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Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki
[J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2022, vol. 26, no. 2, pp. 368-379 ISSN: 2310-7081 (online), 1991-8615 (print) d https://doi.org/10.14498/vsgtu1917
MSC: 34C25, 34B15
Periodic solutions for an impulsive system of integro-differential equations with maxima
T. K. Yuldashev
National University of Uzbekistan named after Mirzo Ulugbek,
4, Vuzgorodok, Universitetskaya st., Tashkent, 100174, Uzbekistan.
Abstract
A periodical boundary value problem for a first-order system of ordinary integro-differential equations with impulsive effects and maxima is investigated. A system of nonlinear functional-integral equations is obtained and the existence and uniqueness of the solution of the periodic boundary value problem are reduced to the solvability of the system of nonlinear functional-integral equations. The method of successive approximations in combination with the method of compressing mapping is used in the proof of one-valued solvability of nonlinear functional-integral equations. We define the way with the aid of which we could prove the existence of periodic solutions of the given periodical boundary value problem.
Keywords: impulsive integro-differential equations, periodical boundary value condition, nonlinear kernel, compressing mapping, existence and uniqueness of periodic solution.
Received: 16th March, 2022 / Revised: 25th April, 2022 / Accepted: 23rd May, 2022 / First online: 30th June, 2022
1. Problem Statement
The mathematical models of many problems of modern sciences, technology, and economics are described by differential and integro-differential equations, the solutions of which are functions with first-kind discontinuities at fixed or non-fixed times. Such differential and integro-differential equations are called equations with impulsive effects. Various publications are appearing on the study of differential
Differential Equations and Mathematical Physics Short Communication
© Authors, 2022
© Samara State Technical University, 2022 (Compilation, Design, and Layout) Q ©® The content is published under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/) Please cite this article in press as:
Yuldashev T. K. Periodic solutions for an impulsive system of integro-differential equations with maxima, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2022, vol. 26, no. 2, pp. 368-379. EDN: TYZLDB. DOI: 10.14498/vsgtu1917. Author's Details:
Tursun K. Yuldashev https://orcid.org/0000-0002-9346-5362 Dr. Phys. & Math. Sci.; Professor; Uzbek-Israel Joint Faculty; e-mail: [email protected]
and integro-differential equations with impulsive effects, describing many natural and technical processes (see for instance [1-13]).
As is known, in recent years, interest in the study of differential and integro-differential equations with periodical boundary conditions has increased. In particular, in the works [14-17], periodic solutions of differential equations with impulsive effects are studied.
In this paper, we investigate a periodical boundary value problem for a system of first-order integro-differential equations with impulsive effects, nonlinear kernel depending on construction of maxima. The questions of existence and uniqueness of the solution to the periodical boundary value problem are investigated. We note that differential and integro-differential equations with maxima have singularity in the study of the questions of solvability [18].
On the interval [0,T] for t = ti (i = 1,2,... ,p) we consider the questions of existence and constructive methods of calculating the periodic solutions of the system of nonlinear ordinary first-order integro-differential equations with impulsive effects and maxima
x'(t) = f(t,x(t),J K(t,s, max{i(T): t e [Ai(s), \2(s)] })ds^J . (1) We study the integro-differential equation (1) with periodic conditions
{
x(t) = <p(t), t e 0], x(0) = x(T),
(2)
and nonlinear impulsive effect
x(t+) - x(tr)= Fi(x(ti)), i = 1, 2,...,p, (3)
where 0 < i < T, i = ti, i = 1, 2,... ,p; 0 = t0 < t1 < ■■ ■ < tp < tp+1 = T x, y e X; X is the closed bounded domain in the space Rra, dX is its border f e Rra, -to < Ai(i) < \2(t) < t, <p(t) e C((-to,0],Rra); ^(0") = z(0+) x(t+) = lim x(t.i + v), x(t") = lim x(ti — v) are the limits of the function on
v ^0+ v
the right and left sides x(t) at the point t = ii, respectively. The function f is T-periodic Fi = F+, U+p = ti + T,
/ \K(t, s, x)\ds < to. J—<x
By C([0,T], Rra) denoted the Banach space, which consists continuous vector function x(t), defined on the segment [0,T], with the norm
«x(i)B =
\
Emax | Xj (i)|.
