Memoirs on Differentil Equtions nd Mthemticl Physics Volume 25, 22, 1 14 M. Tvrdý DIFFERENTIAL AND INTEGRAL EQUATIONS IN THE SPACE OF REGULATED FUNCTIONS
Abstrct.???. 2 Mthemtics Subject Clssifiction.???. Key words nd phrses:???.
3 Prefce This text is intended s self-contined exposition of generlized liner differentil nd integrl equtions whose solutions re in generl regulted functions (i.e. functions which cn hve only discontinuities of the first kind). In prticulr, the problems studied below cover s their specil cses liner problems for systems with impulses. (For representtive surveys of results concerning systems with impulses see e.g. [4], [6] or [58].) Essentilly, the text is collection of the ppers [48], [49], [5], [51] nd [52]. In comprison with the originl versions of these ppers, the nottion used ws unified nd the common preliminries were summrized in Chpter 1. Furthermore, some minor improvements of the exposition nd corrections of severl misprints were included. Chpter 2 is compiltion of the ppers [48] nd [49]. In this chpter the properties of the Perron-Stieltjes integrl with respect to regulted functions re investigted. It is shown tht liner continuous functionls on the spces G L [, b] of functions regulted on [, b] nd left-continuous on (, b) nd G reg [, b] of functions regulted on [, b] nd regulr on (, b) my be represented in the form Φ(x) = q x() + p(t) d[x(t)], where q R nd p(t) is function of bounded vrition on [, b]. Some bsic theorems (e.g. integrtion-by-prts formul, substitution theorem) known for the Perron-Stieltjes integrl with respect to functions of bounded vrition re extended to more generl cse. In Chpter 3 (cf. [52]) the continuous dependence of solutions to liner generlized differentil equtions of the form x(t) = x() + d[a k (s)] x(s), t [, 1] on prmeter k N is discussed. In prticulr, known results due to Š. Schwbik [41] nd M. Ashordi [1] re extended or mended. Boundry vlue problems of the form x(t) x() M x() + d[a(s)] x(s) = f(t) f(), t [, 1], K(τ) d[x(τ)] = r nd the corresponding controllbility problems re delt with in Chpter 4. (This chpter is bsed on the pper [49].) The djoint problems re given in such wy tht the usul dulity theorems re vlid. As specil cse the interfce boundry vlue problems re included. In contrst to
4 the erlier ppers (cf. e.g. [54], [46], [47], [43], [44] nd the monogrph [45]) the right-hnd side of the generlized differentil eqution s well s the solutions of this eqution cn be in generl regulted functions (not necessrily of bounded vrition). Similr problems in the spce of regulted functions were treted e.g. by Ch. S. Hönig [15], [17], [16], L. Fichmnn [9] nd L. Brbnti [5], who mde use of the interior (Dushnik) integrl. In our cse the integrl is the Perron-Stieltjes (Kurzweil) integrl. In Chpter 5 (cf. [51]) we investigte systems of liner integrl equtions in the spce G n L of n-vector vlued functions which re regulted on the closed intervl [, 1] nd left-continuous on its interior (, 1). In prticulr, we re interested in systems of the form x(t) A(t) x() B(t, s) d[x(s)] = f(t), where the n-vector vlued function f nd n n n-mtrix vlued function A re regulted on [, 1] nd left-continuous on (, 1) nd the entries of B(t,.) hve bounded vrition on [, 1] for ny t [, 1] nd the mpping t [, 1] B(t,.) is regulted on [, 1] nd left-continuous on (, 1) s the mpping with vlues in the spce of n n-mtrix vlued functions of bounded vrition on [, 1]. We prove bsic existence nd uniqueness results for the given eqution nd obtin the explicit form of its djoint eqution. A specil ttention is pid to the Volterr (cusl) type cse. It is shown tht in tht cse the given eqution possesses unique solution for ny righthnd side from G n L, nd its representtion by mens of resolvent opertors is given. The results presented cover e.g. the results known for systems of liner Stieltjes integrl equtions x(t) d s [K(t, s)] x(s) = g(t) or x(t) d s [K(t, s)] x(s) = g(t). The study of such equtions in the spce of functions of bounded vrition ws initited minly by Š. Schwbik (see [35], [38] nd [45]).
