On the Convergence of Solutions of Certain Generalized Functional-Differential Equations

UNIVERSITATIS MAEIAE CURIE-SKŁODOWSKA LUBLIN - POLONIA

VOL. XXIX, 22 8ECTIO A 1975

Instytut Matematyki, UniwersytetMarii Curie-Sklodowskiej Lublin

WOJCIECH ZYGMUNT

On the Convergence of Solutions of Certain Generalized Functional- Differential Equations

0 zbieżności rozwiązań pewnych równań kotyngensowo-funkcjonałowych О сходимости решений некоторых дифференциально-функциональных включений.

In this paper we show that the main results of J. Błaż [2] may be extended for generalized functional-differential equations. We shall prove three theorems which are the counterpart of theorems 1—3 in [2].

I. We accept the following notations and symbols:

r< 0 is a fixed real number, R+ = [0, oo), Rn denotes a n-dimonsional Euclidean space with the usual norm |a?| — ж?)1/2, where x =(ж1;x2,...

<-i

...,xn), 0 denotes the originof Rn and {6}denotes thesubsetof 72", whose unique element is 0.

For А, В<= Rn

a(x, A) = inf |ж— y\

VtA

d(A, B) = max{supa(x,B),sup a(y, A)}.

xeA ycB

ConvE" is the family of all convex compact and nonempty subsets of R". This family is metrized by the Hausdorff distance d. G is the space of all continuous functions cp-. [r, oo)->Rn with topology defined by an almost uniform convergence on [r, oo)(i.e. an uniform convergence on each compact subinterval of interval [r, oo)). [ę?]B denotes the functiong>

localized to the interval [r, ■»], ||<p||„ = max |/>(<?)|.

r<«t>

The set of all functions [дз]е, where <peC and » > 0, will be denoted by (£.

In this set set we introduce the metric as follows: by the distance two functions [q?and [y]w we mean the distance of graph of these functions (the graph being a subset ofR xRn) in the Hausdorff sense (the so-called graph topology).

184 Wojciech Zygmunt

Let F be a multivalued mapping (in the abbreviation m.v.m.), F-. R+ x G^ConvjR71, let v: R+ -+R+ and let [£]„e<£.

We shall investigate the existence of solutions for two problems concerning the generalized functional-differential equations:

(1)

99 (Z)e-F(Z, [ç>]v(|)), 0 Z,

<p(Z) = f(Z), r<Z<0, and

(><)

(pMi(0 = £ (0> r «C Z < 0,

where is an arbitrary, but fixed, positive number. By a solution of (1) we meananyfunction tp e G, which isabsolutely continuous on each compact subinterval of the interval R+ = [0, oo), (p'(t)eF(t, [9?]^,)) a.e. / >0 (the abbreviation a.e.Z is used for for almost every t in the Lebesque measure sense), <p(t) = f (Z), r < t< 0. Similarly, a solution of (ill,) is any function g>M.eG which is absolutely continuous on [0, JfJ, <f>'Mi{t')eF(t, [95Mi],(()) a.e’.Z, 0^ Z^ Jf,-, yw.(Z) =^.(df,.) for M{ and = f(Z)' for r^Z^O.

II. Assume the following:

1° The function v is continuous and v(t)^t, t^O.

2° The m.v.m. F satisfies conditions

a) -^(b is Lebesque measurable for each [<??]„«(£, *)

b) F(t, •) is continuous for each t> 0 and there exists a continuous function L: R+->R+ such that

[V’]v(()))<i(0ll7’-V’ll^) for each <>0, 3° There exists a constant k, k > 1, such that

d(-F(Z, [0]„0), {0}) < kL(t), t 0.

4° The following inequality holds

k80J ^L(s}ds

e ‘

sup --- --- = q< 1 **) 0« k

*) We say that a m.v.m. G: -K+->conv(.Rn) is Lebesque measurable iff the set {teR+: G(t)r\B #=0} is Lebesque measurable for each closed subset B c. Rn.

**) Throughout this paper integrals are understood in the Lebesque sense.

III. Let p be constant such that

1-3

Denote by C* the family of all functions <peC satisfying the condition llyll = sup

o<i llç’llj

t kf L(s)ds ) 0

It is easy to verify that the set C* with a norm || * || is a complete metric space.