j=i
By PC([0, T], Rra) is denoted the following linear vector space:
PC ([0,T], Rra) = {x : [0,T] ^ Rra; x(t) e C ((U,ti+1], Rra), i = 1, 2,...,p},
where x(t+) and x(t~) (i = 0,1,... ,p) exist and bounded; x (t~) = x (ti). Note, that the linear vector space PC([0, T], Rra) is Banach space with the following norm:
Wx^Wpc = \\x(t)\\c{{ti,ti+l]), i = 1,2^. .,p}.
Formulation of problem. Find the T-periodic function x(t) £ PC([0,T],Rra), which for all t £ [0,T], t = ti (i = 1, 2,... ,p) satisfies the system of differential equations (1), periodic condition (2) and for t = ti (i = 1, 2,... ,p) 0 < t\ <t2 < ■ ■ ■ < tp < T satisfies the nonlinear limit condition (3) and goes through x0 at t = 0.
2. Reduction to Functional-Integral Equation
Let the function x(t) £ PC([0, T], Rra) be a solution of the periodic boundary value problem (1)-(3). Then by integration of the equation (1) on the intervals (0,ii], (ti,t2], ..., (tp,tp+1], we obtain the following:
rt 1 ft 1
/ f (s,x,y) ds = / x'(s) ds = x(t-) — ^(0+), t £ (0,ii|, J 0 J 0
ft 2 rt2
/ f (s,x,y) ds = / x'(s) ds = x(t-) — x(t+), t £ (t\,t2], Jti Ju
ftp+1 rtp+1
/ f(s,x,y) ds =\ x'(s) ds = x(t-+1) — x(t+),t £ (tp,tp+i],
J t^ J t -n
where
f (s,x,y) = f(t,x(t), J K(t,s, max{x(T) : t £ [Ai(s), \2(s)}})ds^j .
Hence, taking ^(0+) = «(0), x(t-+1) = x(t) into account, on the interval [0,T] we have
/ f (s, x, y) ds = [x(ti) - z(0+)] + [x(t2) - x(t+)] + ■■■ + [x(t) - x(t+)] = J 0
= -z(0) - [x(t+) - x(ti)] - [x(t+) - x(t2)]-----[x(t+) - x(tp)] + x(t).
Taking into account the condition (3), we rewrite the last equality as
x(t)= ®(0)+ / f (s,x,y) ds + V Fi(x(ti)). (4)
0
We subordinate the function x(t) £ PC([0,T], Rra) in (4) to satisfy the periodic condition (2):
¡■t
x(T)= x(0)+ f (s,x,y) ds + V Fl(x(ti)). 0
0 0 <ti<T
Hence, taking the condition (2) into account, we obtain the following:
rT 10
fT
/ f( s,x, y)ds + V F(x(U)) = 0. J о nJT".^
0< ti<T
Consequently, the integro-differential equation (1) can be written as
'(t) = f (t,x(t), J К(t,s, max{x(r) : т G [Xi(s), A2(s)j })ds^ -
^J f(t,x(t), J К(t,s, max{x(r) : т G [\i(s), \2(s)]})ds^Jdt —
1 p
^Fi (x(U)). (5)
Then by integration of equation (5) into the intervals (0, i1], (t 1,12], ..., (tp, tp+1] instead of (4) we obtain the following system of equations:
x(t)=x0 + j f^s,x(s),J K(s,6, max{s(r) : t e [ \1(0),\2(0)]})de^ -
- Uof ie,X(()),£ K ^ max{x(r) : T e MO, A2(0]}K) M-
1 P 1
7^Y,pi(x(ti))ds + E Fi(x(fi)). (6)
i=1 -J 0< ti<t
3. Preliminaries
Lemma 1. For the equation (6) the following estimate is true:
\\x(t) - x0\\pc < M11 + 2M2P, (7)
where
Mi = II/(t ,x(t), y(t))\\, M2 = max \\F(t ,x(t))\\.
