• Nie Znaleziono Wyników

On weak partial differential inequalities of first order with Volterra’s operator

N/A
N/A
Protected

Academic year: 2021

Share "On weak partial differential inequalities of first order with Volterra’s operator"

Copied!
7
0
0

Pełen tekst

(1)

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIV (1984) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO

Séria I: PRACE MATEMATYCZNE XXIV (1984)

Ewa Dolinska (Rzeszôw)

On weak partial differential inequalities of first order with Volterra’s operator

In this paper we prove the theorem on strong differential inequalities of first order with Volterra’s operator and the theorem on weak differential inequalities of the same type. The proof of the second theorem is similar to P.

Besala’s proof for partial differential inequalities of first order (see [1], Remark 59.1). I thank Mr K. Zima for help in my work.

I. Introduction. Let En+1 denote the space of points P(t, x 1, . . x n), D im£ " +1 = n + 1. Let H be a subset of the space En+1 of the form

H = {t0 ^ t < t0 + a, ai + L{t — t0) ^ x t ^ b , - L ( r - t 0), i = 1, 2 ,..., n}, where t, a, ah bh L are such real numbers that L > 0, a, < 0 < a ^

i = 1, 2 ,..., n. Let G be a subset of the half-space En+1 (t ^ t0) such that

G n H = [t = t0, at ^ xt ^ bh i = 1 ,2 ,..., n}.

We introduce the notation:

X = (xu x n), U : = {ul , Q : = (q1, ..., qn),

Q (zv) : = (dzv/d x 1, . . . , dzv/dxn) . Let (z1, ..., zm) < (v1, ..., vm) denote that z* < vl for all i (i = 1 , 2 , . . m). I^et f J(t, X , U, Q), j = 1 ,..., m be functions such that:

(i) The function f j {j = 1 ,..., m) is defined on a such set that its projection on the space (t , V) contains the set H.

(ii) The function f j (j = 1 ,..., m) is increasing with regard to the variable U (if U < Ü, then p { t , X , U, Q) ^ p ( t , X , Ü, Q)).

(iii) The function f j = satisfies Lipschitz’s condition with regard to the variable Q :

X , U , X , U, ë ) K L £ lft-9,1,

where L is from the definition H.

(2)

Let uj (t, X ), X ), j = 1 , m, be defined and continuous in the set G u H, having the continuous derivative in the set H. Let, moreover, the operator A (t, X , U) satisfy Volterra’s condition: if Ui (г, X) = U2{t, X) for (r, X ) e G u # t, then A (t, X , U x) = A (t, X , U2), where Ht = {(r,X ):

t0 ^ t < t ) .

We assume that A is an operator increasing with regard to U : if Ut < U2, then A ( t ,X , U\) ^ A (t, X , U2).

II. A system of strong partial differential inequalities of first order. Now we prove the theorem on strong partial differential inequalities:

Th e o r e m 1. We assume that :

(i) u>(t,X) < if (t, X) for ( t,X ) e G , 7 = 1 , 2 ,. .. ,m ,

(ii) u{(t, X) ^ f j (t, X ,A ( t , X , U),Q(uj)) for (t , X )e H , j = 1 ,..., m, (iii) vi(t, X) > f j (t, X , A (t, X , V), Q M ) for ( t , X ) e H , j = l , . . . , m .

Then uJ(t, X) < i f f , X), j = 1 ,..., m, in the whole set H.

P ro o f. Since (i) is satisfied and uj , i f , j = 1 ,..., m, are continuous in G u H the set of elements t(t0 ^ t0 + a) such that uj (t, X) < v*(t, X) in the intersection H n [ ( f , I ) : t 0 < t < 7} is not empty. Let t* denote its least upper bound. We want to show that t* = t0 + a. Let us suppose that it is not true.

Therefore t* < t0 + a. Then there exists a к and a point X* such that uv(f, A) ^ vv(t, X) for t0 < t ^ t*, v = 1, 2, ..., m,

^ uk{t*,X*) = vk(t*,X *).

