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NANOSYSTEMS: Yuldashev T. K. Nanosystems:
PHYSICS, CHEMISTRY, MATHEMATICS Phys. Chem. Math., 2022,13 (2), 135-141.
http://nanojournal.ifmo.ru
Original article DOI 10.17586/2220-8054-2022-13-2-135-141
Periodic solutions for an impulsive system of nonlinear differential equations with maxima
T. K. Yuldashev
National University of Uzbekistan, University street, 4, NUUz, Tashkent, 100174, Uzbekistan [email protected]
Corresponding author: T. K. Yuldashev, [email protected]
Abstract In this paper, a periodical boundary value problem for a first order system of ordinary differential equations with impulsive effects and maxima is investigated. We define a nonlinear functional-integral system, the set of periodic solutions of which consides with the set of periodic solutions of the given problem. In the proof of the existence and uniqueness of the periodic solution of the obtained system, the method of compressing mapping is used.
Keywords impulsive differential equations, periodical boundary value condition, successive approximations, existence and uniqueness of periodic solution.
For citation Yuldashev T. K. Periodic solutions for an impulsive system of nonlinear differential equations with maxima. Nanosystems: Phys. Chem. Math., 2022,13 (2), 135-141.
1. Introduction
The dynamics of evolving processes is often subjected to abrupt changes such as shocks, harvesting, and natural disasters. Often these short-term perturbations are treated as having acted instantaneously or in the form of impulses. Mathematically, this leads to an impulsive dynamical system. So, differential equations, the solutions of which are functions with first kind "discontinuities" at fixed or non-fixed times, have applications in biological, chemical and physical sciences, ecology, biotechnology, industrial robotic, pharmacokinetics, optimal control, etc. [1-5]. In particular, such kind of problems appear in biophysics at micro- and nano-scales [6-10]. Such differential equations with "discontinuities" at fixed or non-fixed times are called differential equations with impulsive effects. A lot of publications of studying on differential equations with impulsive effects, describing many natural and technical processes, are appearing [11-21].
As is known, in recent years the interest in the study of differential equations with periodical boundary conditions has increased. In the works [22-26] periodic solutions of the differential equations with impulsive effects are studied. In [27] the problems of bifurcation of positive periodic solutions of first-order impulsive differential equations are studied. In [28] the problems of stability of periodic impulsive systems are studied.
In this paper, in contrast to works [22-26], we study a periodical boundary value problem for a system of first order differential equations with impulsive effects and "maxima". The questions of the existence and uniqueness of the solution to the periodical boundary value problem are studied. In addition, the technique used in our work is constructive (see [5]) and allows the practical calculation of periodic solutions of nonlinear dynamical systems with deviations, including those with maxima.
2. Problem statement
On the interval [0, T] for t = tj, i = 1,2, ...,p we consider the questions of existence and constructive methods of finding the periodic solutions of the system of nonlinear ordinary first order differential equations with impulsive effects and maxima
x'(t) = f (t, x(t), max {x(t) |t G [t - h,t] }), 0 <h = const. (1)
We study equation (1) with periodic condition
x(t) = ^(t), t G [-h, 0],
(2)
x(0) = x(T) and nonlinear impulsive effect
x (t+) - x (tr) = Fi (x (tj)), i = 1, 2,...,p, (3)
where 0 < i < T, i = ti, i = 1,2, ...,p, 0 = t0 < ti < ... < tp < tp+i = T, x, y g X, X is a closed bounded domain in the space Rn, dX is its border, f g Rn, x (t+~) = lim x (ti + v), x (tr) = lim x (ti — v) are right-hand
© Yuldashev T. K., 2022
side and left-hand side limits of function x(t) at the point t = respectively. The function f is T-periodic, Fi =
ti+p ti + T.
C ([0, T], Rn) is the notation of 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(t)
\
Emax I xo (t) I.
0<t<T j
j=i < <
PC ([0, T], Rn) is the notation of the linear vector space
PC ([0,T], Rn) = {x : [0, T] ^ Rn; x(t) e C ((ti,ti+i], Rn), i = 1,...,p} ,
where x (t+) and x (t-~) (i = 0,1, ...,p) exist and are bounded; x (t-~) = x (ti). Note, that the linear vector space PC ([0, T], Rn) is Banach space with the following norm
x(t) Wpc = max{II x llc((ti,ti+1]) , * = 1 2
Formulation of problem. To find the T-periodic function x(t) G PC ([0, T], Rn), whichforall t G [0,T], t = * = 1,2, ...,p satisfies the system of differential equation (1), periodic condition (2) and for t = tj, i = 1,2, ...,p, 0 < ti < t2 < ... < tp < T satisfies the nonlinear limit condition (3) and goes through x0 at t = 0.