Proof. We rewrite the equation (6) as follows:
x(t) — x0 =
Jo
f (S, x(S), y(s)) —
1 fT 1 p
1 f(e,x(d), y(d))dd — ~Y,Fi(x(ti)) ds + Fi(x(ti)) '0 i=i
0< ti<t
= So ^(S,x(s), y(s))ds — T So ,x(s), y(s))ds —
x
T
,-t , p
f( s ,x(s), y(s))ds — -Y^FMU)) + Y. Fi(x(U)). Jt i=i 0<ti<t
Hence, this implies that the following estimate is true:
\\x(t) -xo\\pc < a(t)\\f(t,x(t), y(t))\\ + 2p max \\Fi(t,x(t))\\, (8)
l^i^p
where a(t) = 2t(\ — t/T). It is easy to check that from (8) follows (7). Lemma 1 is proved. □
Remark. T-periodic solution xv(t) = ^(t) of the system (1) with the initial value function (p(t) in the initial set (—x>, 0] is defined by the initial value function ip(t), which is a periodic continuation of the solution ^(t) in the initial set (—x>, 0].
Lemma 2. For the difference of two functions with maxima, we have the following estimate:
\\ max{x(r) : t £ [Xi(t), A2CO]} — max{y(t) : t £ [Ai(i), \2(t)]}\\ <
d
< \\x(t) — y(t)\\ +2h -[x(t) — y(t)] , (9)
where
h = sup l\i(t) — \2(t)l
-<x>< t^T
Proof. We use obvious true relations:
max{x(r) : t £ [Xi(t), A2W]} = max{[x(r) — y(r)+y(r)] : t £ [Xi(t), A2W]} < < max{[x(r) — y(r)] : t £ [\i(t), X2(i)]} + max{y(T) : t £ [Xi(t), \2(t)]}.
Hence, we obtain the following:
max{x(r) : t £ [\i(t), A2W]} — max{x(r) : t £ [\i(t), A2W]} <
< max{[x(r) — y(r)] : r £ [Ai(t), A2(t)]}. (10)
We denote by ti and t2 the points of the interval [Ai(i), A2(i)], on which the maximums of the functions x(t) and y(t) are reached:
max {x(r) It £ [Ai(t), A2(t)] } = x(ti), max {y(t) It £ [Ai(i), A2(t)] } = y(ti),
max {[x(r) — y(r)] It £ [Ai(t), A2W] } = x(t2) — y(t2). Then, taking (10) and last equalities, we have
\\ max{x(r):T £ [Ai(t), A2(t)]} — max{y(t) : t £ [Ai(t), A2(t)]} — x(t)+ y(t)\\ < < \\[x(t) — y(t)] — [x(ti) —y(ti)]\\ + \\[x(t2) — y(t2)] — [x(ti) —y(ti)]\\. (11)
From another side, it is obvious that the estimate is valid:
\\[x(i) — y(t)] — [x(f) —
^ h
dt [x(f *) — y(f *)]
^ h
dt [x(f) —y(f)]
(12)
where t, t e [A1(i), A2(i)], t* e (t,t). From the estimates (11) and (12) we come to the following estimate:
\ max{x(r):r G [\i(t), \2(t)]} — max{y(т) : т G [\i(t), \2(t)]} — x(t)+ y(t)\\ <
^ 2h
dit [x(f)—y(f)]
Therefore, it is easy to check that the inequality (9) and we complete the proof of Lemma 2. □
4. Main Results
Theorem 1. Assume that for all t e [0, T], t = ti (i = 1, 2,..., p) the following conditions are satisfied:
1) \\f (t ,x(t), y(i))\\ SM1 < to, max \\Fi(t ,a:(i))\\ S M2 < to;