Chpter 1 Preliminries 1.1. Bsic notions Throughout this text we denote by N the set of positive integers, R is the spce of rel numbers, R m n is the spce of rel m n-mtrices, R n = R n 1 stnds for the spce of rel column n-vectors nd R 1 1 = R 1 = R. For mtrix A R m n, rnk(a) denotes its rnk nd A T is its trnspose. Furthermore, the elements of A re usully denoted by i,j, i = 1, 2,..., m, j = 1, 2,..., n, nd the norm A of A is defined by We hve nd x = A = ( i,j ) i=1,2,...,m j=1,2,...,n n i=1 A = nd mx j=1,2,...,n i=1 m i,j. A T = ( j,i ) j=1,2,...,n i=1,2,...,m for A R m n x i, x T = (x 1, x 2,..., x n ) nd x T = mx j=1,...,n x j for x R n. Furthermore, for mtrix A R m n, its columns re denoted by [j] (A = ( [j] ) j=1,2,...,n ). Obviously, A = mx j=1,2,...,n [j] for ll A R m n. The symbols I nd stnd respectively for the identity nd the zero mtrix of the proper type. For n n n-mtrix A, det (A) denotes its determinnt. If < < b <, then [, b] nd (, b) denote the corresponding closed nd open intervls, respectively. Furthermore, [, b) nd (, b] re the corresponding hlf-open intervls. 5
6 The sets d = {t, t 1,..., t m } of points in the closed intervl [, b] such tht = t < t 1 < < t m = b re clled divisions of [, b]. Given division d of [, b], its elements re usully denoted by t, t 1,..., t m. The set of ll divisions of the intervl [, b] is denoted by D[, b]. Given M R, χ M denotes its chrcteristic function (χ M (t) = 1 if t M nd χ M (t) = if t M.) Finlly, if X is Bnch spce nd M X, then cl(m) stnds for the closure of M in X. 1.2. Functions Regulted functions 1.2.1. A function F : [, b] R m n which hs limits F (t+) = lim F (τ) τ t+ Rm n nd F (s ) = lim F (τ) Rm n τ s for ll t [, b) nd ll s (, b] is sid to be regulted on [, b]. The set of ll m n-mtrix vlued regulted functions on [, b] is denoted by G m n [, b]. For F G m n [, b], we put F ( ) = F () nd F (b+) = F (b). Furthermore, for ny t [, b] we define + F (t) = F (t+) F (t), F (t) = F (t) F (t ) nd F (t) = F (t+) F (t ). (In prticulr, we hve F () = + F (b) =, F () = + F () nd F (b)= F (b).) We shll write G n [, b] insted of G n 1 [, b], G 1 1 [, b]= G[, b]. Obviously, F G m n [, b] if nd only if ll its components f ij : [, b] R re regulted on [, b] (f ij G[, b] for i = 1, 2,..., m, j = 1, 2,..., n). It is known (cf. [15, Corollry 3.2]) tht if F G m n [, b], then for ny ε > the set of points t [, b] such tht + F (t) > ε or F (t) > ε is finite. Consequently, for ny F G[, b] the set of its discontinuities in [, b] is countble. The subset of G m n [, b] consisting of ll functions regulted on [, b] nd left-continuous on (, b) will be denoted by G m n [, b]. L The set of ll functions F G m n [, b] which re regulr on (, b), i.e. 2 F (t) = F (t ) + F (t+) for ll t (, b), will be denoted by G m n reg [, b]. We define F = sup F (t) for F G m n [, b]. t [,b] Clerly, F < for ny F G m n [, b] nd when endowed with this norm, G m n [, b] becomes Bnch spce (cf. [15, Theorem 3.6]). As G m n L [, b] nd G m n reg [, b] re closed in G m n [, b], they re lso Bnch spces.
7 Functions of bounded vrition 1.2.2. For F : [, b] R m n its vrition vr b F on the intervl [, b] is defined by vr b F = sup d D[,b] m F (t j ) F (t j 1 ) j=1 (the supremum is tken over ll divisions d = {t, t 1,..., t m } D[, b] of [, b]). If vr b F <, we sy tht the function F hs bounded vrition on the intervl [, b]. BV m n [, b] denotes the Bnch spce of m n-mtrix vlued functions of bounded vrition on [, b] equipped with the norm F BV m n [, b] F BV = F () + vr b F. Similrly s in the cse of regulted functions, We shll write BV n [, b] insted of BV n 1 [, b] nd BV[, b] insted of BV 1 1 [, b]. F BV m n [, b] if nd only if f ij BV[, b] for ll i = 1, 2,..., m nd j = 1, 2,,..., n. A function f : [, b] R is clled finite step function on [, b] if there exists division {t, t 1,..., t m } of [, b] such tht f is constnt on every open intervl (t j 1, t j ), j = 1, 2,..., m. The set of ll finite step functions on [, b] is denoted by S[, b]. It is known tht S[, b] is dense in G[, b] (cf. [15, Theorem 3.1]). It mens tht f : [, b] R is regulted if nd only if it is uniform limit on [, b] of sequence of finite step functions. A function f : [, b] R is clled brek function on [, b] if there exist sequences {t k } k=1 [, b], {c k } k=1 R nd {c+ k } k=1 R such tht t k t j for k j, c k = if t k =, c + k = if t k = b, nd ( c k + c+ k ) < k=1 f(t) = c k + c + k for t [, b] (1.2.1) t k t t k <t or equivlently f(t) = c k χ [t k,b](t) + c + k χ (t k,b](t) for t [, b]. k=1 Clerly, if f is given by (1.2.1), then + f(t k ) = c + k nd f(t k ) = c k for k N. Furthermore, for ny such function we hve nd f() =, f(t ) = f(t) = f(t+) if t [, b] \ {t k } k=1
8 vr b f = ( c k + c+ k ). k=1 The set of ll brek functions on [, b] is denoted by B[, b]. Notice tht neither S[, b] nor B[, b] re closed in G[, b]. It is known tht for ny f BV[, b] there exist uniquely determined functions f C BV[, b] nd f B BV[, b] such tht f C is continuous on [, b], f B is brek function on [, b] nd f(t) = f C (t) + f B (t) on [, b] (the Jordn decomposition of f BV[, b]). In prticulr, if W = {w k } k N is the set of discontinuities of f in [, b], then f B (t) = ( f(w k ) χ [wk,b](t) + + f(w k ) χ (wk,b](t) ) on [, b]. (1.2.2) k=1 Moreover, if we define f B n (t) = n ( f(w k ) χ [wk,b](t) + + f(w k ) χ (wk,b](t) ) (1.2.3) k=1 for t [, b] nd n N then lim n f B n f B BV = (cf. e.g. [45, the proof of Lemm I.4.23]). Obviously, S[, b] B[, b] BV[, b] G[, b]. For more detils concerning regulted functions or functions of bounded vrition see the monogrphs by G. Aumnn [3], T. H. Hildebrndt [14] nd Ch. S. Hönig [15] nd the ppers by D. Frňková [1] nd [11]. 1.2.3. As usul, the spce of m n-mtrix vlued functions continuous on [, b] is denoted by C m n [, b] nd the spce of m n-mtrix vlued functions Lebesgue integrble on [, b] is denoted by L m n 1 [, b]. For given F L m n 1 [, b] nd G C m n [, b], the corresponding norms re defined by F L1 = F (t) dt nd G C = g = sup G(t). t [,b] Agin, C n 1 [, b] = C n [, b], C 1 1 [, b] = C[, b], L1 n 1 [, b] = L n 1 [, b] nd L1 1 1 [, b] = L 1 [, b]. Moreover, the spce of m n-mtrix vlued functions bsolutely continuous on [, b] is denoted by AC m n [, b], AC n 1 [, b] = AC n [, b], AC 1 1 [, b] = AC[, b], nd F AC = F () + F L1 for F AC n n [, b].
