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Vestnik KRAUNC. Fiz.-Mat. nauki. 2022. vol. 38. no. 1. P. 40-53. ISSN 2079-6641
MSC 34B37, 34B15 Research Article
On a nonlocal problem for impulsive differential equations with
mixed maxima
T.K. Yuldashev1'2
1 National University of Uzbekistan, 100174, Tashkent, Universitetskaya street, 4, Uzbekistan
2 V. I. Romanovskii Institute of Mathematics, Academy of Sciences of Uzbekistan, 100174, Tashkent, Universitetskaya street, 4-B, Uzbekistan
E-mail: [email protected]
A nonlocal boundary value problem for a first order system of ordinary integro-differential equations with impulsive effects and mixed maxima is investigated. The boundary value problem is given by the integral condition. The method of successive approximations in combination it 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 solutions on the right-hand side of the boundary condition is showed.
Key words: impulsive integro-differential equations, nonlocal boundary condition, mixed maxima, successive approximations, existence and uniqueness of solution, continuous dependence of solution.
d DOI: 10.26117/2079-6641-2022-38-1-40-53
Original article submitted: 02.03.2022 Revision submitted: 21.04.2022
For citation.Yuldashev T.K. On a nonlocal problem for impulsive differential equations with mixed maxima. Vestnik KRAUNC. Fiz.-mat. nauki. 2022,38: 1,40-53. d DOI: 10.26117/2079-6641-2022-38-1-40-53
The content is published under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0/deed.ru)
© Yuldashev T.K., 2022
Introduction. Differential equations with maxima
One of the interesting field of the theory of functional-differential equations is the differential equations with maxima. The qualitative theory of differential equations with maxima has singularities on the theoretical investigations. On the interval [0, T] we begin to consider the functional-differential equations of the following form
x'(t) = f (t,x(t),max{x(t) |t e [hi(t);h2(t)]}), Funding. The work was done without financial support
where h (t) < h,2(t), t e [0;T]. This type functional-differential equations are called as differential equations with maxima. The set of increasing solutions of these differential equations with maxima coincides with the set of increasing solutions of the following differential equations with deviation
x'(t) = f (t,x(t),x[h2(t)]), t e [0,T].
The set of decreasing solutions of differential equations with maxima coincides with the set of decreasing solutions of the following differential equations with deviation
x'(t) = f (t,x(t),x[h (t)]), t e [0,T].
The following type differential equations
x'(t) = f (t,x(t),max{x(t) |t e [hi(t) : |: Mt)]}), t e [0,T], (1)
where [h1 (t) : |: h2(t)] = [min^ (t);h2(t)}; max{H1 (t);h2(t)}], we call as a differential equation with mixed maxima. We suppose that there exist some points ti e (0,T), i = 1,2, ...,p, at which h1(t) = h2(t). Then on the interval Op = [0,t1] U [t2,t3] U [t4,t5] U... U [tp-i,tp] the differential equation with mixed maxima (1) has the form
x'(t) = f (t,x(t),max{x(t) |t e [h1 (t);h2(t)]}). (2)
On the complement interval Op = [t1,t2] U [t3,t4] U [t5,t6] U ... U [tp,T] the differential equation with mixed maxima (1) has the form
x'(t) = f (t,x(t),max{x(t) |t e [h2(t);h1 (t)]}). (3)
The set of solutions of the differential equation with mixed maxima (1) on the interval [0,T] coincides with the union of sets of the solutions of two differential equations (2) and (3) on the intervals Op and Op, respectively. At the points t1,t2,t3,...,tp_1,tp the solutions of differential equation (1) with mixed maxima have discontinuities depending from the posed problem for (2) and (3).
Example 1. On the interval [0,oo) we consider the following differential equation with mixed maxima
x'(t) = 2max^x(t) te t: |: Vt|,t e [0,oo). (4)
On the interval [0,1] the differential equation (4) with mixed maxima has the form
x'(t) = 2max|x(t) te t; Vt |,t e [0,1]. (5)
On the interval [1,oo) the differential equation (4) with mixed maxima has the following form
x'(t) = 2maxjx(T) te Vt;t|,t e [1^). (6)
The genereal form of increasing and decreasing solutions of the differential equation (5) with maxima on the interval [0,1] have the form
x(t) = W » 2t
/A■ t2, t g [0,1], A > 0, \ A■ e2t, t g [0,1], A < 0,
where means increasing solutions and \ means decreasing solutions.
