we introduce a function K1(x,y), that is definite and continuous on R+ x R+, and
0 < Ki(x,y) < Ko(y — x), 0 < x < y< (1.9)
oo
xJk i(x,y)dy e L?(R+), (1.10)
X
where L0(R+) — is the space of summable functions on R+, having zero limit at
The main result of this work is the following. Theorem. Let exist positive numbers £ and (0 < £ < rj) such that
a) the function u(x, t, z) for each fixed (x, t) e R+ x R+ monotonically
increases by z on interval [0,^],
b) the function u(x,t,z) satisfies Caratheodory's condition by argument z
on set R+ x R+ x [0, rj], i.e for each fixed z e [0, rj] the function u(x, t, z) is measurable by the set of arguments (x, t) on set R+ x R+, and almost for all (x,t) e R+ x R+ is continuous by z on interval [0,^],
c) for arbitrary measurable function f(x), such that 0 < <p(x) < x e R+
oo
the function J u(x,t,<p(t))dt is measurable by x on R+, o
d) for kernel u(x,t,z), the minorant in the sense of Krasnoselsky is the
function X(x)(K0(y — x) — (x,y))z at 0 < x < y < z e [0,£],
i.e
u(x,t,z) > X(x)(K0(y — x) — (x,y))z, 0 < x < y < z e [0,£].
Then under conditions (1.2), (1.3), the equation (1.1) possesses nonnegative nontrivial measurable and bounded .solution f (x), moreover, f (x) < r], x e R+.
At the end of the work, specific examples of the Urysohn's kernel are given, which have a direct application in natural science.
2. Proof of the theorem
First of all, along with the equation (1.1), consider the following auxiliary inhomogeneous Volterra integral equation with a variable lower limit
oo
F(x)= g(x) + \(x)j(Ko(y - x) - Ki(x,y))F(y)dy, x e R+ (2.1)
X
and with respect to the unknown measurable function F(x). Here the free term g(x) permits of the form
o
g(x) := 1 - X(x) + \(x) j Ki(x, y)dy, x e R+. (2.2)
X
Observe that from conditions (1.7)-(1.10) it follows that
o
J xjg(x)dx < j = 0,1, (2.3)
0
lim g(x) = 0. (2.4)
Direct checking can be verified that F0(x) = 1 is the solution of the equation (2.1).
Below we show that equation (2.1), in addition to the trivial solution F0(x) = 1, possesses nonnegative summable and bounded solution F1 (x). Furthermore F1(x) < F0(x) and lim F1(x) = 0. To this end, consider the
following Volterra conservative nonhomogeneous integral equation with the kernel depending on difference of arguments.
o
<p(x) = g(x) + J K0(y - x)ip(y)dy, x e R+ (2.5)
X
with respect to the sought function f(x). Due to the properties (2.3), (2.4),
(1.4) and (1.5) according to the results of the work [1] (see theorem 3.1 and lemma 3.6, pp. 191-193), the equation (2.5) has nonnegative summerable on R+ solution f(x). Taking into account (1.4), (1.7), (2.2) and (2.4) from
(2.5) we get
g(x) < ф) < g(x) + ( sup Ko(x)) <p(t)dt. (2.6)
жек+ J
Hence, it can be stated that
<p e Li(R+) n M(R+) and lim <p(x) = 0. (2.7)
We introduce the following simple iterations for equation (2.7)
F<«+1>(x)= ,(,) + A(x) fat — ,) — KiM)F<«■(,)*
X
F (0)(x) = g(x), n = 0,1, 2,..., x e R+.
(2.8)
Using (1.9), (1.7), (1.4), (1.5) and (2.5), by the method of mathematical induction it is not difficult to establish the validity of the following statements
F(n) (x) t by n, (2.9)
F(n)(x) < <p(x), n = 0,1,2,..., x e R+, (2.10)
F(n) (x) < 1, n = 0,1, 2,..., x e R+, (2.11)
where <p(x) — is the solution of integral equation (2.5) with the properties (2.7).