9 Nottion 1.2.4. If [, b] = [, 1], we write simply G insted of G[, 1]. Similr bbrevitions re used for ll the other symbols for function spces introduced in this chpter. Functions of two rel vribles 1.2.5. Let < < b <, < c < d < nd let F : [c, d] [, b] R m n. If t [c, d] nd s [, b] re given, then the symbols vr b F (t,.) nd vrd c F (., s) denote the vritions of the functions nd F (t,.) : τ [, b] F (t, τ) R m n F (., s) : τ [c, d] F (τ, s) R m n, respectively. Furthermore, for s [, b] we put nd 1 F (τ, s) = F (τ, s) F (τ, s) if τ (c, d], 1 F (c, s) = + 1 F (τ, s) = F (τ+, s) F (τ, s) if τ [c, d), + 1 F (d, s) =. Similrly, for t [c, d] we put nd 2 F (t, σ) = F (t, σ) F (t, σ ) if σ (, b], 2 F (t, ) = + 2 F (t, σ) = F (t, σ+) F (t, σ) if σ [, b), + 2 F (t, b) =. The symbol v [c,d] [,b] (F ) stnds for the Vitli vrition [c, d] [, b] defined by = sup D of F on v [c,d] [,b] (F ) = m F (t i, s j ) F (t i 1, s j ) F (t i, s j 1 ) + F (t i 1, s j 1 ) <, i,j=1 where the supremum is tken over ll net subdivisions D = { c = t < t 1 < < t m = d; = s < s 1 < < s m = b } of the intervl [c, d] [, b]. We sy tht the function F hs bounded Vitli vrition on [c, d] [, b] if v [c,d] [,b] (F ) <. Moreover, F is sid to be of strongly bounded vrition on [c, d] [, b] if v [c,d] [,b] (F ) + vr b F (c,.) + vr d cf (., ) <. The set of n n-mtrix vlued functions of strongly bounded vrition on [c, d] [, b] is denoted by SBV n n ([c, d] [, b]). If no misunderstnding cn rise, insted of v [c,d] [,b] (F ) we shll write simply v (F ) nd insted of SBV n n ([, 1] [, 1]) we shll write SBV n n. (For the bsic properties of the Vitli vrition nd of the set SBV, see [14, Section III.4] nd [45, Section I.6].)
