ZOFIA MUZYCZKA
ON THE EXISTENCE OF SOLUTIONS
OF THE DIFFERENTIAL EQUATION WITH ADVANCED ARGUMENT
A b s t r a c t . In th is p a p e r th e d iffe re n tia l-fu n c tio n a l e q u a tio n
= /(t, tp), t e R+
9>(0) = n
w ith an u n b o u n d e d a d v a n c e d a rg u m e n t is discovered. U n d e r su ita b le assu m p tio n s, u sin g S c h a u d e r F ix e d P o in t P rin c ip le , a th e o re m on ex iste n c e of so lutions in
a sp ecial fu n c tio n a l-sp a c e is pro v ed .
1. I n t r o d u c t i o n . L et E denote a set of functions <p: R + -*• R n, w here R+ — [0, +0 0) and R n is the euclidean space w ith the norm |*|. Assume th a t the m apping / : R + X E -*■ R n is given and th a t rj e Rn- Consider the differential equation
(1) <p\t) = f ( t , y ) , t e R +
w ith the initial condition
(2) <p{ 0) = rj.
In m ost papers on the existence of a solution of th e Cauchy problem (1)—
(2) (e.g. see [2], [4], [7]) o r the Nicoletti type problem (e.g. see [3], [5], [6]) certain conditions are imposed w hich bound th e advance of th e a r gum ent b y assum ing th a t th e re exists a function d : R + -*• R +, such th a t (3) f(t, q>) = f(t, yj) w henever <p(u) — y>{u) for u e [0, t+d.(t)],
th a t is by assum ing th a t the value of th e function f(t, tp) depends only on the value of th e function <p for u e [0, t+ d (t)]. Boundless advance of th e argum ent of th e function 95 was adm itted in papers [3] and [7] w here (under th e assum ption th a t the function 95 satisfies Lipschitz condition and basing on the Banach F ixed P oint Theorem) th e existence and uni-
Received. N ovem ber 06, 1979.
AMS (MOS) subject classification (1980). Primary 34K10.
queness of th e solution of th e equation under consideration w ere proved in appropriate classes of functions.
In th e presen t note w e prove by m eans of th e Sdhauder F ixed Point Theorem th e existence of a solution of th e problem (1)—(2), w ithout assum ing the restriction (3). We vise for th a t some ideas of p ap er [2] as w ell as of p aper 13].
2. A s s u m p t i o n s . Take on th e following assum ption:
( A ) There exists a locally integrable function L : R + -*• R + and constants
k > 1, X > 0 such th a t for a rb itra ry (t, <p) € R + X E w e have (4) |/{t, 9>)| ^ L(t) su p J|?j(s)| exp k J L(t) d t —Asj : s e R +,
P u t
> t \ .
(5) and (6)
M = sup | |9>(t)| exp k j L(r) d t—it j : t ^ 0
j,
<p G E0 = 99€JS:|95<t)K a exp | k J L(t) dz w here th e constant a satisfies th e condition
(7) k —1*k M
Notice th a t 0 is a non-em pty, closed, 'bounded and convex set.
3 . T h e o r e m o n e x i s t e n c e .
THEOREM. If the assumption {A) is satisfied, then the problem (1)—(2) has at least one solution in the class 0.
P r o o f . Introduce the transform ation T defined for <p e 0 by the form ula
(8) (Tip) (t) = v + j f(s, tp) ds, t e R+.
0
We sh all show th a t th e transform ation T m aps th e s e t 0 in itself, i.e.
th a t T (0)<Z 0. Indeed, let < p € 0 , th e n following (8), (4), (6), (7) we have the estim ations
!(T*p) (t)l ^ M + { If(«, <P)\ ds ^
0
^ [17] + J L(s) sup \<p(r)\ exp k J L(t) d r —).r j : r e R +, r ^ s ds ^ \rj\ -f
+ J L(s) sup |a exp ^ k j L ( r ) d zj exp | —k j L(t) d r —Xr j : r e R +, r ^ s j d s ^
$0
M + J L(s) a exp |fc J £(T)drjsuip {exp (—Ar): r e R +, r ^ s } d s
|j?| + — J k L(s) exp J L ( t ) dt j ds ^ |j??| + exp |fc J L(r) dr
i a exp I k: j L(t) dr j.
