CC BY
15
Chelyabinsk Physical and Mathematical Journal. 2022. Vol. 7, iss. 1. P. 113-122.
DOI: 10.47475/2500-0101-2022-17108
NONLOCAL PROBLEM FOR A NONLINEAR SYSTEM OF FRACTIONAL ORDER IMPULSIVE INTEGRO-DIFFERENTIAL EQUATIONS WITH MAXIMA
T.K. Yuldashev", Kh.Kh. Saburovb, T.A. Abduvahobovc
National University of Uzbekistan, Tashkent, Uzbekistan
"[email protected],, [email protected], [email protected]
A nonlocal boundary value problem for a system of ordinary integro-differential equations with impulsive effects, maxima and fractional Gerasimov — Caputo operator is investigated. The boundary value condition is given in the integral form. The method of successive approximations in combination with the method of compressing mapping is used. The existence and uniqueness of a solution of the boundary value problem are proved.
Keywords: impulsive integro-differential equation, Gerasimov — Caputo operator, nonlocal boundary condition, successive approximations, unique solvability.
1. Introduction
Many problems in modern sciences, in technology and in economics are described by differential equations, the solution of which is function with first kind discontinuities at fixed or non-fixed times. Such differential equations are called differential equations with impulse effects [1-5]. In recent years the interest in the studying of differential equations with nonlocal boundary value conditions is increasing (see, for example, [6-18]). Also a lot of publications of studying on differential equations with impulsive effects, describing many natural and practical processes, are appearing [19-27].
In this paper, we investigate a nonlocal boundary value problem for a system of ordinary fractional order nonlinear integro-differential equations with impulsive effects and maxima. The questions of the existence and uniqueness of the solution of the boundary value problem, as well as the stability of the solution on the right-hand side of the boundary condition, are investigated. We note, that the differential equations with maxima play an important role also in solving control problems of market economy [28]. In [29], it is showed that the differential equations with maxima have singularities in one value solvability.
Fractional calculus plays an important role in the mathematical modeling of many problems in scientific and engineering disciplines [30]. In [31] it is considered problems of continuum and statistical mechanics. In [32] the mathematical problems of Ebola epidemic model are investigated. In [33; 34] the fractional model for the dynamics of tuberculosis infection and novel coronavirus (nCoV-2019), respectively, are studied. The construction of various models of theoretical physics by the aid of fractional calculus is described in [35, vol. 4, 5; 36; 37]. A detailed review of the application of fractional calculus in solving problems of applied sciences is given in [38-40]. In [41], it is considered an inverse problem for a mixed type integro-differential equation with fractional order Gerasimov — Caputo operators. In the direction of applications of fractional derivatives to solving differential equations interesting results were obtained also in [42-44].
We recall some basic terms of fractional integro-differentiation. Let (to, T) C R+ := [0, to) be an interval on the set of positive real numbers, where 0 < to < T < to. The Riemann — Liouville a-order fractional integral of a function n(t) is defined as follows:
t
I0tn(t) = /(t - s)a-1n(s)ds, a > 0, t e (to,T), t0
where r(a) is the Gamma-function. Let n — 1 <a < n, n e N. The Riemann — Liouville a-order fractional derivative of a function n(t) is defined as follows:
dn
DOtn(t) = ^im-an(t) t e (to,T).
The Gerasimov — Caputo a-order fractional derivative of a function n(t) is defined by
t
Cd:= inr= rn—-)j (t—P+t. *e («0,T).
to
These derivatives are reduced to the n-th order derivatives for a = n e N:
dn dtn
Dtotn(t) = cDtntn(t) = n(t), t e (to, T).
In this paper, we use the case 0 < a < 1 for a Gerasimov — Caputo type fractional operator.
2. Problem statement
On the segment [0,T] for t = tj, i = 1, 2,...,p and 0 < a < 1 we consider the
following fractional order system of nonlinear integro-differential equations
( T
CDt«tx(t) = f t,x(t), / 6 (t, s, max {x(t)|t e [ATs,A2s]}) ds with nonlocal boundary value condition
T
Ax(0)^y K(t,s) x(s) ds = B(t,x(t)) (2)
o
and nonlinear impulsive effect
x (t+) — x (t-) = Ij (x (tj)) , i = 1, 2,...,p, (3)
where 0 = t0 < tT < ■ ■ ■ < tp < tp+1 = T, A e Rnxn is a given matrix, K(t, s) is a
T
given n x n-dimensional matrix function and det Q(t) = 0, Q(t) = A + J K(t,s)ds,
o
f : [0, T] x Rn x Rn — Rn, 6 : [0, T]2 x Rn — Rn, Ij : Rn — Rn, B : [0, T] x Rn — Rn
are given functions; 0 <A T <A 2 < 1, x ft+) = lim x (tj + h), x ft-) = lim x (tj — h)
v ' v '
are right-hand sided and left-hand sided limits of function x(t) at the point t = tj, respectively.