1 1.3. Integrls nd opertors Perron-Stieltjes integrl 1.3.1. The integrls which occur in this text re Perron - Stieltjes integrls. We will work with the following definition which is specil cse of the definition due to J. Kurzweil [19]: Let < < b <. The couples D = (d, ξ), where d = {t, t 1,..., t m } D[, b] is division of [, b] nd ξ = (ξ 1, ξ 2,..., ξ m ) R m is such tht t j 1 ξ j t j for ll j = 1, 2,..., m re clled prtitions of [, b]. The set of ll prtitions of the intervl [, b] is denoted by P[, b]. An rbitrry positive vlued function δ : [, b] (, ) is clled guge on [, b]. Given guge δ on [, b], the prtition (d, ξ) of [, b] is sid to be δ-fine if [t j 1, t j ] (ξ j δ(ξ j ), ξ j + δ(ξ j )) for ny j = 1, 2,..., m. For given functions f, g : [, b] R nd prtition D = (d, ξ) P[, b] of [, b] let us define m S D (f g) = f(ξ j ) [g(t j ) g(t j 1 )]. j=1 We sy tht I R is the Kurzweil integrl to b nd denote I = f(t) d[g(t)] or I = of f with respect to g from f dg if for ny ε > there exists guge δ on [, b] such tht I S D (f g) < ε for ll δ fine prtitions D of [, b]. For the definition of the Kurzweil integrl it is necessry to mention the fundmentl fct tht given n rbitrry guge δ on [, b], the set of ll δ-fine prtitions of [, b] is non-empty (Cousin s lemm). The Perron-Stieltjes integrl with respect to function not necessrily of bounded vrition ws defined by A. J. Wrd [55] (cf. lso S. Sks [32, Chpter VI]). It cn be shown tht the Kurzweil integrl is equivlent to the Perron-Stieltjes integrl (cf. [36, Theorem 2.1], where the ssumption g BV[, b] is not used in the proof nd my be omitted). Consequently, the Riemnn-Stieltjes integrl (both of the norm type nd of the σ-type, cf. [14]) is its specil cse. The reltionship between the Kurzweil integrl, the σ-young-stieltjes integrl nd the Perron-Stieltjes integrl ws described by Š. Schwbik (cf. [36] nd [37]). It is well known (cf. e.g. [45, Theorems I.4.17, I.4.19 nd Corollry I.4.27] tht if f G[, b] nd g BV[, b], then the integrl f dg
11 exists nd the inequlity f dg f vrb g is true. The Kurzweil integrl is n dditive function of intervl nd possesses the usul linerity properties. For the proofs of these ssertions nd some more detils concerning the Kurzweil integrl with respect to functions of bounded vrition see e.g. [19], [21], [4] nd [45]. For mtrix vlued functions F : [, b] R p q nd G : [, b] R q n such tht ll integrls f i,k (t) d[g k,j (t)] (i = 1, 2,..., p; k = 1, 2,..., q; j = 1, 2,..., n) exist (i.e. they hve finite vlues), the symbol F (t) d[g(t)] (or more simply stnds for the p n mtrix M with the entries F dg) m i,j = q k=1 f i,k d[g k,j ] (i = 1, 2,..., p; j = 1, 2,..., n). The integrls d[f ] G nd F d[g] H for mtrix vlued functions F, G nd H of proper types re defined nlogously. Liner opertors 1.3.2. For liner spces X nd Y, the symbols L(X, Y) nd K(X, Y) denote the liner spce of liner bounded mppings of X into Y nd the liner spce of liner compct mppings of X into Y, respectively. If X = Y we write L(X) nd K(X). If A L (X, Y), then R (A ), N (A ) nd A denote its rnge, null spce nd djoint opertor, respectively. For P Y nd A L(X, Y), the symbol A 1 (P ) denotes the set of ll x X for which A x P.
Chpter 2 Regulted Functions nd the Perron-Stieltjes Integrl 2.1. Introduction This chpter dels with the spce G[, b] of regulted functions on compct intervl [, b]. It is known tht when equipped with the supreml norm G[, b] becomes Bnch spce, nd liner bounded functionls on its subspce G L [, b] of functions regulted on [, b] nd left-continuous on (, b) cn be represented by mens of the Dushnik-Stieltjes (interior) integrl. This result is due to H. S. Kltenborn [18], cf. lso Ch. S. Hönig [15, Theorem 5.1]. Together with the known reltionship between the Dushnik- Stieltjes integrl, the σ-young-stieltjes integrl nd the Perron-Stieltjes integrl (cf. Ch. S. Hönig [16] nd Š. Schwbik [36], [37]) this enbles us to see tht Φ is liner bounded functionl on G L [, b] if nd only if there exists rel number q nd function p(t) of bounded vrition on [, b] such tht Φ(x) = q x() + p(t) d[x(t)] for ny x G L [, b], where the integrl is the Perron-Stieltjes integrl. We will give here the proof of this fct bsed only on the properties of the Perron-Stieltjes integrl. To this im, the proof of existence of the integrl f(t) d[g(t)] for ny function f of bounded vrition on [, b] nd ny function g regulted on [, b] is crucil. Furthermore, we will prove extensions of some 12
13 theorems (e.g. integrtion-by-prts nd substitution theorems) needed for deling with generlized differentil equtions nd Volterr-Stieltjes integrl equtions in the spce G[, b]. Finlly, representtion of generl liner bounded functionl on the spce of regulr regulted functions on compct intervl is given. Since we will mke use of some of the properties of the σ-riemnn- Stieltjes integrl, let us indicte here the proof tht this integrl is included in the Kurzweil integrl. (For the definition of the σ-riemnn- Stieltjes integrl, see e.g. [14, Sec. II.9].) 2.2. Preliminries Proposition 2.2.1. Let f, g : [, b] R nd let I R be such tht the σ-riemnn-stieltjes integrl σ Then the Perron-Stieltjes integrl f dg exists nd equls I. s well. Proof. Let f dg exists nd equls I, σ f dg = I R, i.e., for ny ε > there is division d = {s, s 1..., s m } D[, b] of [, b] such tht S D (f g) I < ε is true for ny prtition D = (d, ξ) P[, b] of [, b] such tht d D[, b] is refinement of d. Let us define { 1 2 min{ t s j ; j =, 1,..., m } if t d, δ ε (t) = ε if t d. Let prtition D = (d, ξ) P[, b], d = {t, t 1,..., t m }, ξ = (ξ 1, ξ 2,..., ξ m ) T, be given. Then D is δ ε fine only if for ny j = 1, 2,..., m there is n index i j such tht s j = ξ ij. Furthermore, we hve S D (f g) = m [ [f(ξj ) [g(t j ) g(ξ j )] + f(ξ j ) [g(ξ j ) g(t j 1 )] ]. j=1