We sh all show the com plete continuity of the transform ation T. L et us choose a num ber s > 0 and p u t
<9) M(e) = In ~ .
Then for t e ( /u(c), + °°), q>u 9 ^ 6 in v irtu e of (8), (4), (6) we obtain the estim ations
(TVl) ( t ) - ( T ^ ) ( t ) -
J /(*, <Pi) d s - j f(s, xp2) ds < J |f(s, 95,)| ds+ j |/(s, <p2)| ds <
2 J L(s) sup a exp I k J L(r) d r—Ar j : r e R +, r ^ s
— e x p / f c j L(r)diJ.
Hence
(10) K^Vi) (t)—(T<p2) (t)| exp k J L(t) d r - A t j 2a exp (-At).
B ut according to (9) for te(ju(e), + o o) we get inequality 2 a
k exp (—At) < s, which together w ith (10) gives the estim ation
(11) |(T^) (t)—(Ttp2) (t)| exp k J L(t) dr —It j < e, te(iu{e), +oo).
Assume now th a t t e [0, /<({)], then
| ( T ^ ) ( t ) - C Z > 2) ( t ) K J |/(s, p j - f t s , cp2)\ ds < g 0
if only | ! 9^21! is sufficiently small. Hence
(12) ICZVj) (t)—(T<p2) (t)| exp k j L(r) d r —At j < e, t e [0, ,u(e)].
91
From th e inequalities (11) and (12) it results th a t \T p i~ T p 2\ if only
\<Pt~<pA is sufficiently email. The last inequality m eans th a t T is con
tinuous.
The compactness of the set T (^ ) rem ains to be dem onstrated. F ix a num ber /?, 0 < /? < + ° ° , th en for a rb itra ry t u t2 G [0, /3], such th a t t\ < t2, according to (8), (4), (6) we have
\(T<p) M - i T t p ) (tol < J |/(« , tp)I ds <
ti
^ ~ J k L(s) exp |/c J LĄr) d rj ds ■— exp ^fc J L(t) d tj (t2 — ti).
Using the above estim ation we get for p e<P and t e [0, /?],
|(Tp) (t)| < l(Tp) ( t) - ( T p ) (0)1 + \(T<p) (0)! < exp L(r) d t j + ]tJ\.
From the obtained estim ation and from Arzela Theorem for righthand open intervals (see [1]) the compactness of the set T (^ ) is concluded.
The proposition of th e theorem w hich is being proved resu lts di
rectly from th e Schauder Fixed P o in t Theorem.
R EFER EN C ES
[1] A. B IE L E C K I, R ó w n a n ia ró żn ic zk o w e z w y c z a jn e i p e w n e ich u o gólnienia, W a r
szaw a 1961.
[2] A. B IE L E C K I, C ertaines co n d itio n s su ffisa n te s p o u r V existen ce d’u n e solution d e I’eq u a tio n <p'(t) = f(t, q>(t), q>(v(t))), F o lia Soc. Sci L u b lin e n sis 2 (1962), 70—
73.
[3] A. B IE L E C K I, J . BŁA Ż, Vber eine V era llg e m e in e ru n g der N ic o le tti-A u fg a b e filr F u n k tio n a l-D iffe re n tia lg le ic h u n g e n m it v o r e ile n d e m A r g u m e n t, M o n atsh . M ath. 88 (1979), 287—291.
[4] J . BŁA Ż, S u r V existen ce d ’u n e solution d ’u n e eq u a tio n d iffe re n tie lle a a rg u m e n t avance, A nn. Polon. M ath . 15 (1964), 1—8.
[5] J . BŁA Ż, Vber die N ic o le tti-A u fg a b e filr F u n k tio n a l-D iffe re n tia lg le ic h u n g e n m it v o re ile n d e m A rg u m e n t, Arch., M ath. 27 (1976), 529—534.
[6] J. BŁA Ż, W. W ALTER, Vber F u n k tio n a l-D iffe re n tia lg le ic h u n g e n m it v o r e ile n d e m A r g u m e n t, M onatsh. M ath . 82 (1976), 1—16.
[7] T. D ŁOTK O, O istn ie n iu ro zw ią za ń p ew n eg o ró w n a n ia ró żn iczk o w eg o z w y p rze d za ją c ym a rg u m e n te m , Z eszyty N au k o w e W S P w K ato w ic ac h n r 4 (1964), 79—83.