The genereal form of increasing and decreasing solutions of the differential equation (6) with maxima on the interval [1, oo) have the form
x(t) = ï x n 2t
\B ■ t2, t G [1,oo ) B <0, /B ■ e2t, t G [1,oo ), B >0.
Therefore, the genereal form of increasing and decreasing solutions of the differential equation (4) with mixed maxima on the interval [0,oo) have the form
x(t) = {
/A■ t2, t G [0,1], A > 0, \ A■ e2t, t G [0,1], A < 0;
\B ■ t2, t G [1,oo) B <0, /B ■ e2t, t G [1,oo), B >0.
If we do not specify a continuous gluing condition at a point t = 1, then naturally, the solution of the differential equation (4) with mixed maxima suffers a discontinuity of the first kind at this point. For example, if we solve the differential equation (4) with condition x(0) = —2 on the first interval [0,1] and solve the differential equation (4) with condition x(1 ) = 2 on the second interval [1,oo), then we have corresponding solutions x(t) = —2e2t on [0,1] and x(t) = 2e2(t—1 on [1,oo). So, from these solutions we have
limx(t)= lim (—2e2t) = — 2e2, lim x(t) = lim (2e2(t—= 2.
xitj = lim —2e = — 2e , um x t->1—0 t->1 — 0\ / t->1+0 ' ' t->1+0
Consequently, for these solutions we obtained discontinuity
lim x(t) — lim x(tj = 2 + 2ez.
t—>1 +0 t—>1—0
Example 2. On the interval [0,oo) we consider the following differential equation with mixed maxima
/( )= et e(1+(—1)[t]jt +1 x(t) (et + 1)z ' e(1+(—1)[t])t X
x max|x(t) t g [t: |: (j +(—1)[t]) t] } , t G [0 ,°°), (7)
where [t] is the integer part of t.
On the interval O = [0,1] U [2,3] U [4,5] U... the differential equation (7) with mixed maxima has the form
2et
x/(t) = . t , 1a2max{x(t) |t g [0; t]}. (8)
(et+1)2
On the complement interval O = [1,2] U [3,4] U [5,6] U ... the differential equation (7) with mixed maxima has the following form
e2t +1
x/(t) = "tTTTTa max{x(t) It G [t; 2t]}. (9)
et(et +1 )2
The general form of increasing and decreasing solutions of the differential equation (8) with maxima on the interval O = [0,1] U [2,3] U [4,5] U... have the form
xft), /x(t) = Ci ■ e-2(e +1J- , t g Ob Ci>0, X( J S \x(t) = Cj ■ ete+T, t g Ob Cj<0.
The general form of increasing and decreasing solutions of the differential equation (9) with maxima on the interval O = [1,2] U [3,4] U [5,6] U... have the form
x(t) =
x(t)= Di ■ ^t+r, t G O2, Di>0,
\x(t) = Dj (1 + e—■ e—2e t(et+1) 1, t G 02, Dj < 0.
Therefore, general form of increasing and decreasing solutions of the differential equation (7) with mixed maxima on the interval [0,oo) have the form
x(t) = <
/*x(t) = Ci ■ e \x(t) = Cj ^ /x(t)= Di
2(et+1)-1, t G Oi, Ci>0, et+r, t G Oi, Cj <0;
e
\x(t)= Dj (1 + e-t)
et+r, tG O2, Di>0,
t^ „-2e-t(et+1)-1
e
, t G O2, Dj < 0.
It is required to set conditions at each of points tk = t| ,t2,t3,...,tn,... If we do not specify a continuous gluing conditions at these points, the solution of the differential equation (7) with mixed maxima suffers a discontinuity of the first kind at these points.
There are many examples of differential equations, where the maxima are mixed many times. For example, in the following example of the differential equation, the maxima will mix infinitely many times:
x'(t) = f (t,x(t),max{x(t) |t g [1 +sin t : | : 1 +cos t]}), t G (—oo; oo).
Many problems in modern sciences, technology and economics are described by differential equations, the solution of which is functions with first kind discontinuities at fixed or non-fixed times. Such differential equations are called differential equations with impulse effects. Also a lot of publications of studying on differential equations with impulsive effects, describing many natural and practical processes, are appearing [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. In recent years the interest in the studying of differential equations with nonlocal boundary value conditions is increasing (see, for examples, [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]).