We have two possible cases:

C a s e 1. Let (t*,X *) be an interior point of the set Я . The function uj (t*, X) — i f f * , X) of the variable X attains, by (1), the maximum in X*. Since (f*, X*) is an interior point of H and this function has a derivative with regard to X at this point, we have

(2) Q(uj (t*, X * )-V (t* , X*)) = 0 or Q(uj (t*, X*)) = Q ^ f * , X*)).

We consider the function uj (t, X*) — i f f , X*) of the variable t. It attains its maximum, in the interval ( t 0,t* ) , at the point t*. Therefore,

(3) u{(t*, X*) — v{(t*,X*) ^ 0.

On the other hand, from (ii), (iii), and (2)

(4) X *) X *, A(t*, X *, U), * * )))-

- / '( г * , X * ,A (t* , X *, V), < 2 И '* , A-*)))

= / J'(r*, X * ,A (t* , X*, U),Q(uJ( t \ X * )))- - f J(г*, X*, A(t*, X *, V), Q(u‘ (t», **))).

Since A is the Volterra operator, A(t*, X*, U) depends only on a behaviour of

(3)

Weak partial differential inequalities 209

U in the set G u H t. Moreover, A is increasing with regard to U. Since for any (t, X )gG и H{!)

uj (t,X ) ^ v>(t, X), 7 = 1, . . . , m, therefore,

(5) A (t* ,X * ,U ) ^ A (t* ,X * , V).

From (4), (5), and (ii) we have

(6) u {(t* ,X * )-v i(t* ,X * ) < 0

which contradicts (3).

C a s e 2. Let (t*, X*) be a point on a side of the set H. Then we can assume that

x* = bp- L ( t * - t 0), p = 1, 2, . . s,

(7) x* = aq + L ( t * - t 0), q = s + l , s + 2, ... ,s + r,

at + L(t* — t0) < x* < bt — L(r* — t0), l = s + r + 1 ,..., n.

(In the case where we change indexes a proof is analogous.) We consider the function

I d ( f •t' у y Y y*^^ —

U y i ) A J , A 2j . • ' X p — 1 9 • A ' p j *Л р + 1 9 • • *9 •А' И /

__ « J / f y*^ y* V y*^\

U 9 Л 1 ) Л 2 9 ' * *9 ■ A ' p — 1 9 Л р 9 *Л ' р + 1 9 * • *9 ‘A ' r t /

of the variable xp, ap + L (t — t0) ^ xp ^ bp — L(t — t0).

By (1) and (7) this function attains maximum at x* = bp —L (t* — t0).

Therefore

(8) uip(t* ,X * )-v iXp(t* ,X * )> 0, p = 1, 2 ,..., s.

We have also:

(9) ^ 4 ,(* * ,* * ) < 0 , # = s + l , . . . , r ,

< (* * , Х*)-и>Х1(1*,Х*) = 0, l = s + r + 1 ,..., n.

For t0 ^ t ^ t* we consider the function K (t) := uJ (t, £>! — L(t —10), ...

..., bs — L(t — t0), as+l + L ( t - t 0) , ..., as+r + L (t-to ), x*+r+1, ..., х*)-!У(г, bx- - L ( t - t 0) , . . . , b s- L ( t - t 0), as+1f f L ( t - t 0) , . . . , a s+p + L { t - t 0),x*+r+1,.. ., x * ) of the variable t. This function attains, by (1) and (7), maximum at t*.