3. Reduction to an functional-integral equation
Let the function x(t) g PC ([0, T], Rn) is a solution of the periodic boundary value problem (1)-(3). Then by integration of the equation (1) on the intervals: (0, t1], (t1, t2], .. ., (tp, tp+1], we obtain:
Jf (s, x, y) ds = J x'(s) ds = x(t- ) - x(0+), t G (0,t1], 00 t2 t2
j f (s, X, y) ds = j x'(s) ds = X (t-) — X (t+) , t G (t1, Î2] :
t1
tp+1 tp+1
J f (s, x,y) ds = J x'(s) ds = x (tp+1) - x (t+) , t e (tp,tp+i], ip ip
where f (s, x, y) = f (t, x(t), max {x(t) |t e [t — h, t] }).
Hence, taking x(0+) = x(0), x(t-+1) = x(t) into account, on the interval (0, T] we have
t
J f (s,x,y) 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 (tp)] + x(t). Taking into account the condition (3), the last equality we rewrite as
s(t) = x(0) + / f (s,x,y) ds + ^ Fj (x (ti)). (4)
0 0<ti<t
We subordinate the function x(t) g PC ([0, T], Rn) in (4) to satisfy the periodic condition (2):
T
x(T)= x(0) + i f (s,x,y) ds + ^ Fj (x (tj)). ^ 0<ti<T
0
Hence, taking the condition (2) into account, we obtain
T
(x
0<ti<T
if (s,x,y) ds + ^ Fj (x (tj)) = 0.
t
Consequently, the differential equation (1) one can write as
:'(t) = f (t, x(t), max {x(t) |t e [t — h, t] }) —
1 T 1 p
— f (t,x(t), max {x(t) |t e [t — h,t] }) dt — Fi (x (ti)). (5)
n i=1
Then by integration of the equation (5) on the intervals: (0, t1], (t1, t2], .. ., (tp, tp+1], instead (4) we obtain the following equation:
(t) = x0 + / f (s, x(s), max {x(t) |t e [s — h, s] }) —
1 T 1 p 1 — y/ f (0,x(0), max {x(t ) |t e [0 — h, 0] }) d0 — - ]T Fi (x (ti)) ds + ]T Fi (x (ti)). (6)
n i=1 -J 0<ti<t
Lemma 1. For solution of equation (6) one has the following estimate
' T
x(t) — xo ||pC < M 77+2p),
where M = max || f (t, x(t), y(t)) ||; max || Fi (t, x(t)) I 1<i<p
Proof. We rewrite the equation (6) as
t T p i x(t) — xo =i [f (s,x(s),y(s)) — 1f f (0,x(0),y(0)) d0 — T £ Fi (x (ti)) ds + £ Fi (x (ti)) n n i=1 J 0<ti<t
(7)
= f (s,x(s),y(s)) ds—^ f (s,x(s),y(s)) ds—^ f (s,x(s),y(s)) ds—^ £ Fi(x (ti))+ £ Fi(x (ti)).
_1 +
,x(s),y(s)) ds—T J f (s,x(s),y(s)) ds—— I f (s,x( 00t Hence, implies that there is true the following estimate
|| x(t) — x0 ||pC < a(t) • || f (t, x(t), y(t)) || + 2p • max || Fi (t, x(t)) || ,
0<ti<t
1<i<p
(8)
t
where a(t) = 2t ^ 1 — — J. 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 initial value function y>(t) on the initial set [—h, 0] is defined by the initial value function y>(t), which is periodical continuation of the solution ^(t) into initial set [—h, 0]. Lemma 2. For the difference of two functions with maxima there holds the following estimate
| max {x(t) |t e [t — h, t] } — max {y(r) |t e [t — h, t] } || < || x(t) — y(t) || + 2h
d
d^ [x(t) — y(t)]
(9)
Proof. We use obvious true relations
max {x(t) |t e [t — h, t] } = max {[x(t) — y(r) + y(r)] |t e [t — h, t] } <
< max { [x(t) — y(t)] |t e [t — h, t] } + max {y(t) |t e [t — h, t] } .