In this paper we consider the questions of existence and uniqueness of solution of the nonlocal boundary value problem for an impulsive system of integro-differential equations with mixed maxima.
1 Nonlocal boundary value problem
1.1 Problem statement
On the segment [0,T] for t = ti, i = 1,2,...,p, we consider the following first order system of nonlinear integro-differential equations
T
. /
x'(t) = f I t,x(t),
0 (t,s,max {x(t)|t g [Ai(s) : | : Â2(s)]}) ds I (10)
0
with nonlocal boundary value condition
T
Ax(0) +
K(t,s)x(s) ds = B(t) (11)
and nonlinear impulsive effect
x (t++) -x (t—) = Ii (x(ti)), i = 1,2, ...,p, (12)
where 0 = t0 < t1 < ... < tp < tp+1 = T, A G Rnxn is given matrix, K(t,s) is given n x nT
dimensional matrix function and det Q(t) = 0, Q(t) = A + J"K(t,s)ds, f : [0,T] x Rn x
0
Rn —> Rn, 0 : [0,T]2 x Rn —> Rn, Ii: Rn —> Rn are given functions; 0 < A1 (t), A2(t) < T,
x (t+) = lim x (ti + h), x (t ) = lim x (ti — h) are right-hand sided and left-hand sided v w h->0+ v w h^0-limits of function x(t) at the point t = ti, respectively.
By C ([0,T], Rn) denoted the Banach space, which consists of continuous vector functions x(t), defined on the segment [0,T], with values in Rn and with the norm
x=
\
n
Lmax I Xi (t) I.
0<t<T1 J 1
j=i < <
By PC ([0,T], Rn) denoted the linear vector space
PC([0,T],Rn)= {x: [0,T] -> Rn; x(t) G C((ti,ti+1],Rn), i = 1,...,p},
where x (t+) and x (t—) (i = 0,1, ...,p) exist and bounded; x (t—) = x (tj.
Note, that the linear vector space PC ([0,T], Rn) is Banach space with the following norm
I x ||PC = max
lx |c((ti,ti+1]), 1 = !,2,...,p}.
Problem. To find the 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 (10), nonlocal integral condition (11) and for t = ti i = 1,2, ...,p, 0 < ti < t2 < ... < tp < T satisfies the nonlinear limit condition (12).
1.2 Reduction to a nonlinear functional-integral equation
Let the function x(t) g PC ([0,T], Rn) is a solution of the nonlocal boundary value problem (10)-(12). Then by integration of the equation (10) on the interval t g (0, ti+i ], we obtain
f(s,x,y) ds =
x'(s) ds = [x (t1)— x(0+)] + [x (t2)— x(t+)] +... + [x(t)— x(t+)] =
= -x(0J- [x(t+)-x (ti)]- [x(t+)-x (t2^- ... - [x (t+) — x (ti)] + x(t) Taking into account the condition (12), the last equality we rewrite as
x(t) = x(0) +
f(s,x,y) ds + !i (x(ti)).
0<ti<t
(13)
We subordinate the function x(t) e PC ([0,T], Rn) in (13) to satisfy the boundary value condition (11):
A +
K(t,s)ds
x(0) =
= B(t) —
K(t,s)
f(0,x,y)d0ds —
00
K(t,s) ^ Ii(x(ti)) ds.
0<ti<t
(14)
By virtue of det Q(t) = det
A ^ K(t,s)ds
0
= 0, the equality (14) we rewrite as
x(0) = Q—'(t)
B(t) —
K(t,s)
f(0,x,y ) d0ds —
K(t,s) ^ Ii (x(ti)) ds
0<ti<t
(15)
Substituting equality (15) into representation (13), we obtain
x(t) = Q—1 (t)
B(t) —
K(t,s)
f(0,x,y) d0ds —
00
K(t,s) ^ Ii (x(ti)) ds
0<ti<t
+
+
f(s,x,y)ds + ^ Ii(x(ti)). (16)
0<ti<t
Since the following equalities hold
T s
TT
K(t,s)
f(0,x,y) d0ds =
K(t,0) d0f(s,x,y ) ds,
00
0 s
K(t,s) ^ Ii(x(ti)) ds = ^
0<ti<t 0<ti<T+t.