Therefore,

dK(t) dt t=t

(4)

dK(t)

dt = u {(t* ,X * )-v i(t* ,X * ) + ( - L )

Z +

p= l

+ L Z ( u U t* ,X * ) ~ d ( t * ,X * ) )

q — s + 1

= «/((*, X * )-v j{ t* , X * ) - L [ £ (u>Xp(t*, X * ) - v i p(t*, X * ))-

P = 1 s + r

- X (u i< t* ,X * )-v i(t* ,X * ))] .

q = s + 1

Therefore

(10) u i(t* ,X * )-v j{ t* ,X * )

> L [ X ( < ( ' * .x * ) - K p ( r * , x * ) ) ~ x № , ( t * , x * ) - H ' ( t * , x * ) ) ] .

p = 1 q = s+ 1

On the other hand, we have from (ii) and (iii):

X * )-v{(t* , X*) X * ,A (t* , X*, U ),Q (uJ(t*, A"*))) — X*, A(t*, X •, V),Q(v>(t*, A*)))

= [ / J (f*, X * ,A (t* , X », U),Q(u>(t*, * * ) ) ) - X*, A(t*, X*, V),Q(u*(t*, **)))] + + [ f J(t* ,X * ,A {t* ,X * ,V ),Q (u > (t* , A-*)))- - P { f , X *, A(t*, X*,V),Q (v> ( t*, A*)))].

The part in the first brackets is non-positive by properties of the operator A and the function f j .

From (iii) and (8), (9) we obtain

u((t*, X * )-p i(t* , X * ) < L [ X (uLp(t*, X*)-v>Xp( S , X * ))-

p = 1

- I AT*))]

q — s + i

which contradicts (10).

The theorem is proved.

III. A system of weak partial differential inequalities of first order. We prove the theorem on weak partial differential inequalities of first order:

Theorem 2. We assume that

(i') ul (ti X ) ^ v i { t , X ) , j = l, ...,m for (t, X )e G ,

(ii) u/(f, X) a p ( t , X , A ( t , X , ( i ) , 2 ( 4 J = 1 ,.... rn, (t , X ) e H ,

(5)

Weak partial differential inequalities 211

(ÜÏ) u j ( t , X ) ^ f i ( t , X , A ( t , X , V ) , Q H ) , j = l , . . . , m , (t , X ) e H .

Suppose that there exists a sequence { Wv(t, X)}, v = l , 2 , ... ( Wv

= (wVl, y/ 2, ..., w "}), which satisfies the conditions:

(iv) Functions wv(t, X) , j = 1 ,..., m, are continuous and positive in G u H . (v) lim wv(r, X) = 0 for (t, X )eG u H.

GO

(vi) vi(t, X) + w?(t, X) X , A (t, X , V )+ W '(t, X), Q(v< + w '% j = (t, X) e H.

Let, moreover,

(vii) A(t, X , V ) + W v( t , X) > A ( t , X , V + W v), v = 1 ,2 ,...

Then

uj (t, X) ^ |У(f, X), j — 1 ,..., m, in the set H.

P ro o f. Let V(t, X ) : = V(t, X ) + W v{t, X). Then f ‘ (t, X , A (t, X , V)+ W ’ (t, X ),Q (v‘ + w'’<))

X , A (t, X , V+ W% Q(v< + ^ ) ) . This follows from (7) and (ii). Therefore, from (vi) and this inequality, we infer

v i(t,X ) + wvfi ( t , X) > f J ( t , X , A { t , X , W v),Q (vi + wv-i)) or

Щ(1,Х) > f j ( t , X , A ( t , X , V),Q(V)).

From the theorem on strong inequalities:

uj (t, X) < v>(t, X) for (t , X ) e H u G , j = 1, 2 ,..., m, or

uj {t , X) < vi {t,X ) + wVj( t , X) for { t , X ) e G v H , j = 1 ,2 ,..,, w.

Passing in this inequality, to the limit as v -> oo, we get uj {t, X) ^ гУ(г,Х), j = 1, ..., m, {t, X ) e G u H .

IV. Examples of the operator A (t, X , U) which is increasing with regard to the variable U and satisfies Volterra’s condition and property

(*) A (t , X , V + W ) ^ A (t , X , V) + W(t, X).

( 1 ) r 4 ( a , t / ) : = f l / ( f 1 + M ) , p > 0 , t g( 0 , 1 ) ,

(2) A{t, X , U) := U { t-o t,X ), a > 0.

