On certain differential equations with deviated argument

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ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I : СОММЕ NT ATIONES MATHEMATICAE X I I (1968)

M . Kw a p i s z (G d a ń s k )

On certain differential equations with deviated argument

1. The purpose of this paper is to discuss existence and uniqueness problems for differential equation

(A) x'(t) = f ( t , a? 00» a?(9h(0), a(<Pn(t))t x ' {t), x'[ щ Щ , . . . , x’(ipm(t))) with initial condition a?(

0

)

= 0

(a general initial condition x(t0) = c can be reduced to the condition considered here). An equation of this type was considered by E. Driver [3] under the assumption and ipj(t) < t. More general equation of differential-delay type was investi

gated in paper [4]. In this paper we shall not assume the equation (A) to be of a delay type.

This paper is concerning with a problem posed by B. Bellman [1].

He asked: what aditional conditions on the solution ensure uniqueness of a real solution of the problem

x'(t) = g(t, x(t), x ( h ) t . .. , x(tN)), (B) a?(

0

) = c,

0

< ti < t

²

< ... < tN ?

It may happen an equation of type (B) has no solution; e.g., there is no solution of the equation x' (t) = a{t)x(t

1

)-\-b(t),x(

0

) =

0

,withj a(s)ds = h

1

,

о J b(t)dt Ф 0. It is evident that if the function f ( t , u Q, . .. , un, v0, h

0

does not depend on % and <pi(t) = t{, then the equation (A) reduces to the equation (B).

In view of these facts it is necessary to give some conditions which ensure the existence of a solution of the problem (A). We shall give also some conditions ensuring the uniqueness of solutions of the problem (A).

2. We shall prove the existence theorem for the problem (A) in a special case. We introduce

Assumption

Hx. Suppose

th at

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1° the function/(ź ,

u0, un)

is defined and continuous for 2e<0, u>,

Щ€(—

oo, + oo),

i =

0 , 1 ,

Щ,

. . . ,

Un)e(

— o o , + oo),

2° the functions

w{t)

are defined and continuous for

te(0, a

>;

(pi(t)e(0, a

>,

3° there exists an r0 > 0 such th at sup |/(/, w0, • • •» wn)l < r0/a for 2e<0, a ) , |w*| < r0.

Th eo eem 1.

Under the assumption

H j

there exists at least one solution of the initial-value problem

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x'(t) = f(t, x(t), x{<px{t)),

. . . ,

x((pn(t))),

* a ? ( 0 ) = 0 ;

moreover

,

this solution is defined for all

tfe<0,u>.

P r o o f . L e t us consider the Banach space C<0, u> of real-valued continuous functions defined on the interval <0,

a},

with ||a?(i)||

= max{|a?(i)|: 0 <

t

<

a).

I t is obvious th at the problem (1) is equivalent to the following integral equation

t

(2)

x{t) = . f f( s , x(s),x(<Pl(s)), .. . ,x ( p n(s)))ds

0

considered in

C

<0,

ay.

If

T

is the operator defined by right-hand side of the equation (2) and

K 0

=

[x(t): x(t)

e(7<0, a>, ||ж(1)|| < r 0], then we infer easily, from the assumption H j, th at

T

is continuous, com pact and

T ( K

0) c:

K 0.

Now, Theorem 1 is implied by Schauder fixed-point theorem [5].

3.

In this paragraph we are going to consider the initial value problem

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x'{t) = f ( t , x{t), x((Pl{t)), ...,x(<pn(t)), x'(t), x'lf^t)),

. . . ,

X'(ipm{t)))

with

x(0) =

0, under the following

As s u m p t io n H 2 . Suppose th at

1° the function

f ( t , u 0, . . . , u n,v0, . . . , vm)

is defined and continuous for

te(0, а}, Щ, Vje(—

oo, + o o ) ; / e ( — oo, + oo),

2° the functions

cpi{t

) ,

ipj-(t

) ,

i =

1 , . . . ,

n, j

= 1, . . . , m, are defined and continuous for ie<0, a> and

<pi(t

) ,

ip}-(t)

e<0,

a},

3° for any ^ , / 2e<0, u> the following inequality holds:

4° there exists an r0 > 0 such th at

sup|/(ź,

Щ,

. . . ,

un, v

0, . . . ,

vm)\

< r0 for f €<0, u>,

\Ui\^r0, \Vj-\

< r 0,

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5

° there exist non-negative constants /q, j =

⁰

, m, such that the inequalities

\f(tj

U^q, . . . , Un > « 0 , • • • ? ^m)

f

• • • j ^n i Voi • • • > ^m) I

m m

^ У (ij i % v3 u № ~ У № < i

j

/=0 ?=0

are satisfied for any t e <

⁰

, ay, щ, v3-, e{ — oo, + oo).

T

^heorem

2. I / Йе assumption H

2

is satisfied, then there exists at least one solution of the problem (3), this solution is defined in the interval

<

0

, a}.