K(t,s) dsIi (x(ti)),
from presentation (16) we obtain
T T
x(t) = Q—1 (t)B(t) — Q—1 (t)
K(t,9) d0f(s,x,y) ds—
0 s T
Q—1(t) ^
K(t,s) dsIi (x(ti)) +
0<ti<tt ui
f(s,x,y) ds + ^ Ii (x(ti)). (17)
0<ti<t
After some simplifications in representation (17) we obtain that the following equalities hold:
t TT
1
f(s,x,y) ds — Q—1 (t)
K(t,0) d0f(s,x,y) ds =
0
0s
= Q—'(t)
A +
K(t,0)d0 I f(s,x,y) ds—
00
T T
— Q—1 (t)
K(t,0) d0f(s,x,y) ds; (18)
0s
Ii(x(ti)) — Q—1 (t)
0<tt<t 0<ti<Tt
H
K(t,s) dsIi (x (ti)) =
ti
Q—1(t^ I A +
K(ti,s)ds I Ii(x(ti)) —
0<ti<^ y
X Q—1 (ti)
t<ti+i <T
K(ti,s) dsIi (x(ti)). (19)
ti
Taking into account (18) and (19), from the presentation (17) we obtain the nonlinear equation
(t)= J(t; x) = Q—11(t)B(t)+ G (t,ti) Ii (x (ti)) +
0<ti<t T / T
+
G(t,s)f ( s,x(s),
0 (s,0,max {x(t)|t g [Ä1 (0) : |: Ä2(0)]}) d0 I ds (20)
for t G (ti,ti+1], i = 0,1,...,p, where
G(t,s) = ^
Q—1(t^A + JK(t,0)d0j, 0 < s < t,
t
—Q—1(t^K(t,0)d0, t < s < T.
1.3 The questions of one value solvability
Theorem. Suppose the following conditions are fulfilled:
' t '
1 r + ^nr + Wd /+ o n-1
1 ). Mf = max
0<t<T
f t,Q—1(t)B(t),J©(t,s,Q—^^(s)) ds
0
< oo ;
2). mr = max max I L (Q Vt)B(t)) I < oo;
0<t<Tie{1,2,...,p}' v n
3). For all t e [0,T], x,y e Rn holds
|f(t,X1 ,y1) - f(t,X2,y2)l < M1 (t) | X1 -X2 | + M2(t) |y1 -y2 I;
4). For all t,s e [0,T]2, x e Rn holds
|0(t,s,x1) — 0(t,s,x2)| < Ms(t,s) |x1 — x21;
5). For all x g Rn, i = 0,1, ...,p holds
|Ii(x1 ) — Ii(x2)| < mi |x1 — x21;
6). p = S1 + S2 + S3 < 1, where T
S1 = max
0tT
|G(t,s)| M1(s) ds, S2 = max
0tT
|G(t,s)| M1 (s)
M3(s,0) d0 ds,
S3 = max Y" | G (t, ti) | ■ mi.
0tT
i=1
Then the nonlocal boundary value problem (10)-(12) with mixed maxima has a unique solution x(t) G PC ([0,T],Rn). This solution can be found by the following iterative process:
xk(t) = J(t;xk-1), k = 1,2,3,... x0 (t)= Q-1 (t)B(t), t e (ti,ti+1), i = 0,1,2,...,p.
In addition, for this solution it is true the following estimate
||x1 (t)— x2(t)||pC <
Q—1 (t) I
1 — P
|B1 (t) — B2(t)||
Proof. We consider the following operator
J : PC([0,T];Rn^ PC ([0,T] x Rn),
(21)
defined by the right-hand side of equation (20). Appling the principle of contracting operators to (20), we show that the operator J, defined by equation (20), has a unique fixed point.
For the zero approximation of the iteration process (21) we easily obtain that
x0(t)
<
Q (t)B(t) < Q (t)
B(t) || < 00.