By C ([0,T]; Rn) the Banach space is denoted, which consists of continuous vector functions x(t), defined on the segment [0,T], with values in Rn and with the norm
x =
\
> max x, (t) .
0<t<T j j=i " "
By PC ([0,T]; Rn) denote the linear vector space
PC ([0, T ]; Rn) = {x : [0, T] ^ Rn; x (t) G C ((¿¿,¿¿+1]; Rn) , i = 1,...,p} ,
where x (t+) and x (t-), i = 0, 1,... ,p, exist and bounded; x (t-) = x (¿¿). Note that the linear vector space PC ([0,T]; Rn) is Banach space with the norm
II x IIpc = max { II x llc((ii,ii+ij) , i = 1, 2,... ,p} .
Formulation of problem. To find a function x(t) G PC ([0, T]; Rn), which for all t G [0, T], t = ti, i = 1, 2,... ,p satisfies the integro-differential equation (1), nonlocal integral condition (2) and for t = ti, i = 1, 2,... ,p, 0 < t1 < t2 < ■ ■ ■ < tp < T satisfies the nonlinear limit condition (3).
3. Reduction to an integral equation
Let the function x(t) G PC ([0,T]; Rn) is a solution of nonlocal boundary value problem (1)-(3). Then by integration of equation (1) on the intervals (0,t1], (t1,t2], ..., (tp,tp+1], we obtain:
ti ti P^O) /(t1 -s)a_1/(s,x,y) ds = 1oatilo1-V(t) = J x'(s) ds = x(t-) —x(0+), t G M , 0 0
t2 t2 (t2 — s)a-1/(s,x,y) ds = x'(s) ds = x(t-) — x (t+) , t G (t1,t2] ,...,
r(a)
ti ti
tp+i tp+i
(tp+1 - s)a-1/ (s,x,y) ds = x'(s) ds = x (t-J - x (t+) , t G (tp,tp+i]
r(a)
tp tp
Hence, taking x(0+) = x(0), x(t-+1) = x(tk+1) into account, on the interval (0,T] we have
t
1 ' ' \a-1
(t - s)a /(s,x,y) ds
r(a)./
0
= [x (ti) - x (0+)] + [x (t2) - x (t+)] + ■ ■ ■ + [x(t) - x (t+)] =
= -x(0) - [x (t+) - x (ti)] - [x (t+) - x (t2)]-----[x (t+) - x (ti)] + x(t).
Taking into account condition (3), the last equality we rewrite as
t
x(t) = x(0) + pfr /(t - s)a-1/(s,x,y) ds + ^ (x (ti)). (a) */ 0<t<t
0 0<ti<t
We subordinate the function x(t) G PC ([0,T], Rn) in (4) to satisfy boundary value condition (2). Substituting presentation (4) into condition (2), we obtain the equality
T
T
A + J K(t,s)ds
0
\ot— 1
x(0) = B (t,x(t))-
T
r(Oy J K (t, s) j (s - 0)a-1f (0, x, y)d0ds -J K (t, s) J] / (x (ti)) ds. (5)
0 ^Ct^ ^ct
By virtue of det Q(t) = det
T
A + / K(t,s)ds
0
= 0, equality (5) one can rewrite as
x(0) = Q-1 (t)
T
B(t, x(t)) - — J K (t, s) y (s - 0) a-1f (0, x, y) d0ds-
T
K(t,s) ^ / (x (ti)) ds
0<ti<t
(6)
Substituting equality (6) into representation (4), we obtain
x(t) = Q (t)
B(t,x(t)) K(t,s) /(s - 0)a-1f (0, x,y) d0ds-
r(a)
T
K(t,s) ^ / (x (ti)) ds
0<ti<t
00 t
1
+ fM /(t - s)a-1 f (s,x,y)ds + J] / (x (ti)). (7)
( ) J 0<ti<t
Since the following equalities hold
T s T T
Jk(t,s^(s - 0)a-1f (0, x,y) d0ds = J j K(t,0) d0 (t - s)a-1f (s, x,y) ds,
0 0 0 s
T
T
K (t, s) ^ / (x (ti)) ds = ^ / K (t, s) ds / (x (ti))
0<ti<t
0<ti<T'
from presentation (7) we obtain
x(t) = Q-1(t)B(t,x(t)) - Q-1(t)
T T
r(a)
K(t, 0) d0 (t - s)a-1f (s, x, y) ds-
T
0s t
Q-1(t) ^ K(t,s) ds/ (x (ti)) + f-— /(t - s)a-1f (s,x,y) ds + ^ / (x (ti))
0<ti<T^ ( ) ^ 0<t<t
(x + I (t - s)
(a)
0
0<ti<t
s
Ts
1
After some simplifications in (8) we obtain that the following equalities hold:
t T T
1 f(t - s)a-1f (s,x,y)ds - Q-1(t)-^- I /*K(s,0)d0 (t - s)a-1/(s,x,y)ds =
Г (a) J ^ ^ v'f(a)
0 0 s
t / s
= Q-1(t)f(^/ |A + У K(t,0)dfl| (t - s)a-1/(s, x,y) ds-00 T T
-Q-1(t)f(^//K(t,0) d^(t - s)a-1/(s,x,y) ds; (9)
0s
T
^ I (x (ti)) - Q-1(t) ^ f K (t, s) ds/ (x (ti)) =
0<ti<t 0<ti<T^
= Q-1(t) ^ (a + f K (ti ,s)ds) /i (x (ti)) - ^ Q-1(t) / K (t,s) ds/i (x (ti)).