Hence, we obtain
max {x(t) |t e [t — h, t] } — max {x(t) |t e [t — h, t] } < max { [x(t) — y(t)] |t e [t — h, t] } . (10) We denote by t1 and t2 the points of [t — h, t], on which the maximums of the functions x(t) and y(t) are reached: max {x(t) |t e [t — h, t] } = x(t1), max {y(T) |t e [t — h, t] } = y(t1), max {[x(t) — y(t)] |t e [t — h,t] } = x(t2) — y(t2). Then, taking (10) and last equalities, we have
|| max {x(t) |t e [t — h, t] } — max {y(T) |t e [t — h, t] } — x(t) + y(t) || <
< || [x(t) — y(t)] — [x(t1) — y(t1)] | + || [x(t2) — y(t2)] — [x(t1) — y(t1)]
From another side, it is obvious that, there holds the estimate
[x(t) — y(t)] — [x(t) — y(t)] II < h
d~t [x(f) — y(f)]
h
[x(t) — y(t)]
(11)
(12)
x
where t, t g [t - h, t], t* g (t, i). From the estimates (11) and (12) we come to the following estimate
|| max {x(t) |t G [t - h, t] } - max {y(r) |t G [t - h, t] } - x(t) + y(t) || < 2h
Therefore, it is easy to check that there holds the inequality (9) and we complete the proof of the Lemma 2. By the BD we denote the Banach space of functions on the interval [0, T] with the norm
y x(t) ||bd < y x(t) IIpc + h y x'(t) IIpc . Theorem 1. Assume that for all t g [0, T], t = ti, i = 1,2, ...,p are fulfilled the following conditions:
1. max ( || f (t, x(t), y(t)) | ; max || Fi (t, x(t)) || 1 = M < rc;
i i<i<p i
2. | f (t, xi, yi) - f (t, X2, y2) | < Ll [|| xi - X2 I + I yi - y2 ||];
3. || Fi (xi) - Fi (x2) || < L2 || xi - x2 ||;
(T \
4. The radius of the inscribed ball in X is greater than M — + 2p ;
n 2
5. + ^ + L2p(^2+ < 1.
If the system (1) has a periodic solution for all t g [0, T], t = ti, i = 1, 2, ...,p, then this solution can be founded by the system of nonlinear functional-integral equations
t
x(t, xo) = xo + j f (s, x(s, xo), max {x(r, xo) |t G [s - h, s] }) -
0
T
1 T 1 p 1
- 1 f (0,x(0,xo), max {x(r,xo) |t G [0 - h, 0] }) d0 - Fj (x (tj,xo)) ds + ^ Fj (x (tj,xo)). (13)
0 i=i -J 0<ti<t
Proof. We will show that the right-hand side of the system of equations (13) as an operator maps a ball with radius
(T \
M • I — + 2p 1 into itself and is a contraction operator. So, according to the lemma 1, from (7) we have
|| x(t,xo) - xo ||pC < M^| + 2p). (14)
From the equation (5) we obtain
|| x'(t,xo) < M (2+ P). (15)
We consider a difference x(t, xo) - $(t, xo), where functions x(t, xo) and $(t, xo) satisfy the system of equations (13). By the conditions of the theorem, from (13) we have
t
|| x(t, xo) - $(t, xo) || < Li J j || x(s, xo) - $(s, xo) || +
o
T
+ || max {x(t, xo) |t G [s - h, s] } - max {$(r, xo) |t G [s - h, s] } || + 1 J [|| x(0, xo) - $(0, xo) || +
max {x(t, x0) |t g [0 - h, 0] } - max {$(t, x0) |t g [0 - h, 0] } ||] d0 jds+
p
+ Y, L2 || x (ti,xo) - â (ti,xo) || + Y, L2 || x (ti,xo) - â (ti,xo) || <
i=i 0<ti<t
"'PC
< 2a(t)Li II x(t, xo) - â(t, xo) ||PC + h || x'(t, xo) - â'(t, xo) ||PC + 2pL2 || x(t, xo) - â(t, xo) ||PC <
< 2 ( Lit + pL^ || x(t,xo) - â(t,xo) ||PC + LiTh || x'(t,xo) - â'(t,xo) ||PC . (16)
I PC ' 0 || ^ V1-; •''Uy - " \u> ^J || PC
T 2
Similarly, by the assumptions of the theorem 1, from (5) we have
H x/(t, xo) - xo) || PC < (2L1 + L2 p) || x(t,xo) - ^(t,xo) || PC + 2L1h || x/(t, xo) - xo) || PC . (17) Multiplying both sides of (17) to h and the result adding to (16), we obtain
|| x(t, xo) - 0(t, xo) ||bd < P • || x(t, xo) - 0(t, xo) HBD , p = 2LW 2 + M + L2P ( 2 + h j . (18)
According to the last condition of the theorem 1, p < 1. So, from the estimate (18) we deduce that the operator on right-hand side of (13) is compressing. From the estimates (14), (15) and (18) implies that there exists unique fixed point x(t, x0) G BD. The theorem 1 is proved.