Therefore there exists pointwise limit of sequences
[F(n)(x)}o°=0 : lim F(n)(x) =: Fi(x),
n—too
moreover limit function in accordance to B. Levi's theorem (see [6]), satisfies the equation (2.1). In view of (2.7) from (2.9)-(2.11) it follows that
g(x) < Fi(x) < <p(x), x e R+, (2.12)
Fi e Li(R+), Fi(x) < 1, x e R+, lim Fi(x) = 0. (2.13)
Thus, due to (2.12) and (2.13), we can state that Fi(x) ^ 1, x e R+. It is obvious that S*(x) := 1 — Fi(x) > 0, S*(x) ^ 0 will be solution of the linear homogeneous equation
oo
S(x) = \(x) j(Ko(t — x) — Ki(x, t))S(t)dt, x e R+. (2.14)
X
Moreover, due to (2.12)-(2.13), the function S*(x) possesses the following properties
0 < 5*(x) < 1, S*(x)^ 0, x e R+, (2.15)
lim 5*(x) = 1, 1 — 5* e Li(R+). (2.16)
Now let's go back to the study of the initial nonlinear integral equation (1.1). We introduce the following special iterations for (1.1):
CO
fn+\(x) = u(x,t, fn(t))dt, X e R+, 0
1* (
(2.17)
fo(x)= £S*(x), n = 0,1, 2,.... By induction we prove that
fn(x) f by n, (2.18)
fn(x) < V, x e R+, n = 0,1,2,.... (2.19)
Observe that from (2.15) and £ < condition follows the inequality fo(x) < x e R+. Now we show fl(x) > fo(x). Indeed, due to monotonic-ity of Urysohn's kernel u(x, t, z) by z on [0, rf], and taking into consideration (1.7), (2.14), condition d) from (2.17) we gate
oo oo
fi(x) = ju(x,t,iS*(t))dt > ju(x,t,iS*(t))dt >
0 x
o
> &(x) J(Ko(t - x) - Ki(x,t))S*(t)dt = *(x) = fo(x).
x
Assume that fn(x) > fn-i(x) and fn(x) < ri, x e R+ for some natural n. Using monotonicity of the function u(x, t, z) by z, and also inequality (1.3), from (2.17) we obtain
o
fn+\(x) >ju(x,t, fn-i(t))dt = fn(x), x e R+,
fn+l(x) < ju(x,t,rj)dt < 'q, x e R+. 0
Thus (2.18) and (2.19) are proved. Using c) and the Caratheodory's conditions, taking into account M.A. Krasnoselskii theorem (see [7]), by induction in n it is easy to check that each element of sequence{/ra(^)}00Lo represents measurable function on R+.
Thus, due to (2.18) and (2.19) the sequence of measurable functions {fn(^)}0Lo has a pointwise limit: lim fn(x) = f (x), and
*(x) < f (x) < v, X e R+. (2.20)
Finally, based on the B. Levi's limit theorem and Caratheodory condition, we conclude that f (x) satisfies the equation (1.1). Theorem is proved.
Remark 1. We consider the following particular representation of Ury-sohn's kernel
u
(x,t,z) = A(x)(K(x-t) --Kl(x,y))G(z), (x,t,z) G R+ x R+ x R, (2.21)
where K(x) is defined on set R, even and bounded function, moreover
CO <
0 < K e Li (R), J K(x)dx = 1, J xK(x)dx <
K(x - y) > 2Ki(x,y), (x,y) e R+ x R+,
and function G is the monotonic increasing convex up and continuous function on interval [0,n], G(u) > u,u e [0,n], G(n) = n and G(£) = 2£, 0 <£ <n (see fig. 1).
Figure 1.
Note that with such a representation of the Urysohn's kernel, all conditions of the proved theorem are automatically satisfied.
If Urysohn's kernel admits representation (2.21), then from result of the work [14] follows that the solution of equation (1.1) possesses the following additional properties
(2.22)
lim f(x) = n,
n - f G Li(R+ ).
(2.23)
Remark 2. It should be also noted that the proved result generalizes and complements the existence theorems for the solution of the Urysohn's equation reported by the works [11]- [13].
KH. A. KHACHATRYAN, H. S. PETROSYAN 3. Application in p-adic string theory
In the dynamic theory of p-adic open-closed strings, the following nonlinear integral equation with Gaussian type kernel depending on sum and difference of arguments are encountered (see [16]):
BP(x) = I (e-(x-t)2 - e-(x+t)2)B(t)dt, x e R+ (3.1)
■к
with respect to sought bounded function B(x), where p > 2 - odd number. In this theory it becomes necessary to construct nonnegative nontrivial and bounded solution for equation (3.1), satisfying the limit relation lim B(x) = 1.
By simply substitution f (x) = Bp(x), the problem of finding such solution is reduced to solve the problem of Hammerstein type integral equation
f (x) = —^ f (e-(x-t)2 - e-(x+t)2)f p (t)dt, x e R+, (3.2)
v^ j 0
with boundary condition
lim f(x) = 1. (3.3)
By direct verification, one can state that the Urysohn's kernel u(x,t,z) = —1—(e-(x-t)2 - e-(x+t)2)zp
satisfies all the conditions of the theorem, which we have proved by choosing
p
as a ^ = 1, £ = (1)p-1 , and
K0(x) = e-x2, Ki(x,y) = e-(x+y)2, X(x) = 1, x G R+,y G R+. J-K Jn
Based on Remark 1 and on the above proved theorem we can assert that solution of the boundary value problem (3.2), (3.3) possesses property 1 - f e Li(R+).
