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NANOSYSTEMS: Yuldashev T.K., Fayziev A.K. Nanosystems:
PHYSICS, CHEMISTRY, MATHEMATICS Phys. Chem. Math., 2022,13 (1), 36-44.
http://nanojournal.ifmo.ru
Original article DOI 10.17586/2220-8054-2022-13-1-36-44
On a nonlinear impulsive system of integro-differential equations with degenerate kernel and maxima
Tursun K. Yuldashev1", Aziz K. Fayziev2'6
1National University of Uzbekistan, Tashkent, Uzbekistan 2Tashkent State Technical University, Tashkent, Uzbekistan
°[email protected], [email protected]
Corresponding author: Tursun K. Yuldashev, [email protected]
Abstract A nonlocal boundary value problem for a system of ordinary integro-differential equations with impulsive effects, degenerate kernel and maxima is investigated. The boundary value problem is given by the integral condition. The method of successive approximations in combination with the method of compressing mapping is used. The existence and uniqueness of the solution of the boundary value problem are proved. The continuous dependence of the solution on the right-hand side of the boundary value condition is shown. Keywords impulsive integro-differential equations, nonlocal condition, successive approximations, existence and uniqueness, continuous dependence of solution
For citation Yuldashev T.K., Fayziev A.K. On a nonlinear impulsive system of integro-differential equations with degenerate kernel and maxima. Nanosystems: Phys. Chem. Math., 2022,13 (1), 36-44.
1. Introduction
Many problems in modern sciences, technology and economics are described by differential equations, the solutions of which are functions with first kind discontinuities at fixed or non-fixed times. Such differential equations are called differential equations with impulse effects [1-8]. As is known, in recent years the interest in the study of differential equations with nonlocal boundary conditions has increased (see, for example, [9-17]). In particular, in [15] a physical situation in which a non-metallic conductor is in contact with a perfect conductor is studied. In [16], the problems of mathematical models in reaction-diffusion systems are considered. In [17], the nonlocal conditions are used in the theory of phase transitions.
In [18-24] the problems of solvability for some type of integro-differential equations with degenerate kernel were considered. Also, a lot of publications of studying on differential equations with impulsive effects, describing many natural and technical processes, are appearing [25-35].
In this paper, we investigate a nonlocal boundary value problem for a system of first order Fredholm integro-differential equations with impulsive effects, degenerate kernel and nonlinear maxima. The questions of the existence and uniqueness of the solution to the boundary value problem, as well as the continuous dependence of the solution on the right-hand side of the boundary condition, are investigated. In [36], it is justified that the theoretical study of differential equations with maxima is relevant.
We consider the following system of Fredholm integro-differential equations:
T
x'(t) = \j H(t,s)x(s)ds + f (t,x(t), max {x(t)|t g [hi; h2]} ),
for t g [0, T], t = ti, i = 1, 2, ...,p with nonlocal boundary value conditions:
T
Ax(0) + J K(t)x(t)dt = B,
(1)
(2)
x (t+) - x (tr) = Ii (x (tj)), i = 1, 2, ...,p,
(3)
and impulsive effect:
m
where H(t,s) = ^ ak(t)bk(s), 0 = to < ti < ... <tp < tp+1 = T, A e Rnxn,K(t) G Rnxn are given matrix and
k= 1
T
det Q = 0,Q = A + J K (t) dt, f : [0, T] x Rn x Rn ^ Rn, I, : Rn ^ Rn are given functions; 0 < h1 < h2 < t,
© Yuldashev T.K., Fayziev A.K., 2022
hj = hj(t,x(t)), j = 1, 2, A is real nonzero parameter, x (t+~) = lim (xj + h), x (t— = lim (tj — h) are right-
sided and left-sided limits of function x(t) at the point t = tj, respectively. Every system of functions {afc(t)}m=i and (s)}m=1 are linearly independent.
2. Reduction to an integral equation
Here are some notations that will be used below. We denote by C ([0, T], 1") the Banach space, which consists of continuous functions x(t) € 1" on the segment [0, T] with the norm:
\
Emax |x„(t)I.
te[0,T] 1 j
j=i
Since we consider the integro-differential equation (1) with impulsive effect at the points tj, i = 1,2, ...,p, use the following linear space:
PC([0,T], 1") = {x : [0,T] ^ 1"; x(t) € C((tj,tj+1], 1"),i = 1,...,p},
where x (t+) and x (t— (i = 0,1, ...,p) exist and bounded; x (t— = x (tj).