From the estimate (18) it is easy to obtain that for x0, x0 g X holds
II x(^ x0) - x(t x0) ||BD < 1 X0 - X 0 1 . We note that the theorem 1 one can proof by the method of successive approximations, defining iteration process as
t
x0(t, x0)= x0, xfc+i(t, x0) = x0 + J f (s, xfc(s, x0), max |xfc(t, x0) |t G [s - h, s] }) -
0
T
1 T 1 p
- T f (0,xfc(0,x0), max jxfc(t,x0) |t G [0 - h, 0] }) d0 - ^ ^ F (xfc (tj,x0))
ds+
+ Fi (xfc (ti,xo)). (19) 0<ti<t
Now we will show the existence of periodic solutions of the system of impulsive differential equations (1). We introduce designations:
T
1 T 1 p
A(xo) = — f (i,xœ(i,xo), max jxTO(r, xo) |t G [t - h,t] }) dt + — ^ Fi (xœ (ti,xo)), (20)
^ — 1
0 j=1 T
1 T 1 p
Afc(x0) = T f (t, xfc(t, x0), max jxfc(T, x0) |t G [t - h,t] }) dt + (xfc (tj,x0)), (21)
0 i=1 where x(t, x0) = lim xk(t, x0) = xTO(t, x0) is solution of the nonlinear system (13). Therefore, xTO(t, x0) is the
fc^w
solution of the system of impulsive differential equations (1) for A(x0) = 0 going through x0 at t = 0. Consequently, the questions of existence of solution of the system of impulsive differential equations (1) we reduce to the questions of existence of zeros of function A(x0) and we solve this problem, finding zeros of the function Ak(x0). Theorem 2. Assume that
1. All conditions of the theorem 1 are fulfilled;
2. There is a natural number k such that the function Ak(x0) has an isolated singular point Ak(x0) = 0, index of which is nonzero;
3. There is a closed convex region X0 c X, containing a single singular point such that on the its border d X0 is fulfilled estimate
inf || Afc(x) || > —(22)
xEdlo 1 - p
Then the system of impulsive differential equations (1) has a periodic solution for all t g [0, T], t = tj, i = 1,2, ...,p that x(0) G X0.
Proof. Let us consider families of everywhere continuous on d X vector fields
V(<7,x0) = Afc(x0) + a(A(x0) - Afc(x0)),
which connect the fields
V(0,x0) = Afc(x0), V(1,x0) = A(x0).
We note that there is true the estimate
| A(x0) - Afc(x0) |< —p-. (23)
1 - P
Therefore, the vector field V (a, x0) does not vanish anywhere on d X0. Indeed, from (22) and (23) implies that
|| V(a, x0) | > | Ak(x0) | - || A(x0) - Afc(x0) | > 0. (24)
The fields Ak(x0) and A(x0) are homotopic on dX and the rotations of the fields homotopic on the compact are equal. Hence, taking into account (24), we conclude that the rotation of the field A(x0) on the d X0 is equal to the index of the singular point x0 of the field Ak (x0) and nonzero. Consequently, the vector field A(x0) on the X0 has a singular point x0, for which A(x0) = 0. Therefore, the system of impulsive differential equations (1) has a periodic solution for all
t G [0, T], t = tj, i = 1, 2, ...,p that x(0) G X0. In addition, we note that for x0, x0 G X from (20) and (21) we have
p-
A(xo) ypo < M (l + I),
|| A(xo) - A(xo) ||bd < 1 x° - xo 1.
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 differential equations (1) with periodical boundary value condition (2) and with nonlinear condition (3) of impulsive effects for t = ti, i = 1, 2,... ,p, 0 < t1 < t2 < • • • < tp < T. The nonlinear right-hand side of this equation consists of the construction of maxima. The questions of 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 g [0, T], t = ti, i = 1,2, ...,p, then it is proved that this solution can be founded by the system of nonlinear functional-integral equations (13). The questions of existence of solution of the system of impulsive differential equations (1) we reduce to the questions of existence of zeros of function A(xo) and we solve this problem finding zeros of the function Ak (xo) in (21).
The results obtained in this work will allow us in the future to investigate another kind periodical boundary value problems for the heat equation and the wave equation with impulsive actions.
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Submitted 8 March 2022; revised 26 March 2022; accepted 2 April 2022
Information about the authors:
T. K. Yuldashev - National University of Uzbekistan, University street, 4, NUUz, Tashkent, 100174, Uzbekistan; [email protected]
Conflict of interest: the authors declare no conflict of interest.