14 Consequently, for ny δ ε fine prtition D = (d, ξ) of [, b] the corresponding integrl sum S D (f g) equls the integrl sum S D (f g) corresponding to prtition D = (d, ξ ), where d is division of [, b] such tht d d. Hence S D (f g) I < ε. This mens tht the Kurzweil integrl f dg exists nd f dg = σ f dg = I is true. To prove the existence of the Perron-Stieltjes integrl f dg for ny f BV[, b] nd ny g G[, b] in Theorem 2.3.8 the following ssertion is helpful. Proposition 2.2.2. Let f BV[, b] be continuous on [, b] nd let g G[, b], then both the σ-riemnn-stieltjes integrls σ exist. f dg nd σ g df Proof. Let f BV[, b] which is continuous on [, b] nd g G[, b] be given. According to the integrtion-by-prts formul [14, II.11.7] for σ- Riemnn-Stieltjes integrls, to prove the lemm it is sufficient to show tht the integrl σ g df exists. First, let us ssume tht n rbitrry τ [, b] is given nd g = χ [,τ]. Let us put { {, b} if τ = or τ = b, d = {, τ, b} if τ (, b). It is esy to see tht then for ny prtition D = (d, ξ) such tht d d = {t, t 1,..., t m } we hve τ = t k for some k {, 1,..., m} nd S D (g f) = { f(τ) f() if ξk+1 > τ, f(t k+1 ) f() if ξ k+1 = τ. Since f is ssumed to be continuous, it is esy to show tht for given ε > there exists division d of [, b] such tht d d nd S D (g f) [f(τ) f()] < ε
15 is true for ny prtition D = (d, ξ) of [, b] with d d, i.e. σ χ [,τ] df = f(τ) f() for ll τ [, b]. By similr rgument we could lso show the following reltions: σ χ [,τ) df = f(τ) f() for ll τ (, b], nd σ σ χ [τ,b] df = f(b) f(τ) for ll τ [, b], χ (τ,b] df = f(b) f(τ) for ll τ [, b). Since ny finite step function is liner combintion of functions χ [τ,b] ( τ b) nd χ (τ,b] ( τ < b), it follows tht the integrl σ g df exists for ny f BV[, b] continuous on [, b] nd ny g S[, b]. Now, if g G[, b] is rbitrry, then there exists sequence {g n } k=1 S[, b] such tht lim g n g =. n Since by the preceding prt of the proof of the lemm ll the integrls σ g n df hve finite vlue, by mens of the convergence theorem [14, Theorem II.15.1] vlid for σ-riemnn-stieltjes integrls we obtin tht the integrl σ g df exists nd the reltion lim σ g n df = σ n holds. This completes the proof. g df R A direct corollry of Proposition 2.2.2 nd of [14, Theorem II.13.17] is the following ssertion which will be helpful for the proof of the integrtionby-prts formul, Theorem 2.3.15. (Of course, we could prove it s well by n rgument similr to tht used in the proof of Proposition 2.2.2.) Corollry 2.2.3. Let f BV[, b] nd g G[, b]. Let + f(t) + g(t) = f(t) g(t) = for ll t (, b).
16 Then both the σ-riemnn-stieltjes integrls exist. σ f dg nd σ g df 2.3. Perron-Stieltjes integrl with respect to regulted functions In this section we del with the integrls f(t) d[g(t)] nd g(t) d[f(t)], where f BV[, b] nd g G[, b]. We prove some bsic theorems (integrtion-by-prts formul, convergence theorems, substitution theorem nd unsymmetric Fubini theorem) needed in the theory of Stieltjes integrl equtions in the spce G[, b]. However, our first tsk is the proof of existence of the integrl f dg for ny f BV[, b] nd ny g G[, b]. First, we will consider some simple specil cses. nd Proposition 2.3.1. Let g G[, b] nd τ [, b]. Then χ [,τ] dg = g(τ+) g(), (2.3.1) χ [,τ) dg = g(τ ) g(), (2.3.2) χ [τ,b] dg = g(b) g(τ ), (2.3.3) χ (τ,b] dg = g(b) g(τ+) (2.3.4) χ [τ] dg = g(τ+) g(τ ), (2.3.5) where χ [) (t) χ (b] (t) nd the convention g( ) = g(), g(b+) = g(b) is used. Proof. Let g G[, b] nd τ [, b] be given. ) Let f = χ [,τ]. It follows immeditely from the definition tht τ f dg = g(τ) g().
17 In prticulr, (2.3.1) holds in the cse τ = b. Assume τ [, b). Let ε > be given nd let δ ε (t) = { 1 2 τ t if τ < t b, ε if t = τ. It is esy to see tht ny δ ε fine prtition D = (d, ξ) of [τ, b] must stisfy ξ 1 = t = τ, t 1 < τ + ε nd S D (f g) = g(t 1 ) g(τ). Consequently, nd τ f dg = g(τ+) g(τ) f dg = τ f dg + τ f dg = g(τ) g() + g(τ+) g(τ) = g(τ+) g(), i.e. the reltion (2.3.1) is true for every τ [, b]. b) Let f = χ [,τ). If τ =, then f, g(τ ) g() = nd (2.3.2) is trivil. Let τ (, b]. For given ε >, let us define guge δ ε on [, τ] by δ ε (t) = { 1 2 τ t if t < τ, ε if t = τ. Then for ny δ ε fine prtition D = (d, ξ) of [, τ] we hve t m = ξ m = τ, t m 1 < τ ε nd S D (f g) = g(t m 1 ) g(). It follows immeditely tht τ nd in view of the obvious identity f dg = g(τ ) g() τ f dg =, this implies (2.3.2). c) The remining reltions follow from (2.3.1), (2.3.2) nd from the equlities χ [τ,b] = χ [,b] χ [,τ), χ (τ,b] = χ [,b] χ [,τ] nd χ [τ] = χ [,τ] χ [,τ).