It is obvious, that the linear space PC ([0, T], 1") is Banach space with the following norm:
IPC
= max{|M|C((ti,ti+i]) > » = 1> 2,...,p}.
Formulation of problem. To find the function x(t) € PC ([0,T], 1"), which for all t € [0,T], t = tj, i = 1,2, ...,p satisfies the integro-differential equation (1), nonlocal integral condition (2) and for t = tj, i = 1,2, ...,p, 0 < t1 < t2 < ... < tp < T satisfies the limit condition (3).
Let the function x(t) € PC ([0, T], 1") is a solution of the problem (1)-(3). Then we rewrite the Fredholm integro-differential equation (1) as:
T
/m
ak(t)6k (s)x(s)ds + /(t, x(t), max {x(t)|t € [h1; h2]} ).
0
By the designation:
T
Cfc = J 6fc(s)x(s)ds,
0
the last integro-differential equation we rewrite in the following form:
m
x'(t) = A^afc (t)cfc + /(t,x(t), max {x(t )|t € [h1; h2]} ). k = 1
Then, by integration of the last equation on the interval t € (0, tj+1], we obtain:
t r "1
m
/ A^afc(s)cfc + /(s, x(s), •) ds =
k=1
t
= j x'(s)ds = [x (ti) - x (0+)] + [x (t2) - x (t+)] + ... + [x (t) - x (t+)] =
0
= -x (0) - [x (t+) - x (ti)] - [x (t+ ) - x (t2)] - ... - [x (t+) - x (ti)] + x (t) .
Taking into account the integral condition (2) in the last equality, we obtain:
t
xW = x(°) + /
A^afc (s)cfc + f (s,x(s), •)
k=1
ds + ^ (x (ti)) .
0<ti<t
(4)
x=
x
Let the function x(t) € PC ([0, T], 1") in (4), satisfies the boundary value conditions (2):
A + J K(t)dt
x(0)
t t
= B — / K <«/
A^afc(s)cfc + /(s,x(s), •)
fc = 1
00
Since det Q = 0, from the equality (5) we have:
t t
dsdt — K(t) j I (x (tj)) dt. (5)
0<t»<t
x(0) = Q
1
B -
//* I I V n
K(t) / Ajafc(s)cfc + /(s,x(s), •) dsdt — K(t) j Ij (x (tj)) dt
0 0 k = 1 -I ^ 0<ti<t
(6)
Substituting the equality (6) into representation (4), we obtain:
t t „,
x(t) = Q
1
B -
00
f K(t) / A jj ak (s)ck + /(s,x(s), •) dsdt — f K(t) j dt +
^ L %=1 -I 0 0<ti<t -
t
+ [ /(s,x(s), Ods + j Ij (x (tj)) . (7) 0<ti<t
Since the following equalities hold:
t t
K(t)
00
A^afc(s)ck + /(s,x(s), •)
fc = 1
T T
dsdt = j j K(s)ds
0t
A j afc(t)cfc + /(t,x(t), •)
fc=1
dt,
0
from presentation (7) we obtain:
T T
x(t) = Q-1B — Q"1 J J K(s)ds
0t
f K(t) j Ij (x (tj)) dt = j f K(t)dtlj (x (tj)),
0<ti<t nw.^W
0<ti<Tl
A j afc (t)cfc + /(t,x(t), •)
fc = 1
dt
T t
— Q"1 j f K(t)dtlj (x (tj))+ f
0<ti<t
afc(s)cfc + /(s, x(s), •)
fc=1
ds + j Ij (x (tj)). (8)
0<ti<t
Let us make some simplifications in representation (8). Then the following equalities hold:
A^afc(s)cfc + /(s,x(s), •)
fc=1
ds — Q
"^y K(s)ds
0t
A j afc(t)cfc + /(t,x(t), •)
fc=1
dt =
t / e
Q"^ [a + J K (s)ds
00
A j afc (0)cfc + /(0,x(0), •)
fc=1
d<9-
T T
—Q
^ J J K(s)ds
0t
A j afc(0)cfc + /(0,x(0), •)
fc=1
d<9; (9)
0<ti<t
j Ij (x (tj)) — Q"1 j f K(t)dt/j (x (tj)) = 0<ti<T/.