1S iSSp
2) \\f(t ,X1, V1) - f(t ,X2, V2)\\ s L1[\\X1 -X2\\ + \\V1 - y2\\],
3) \\K(t,s,x1)-K(t,s,x2)\\ - x2\\, 0 < sup L2(s)ds < to;
t J — <x
4) \\Fi(t, X1) -Fi(t,X2)\\ <Ls\;c1 - X2W;
5) the radius of the inscribed ball in X is greater than M1T/2 + 2M2p;
6) p < 1, where
p = max{L1(1 + M3)(1 + + PLs(2 + 1), 2LM(T +
If the system (1) has a solution for all t e [0, T], t = ti (i = 1, 2,..., p), then this solution can be founded by the system of nonlinear functional-integral equations
x(t ,X0) =X0 +
+
0
jf f^s, x(s, X0), j К {s, 9, max{x( т, X0) : т G [Ai(0), A2(9)]})d9 ^ — — Uof {9, x(°, X0),f К{0, С, max{x(т, X0) : т G [Ai(0, A2(0]})d^ d9 —
, x(
0
1 p
TY,Fi(x(ti,X0)) ds + Y, Fi(x(ti,x0)). (13) i=1 -1 0<ti<t
Proof. The theorem we proof by the method of successive approximations, defining the iteration process as
X0(t, X0) = X0, xk+i(t, X0) = X0 +
+f
0
1 fT
T. 0
fys,Xk(s,X0), J K{s,9, max{xk(т,Х0):т G [Ai(9), A2(9)]})d9j — e,Xk (9 ,X0 ),Je К (в, С, max{xk (т ,Х0):т G [Ai(0,A2(0]})d^ dB—
1 P 1
T^Fi(xk(ti,X0)) ds + Fi(xk(ti,X0)). (14) i=1 1 0< ti<t
We will show that the right-hand side of the system of equations (13) as an operator maps a ball with radius MiT/2 + 2M2p into itself and is a contraction operator. So, according to Lemma 1, from (7) and (14) we have
\\xk+i(t,xo) —xo\\pc < MiT- + 2M2P. (15)
From the system of integro-differential equations
x' (t,x0) = f ^t,x(t ,x0), J K (t,s, max[x(T, x0) : t £ [Ai(s), A2(s)]})ds^ —
J f^t,x(t,x0), J K (t,s, max[x(T,x0 ):t £ A (s), A2(s)]})ds^j dt —
T. ,o
1 P
(16)
i=i
we obtain the following:
\\x'k+i(tx)\\pc < 2Mi + Tm2. (17)
We consider a difference xk+i(t,x0) — xk(t,x0) of two approximations, where the functions xk+\(t,x0) and xk(t,x0) are defined from the approximations of the system of equations (14). By the conditions of the theorem, from (14) we have
\\xk+i(t,xo) —xk(t,xo)\\ ^ Li J |\xk(s,xo) — xk-i(s,xo) \\ +
+ ¡S L2 (d)\\max{xk (r,xo):r G [Ai(0), A2W]} -
— max[xk-i(T,xo) : T G [Xi(d), X2(d)]}\dd + \\xk(d,xo) -xk-i(0,xo)\\ +
' — oo
1 '-T
+ T J0
+ Î L2(0\max{xk(r,xo):r G [\i(0,HO]}-
' — oo
- max{xk-i(r,xo) :t G [Ai({), A2CO]}\\d{
dd >ds +
+ 2^L3\xk(U,xo) - xk-i(ti,xo)\\ ^
i=i
< a(t)Li [(1 + M3)\\xk(t, xo) - xk-i(t, xo)\\pc +
+ 2hM3\\x'k(t,xo) -x'k-i(t,xo)\\pc] +2pL3\\xk(t,xo) - xk-i(t,xo)\\pc ^ < (Li(1 + M3)T. + 2pL3^\\xk(t,xo) -xk-i(t,xo)\\pc +
+ 2LiM3Th\\x'k(t,xo) -xk_i(t,xo)\\pc, (18)
where
M3 = sup L2(s)ds < ж.