We now state the following theorem:

Theorem 1. Ifthe hypotheses 1° — 4° are fulfilled, then the problem (1) has at least one solution which belongs to C*.

Proof. Let us consider the m.v.m. r defined in C* by formula

r<p =

i

£(0)+ fx(s)ds, 0

Ç(t), r<t<0.

t > 0, where a? is a Lebesque measu-' rable selector of F(-, [<p]v(.))

It follows from Bridgland’s Lemma [3, Lemma 2.8] (cf also [4]) that the m.v.m. F(-, [<£>]„(.)): R+->ConvRn is measurable. Then in view of Kura- towski-Ryll-Nardzewski theorem [6] there exists a measurable selector x of F(-, [<p],,(.)). Thus Frp is nonempty for each <peC*. Using the Bridgland’s theorem [3, Theorem 3.1] we conclude that rq> is closed in G* for gpeC*.

To show the inclusion r<p <=. G*, cpeG*, first let us observe that d(U(<, [?’]^i)), {3}) d(F(t, Dp]^)), F{t, [0]^))) + d[F(t, [0]^), {0})

O||^0 + hL(t) — L(t) ||9?||»(j) +kL(t), and let us choose arbitrary peTcp. We have for / >0

<

cp(t) = f (0) + J x{s')ds.

0

Since x{t)eF(t, [9>]H0) a.e. t 0, then following closelyas in ([2], see the proof of Theorem 1]) we obtain for <>0

(3)

( t J

\r(t)\ |f(O)|+ J \x(s)\ds^ |f(0)| + f (L(s)\\p\\^+ kL(s))ds^pe0 ’ .

0 0

186 Wojciech Zygmunt Obviously for r < t< 0

(4) 1^(01 = \W)\ < ||£||o,

So in view of (3) and (4) Ill’ll P- Consequently ry <=. C*. Now we shall prove that f is a contraction with constant q,i.e. that D(Ty, rip)< q\\y— y||

for each y, iptC*, where D is the Hausdorff metric (in a family of all nonempty closed subsets of C*) generated by the norm ||-||.

Let y,ipeC*, y ip, and let ytTy. Then, for t > 0, y(t) = £(0)+ J x(s)ds,

0

where x is measurable and x{t)eF(t, [9?],^) a.e. t 0. Since d(F(t, [99]^), F(t,M^i})) <L(t)\\y-ip\\^t), there is yt'F(t, such that \x(t)-yt\

<L(t)\\y-y\\^t).

Let us put K(t) = {yteBn: \x(t)— yt)^L(t)\\y— ip\\^t)}. K(t) is a non­ empty closed convex set and the m.v.m. K: R+^-coavBn is Lebesque measurable. Then the m.v.m. G: J2+->convJB’1 defined by G(t) =6

= F(t, [ip)rW)CiK(t) is also Lebesque measurable (cf for example [4]).

Let z be a measurable selector for G. Then we have z(t)eF(t, [y]^) a.e.

/> 0 and

\x(t)-z(t)\ < L(t)\\y-ip\\<l} a.e. <>0.

Now define a function ip: [r, oo)->j2n by

f(<) =

f(0)+fz(s)ds, 0, 0

f($),

Obviously yePy and — =0 for For t^0

t t

\y(t)-ip{t)\<f ix(s)-z(s)ids< f L(s)Hy-y^(3)ds

0 0

and further identically as in [2, see the proof of Th. 1] we obtain the inequality

k I L(s)dst

|p(J)-y(0K 2ll9

’

’

°

Hence ||£—ylK glfo —y||.

From this, and the analogues inequality obtained by interchanging the roles of y and ip, we get D(ry, Tip) q\\y —ip\\.

So we see that the m.v.m. r fulfills all hypotheses of the contraction principle of Covitz and Nadler [5, Corollary 3] (r maps the complete metric space G* into the family of all nonempty closed subset of G* and is the contraction with constant q < 1) Therefore, there exists a function cpeG* such that cpeTcp what means that

<p'(t)eF(t, [99]^) a.e. <>0,

<p(t) - f(f), r^<<0.