= Q"1 j (a + / K (t)dt) Ij (x (tj)) — j Q"1 / K (t)dt/j (x (tj)). (10) 0<ti<t \ J I t^t ^rj, J
Taking into account (9) and (10), from the presentation (8) we obtain the following integral equation:
T T
m
x(t) = Q-1B + ^ G (ti) I, (x (ti))+ X ^ G(s)ak (s)ckds + G(t,s)f(s,x(s), -)ds, (11)
0<ti<t k=10 0
for t g (ti,ti+1] ,i = 0,1, ...,p, where:
G(t) = <
Q—i + J K(s)dsj , 0 < s < t,
T
-Q-1 J K(s)ds,
t < s <T.
Substituting the equation (11) into designation:
T
Ck = J bk(s)x(s)ds,
we obtain the following linear system of algebraic equations (LSAE):
m
Ck + X^ckj= ^1k + ^2k (f,Ii), k = 1,m, j=1
(12)
where:
T T
$kj (f,Ii) = J bk(s) J G(0)aj(d)dMs, ^ik = Qr1B J bk(s)ds,
o o
T T
^2k (f,Ii)= f bk (s) ÎG(0)f (0,x(0), max {x(r)|r G [hi; h2]} )d0 + ^ G (tj) Ij (x (tj))
0<ti<t
ds,
k = l,m,hi = hl(t,x(0)),l = 1, 2. (13) The LSAE (12) is uniquely solvable for any finite right-hand side, if the following Fredholm condition is satisfied:
1 + A$fcU A$fci2 ... A$fcim
A$fc21 1 + A$fc22 ... A$fc2m
Ak (A)
kmi
km2
... 1 + A$kmm
= 0.
(14)
Consider such regular values of parameter X, for which condition (14) is satisfied. Then, solving LSAE (12), we obtain:
A1k(X) , A2k (X,f,Ii)
where:
Aifc (A)
1 + A$ii
Ck
A(A) + A(A)
(15)
i(j—i)
i(j+i)
2i
... A$2(ï —i) ^2 A$2(j+i)
A$mi ...
>(j—i) ^ l m A^m(j+i)
im
.. 1+
I = 1, 2.
(16)
Substituting equality (15) into representation (11), we obtain the following new presentation of solution:
s(t) = 8(t; x) = xo + A ^
k=i
Aik(A) + A2k (A,f,Ij) L A(A) + A(A)
Xik +
T
+ i G(s)f (s,x(s), max {x(r)|r G [hi; h2]} )ds + ^ G (tj) Ij (x (tj)), (17)
0<ti<t
A$2m
where:
X0 = Q 1B, xik = /G(s)afc(s)ds, k = 1,m, h; = h;(t,x(s)), l = 1, 2.
0
3. The questions of one value solvability
Theorem. Suppose the following conditions are fulfilled:
1) For all t G [0, T], x, y G Rn holds:
|f (t, xi, yi) - f (t, x2, y2)| < Mi(t) |xi - x21 + M2(t) |yi - y21 ;
2) For all t G [0, T], x G Rn holds:
|hj(t,xi) - hj(t,x2)| < M3j(t) |xi - x2| ,j = 1, 2;
3) For all x G Rn,i = 0,1, ...,p holds:
|1i(xi) - Ii(x2)| < mi |xi - x21 ;
4) p = S1 + S2 < 1, where:
T
m „
Si = |A| ^ |xik| • |A2k(A)| / |G(s)| [Mi(s) + M2(s) (1 + Mf (Msi(s) + Ms2(s)))] ds,
k=i ^
S2 = |A| ^ |xifc| • |i2k(A)|E |G(ti)| mi,
k=1
A2k ( A)
1 + A$ii ... A$i(i_i) ' 2i A$i(i+i) A$2i ... A$2(i_i) ' 22 A$2(i+i)
A$im A$2m
A$mi ... A$m(i_i) ' 2m A$m(i+1) ... 1 +
'2k = J bk (s)ds.
0
Then, the nonlocal boundary value problem (1)-(3) has a unique solution x(t) g PC ([0, T], Rn) for the regular values of parameter A. This solution can be found from the following iterative process:
V(t) = 0(t; xj _1), j = 1, 2, 3,... x0(t) = xo = Q-iB, t G (ti,ti+i), » = 0,1, 2, ...,p.