(22)
Taking first and second conditions of the theorem and estimate (22), for the first difference of the approximations (21) we have the following estimate
x1(t)- x0(t) T
<
p
max Y |G(t,ti)| Ii(x°(ti))
0<t<T
i=1
+
+ max 0<t<T
|G(t,s)|
0
f I s,x0(s),
0(s,9,max |x0(t)|t g [Ai (9) : | : A2(9)]}) d9
ds
< S0 (Mf + mi) <00, (23)
where
S0 = max
0tT
|G(t,s)| ds + max Y |G(t,ti)|, 0tT
i=i
Mf =
0tT
f I t,Q (t)B(t),
mI = max max
0<t<Tie{1,2,...,p}
m t,s,Q-1 (s)B(s) ) ds
Ii (Q-1 (t)B(t))
Then, by the third, fourth and fifth conditions of the theorem, for difference of arbitrary consecutive approximations and arbitrary t G (ti,ti+i] we have
xk(t)-xk-i (t)
<
<
|G(t,s)|
f I s,xk-1(s),
0 (s,9,max |xk-1 (t)|t g [A1 (9) : | : A2(9)]}) d9 I -
-f I s,xk-2(s),
0(s,9,max{xk-2(T)|T g [A1 (9) : | : A2(9)]}) d9
0
ds+
+ YY_|G(t,ti)| | I^xk-1 (ti)) - Ii (xk-2(ti)) i=1
<
<
|G(t,s)|
M1 (s) ■
xk-1(s)- xk-2(s)
+ M2(s)
Ms(s,9)x
x
max jxk-i (t)|t g [Ai (9) : | : A2(9)]} -max {xk-2(T)|T G [Ai (9) : | : A2(9)]}
d9
ds+
+ ^ |G(t,ti)| ■ mi ■
i=i
xk-i(ti)- xk-2(ti)
k-2
Hence, by passing to the norm, we obtain
xk(t)-xk-i (t)
PC
< Si
xk-i(t)- xk-2(t)
PC
+
S2
max {xk-i(T)|T G [Ai(t) : | : A2(t)]}-max {xk-2(T)|T G [Ai(t) : | : A2(t)]}
+ S3
xk-i(t)- xk-2 (t)
k-2
PC
< p
xk-i (t)-xk-2(t)
k-2
PC
+
PC
, (24)
where p = Si + S2 + S3, T
Si = max
0<t<T
|G(t,s)| Mi(s) ds, S2 = max
0tT
|G(t,s)| Mi(s)
M3(s,9) d9 ds,
0
S3 = max y | G (t, ti) | ■ mi.
0tT
i=1
According to the last condition of the theorem, p < 1. Therefore, from the estimate (24) we have
< xk-1(t)- xk-2(t) . (25)
xk(t)-xk-i (t)
PC
PC
It follows from (25) that the operator J on the right-hand side of the equation (20) is contracting. According to fixed point principle, taking into account estimates (22)-(25), we conclude that the operator J has a unique fixed point. Consequently, the nonlocal boundary value problem (10)-(12) 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 (10)-(12) on the right-hand side of condition (11). Let Bi (t), B2(t) G Rn are two different vector functions and xi(t),x2(t) G PC([0,T],Rn) are corresponding solutions of the problem (10)-(12). Then from the equation (20) we have
xi (t) -x2(t) = Q-i (t) [Bi (t) - B2(t)] +
+
G(t,s)
f I s,xi(s),
-f I s,x2(s),
0(s,9,max{xi(t)|tg [Ai(9) : | : A2(9)]}) d9 | -
ds+
0(s,9,max{x2(t)|t g [Ai (9) : | : A2(9)]}) d9
P
+ Y_ G(t,tk) [Ik(xi (tk)) - Ik(x2(tk))]. (26)
k=i
Now, using the conditions of the theorem, similarly to the estimate (24) from (26) we obtain
lx (t) -X2(t)| < Q-' (t) [Bi (t) - B2(t)] +
|G(t,s)| ■ [Mi (s) ■ |xi(s)-X2(s) l ds+
+ ^ | G(t,ti) | ■ mi ■ | xi (ti) - X2(ti) | + M2(s)
i=1
Ms(s,9)x
X |max{xi(T)|T g [At (9) : |: A2(0)]}-max{x2(t)|t g [At (9) : |: A2(9)]}| d0] ds or, passing to the norm, we obtain from last that
llxi (t) -X2(t) IIpc < || Q- (t) | ||Bi (t) - B2(t) II + P ■ ||xi (t) - X2(t) IIpc .
According to the last conditions of the theorem, p < 1. So, from the last inequality follows that
|xi (t) -X2(t) IIpc < (1 - P)-' Q-'(t)
Bi(t)- B2(t) ||.