This completes the proof of our Theorem.

IV° Let

Om, = {<P«C: ?>(<)= •pW) forOJf,-}, C*/; = {(peG*: cp(/) =^(df,) for / M}.

Similarly as C* in Theorem 1, the set G*M. with a norm ||• || given by (2) is a complete metric space.

Define on C*M the m.v.m. r by formula t

f(0)+ 0<<<dfn

0 wherexM. is a

measurable se­

lector of Jf,.

£(0)+ f xM.(s)ds, M^t,

0 _^(■» Dp-jf,-]►(.>) >

f(f),

Considering this mapping in the same way as in previously section we get the following

Theorem 2. If hypotheses 1° — 4° are fulfilled, the problem (M{) has at least one solution which belongs to C*ir..

Remark 1. If F is a single-valued mapping, then the problems (1) and (dfj have exactly one solution. This it follows immediately from proof of these theorems.

V° In this section we prove a theorem which is a generalization of Theorem 3 in [2].

Theorem 3. Let {df,}^ be an increasing sequence of real numbers such that limdfj = +00.

i-*oo

a) If is asequence of solutionsofproblems (df,) (inG*M respectively), then there exists a subsequence {<PMi}}T-i which is uniformly convergent on each compact subinterval of [r, 00) to a function tp and cp is a solution (in C*) of problem (1).

b) If <p0 is a solution (in C*) of problem (1), then there exists a sequence {'Z’.ujyii °f solutions (in C*M{ respectively) of problems (df,), which is uni­ form convergent on each compact subinterval of [r, 00) to the function <p0.

188 Wojciech Zygmunt Proof, a) It is easy to verify that

i

k JL(s)ds

Pe ° »=1,2,...

and

t

+ J(Z(«)||9,J/.||,(g) + A:Z(8))ds, 0 < t, 0< h, i =1,2,...

0

Since = £(Z) for r<f<0, consequently the functions <pM. are uniformly continuous on each compact subinterval of [r, oo). Thus, by well-known Arzela’s theorem, there exists a subsequence which is almost uniformly convergent on [r, oo) to some function 99. Obviously

<peC*.

To prove that cp is the solution of (1), it suffices to show that q> satisfies the equation (p'(t)eF(t, [^]„(()) almost everywhere on each compact in­ terval [0,T] a R+.

Let us fix arbitrary T> 0 and let us define m.v. mappings G/. [0, T]->

->convR" and G: [0, T]->convR” by formulas Gj(<) = F(t, [?>.afg]»(o) > 0 < t < T,

G(t)=F(t,[v\{l}),

Since <pM converges uniform to

99

13 o«ssr

conclude that limd(Gj(/), G(t)) = 0 on [0,T].

>-»00

From this it follows that

a.e. Z«[O,T].

By virtue of PliS’s Lemma [6, Lemma 1] we get

<p’(t)cG(t) = F(t, [9?]„(0) a.e. Ze[O, T].

Therefore the proof of the part a) is completed.

Remark 2. In the case when F is a single-valued mapping the whole sequence {99.^.}“, of solutions of (II,) (which in view of Remark 1 are unique) converges to a solution of (1).

b) Now let 9>0 be a solution of (1). Let us define

r . . _ fProj^W/F«,[xD) for (Z, [x]b)6jR+ xGand if ^(f) exists, lProj(0/F(Z,[/]„)) otherwise,

(1,/) and

W,/)

where Proj(0/Æ) denotes the metric projection a point zeRn onto a no­

nempty compact convex subset K of Rn, i.e.

Proj(z/A) = {yeK-. \y— z\ = inf|fc-«I}.

kdC Obviously / is the single-valued mapping.

According to the result in [1, Chapter VI, v 3, Th. 3]f is continuous in [%]„ for each fixed teR+ and by Castaing’s theorem [4,Th. 5.1]/is Lebesque measurable in f>0 for each fixed [/] e(£. Moreover /satisfies the hypo­

theses 2° b) and 3°. Therefore the following problems

<p'(t) =/(^»

0<*,

VifffO — [Ç’jqlqo)» 0 t

= £(0>

have, in view of our Eemark 1, exactly one solutions ÿ and yM. and, by our Eemark 2, rpM. converges to y. But the function y0 is the solution of (1,/) too, because

f(t, EM(i)) = Proj(^(0/P(<, [<po],(o)) = v'M a.e. 0.