The operator A fulfils condition (*) for functions W(t, X) non-decreasing with regard to t,

(6)

(3) A{t, X, U) := max U{t , X) .

te <tQ ,t0 + a )

V. Certain properties of the function / implying the existence of the sequence {W v(t, X)} fulfilling the assumptions of Theorem 2. We assume that the operator A satisfies condition (*) for functions W non-decreasing with regard to t.

Ex a m p l e 1. Assume that f0 := 0. In other cases considerations are analogous. Let

where aj (t, U) is a comparison function of the first type (see [1]). Let z Vj(t) be a solution of the equation

with the condition zj (0) = 1/v.

We define the sequence { Wv(t, X)}, v = 1, 2 , in the following way:

Since <rj is a comparison function of the first type the sequence [ Wv(t, X)}

satisfies assumptions (iv)-(vi).

Ex a m p l e 2. Assume that t0 = 0. Let

where <rJ(t, U) is a comparison function of the second type (see [1]). Let z Vj(t) be a solution of the equation

with the condition zj (a) = 1/v, v = 1, 2, . . .

We define the sequence [Wv(t, X)}, v = 1, 2 ,..., in the following way:

where a v = lim wv(t). The sequence { Wv(t, X)} satisfies assumptions (ivH viX t-o +

as is easy to verify.

Ex a m p l e 3. Assume that t0 = 0. Let

z{(t) = aj (t,z) + 1/v, t e ( 0 , a ) , z = (z1, z 2, ..., zm)

for ( t , X ) e H ; for (t , X )eG .

f J(t, X, U , Q H ) - f j ( t , X, Ü , Q (iïJ)) < <Jj (t, U - U ) ,

z{(t) = ffj (t,z), te <0, a )

where l are constans, c > 0, к > 0, Ae(0, 1), k( 1 —Л) < 1 .

(7)

Weak partial differential inequalities 213

We introduce the notation

z l = \ul — u l\.

Then

f ‘ (t, X , V ,Q (u > ))-f‘ (t, X , U , < m i n ^ c - f z y . y A We consider solutions of the equations:

= c (z [ )\ z\ (0) = - ,

V

K W T -7 *i.

We form the sequence { Wv(t, X)}:

( 4 ( 0 = [( — Я + l ) c t ] 1/(-A+1) + 1/v,

t e { 0, - >,

t e ( - , a

wVj(t, X)

'4(0 = 1 -A

tE< 0 , 1 / V > ,

l / l - A + l ) f \ \ ~ k / j X - k + l ' J t k U

+ 1 - I > , t e l - , a ).

This sequence satisfies conditions (iv)-(vi).

References

[1] J. S z a r sk i, Differential inequalites, Monografie Mat. T. 43, PWN, Warszawa 1967.

[2 ] K. Z im a, On differential inequality with a lagging argument, Ann. Polon. Math. 18 (1966), 227- 233.

Cytaty

Powiązane dokumenty

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXII (1981) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO.. Séria I: PRACE MATEMATYCZNE

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXIV (1984) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGOJ. Séria I: PR ACE MATEMATYCZNE

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series 1: COMMENTATIONES MATHEMATICAE XXVI (1986) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGOM. Séria I: PRACE MATEMATYCZNE

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXVIII (1989) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO. Séria I: PRACE MATEMATYCZNE

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXV (1985) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO.. Séria I: PRACE MATEMATYCZNE

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXV (1985) ROCZNIK.I POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO.. Séria I: PRACE MATEMATYCZNE

ANNALES SOCIETATIS MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXVII (1987) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO.. Séria I: PRACE MATEMATYCZNE

ANNALES SOCIETAT1S MATHEMATICAE POLONAE Series I: COMMENTATIONES MATHEMATICAE XXVII (1987) ROCZNIKI POLSKIEGO TOWARZYSTWA MATEMATYCZNEGO.. Séria I: PRACE MATEMATYCZNE