18 Remrk 2.3.2. Since ny finite step function is liner combintion of functions χ [τ,b] ( τ b) nd χ (τ,b] ( τ < b), it follows immeditely from Proposition 2.3.1 tht the integrl nd ny g G[, b]. Other simple cses re covered by f dg exists for ny f S[, b] Proposition 2.3.3. Let τ [, b]. Then for ny function f : [, b] R the following reltions re true nd f dχ [,τ] = f dχ [,τ) = f dχ [τ,b] = f dχ (τ,b] = { f(τ) if τ < b, if τ = b, { f(τ) if τ >, if τ =, { f(τ) if τ >, if τ =, { f(τ) if τ < b, if τ = b f() if τ =, f dχ [τ] = if < τ < b, f(b) if τ = b, (2.3.6) (2.3.7) (2.3.8) (2.3.9) (2.3.1) where χ [) (t) χ (b] (t) nd the convention g( ) = g(), g(b+) = g(b) is used. For the proof see [45, I.4.21 nd I.4.22]. Corollry 2.3.4. Let W = {w 1, w 2,..., w n } [, b], c R nd h : [, b] R be such tht Then h(t) = c for ll t [, b] \ W. (2.3.11) f dh = f(b) [h(b) c] f() [h() c] (2.3.12) is true for ny function f : [, b] R. Proof. A function h : [, b] R fulfils (2.3.11) if nd only if n h(t) = c + [h(w j ) c] χ [wj](t) on [, b]. k=1
19 Thus the formul (2.3.12) follows from (2.3.6) (with τ = b) nd from (2.3.1). Remrk 2.3.5. It is well known (cf. [45, I.4.17] or [4, Theorem 1.22]) tht if g BV[, b], h : [, b] R nd h n : [, b] R, n N, re such tht h n dg exist for ny n N nd lim n h n h =, then h dg exists nd lim n h n dg = h dg (2.3.13) is true. To prove n nlogous ssertion for the cse g G[, b] we need the following uxiliry ssertion. Lemm 2.3.6. Let f BV[, b] nd g G[, b]. Then the inequlity S D (f g) ( f() + f(b) + vr b f) g (2.3.14) is true for n rbitrry prtition D of [, b]. Proof. For n rbitrry prtition D = (d, ξ) of [, b] we hve (putting ξ = nd ξ m+1 = b) m+1 S D (f g) = f(b) g(b) f() g() [f(ξ j ) f(ξ j 1 )] g(t j 1 ) j=1 ( m+1 f(b) + f() + f(ξ j ) f(ξ j 1 ) ) g j=1 ( f() + f(b) + vr b f) g. Theorem 2.3.7. Let g G[, b] nd let h n, h : [, b] R be such tht h n dg exists for ny n N nd lim n h n h BV =. Then h dg exists nd (2.3.13) is true. Proof. Since we hve by (2.3.14) f(b) f() + f(b) f() f() + vr b f, S D ((h m h k ) g) 2 h m h k BV g
2 for ll m, k N nd ll prtitions D of [, b]. Consequently, (h m h k ) dg 2 h m h k BV g holds for ll m, k N. This immeditely implies tht there is n I R such tht It remins to show tht lim n h n dg = I. I = h dg. (2.3.15) For given ε >, let n N be such tht h n dg I < ε nd h n h BV < ε, (2.3.16) nd let δ ε be such guge on [, b] tht S D(h n g) h n dg < ε (2.3.17) for ll δ ε fine prtitions D of [, b]. Given n rbitrry δ ε fine prtition D of [, b], we hve by (2.3.16), (2.3.17) nd Lemm 2.3.6 I S D (h g) I b h n dg + h n dg S D (h n g) + SD (h n g) S D (h g) 2 ε + SD ([h n h] g) 2 ε + 2 h n h BV g 2 ε (1 + g ) wherefrom the reltion (2.3.15) immeditely follows. proof of the theorem. This completes the Now we cn prove Theorem 2.3.8. Let f BV[, b] nd g G[, b]. Then the integrl is true. f dg exists nd the inequlity f dg ( f() + f(b) + vr b f) g (2.3.18)
Proof. Let f BV[, b] nd g G[, b] be given. Let W = {w k } k N be the set of discontinuities of f in [, b] nd let f = f C + f B be the Jordn decomposition of f (i.e., f C is continuous on [, b] nd f B is given by (1.2.2)). We hve lim f B n n f B BV = for f B n, n N, given by (1.2.3). By (2.3.3) nd (2.3.4), = f B n dg = n [ + f(w k ) (g(b) g(w k +)) + f(w k ) (g(b) g(w k ))] (2.3.19) k=1 21 holds for ny n N. Thus ccording to Theorem 2.3.7 the integrl exists nd f B dg The integrl f B dg = lim n fn B dg. (2.3.2) f C dg exists s the σ-riemnn-stieltjes integrl by Proposition 2.2.2. This mens tht f dg exists nd f dg = f C dg + f B dg = f C dg + lim n The inequlity (2.3.18) follows immeditely from Lemm 2.3.6. Remrk 2.3.9. Since f B n dg. [ + f(w k ) (g(b) g(w k +)) + f(w k ) (g(b) g(w k ))] k=1 2 g ( + f(w k ) + f(w k ) ) 2 g (vr b f) <, k=1 we hve in virtue of (2.3) nd (2.3.2) = f B dg = [ + f(w k ) (g(b) g(w k +)) + f(w k ) (g(b) g(w k ))]. (2.3.21) k=1 As direct consequence of Theorem 2.3.8 we obtin