(18)
Moreover, for this solution the following estimate is true:
|xi (t) - x2(t)||pc < (1 - p)-i ||Q II • l|Bi - B
Proof. We consider the following operator:
© : PC([0,T]; 1") ^ PC([0,T] x 1"),
defined by the right-hand side of integral equation (11). Obviously, the fixed point of the operator © is the unique solution to the boundary value problem (1)-(3). Using the principle of contracting operators, we show that the operator © defined by equation (17), has a unique fixed point.
For the zero approximation from (18) we easily obtain that:
On a nonlinear impulsive system ofintegro-differential equations For the first difference from approximation (18) we have the estimate:
x1(t) - x0(t)|| <|A|]T
k = 1
Aifc (A)
A(A)
+
A2k (A,/,/i)
A(A)
|Xifc| +
+ / |G(s)| • |/ (s,x0 (s), max {x0(r)|r G [h° ; h0]})|ds + £ |G(ti)| • (x0(ti))| <
n i=1
<|A| E
k = 1
" Aik(A) A2k (A,/0,/0) ]
_ A(A) + + A(A) _
|X1k | + S (Mf + mi) < TO, (20)
where:
/0 = J G(t)/ (t,x0(t), max {x0(r)|t g [h?; h0]}) dt,
0
t p
/0 = / (x0 (ti)) , S = J |G(s)| ds + £ |G(ti)|,
Mf = max |/ (t,Q-1B,Q-1B)| , m/ = max |/i (Q-1B)|.
te[0,T] 1 1 ie{1,2,...,p} 1 1
Then, by virtue of the conditions of the theorem and (13), (16), for arbitrary t g (ti, ti+1] we have:
|A2k (A,/-1,//-1) - A2k fA,/j-2,/r2'
|xj(t) - xj-1(t)| <|A|]T
k=1
A(A)
+J |G(s)|^ /(s,xj-1(s), max Ixj-1(r)|t g 0
hi-1; hj-1
|X1k| +
-/(s, xj 2(s), maJ xj 2(t)|t g
h1-2; hj-2
ds+
+ E |G(ti)| • |/i (xj-1(ti)) - /i (xj-2(ti)) | < i=1
m t
< |A| EE |X1k| • |A2k(A)| J |G(s)| • [M1(s) • |xj-1(s) - xj-2(s)| +
| - max jx j-2 (t)|t g t
k=1
h1-1 ; h2-1
2 , j-2
hi ; hj
+ M2(s) • | max|xj-1 (t)|t g
m p „
+ |A| E |xik| • |AA2k(A)| • E |G(ti)| • mi • |xj-1(ti) - xj-2(ti)| + |G(s)| • [Mi(s) • |xj-1(s) - xj-2(s)| + i—1 —1 J
k=1
+ M2(s) • | ma^ xj-1 (t)|t g
where hj = hl(t, xj(t)), l = 1,2 and:
hi-1 ; h2-1
2 , j- 2
hi-2; hj
| - ma^xj 2(t) |t
p | |
+ E |G(ti)| • mi • |xj-1 (ti) - xj-2(ti)| , (21)
A2k ( A)
1+ A$11 ... A$i(i-i) 21 A$i(i+i) ... A$im A$21 ... A$2(i-1) ^ 22 A$2(i+1) ... A$2m
A$m1 ... A$m(i-1) ^ 2m A$m(i+1) ... 1 + A$mm
i
<J>2k = J bk (s)ds.
By virtue of third condition of the theorem, we have:
mad xj 1(t)|t g
hi-1; h2-1
| - max jxj 2(t)|t g
2 , j_2
hi_2; hj
<
< +
ma^xj 1(t)|t g mad xj_2(t)|t g
hi_1; h2_1
hi_1; h2_1
hi_1; h2_1
hi_2; h2_2
j- maxj xj 2 (t )|t g | - max jxj_2 (7
< |xj_1(t) - xj_2(t)1 + Mf [|h1 (t,xj_1(t)) - h1 (t,xj_2(t))| + |h2 (t,xj_1(t)) - h2 (t,xj_2(t))|] <
< (1 + Mf (Msi(t) + M32(t)) )|xj_1(t) - xj_2(t)|. (22)
Substituting the estimate (22) into (21), we obtain:
Ixj (t) - xj_1(t)|pc < p • ||xj_1(t) - xj_2 (t) I
PC :
(23)
where p = S1 + S2 and:
T
m „
Si = |A| ^ |xik| • |A2k(A) | / |G(s)| • [Mi(s) + M2(s) (1 + Mf (M3i(s) + M32M))] ds,
k = 1
0
S2 = |A| £ |xik| • |A2k(A)| £ |G(ti)| • mi.
k=1 i=1
According to the last condition of the theorem, p < 1. Therefore, from the estimate (23) we have:
|xj(t) - xj_1(t)|pc < ||xj_1(t) - xj_2(t)|pc .