0<ti<t \ 0 / t<ti+i<T /
1 (10)
Taking into account (9) and (10), from presentation (8) we obtain the nonlinear equation
x(t) = J(t; x) = Q-1(t)B(t,x(t)) + ^ G (t,ti) /i (x (ti)) +
0<ti<t
T / T
+ / G(t,s)(t - s)a-1/ I s,x(s),/ 0(s,0, max |x(r)|r G Л2^]>) d0 1 ds (11) 00 for t G (ti, ti+1 ], i = 0,1,... ,p, where
Q-1(t) ( A + /K(t,0)d0j , 0 < s < t G(t,s)=< 4 0 J
-Q-1(t)/ K (t,0)d0, t<s < T.
s
It is easy to verify that function (11) satisfies problem (1)-(3).
4. The questions of one value solvability
Теорема 1. Suppose the following conditions are fulfilled:
1) Mf = max
0tT
/T
f t, 0J 0 (t, s, 0) ds
< to;
0
2) N = max max |1 (0)| < to; Bq = max |Q-i(t)B (t, 0)| < to;
0<t<Tte{i,2,...,P} 0<t<T
3) for all t G [0,T], x,y G the inequality
|f (t,xi,yi) - f (t,x2,y2)| < Mi(t) |xi - x2| + M2(t) |yi - ^2 |
holds, where 0 < M, (t) G C[0; T], i =1, 2;
4) for all t, s G [0, T]2, x G the inequality
|6(t, s,xi) - 6(t, s,x2)| < M3(t, s) |xi - x21 ,
holds, where 0 < Ms(t) G C[0,T];
5) for all t G [0,T], x G Rn the inequality |B(t,xi) - B(t,x2)| < M4(t) |xi - x2| holds, where 0 < M4(t) G C[0,T];
6) for all x G i = 0,1,...,p the inequality |1i(x1) — ij(x2 )| < N |x1 — x2|, N > 0;
7) p = S1 + S2 + + S4 < 1, where
T
\a-1
Si = / |G(t,s)| (t - s)a-1Mi(s)ds,
0
T
T
1
S2 = max / |G(t,s)| (t - s)a-1Mi(sW M3(s,Ö)dÖds
0<t<T J J0
00 p
S3 = max V |G(t,tj)| N,, S4 = max |Q-1(t)| M4(t).
0<t<T 0<t<T
,=1
Then the nonlocal boundary value problem (1)-(3) has a unique solution x(t) G PC ([0,T]; Rn). This solution can be found by the following iterative process:
xm (t) = J (t; xm-1), m =1, 2, 3,..., x0 (t) = 0, t G (ti,ti+1), i = 0,1, 2,...,p.
'12)
floKa3amejibcrneo. We consider the operator J : PC ([0, T]; Rn) ^ PC ([0,T]; Rn), defined by the right-hand side of equation (11). Appling the principle of contracting operators to (11), we show that the operator J, defined by equation (11), has a unique fixed point. Taking the first and the second conditions of the theorem, for the first difference of the approximations (12) we have the following estimate:
p
lx1(t) - x0(t)|| < maxjQ-1 (t)B (t,x0(t))| + max £ |G(t,t,)| 11, (x0(t*))| +
i=1
T
1
+ max / |G(t,s)| (t - s)
0tT
0
T
f |s,x0(s), / 8 (s,0,max{x0(r)|r G [A^^Ô]}) dÖ '
ds
0
< S0 (Mf + Ni) + BQ < to, (13)
where
T
S0 = max / |G(t, s)| (t - s)a-1ds + max V |G(t,t,)|
0 t T 0 t T
0 i=1
Mf = max
0<t<T
T
f I t,x0(t), / 8 (t,s,x0(s)) ds
Bq = max |Q-1(t)B (t,x°(t)) I , N = max max |l (x°(t)) I .