If we suppose that max || Bi (t) — B2(t) || < 6, where 0 < 6 is small quantity, then from
0<t<T
last estimate we obtain small difference for the solution || xi(t) — x2(t) ||PC < e, where
0 < e = (1 — p)—'1 max II Q—1 (t) || ■ 6. The theorem is proved. □
^kt" 11
2 Conclusion
A nonlocal boundary value problem for a first order impulsive system of ordinary integro-differential equations (10) with mixed maxima is investigated. The boundary value problem is given by the integral condition (11) is studied with nonlinear impulsive condition (12). The existence and uniqueness of the solution of the boundary value problem are proved. In the proof of the theorem the method of successive approximations in combination it with the method of compressing mapping is used. The continuous dependence of the solutions on the right-hand side of the boundary condition (11) is showed.
Remark. If instead condition (11) we consider the following condition
Ax(0) +
K(s)x(s) ds = B,
(27)
where B is constant matrix of redefinition, then by the additional condition
x(t) = C, t = ti, i = 1,2,...,p
(28)
one can consider an inverse problem for the integro-differential system (10) with mixed maxima. Indeed, by virtue of the condition (28), from the presentation (20) we obtain
B = Q(t) C- ^ G (ti) Ii (x (ti))-
0<ti<t
+
0
where
G(s)f ( s,x(s),
G(t) = {
0(s,9,max{x(t)|t g [At (9) : |: A2(6)]}) d9 I ds,
Q-1(t) |a + jK(s)d^, 0 < s < t,
t
-1
-Q-1(t)JK(s)ds, t<s < T.
t
Competing interests. The author declares that there are no conflicts of interest with respect to authorship and publication.
Contribution and responsibility. The author contributed to the writing of the article and is solely responsible for submitting the final version of the article to the press. The final version of the manuscript was approved by the author.
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Yuldashev Tursun KamaldinovichA - D. Sci. (Phys. & Math.), Associate Professor, Professor of the Uzbek-Israel Joint Fac., National University of Uzbekistan, Tashkent, Uzbekistan, ORCID 0000-00029346-5362.
Юлдашев Турсун КамалдиновичА - доктор физико-математических наук, доцент, профессор Узбекско-Израильского объединенного факультета, Национального университета Узбекистана, Ташкент, Узбекистан, СЖСГО 0000-0002-9346-5362.
Вестник КРАУНЦ. Физ.-мат. науки. 2022. Т. 38. №. 1. С. 40-53. ISSN 2079-6641
УДК 517.911 Научная статья
О нелокальной задаче для дифференциальных уравнений с импульсными воздействиями и смешанными максимумами
Т.К. Юлдашев1'2
1 Национальный университет Узбекистана, 100174, Ташкент, ул. Университетская, 4, Узбекистан
2 Институт математики академии наук Узбекистана, 100174, Ташкент, Университетская, 4-B, Узбекистана
E-mail: [email protected]
Исследуется нелокальная краевая задача для системы обыкновенных интегро-дифференциальных уравнений первого порядка с импульсными эффектами и смешанными максимумами. Краевая задача задается интегральным условием. Используется метод последовательных приближений в сочетании с методом сжимающего отображения. Доказаны существование и единственность решения краевой задачи. Показана непрерывная зависимость решений от правой части граничного условия.
Ключевые слова: импульсные интегро-дифференциальные уравнения, нелокальные граничные условия, смешанные максимумы, последовательные приближения, существование и единственность решения, непрерывная зависимость решения.
d DOI: 10.26117/2079-6641-2022-38-1-40-53
Поступила в редакцию: 02.03.2022 Revision submitted: 21.04.2022
Для цитирования. T.K. Yuldashev On a nonlocal problem for impulsive differential equations with mixed maxima // Вестник КРАУНЦ. Физ.-мат. науки. 2022. Т. 38. № 1. C. 4053. DOI: 10.26117/2079-6641-2022-38-1-40-53
Конкурирующие интересы. Автор заявляет, что конфликтов интересов в отношении авторства и публикации нет.
Авторский вклад и ответственность. Автор участвовал в написании статьи и полностью несет ответственность за предоставление окончательной версии статьи в печать. Окончательная версия рукописи была одобрена автором.
Контент публикуется на условиях лицензии Creative Commons Attribution 4.0 International (https://creativecommons.org/licenses/by/4.0/deed.ru)
© Юлдашев Т.К., 2022
Финансирование. Работа выполнена без финансовой поддержки