Thus must be =ç)0.

Similarly the functions are solutions of problems (JT,) because

=/(^» [ÿj/f]r(<))e^’(^ ]»(,)) a.e. <e[0, JI,].

This proves the part b) and finally the proof of our Theorem 3 is com­

pleted.

REFERENCES

[1] Bergo Cl., Topological Spaces, Oliver and Boyd, Edinburgh and London 1963.

[2] Błaż J., On a certain differential equation with deviated argument, Prace Mate­

matyczne I, Silesian University, Katowice (1969) 15-23, (in Polish).

[3] Bridgland T., Trajectory integrals of set-valued functions, Pacific J. Math. 33 (1970), 43-68.

[4] Castaing Ch., Sur les multiapplications mesurables, Revue d’lnf et de Rech.

Op., 1 (1967), 91-126.

[5] Covitz H., Nadler S., Multi-valued contraction mappings in generalised metric space, Israel J. Math., 8 (1970), 5-11.

[6] Kuratowski K., Ryll-Nardzewski Cz., A Oeneral Theorem of Selectors Bull.

Acad. Polon. Sci., Ser. sci. math. astr, et phys., 13 (1965), 397 403.

[7] Pliś A., Measurable orientor fields, ibidem 13 (1965), 565-569.

190 Wojciech Zygmunt

STRESZCZENIE

W pracy podano trzy twierdzenia. Pierwsze dwa — to twierdzenia o istnieniurozwiązaniaw klasie C* równań (1) i (Jf<). Twierdzenie trzecie jest następujące:

a) Jeśli i jest ciągiem rozwiązań (w klasie C*) równań to istnieje podciąg tego ciągu, który na każdym zwartym pod- przedziale przedziału [r, oo) jest jednostajnie zbieżny do rozwiązania (p (w klasie C*) równania (1).

b) Jeżeli <p jest rozwiązaniem (w klasie C*) równania (1), to istnieje ciąg {•PjłĄ.}“ i rozwiązań (w klasie C*) równań (JIt) zbieżny jednostajnie na każdym zwartym podporzedziale przedziału [r, oo) do funkcji (p.

РЕЗЮМЕ

В работе даны три теоремы. Две первые — это теоремы о сущест­

вованию решения в классе С* уравнений (1) и (М{). Теорема третья следующая: Теорема 3: а) Если последовательность решений (в классе С*) уравнений (М{), то из этой последовательности можно выделить подпоследовательность {<Рмц}г^=1, равномерно сходящиеся на каждом компактном интервале луча [г, оо) к решению <р (в классе С*) уравнения (1).

в) Если <р решение (в классе С*) уравнения (1), то существует последовательность решений (в классе С*) уравнений (Л/,), равномерно сходящаяся на каждом компактном интервалелуча [г, оо) к решению (р.

UNI VE RSITATIS MARIAE CURIE-SKŁODOWSKA

VOL. XXVII SECTIO A 1973

1. P. J. Eenigenburg and E. M. Silvia: A Coefficient Inequality for Bazilevié Functions.

Nierówności na współczynniki dla funkcji Bazileviëa.

2. N. K. Govil and V. K. Jain: On the Enestróm-Kakeya Theorem.

O twierdzeniu Enestróma-Kakeyi.

3. L. Grzegórska: Recurrence Relations for the Moments of the so-called Inflated Distributions.

Wzory rekurencyjne na momenty tak zwanych rozkładów “rozdętych”.

4. L. Grzegórska: Distribution of Sums of the so-called Inflated Distributions.

Rozkłady sum tak zwanych rozkładów “rozdętych”.

5. F. Kudelski: Sur quelques problèmes do la théorie des fonctions subordinées.

O kilku problemach w teorii funkcji podporządkowanych.

6. J. Kurek: Construction of an Object of Center-Projective Connection.

Konstrukcja obiektu koneksji środkowo-rzutowej.

7. H. Mikos: Orthogonality in the N-way Nested Classification.

Ortogonalność w N-krotnej klasyfikacji hierarchicznej.