Q °<t<T 1 v w/i' °<t<Tie{i,2,...lP^ v iwy|
Then, by the third, fourth, fifth and sixth conditions of the theorem, for difference of arbitrary consecutive approximations and arbitrary t G (tj,ti+1] we have
|xm(t) - xm-1(t) < |Q-1(t) [B (t,xm-1(t)) - B (t,xm-2(t))]| +
T
+ y |G(t,s)| (t - s) 0
a—1
T
f I s,xm-1(s), / 8 (s,0,max{xm-1(r)|t G [A^^]}) I -
T
ds+
-f |s,xm-2(s)^ 8 (s,0,max{xm-2(r)|r G [A^^]})
p
+ ^ |G(t,ti)| |l (xm-1 &)) - (xm-2(ti))| < |Q-1(t)| M4(t) ■ |xm-1(t) - xm-2(t)| +
i=1
T
+ J |G(t,s)| (t - s)
0
a—1
M1(s) ■ |xm-1(s) - xm-2(s) +
T
M2(s) / M3(s,0) |max{ xm 1(t )|t G [A^, A20]} - max {xm-2(r)|t G [A^^]}| 0
p
+ ^ |G(t, ti)| ■ Ni ■ | xm-1(ti) - xm-2(ti) | .
ds+
i=1
Hence, by the introduced norm we obtain
||xm(t) - xm-1(t)||PC < (S1 + S3 + S4) II xm-1(t) - xm-2(t)||PC +
+S2 11max{xm-1(r)|r G [A^t]} - xm-2(r)|r G [A^t]}||PC <
< p ■ ||xm-1 (t) - xm-2(t)|
PC
:i4)
where p = S1 + S2 + S3 + S4.
According to the last condition of the theorem p < 1. Therefore, from the estimate (14) we have
xm(t) - xm-1(t)^pc < ||xm-1(t) - xm-2(t)|
PC
:i5)
It follows from (15) that the operator J on the right-hand side of the equation (11) is contracting. According to the fixed point principle, taking into account estimates (13)-(15), we conclude that the operator J has a unique fixed point. Consequently, nonlocal boundary value problem (1)-(3) has a unique solution x(t) G PC ([0,T]; Rn). The theorem is proved. □
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Article received 29.12.2021. Corrections received 27.02.2022.
Сведения об авторах Юлдашев Турсун Камалдинович, доктор физико-математических наук, доцент, профессор Узбекско-Израильского совместного факультета высокой технологии и инженерной математики, Национальный университет Узбекистана, Ташкент, Узбекистан; e-mail: [email protected].
Сабуров Хикмат Хажибаевич, кандидат физико-математических наук, доцент, проректор по науке и инновации, Национальный университет Узбекистана, Ташкент, Узбекистан; e-mail: [email protected].
Абдувахобов Тохиржон Акбарали огли, стажер Узбекско-Израильского совместного факультета высокой технологии и инженерной математики, Национальный университет Узбекистана, Ташкент, Узбекистан; e-mail: [email protected].
Челябинский физико-математический журнал. 2022. Т. 7, вып. 1. С. 113-122.
УДК 517.911 Б01: 10.47475/2500-0101-2022-17108
НЕЛОКАЛЬНАЯ ПРОБЛЕМА ДЛЯ НЕЛИНЕЙНЫХ СИСТЕМ ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ ДРОБНОГО ПОРЯДКА С ИМПУЛЬСНЫМИ ВОЗДЕЙСТВИЯМИ И МАКСИМУМАМИ
Т. К. Юлдашев, Х. Х. Сабуров, Т. А. Абдувахобов
Национальный университет Узбекистана, Ташкент, Узбекистан [email protected], [email protected], [email protected]
Исследуется нелокальная краевая задача для системы обыкновенных интегро-дифференциальных уравнений с импульсными воздействиями, максимумами и дробным оператором Герасимова — Капуто. Граничное условие задаётся в интегральной форме. Используется метод последовательных приближений в сочетании с методом сжимающих отображений. Доказаны существование и единственность решения краевой задачи.
Ключевые слова: интегро-дифференциальное уравнение с импульсными воздействиями, оператор Герасимова-Капуто, нелокальное граничное условие, последовательные приближения, однозначная разрешимость.
Поступила в редакцию 29.12.2021. После переработки 27.02.2022.