t J — X
Similarly, by the assumptions of Theorem 1, from (16) we have the following: \\x'k+l(t,xo) -x'k(t,Xo)\\pc <
< (lx(1 + M3) + L3T)\\xk(t, xo) - Xk—i(t, xo)\\pc +
+ 2LlM3h\\xk(t,xo) -xk—i(t,xo)\\pc. (19) Adding both sides of (19) to (18), we obtain
\\ Vk+i(t,xo) - Ук(t,xo) \\pc < р\\Ук(t,xo) -yk—i(t,xo)\\pc, (20)
where
\\yk+i(t,xo) - yk(t,xo)\ = \\xk+i(t,xo) -xk(t,xo)\\ + \\x'k+i(t,xo) - x'k(t,xo)\\,
p = max{Li(1 + M3)(l + + PLs(2 + 1), 2LiM3(T + 1)hJ. According to the last condition of Theorem 1, p < 1. Since
\\xk+i (t ,xo) -xk (t ,xo)\\ ^ \\yk+i(t ,xo) - yk (t ,xo)\\,
from the estimate (20) we deduce that the operator on right-hand side of (13) is compressing. From the estimates (15), (17) and (20) implies that there exists a unique fixed point x(t,x0). Theorem 1 is proved. □
From the estimate (20) it is easy to see that for x0, x0 E X holds
\\x(t, xo) x( t ,xo)\\bd ^ щ-—.
1- p
Now we will show the existence of periodic solutions of the system of impulsive
integro-differential equations (1). We introduce the following designations:
1
xo) = 1 ^ Fi(xx(ti, xo)) +
T
i=l
+1 J f(t,x„(t,xo), j K(t,s, max[xXl(T,xo) : t e [Xi(s), \2(s)]})ds^jdt, 0 (21)
1 P
A k (Xo) = 7^YFi(Xk (t i,X0 )) +
i=l
+ 1/ f(t,Xk(t,Xo), J K(t,s, max{xk(r,x0) : T e [\i(s), \2(s)]})ds^jdt, 0 (22) where x(t,x0) = lim xk(t,x0) = x^(t,x0) is the solution of the non-linear sys-
k—y^o
tem (13). Therefore, x^(t,x0) is the solution of the system of impulsive integro-
differential equations (1) for A(xo) = 0 through xo at t = 0. Consequently, the questions of the existence of a solution of the system of impulsive integro-differential equations (1) were reduced to the questions of the existence of zeros of the function A(xo) and we solve this problem by finding zeros of the function Ak (xo).
Theorem 2. Assume that
1) all the conditions of Theorem 1 are fulfilled;
2) there is a natural number k such that the function Ak(x0) has an isolated singular point x0 that Ak(x°) = 0 and the index of isolated singular point x0 is nonzero;
3) there is a closed convex region X0 C X, containing a single singular point such that on its border dX0 is an estimate fulfilled:
Mpk+i
inf \\Ak(x)\\ > (23)
xedXo I — —
Then the system of impulsive integro-differential equations (1) has a periodic solution for all t £ [0,T], t = U (i = I, 2,...,p) that x(0) £ X0.
Proof. By definition, the index of an isolated singular point x0 of continuous mapping A k(x0) is equal to the characteristic of the vector field, generated by mapping Ak(x0) on a sufficiently small sphere Sn with the center in x°. Since in X0 there is no other singular point, which will be different from x0 and X0 is homeomorphic to the unit ball En, then the characteristic of the vector field A k(xo) on the sphere Sn is equal to the characteristic of this vector field on dX0. The fields Ak(x0) and A(x0) are homotopic on dX0. Let us consider families of everywhere continuous on dX vector fields
V (a, xo) = Ak (xo) + a(A(xo) — Ak (xo)), which connect the fields
V (0, xo) = Ak (xo), V (I, xo) = A(xo). We note that the estimate is true:
Mpk+i
\\A(xo) — Ak(xo)\\ < (24)
I—
Therefore, the vector field V(a,xo) does not vanish anywhere on dXo. Indeed, from (23) and (24) implies that
\\V(a, xo) \\ ^ \\Ak(xo)\\ — \\A(xo) — Ak(xo)\\ > 0. (25)
The fields A k(xo) and A(xo) are homotopic on dX and the rotations of the homotopic fields in the compact are equal. Therefore, taking into account (25), we conclude that the rotation of the field A(xo) on the dXo is equal to the index of the singular point xo of the field Ak(xo) and nonzero. Consequently, the vector field A(xo) on Xo has a singular point xo, for which A(xo) = 0. Therefore, the system of impulsive integro-differential equations (1) has a periodic solution for
all t £ [0,T], t = U (i = 1,2,..., p) that x(0) £ Xo. In addition, we note that for xo, xo £ X from (21) and (22) we have