22 Corollry 2.3.1. Let h n G[, b], n N, nd let h G[, b] be such tht lim h n h =. n Then for ny f BV[, b] the integrls exist nd f dh lim n nd f dh n = f dh n, n N f dh. Lemm 2.3.11. Let h : [, b] R, c R nd W = {w k } k N [, b] be such tht (2.3.11) nd h(w k ) c < (2.3.22) k=1 hold. Furthermore, for n N, let us put W n = {w 1, w 2,..., w n } nd h n (t) = { c if t [, b] \ Wn, h(t) if t W n. (2.3.23) Then h n BV[, b] for ny n N, h BV[, b] nd lim h n h BV =. (2.3.24) n Proof. The functions h n, n N, nd h evidently hve bounded vrition on [, b]. For given n N, we hve { if t W n or t [, b] \ W h n (t) h(t) = c h n (t) if t = w k for some k > n. Thus, nd, moreover, lim h n(t) = h(t) on [, b] (2.3.25) n m (hn (t j ) h(t j )) (h n (t j 1 ) h(t j 1 )) 2 j=1 k=n+1 h(w k ) c holds for ny n N nd ny division {t, t 1,..., t m } of [, b]. Consequently, vr b (h n h) 2 k=n+1 h(w k ) c (2.3.26)
23 is true for ny n N. In virtue of the ssumption (2.3.22) the right-hnd side of (2.3.26) tends to s n. Hence (2.3.24) follows from (2.3.25) nd (2.3.26). Proposition 2.3.12. Let h : [, b] R, c R nd W = {w k } k N be such tht (2.3.11) nd (2.3.22) hold. Then h dg = [h(w k ) c] g(w k ) + c [g(b) g()] k=1 is true for ny g G[, b]. Proof. Let g G[, b] be given. Let W n = {w 1, w 2,..., w n } for n N nd let the functions h n, n N, be given by (2.3.23). Given n rbitrry n N, then (2.3.1) (with τ = b) nd (2.3.5) from Proposition 2.3.1 imply h n dg = Since (2.3.22) yields k=1 n [h(w k ) c] g(w k ) + c [g(b) g()]. k=1 n [h(w k ) c] g(w k ) 2 g h(w k ) c < nd Lemm 2.3.11 implies k=1 lim h n h BV =, n we cn use Theorem 2.3.7 to prove tht h dg = lim n h n dg = [h(w k ) c] g(w k ) + c [g(b) g()]. k=1 Proposition 2.3.13. Let h : [, b] R, c R nd W = {w k } k N fulfil (2.3.11). Then is true for ny f BV[, b]. f dh = f(b) [h(b) c] f() [h() c] (2.3.27) Proof. Let f BV[, b]. For given n N, let W n = {w 1, w 2,..., w n } nd let h n be given by (2.3.23). Then lim h n h =. (2.3.28) n
24 Indeed, let ε > be given nd let n N be such tht k n implies h(w k ) c < ε. (2.3.29) (Such n n exists since h(w k ) c = h(w k ) = + h(w k ) for ny k N nd the set of those k N for which the inequlity (2.3.29) does not hold my be only finite.) Now, for ny n n nd ny t [, b] such tht t = w k for some k > n (t W \ W n ) we hve h n (t) h(t) = h n (w k ) h(w k ) = c h(w k ) < ε. Since h n (t) = h(t) for ll the other t [, b] (t ( [, b]\w ) W n ), it follows tht h n (t) h(t) < ε on [, b], i.e. h n h < ε. This proves the reltion (2.3.28). By Corollry 2.3.4 we hve for ny n N f dh n = f(b) [h(b) c] f() [h() c]. Mking use of (2.3.28) nd Corollry 2.3.1 we obtin f dh = lim n f dh n = f(b) [h(b) c] f() [h() c]. Corollry 2.3.14. Let h BV[, b], c R nd W = {w k } k N (2.3.11). Then (2.3.27) is true for ny f G[, b]. fulfil Proof. By Proposition 2.3.12, (2.3.27) is true for ny f BV[, b]. Mking use of the density of S[, b] BV[, b] in G[, b] nd of the convergence theorem mentioned in Remrk 2.3.5 we complete the proof of this ssertion. Theorem 2.3.15. (Integrtion-by-prts) Let f BV[, b] nd g G[, b]. Then both the integrls f dg + f dg nd g df = f(b) g(b) f() g() g df exist nd + [ f(t) g(t) + f(t) + g(t)]. (2.3.3) t [,b]