8. II. Mikos: Variance Component Estimation in the Unbalanced N-way Nested Classification.

Estymacja komponentów wariacyjnych w niezrównoważonej N-krotnej klasyfikacji hierarchicznej.

9. E. Niedokos: Estimation of Variance Components in Unbalanced Mixed Models.

Estymacja komponentów wariacyjnych w modelach mieszanych nie- ortogonalnych.

10. J. Stankiewicz: The Influence of Coefficients on some Properties of Regular Functions.

Wpływ współczynników na pewne własności funkcji regularnych.

11. Z. Stankiewicz: Sur la subordination en domaine de certains opérateurs dans les classes S (a, fi).

Podporządkowanie obszarowe pewnych operatorów w klasach S(a, p).

12. D. Szynal and W. Zięba: On Infinitely Divisible Generalized Distributions in Rk.

O ogólnych rozkładach nieskończenie podzielnych w JRk.

13. J. Zderkiewicz: Sur la courbure des lignes de niveau dans la classe des fonctions convexes d’ ordre a

O krzywiźnie poziomic w klasie funkcji wypukłych rzędu a.

ANNALE UNIVEKSITATIS MARIAE CU

VOL. XXVIII SECTIO A

Biblioteka Uniwersytetu MARII CURIE-SKLODOWSKIEJ

w Lublinie

n 050 29

CZASOPISMA

1945

1. D. M. Campbell, M. R. Ziegler: The Arg of Finito Order and the Radius of Argument pochodnej i promień prawi rodziny funkcji skończonego rzędu.

2. II. B. Coonce, P. J. Eenigenburg, M. R.

Mocanu Variation II.

Funkcje z ograniczoną wariacją Mocanu II.

3. R. Janicka: An Existence Theorem for an Integro-Differential Equation of Ncutral Type.

Twierdzenie o istnieniu rozwiązań równania całkowo-różniczkowego typu neutralnego.

4. L. Koczan, W. Szapiel: Sur certaines classes de fonctions holomorphes définies par une intégrale de Stieltjes.

O pewnych klasach funkcji holomorficznych określonych całką Stieltjesa.

5. Z. Lewandowski, S. Miller, E. Złotkiewicz: Gamma-Starlike Functions.

Funkcje gamma-gwiaździste.

6. Z. Lewandowski, S. Wajler: Sur les fonctions typiquement réelles bornées.

O funkcjach typowo-rzeczywistych ograniczonych.

7. S. Ruscheweyh: On Starlike Functions.

O funkcjach gwiaździstych.

8. Z. Rychlik: The Convergence of Rosen’s Sériés for the Sums of a Random Number of Independent Random Variables.

O zbieżności szeregów Rosena dla sum niezależnych zmiennych losowych z losową liczbą składników.

9. J. Stankiewicz, J. Waniurski: Some Classes of Functions Subordinate to Linear Transformation and their Applications.

Pewne klasy funkcji regularnych podporządkowanych transformacji ulamkowo-liniowej i ich zastosowania.

10. Z. Stankiewicz: Sur la subordination en domaine de certains opérateurs.

Podporządkowanie obszarowe pewnych operatorów.

11. A. Szybiak: Grassmannian Connections.

Koneksje Grassmannowskie.

12. D. Szynal: Sur une loiforte des grands nombres de variables aléatoires enchaîneés.

O mocnym prawie wielkich liczb dla zmiennych losowych powiązanych w łańcuch Markowa.

13. J. Zderkiewicz: Sur la courbure des lignes do niveau dans la classe. XJ?

O krzywiźnie poziomic w klasie. XJ?

14. W. Zygmunt: On a Certain Paratingent Equation with Deviated Argument.

O pewnym równaniu paratyngensowym z odchylonym argumentem.

15. W. Zygmunt: On Some Properties of a Certain Family of Solutions of Paratin­

gent Equation.

O kilku własnościach pewnej rodziny rozwiązań równania paratyngen- sowego.

UNIWERSYTET MARII CURIE-SKLODOWSKIEJ

LUBLIN

BIURO WYDAWNICTW

Plac Litewski 5 POLAND