The theory of differential and integro-differential equations plays an important role in solving many applied problems. Especially, local and nonlocal periodical boundary value problems for differential and integro-differential equations with impulsive actions have many applications in mathematical physics, mechanics and technology, in particular in nanotechnology. In this paper, we investigate the system of first-order integro-differential equations (1) with periodical boundary value condition (2), with nonlinear kernel and with nonlinear condition (3) of impulsive effects for t = U, i = 1,2,...,p, 0 < ti < 12 < ■ ■ ■ < tp < T. The nonlinear right-hand side of this equation consists of the construction of maxima. The questions of the existence and uniqueness of the T-periodic solution of the boundary value problem (1)-(3) are studied. If the system (1) has a solution for all t £ [0, T], t = ti, i = I, 2,... ,p, then this solution can be proven to be based on the system of nonlinear functional-integral equations (13). The questions of the existence of a solution of the system of impulsive differential equations (1) we reduce to the questions of the existence of zeros of function A(xo) in (21) and we solve this problem by finding zeros of function Ak(xo) in (22).
The results obtained in this work will allow us in the future to investigate other kind of periodical boundary value problem for the heat equation and the wave equation with impulsive actions.
Competing interests. I declare that I have no competing interests. Authors' contributions and responsibilities. I take full responsibility for submit the final manuscript to print. I approved the final version of the manuscript. Funding. Not applicable.
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□
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Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки. 2022. Т. 26, № 2. С. 368-379 ISSN: 2310-7081 (online), 1991-8615 (print) d https://doi.org/10.14498/vsgtu1917
EDN: TYZLDB
УДК 517.968.78
Периодические решения системы интегро-дифференциальных уравнений с импульсными воздействиями и максимумами
Т К. Юлдашев
Национальный университет Узбекистана им. М. Улугбека,
Узбекистан, 100174, Ташкент, Вузгородок, ул. Университетская, 4.
Аннотация
Исследуется краевая задача для системы обыкновенных интегро-дифференциальных уравнений первого порядка с импульсными эффектами и максимумами. Получена система нелинейных функционально-интегральных уравнений и, таким образом, существование и единственность решения периодической краевой задачи сводятся к разрешимости системы нелинейных функционально-интегральных уравнений. Метод последовательных приближений в сочетании с методом сжимающих отображений используется при доказательстве однозначной разрешимости нелинейных функционально-интегральных уравнений. Определим способ, с помощью которого можно будет доказать существование периодических решений данной периодической краевой задачи.
Ключевые слова: интегро-дифференциальные уравнения с импульсными воздействиями, периодическое краевое условие, нелинейное ядро, сжимающее отображение, существование и единственность периодического решения.
Получение: 16 марта 2022 г. / Исправление: 25 апреля 2022 г. / Принятие: 23 мая 2022 г. / Публикация онлайн: 30 июня 2022 г.
Конкурирующие интересы. Я заявляю, что конкурирующих интересов не имею.
Авторская ответственность. Я несу полную ответственность за предоставление окончательной версии рукописи в печать. Окончательная версия рукописи мною одобрена.
Финансирование. Исследование выполнялось без финансирования.
Дифференциальные уравнения и математическая физика Краткое сообщение
© Коллектив авторов, 2022 © СамГТУ, 2022 (составление, дизайн, макет)
3 ©® Контент публикуется на условиях лицензии Creative Commons Attribution 4.0 International (https://creativecommons.org/licenses/by/4.0/deed.ru) Образец для цитирования
Yuldashev T. K. Periodic solutions for an impulsive system of integro-differential equations with maxima, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2022, vol. 26, no. 2, pp. 368-379. EDN: TYZLDB. DOI: 10.14498/vsgtu1917.
Сведения об авторе
Турсун К. Юлдашев А https://orcid.org/0000- 0002- - 9346- 5362
доктор физико-математических наук; профессор; Узбекско-Израильский совместный факультет; e-mail: [email protected]