At the end of the work, we still list some nontrivial examples of the Urysohn's kernels
1) u(x, t, z) = (K (x - t) - К (x + t))G(z), (x, t, z) e R+ x R+ x R,
where К and G satisfy conditions of Remark 1,
2) u(x, t, z) = \(x)K(x - t)G(z), (x, t, z) e R+ x R+ x R,
3) u(x,t,z) = (1 -\{x))K{x-t)G0(z) + \(x)(Ko(t-x) -Ki(x,t))zd(t-x),
(x, t, z) £ R+ x R+ x R, where d — Heaviside function, and Go £ C[0, rj], G0 t on [0, rj], Go(0) = 0, G0(rj) = rj,
4) u(x,t,z) = (1 - \(x))K(x - t)GQ(z) + \(x)K0(t - x)G(z)d(t - x),
(x,t,z) £ R+ x R+ x R.
4. Conclusion
In this paper, we have obtain sufficient conditions for the existence of a nonnegative and bounded solution for class of Urysohn type nonlinear integral equations on half line. The proved theorem of existence has a constructive character. For one particular representation of Urysohn's kernel the limit of constructed solution at infinite is found, as well as has been shown that difference between limit and solution represents summable function on positive part of numerical axis. The obtained results are applied in dynamic theory of p - adic open-closed strings. The proved results generalize and complement the theorems of existence from works [11] -[13]. Specific and applied examples of the Urysohn's kernel are given. The examples illustrate how all conditions of the proved theorem are fulfilled.
References
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Khachatur Khachatryan, Doctor of Sciences (Physics and Mathematics), Professor, Lomonosov Moscow State University, 1, Leninskiye Gory, GSP-1, Moscow, 119991, Russian Federation; Institute of Mathematics of NAS of Armenia, 24/5, Marshal Baghramyan pr., Yerevan, 0019, Republic of Armenia; Yerevan State University, 1, Alex Manoogian st., Yerevan, 0025, Republic of Armenia, tel.: (+374)91300251,
email: khachatur.khachatryan@ysu.am, ORCID iD https://orcid.org/0000-0002-4835-943X.
Haykanush Petrosyan, Candidate of Sciences (Physics and Mathematics), Associate Professor, Lomonosov Moscow State University, 1, Leninskiye Gory, GSP-1, Moscow, 119991, Russian Federation; Armenian National Agrarian University, 74, Teryana st., Yerevan, 0009, Republic of Armenia, tel.: (+374)55022007, email: Haykuhi25@mail.ru, ORCID iD https://orcid.org/0000-0002-7172-4730.
Received 01.02.2021
О разрешимости одного класса нелинейных интегральных уравнений Урысона на положительной полупрямой
Х. А. Хачатрян1'2'3, А. С. Петросян1'4
1 Московский государственный университет им. М. В. Ломоносова, Москва, Российская Федерация
2 Ереванский государственный университет, Ереван, Республика Армения
3 Институт математики НАН Армении, Ереван, Республика Армения
4 Национальный аграрный университет Армении, Ереван, Республика Армения
Аннотация. Исследовано нелинейное интегральное уравнение Урысона на положительной полупрямой. Обозначено, что некоторые частные случаи данного уравнения имеют конкретные применения в различных направлениях современного естествознания. В частности, такие уравнения возникают в кинетической теории газов, в теории р-адических открыто-замкнутых струн, в математической теории пространственно-временного распространения эпидемии и в теории переноса излучения в спектральных линиях. Доказана теорема существования неотрицательных нетривиальных и ограниченных решений. Изучены некоторые качественные свойства построенного решения. Приведены конкретные прикладные примеры ядра Урысона, удовлетворяющие всем условиям доказанной теоремы.
Ключевые слова: уравнение Урысона, монотонность, условие Каратеодори, итерации, ограниченное решение.
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Хачатур Агавардович Хачатрян, доктор физико-математических наук, профессор, Московский государственный университет им. М. В. Ломоносова, Российская Федерация, г. Москва, 119991, ГСП-1, Ленинские горы, 1; Ереванский государственный университет, Республика Армения, 0025, г. Ереван, ул. Алека Манукяна, 1; Институт математики НАН Армении, Республика Армения, 0019, г. Ереван, пр. Маршала Баграмяна, 24/5, тел.: (+374)91300251,
email: khachatur.khachatryan@ysu.am, ORCID iD https://orcid.org/0000-0002-4835-943X.
Айкануш Самвеловна Петросян, кандидат физико-математических наук, доцент, Московский государственный университет им. М. В. Ломоносова, Российская Федерация, г. Москва, 119991, ГСП-1, Ленинские горы, 1; Национальный аграрный университет Армении, Республика Армения, 0009, г. Ереван, ул. Теряна, 74, тел.: (+374)55022007, email: haykuhi25@mail.ru, ORCID iD https://orcid.org/0000-0002-7172-4730.
Поступила в 'редакцию 01.02.2021