Proof. We reduce the problem (3), by substitution x'(t) = u(t), to the following equation

(4) u(t)

t rpi(t) q>n (t)

= f[t , f u(s)ds, j u(s)ds, . .. , J u(s)ds, u(t), u(ipx{t) ) , ..., u(ym(t))j.

0 0 0

Let T be the operator defined by the right-hand side of the equa

tion (4) and K

⁰

be the sphere defined in section 2. From the assumption H

2

it follows that the operator T is continuous in K 0, and T ( K 0) c K 0.

Put

F ( u , t, r)

t <pi(t) g>n (t)

= f [ t , J u{s)ds, f u(s)ds

, . . . ,

J u(s)ds, u( г), и(гр

ⁱ

(т))

, . . . ,

u[xp m (^)j)

0 0 0

for t, r e <0, a } , u e K 0. According to the continuity property of the func

tions f , pi, yjj we have: for any e >

0

there exists 3(e) >

0

such that for any u e K Q and tx, ź

2

e<0, ay, \tx — t2\ < 3 the inequality |F { u , tx, tx) —

—F ( u , t

2

, t x)I < e holds. If the number e > 0 is fixed we consider the set Щ , K l = {u: UeKQ, \u{tx) - u { t 2)\ <£/(

¹

- p ) , \tx- t 2\ < 3(e), tx, t

2

e<

0

, a)}.

Now we can prove the inclusion T { K e0) <=. K

q

. In fact, if u e K „ and z = Tu, then z e K Q and

\z{tx) — z(t2)\ < IF ( u , tx, tx) —F ( u , t2, tx)\ + \F(u, t2, tx) —F ( u , t2, t2j

m

e + ^ м \ и (щ {1 2)) —

u(ipj{tx))\

-f-

[I

q\u

(tx)

и

(t2)

j < e + - j y y

?•=! p

8

1

— [A

Thus, we see that the continuous operator T maps the bounded,

closed, convex and compact set K eQ into itself; therefore, according to

Schauder fixed-point theorem there exists at least one fixed point of T ,

this fixed-point is a solution of the equation (4).

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Be m ar k 1. If а > 1, then Theorem 1 is a special case of Theorem 2.

4. We shall now consider the initial-value problem for the implicit differential equation

(5) / ( / ,

x(t), x(<px(t)), ...,x[cpn{t)), x'(t), х'(щ(г)), ..., x'(ipm(t))) =

0,

with a?(

0

) =

0

.

As s u m p t io n

H3. Suppose that

1

° conditions l°-3° of the assumption H

2

are satisfied,

2

° there exist non-negative constants mx, m

²

such that

0

< mx

f ( t , Uq , . . . , Un , Vq , Vx , • • •, Vm ) f { i f i • • •» ^0 > ^1 ? • • • > ^m)

< --- < yyi-

for any

t e < 0 , a y , Щ, Vj, v0 e ( — oo, -f- oo),

3° there exist non-negative constants /q, j = 1 , . . . , m, such that for any t e

< 0

, a } , щ , %, щ e ( —

o o

, -f-

o o )

, i —

0

, . . . , n, j =

1

, . . . , m, the following inequalities hold:

[j, =

11=1

2

mx-\-m2 + m2 — mx mx-\-m

2

< ¹ ,

\f { t , U0 , . . . , Un , V^q , , . . . , Vm ) f ( t , П 0 , • • • > 'M'n j ^0 ? ^1 ? • • • j ^m) I

^ № I I ? 4° there exist an r

0

> 0 such that

I

2

SUP К --- ; f(t ,

U0 , . . . , Un , V0 , . . . , Vm )

I TOj-f-ma

< r 0 , t e < 0 , a > ,

for

\щ\, \Vj\

< r 0.

Th e o r e m

3. Under the assumption H

3

there exists at least one solution of the initial-value problem (5), defined in the interval <0, a).

Proof. The function

h ( t , U^q , . . . , Un , V^q , . . . , Vm ) — V0 . f { l ? Щ j • • • j ) ^o> • • • ? ^m)

wh + m

2

has all properties pointed in assumption H2, therefore Theorem 3 follows immediatelly from Theorem 2.

Be m ar k

2

. Theorem 3 is a generalisation of the result of the pa

per [

2

], where a theorem on the existence of solutions of the equation

f[t, x(t), x'(t))

= 0

was established.

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5. In this section we are going to establish a theorem on the exist

ence of a unique solution of the initial-value problem (3).

Assumption

H4. Suppose that

1

° conditions l°-3° of the assumption H

2

are satisfied,

2

° there exist non-negative constants Xi, w such that

\f(ty Щ, • • •, Пn j j • • • ? ®т) fiU U0,

^{. ,}^{Un , V}^q , . . •, Vm ) I

n m

^ \Щ Щ\ jW ji |% I

г = 0 7 = 0

for any t€<

0

**, a>, w, Щ, Щ, Vj{ — ° ° ? + °°)>**

3° if ai = max{ę>i(tf): 0 < t < a], then the following inequality holds:

n m

v ~ щ Xi + aX

⁰

-f- fij < 1.

г = 1 j —⁰

Theorem 4.