(24)
It follows from (24) that the operator © on the right-hand side of (17) is contracting. According to fixed point principle, taking into account estimates (19), (20) and (24), we conclude that the operator © has a unique fixed point. Consequently, the nonlocal boundary value problem (1)-(3) has a unique solution x(t) g PC ([0, T], Rn).
Now, let us show the continuous dependence of the solution to the boundary value problem (1)-(3) on the right-hand side of condition (2). Let B1, B2 G Rn are two different constants and x1(t), x2(t) G PC ([0, T], Rn) are corresponding solutions of the problem (1)-(3). Then, we have:
i+\ m /~>_1rD D1 1 \ ^ A2k (A f1,11i) - A2k (A f2,12i)
xi(t) - x2(t) = Q [Bi - B2] + A-Am-xik+
k = 1 ( )
T
+ j G(s) • |f (s, x1(s), max {x1(t)|t g [h1 ; h2]}) - f (s, x2(s), max {x2(t)|t g [h2; h2]}) | ds+
0
P
+ ^ G(ti) [Ii (x 1 (ti)) - Ii (x2 (ti))], (25)
i=i
where h^ = hj (t, xk (t)), j, k =1,2. Now, using the first two conditions of the theorem, from (25) we obtain:
|xi(t) - x2(t)| < Q_1 [Bi - B2] + |A| ]T
k = 1
|A2k (A, fi, Iii) - A2k (A,f2,/2i
A(A)
|xik| +
+ J |G(s)| • [M1(s) • |x1(s) — x2(s)| + M2(s) • |max {x1(r)|t € [h1; h2]} — max {x2(r)|t € [h2; h2]}|] ds+
0
p
+ |G(tj)| • mj • |x1(tj) — x2(tj)| .
j=1
Hence, as in the case of estimation process for (23), we obtain:
yx1(t) — x2(t)||pC < HQ"11| |B — B2II + p • yx1(t) — x2(t)ypo . Since p < 1, from the last inequality, it follows that:
I|x1 (t) — x2(t)||pc < (1 — p)"1 HQ"1 H • IIB1 — B2I .
If we put ||B1 — B2|| < S and e = (1 — p)"1 11Q"11 • S, then, from the last inequality, we obtain ||x1(t) — x2(t)||PC < e. The theorem is proved.
4. Conclusion
The theory of differential equations plays an important role in solving applied problems. Especially, nonlocal boundary value problems for differential equations with impulsive actions have many applications in mathematical physics, mechanics and technology, in particular in nanotechnology.
In this paper, we investigated the system of first order Fredholm integro-differential equations (1) with nonlocal boundary value condition (2) and with condition (3) of impulsive effects for t = ti i = 1, 2, ...,p, 0 < t1 < t2 < ... < tp < T. The kernel of integro-differential equation (1) is degenerate. The nonlinear right-hand side of this equation consists of the construction of nonlinear maxima. The questions of the existence and uniqueness of the solution of the boundary value problem (1)-(3) are studied. The continuous dependence of the solution on the right-hand side of the boundary condition was proved.
The results obtained in this work will allow us in the future to investigate nonlocal boundary value problems for the heat equation and the wave equation with impulsive actions. We hope that our work will stimulate the study of various boundary value problems for partial differential and integro-differential equations with impulsive actions.
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Submitted 28 November 2021; revised 26 December 2021; accepted 28 December 2021
Information about the authors:
Tursun K. Yuldashev - National University of Uzbekistan, Universitet street, 4, NUUz, Tashkent, 100174, Uzbekistan; ORCID 0000-0002-9346-5362; [email protected]
Aziz K. Fayziev - Tashkent State Technical University, Universitet street, 2, TSTU, Tashkent, 100174, Uzbekistan; ORCID 0000-0001-6798-3265; [email protected]
Conflict of interest: the authors declare no conflict of interest.