Under the assumption

H 4

there exists a unique solution of the problem (3), this solution is defined for fe<0, a}.

Proof. The operator T defined by the right-hand side of equa

tion (4) has the contraction property, therefore it has a unique fixed point being the unique solution of the problem (3).

6 . We now want to formulate another theorem on the existence and uniqueness of the solution of the problem (3).

Assumption

H5. Suppose

th at

1° conditions 1°, 2° of the assumption H

4

are satisfied,

2

° there exists a positive real number g such that

m \ П

v

= UoH— - I + /h-ехр(efr) + — Агехр(еуг) < 1,

' ® ' 7 = 1 ^ г = 1

**where y* = max{(pi(t) — t:**

0

< t < a }, = т а x.{y>j(t) — t:

0

< t < a).

Theorem 5.

I f the assumption H

5

is satisfied, then there exists a unique solution of the problem (3), defined for t e ( 0, ay.

Proof. Using the norm ||'w(f

^)||1

= max{[w(<)|exp( — gt): 0 < t < a) it is easy to prove that the operator T considered above is a contraction.

E e m a r k 3. If щ {

1

) i-f. t and ipjil) ^ then ^

0

, у у ^

0

, I^ow the inequality v <

1

results from the following one

v ^ ' ? = 1 c г —1

m

The last inequality is fulfilled if м < 1 and g is sufficiently large.

/ =

⁰

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Be m ar k 4. If щ(1) = t and <pi(t) **0, a>, then fa = 0, y* = tt and**

m n

A

0 1

v — **^0H--- h / 1 H H----/ 1 A*eXp (^i) .**

^ y-=i ^ г=1

m n

Now the inequality v <

1

can be fulfilled if £ & <

1

**and £ A* is suffi-**

7 =

0

i—l

ciently small.

7. In this section we shall deal with the problem (5).

Assumption

H6. Suppose that

1° conditions l°-2° of the assumption H

3

are satisfied,

2

**° there exist non-negative constants A*, щ , i =**

0

, . . . , n , j =

0

, ..., m, such that for any tfe<

0

, а }, щ, щ, щ, fae( — oo, -f oo) the following inequality holds:

\ f { t ) ^ 0 7 • ' • 7 У ' П 7 ® 0 7 ® 1 7 ' • * 7 ^ m ) f ( t 7 ^ 0 7 * • • 7 ^ " П 7 ^ 0 7 ^ X 7 • • • 7 ^ m ) |

п Ш

^ Z{ jU{ u^\ -(- /q \Vj VjJ, 3° if щ = max {

9

**?* (**

2

):

0

< t < a}, then

n m

2 v r 2

г

=0 7 = 1

v i л vn z z m2

A

⁼

У

щ ---- ---

Аг-

⁺

У

■ ■ iij + ал0 — I---

4

- j тх-\-т2 j-J тх-\-т2 mx-\-m2 mx

m2 — mx + w

2

<

1

.

Theorem 6

. Under the assumption H

6

there exists a unique solution of the problem (5), this solution is defined for t e<0, «>.

Proof. In order to prove this theorem it is sufficient to observe that the function

h (

^{1 7}

U

q

, ... , un, V

q

, . .. , vm) v

0 2

■ ■ f (

^{1 7}

Щ, . .. , un, V

q

, .. ., vm)

%-fWa

has all the properties listed in assumption H4. Thus, Theorem

6

is a simple consequence of Theorem 4.

Assumption

H7. Suppose that

1

° conditions

1

°,

2

° of the assumption H

6

are satisfied,

2

° there exists a real number

^q

>

0

such that m 2—mx ______ A

0

mx-\-mz Q{rnxĄ-m2)

I

1

Xi

Q(mx + m2) exp ( gyi) +

+ z

№

mx + m2 e x p (

q

fa) <

1

.

A simple consequence of Theorem 5 is

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Theorem 7.

Under the assumption

H 7

there exists a unique solution of the problem (5), this solution is defined for t e ( 0, a}.

Remarks similar to Remarks 3, 4 also hold true.

R e fe re n ce s

[1] R. B e llm a n , Research Problems, Bull. Amer. Math. Soc. 71 (1965), p. 853.

[2] R. C o n ti, Sulla resoluzione dell' equazione F ( t , x , x ' ) — 0, Ann. Mat. Рига Appl. ser. IV, vol. 48 (1959), pp. 97-107.

[3] R. D. D riv e r, Existence and continuous dependence of solutions of a neutral functional-differential equation, Arch. Rational Mach. Anal. 19 (1965), pp. 149-186.

[4] M. K w a p isz , O pewnej metodzie Tcolejnych przybliżeń i jakościowych zagad

nieniach równań różniczkowo-funkcyjnychi różnicowych w przestrzeni Banacha, Zeszyty Naukowe Politechniki Gdańskiej, Matematyka IV (1965), pp. 4 -7 0 .

[5] W. P o g o r z e ls k i, Równania całkowe i ich zastosowania I I , Warszawa 1958.

On certain differential equations with deviated argument