Proof. The existence of the integrl g df is well known nd the existence b of g df is gurnteed by Theorem 2.3.8. Furthermore, f dg + g df = f(t) d[g(t) + + g(t)] + f(t) d[ + g(t)] + 25 g(t) d[f(t) f(t)] g(t) d[ f(t)]. It is esy to verify tht the function h(t) = + g(t) fulfils the reltion (2.3.11) with c = nd h(b) =. Consequently, Proposition 2.3.13 yields Similrly, by Corollry 2.3.14 we hve Hence f dg + g(t) df = f(t) d[ + g(t)] = f() + g(). g(t) d[ f(t)] = f(b) g(b). f(t) d[g(t+)] + g(t) d[f(t )] + f() + g() + f(b) g(b). (2.3.31) The first integrl on the right-hnd side cn be modified in the following wy: f(t) d[g(t+)] = f(t ) d[g(t+)] + f(t) d[g(t+)]. (2.3.32) Mking use of Proposition 2.3.12 nd tking into ccount tht g 1 (t) = g(t) on [, b] for the function g 1 defined by g 1 (t) = g(t+) on [, b], we further obtin Similrly, g(t) d[f(t )] = f(t) d[g(t+)] = = t [,b] g(t+) df(t ) f(t) g(t). (2.3.33) + g(t) d[f(t )] g(t+) d[f(t )] + g(t) f(t). (2.3.34) t [,b]
26 The function f(t ) is left-continuous on [, b], while g(t+) is right-continuous on [, b). It mens tht both the integrls f(t ) d[g(t+)] nd g(t+) d[f(t )] exist s σ-riemnn-stieltjes integrls (cf. Corollry 2.2.3), nd mking use of the integrtion-by-prts theorem for these integrls (cf. [14, Theorem II.11.7]) we get f(t )d[g(t+)] + Inserting (2.3.32)-(2.3.35) into (2.3.31) we obtin f dg + + g(t+)d[f(t )] = f(b )g(b) f()g(+). (2.3.35) g df = f(b ) g(b) f() g(+) t [,b] f(t) [ g(t) + + g(t)] [ f(t) + + f(t)] + g(t) + f() + g() + f(b) g(b) t [,b] = f(b) g(b) f() g() + [ f(t) g(t) + f(t) + g(t)] nd this completes the proof. t [,b] The following proposition describes some properties of indefinite Perron- Stieltjes integrls. Proposition 2.3.16. Let f : [, b] R nd g : [, b] R be such tht f dg exists. Then the function h(t) = is defined on [, b] nd (i) if g G[, b], then h G[, b] nd f dg + h(t) = f(t) + g(t), h(t) = f(t) g(t) on [, b]; (2.3.36) (ii) if g BV[, b] nd f is bounded on [, b], then h BV[, b]. Proof. The former ssertion follows from [19, Theorem 1.3.5]. The ltter follows immeditely from the inequlity m j f dg m [ t f (vr j t j 1 g) ] b = f (vr b g) t j 1 j=1 j=1 which is vlid for ny division {t, t 1,..., t m } of [, b].
27 In the theory of generlized differentil equtions the substitution formul [ ] h(t) d f(s) d[g(s)] = h(t)f(t) d[g(t)] (2.3.37) is often needed. In [14, II.19.3.7] this formul is proved for the σ-young- Stieltjes integrl under the ssumption tht g G[, b], h is bounded on [, b] nd the integrl f dg s well s one of the integrls in (2.3.37) exist. In [45, Theorem I.4.25] this ssertion ws proved for the Kurzweil integrl. Though it ws ssumed there tht g BV[, b], this ssumption ws not used in the proof. We will give here slightly different proof bsed on the Sks-Henstock lemm (cf.e.g.[4, Lemm 1.11]). Lemm 2.3.17. (Sks-Henstock) Let f, g : [, b] R be such tht the integrl f dg exists. Let ε > be given nd let δ be guge on [, b] such tht S D(f g) f dg < ε is true for ny δ fine prtition D of [, b]. Then for n rbitrry system {([β i, γ i ], σ i ), i = 1, 2,..., k} of intervls nd points such tht nd the inequlity is true. β 1 σ 1 γ 1 β 2 β k σ k γ k b (2.3.38) [β i, γ i ] [σ i δ(σ i ), σ i + δ(σ i )], i = 1, 2,..., k, k i=1 [ γi ] f(σ i ) [g(γ i ) g(β i )] f dg < ε (2.3.39) β i Mking use of Lemm 2.3.17 we cn prove the following useful ssertion. Lemm 2.3.18. If f : [, b] R nd g : [, b] R re such tht f dg exists, then for ny ε > there exists guge δ on [, b] such tht m f(ξ j) [g(t j ) g(t j 1 )] j=1 is true for ny δ fine prtition (d, ξ) of [, b]. f dg < ε (2.3.4) t j 1 Proof. Let δ : [, b] (, ) be such tht S D(f g) f dg = m j f(ξ j ) [g(t j ) g(t j 1 )] f dg < ε t j 1 2 j=1 j
28 for ll δ fine prtitions D = (d, ξ) of [, b]. Let us choose n rbitrry δ fine prtition D = (d, ξ) of [, b]. Let γ i = t pi nd β i = t pi 1, i = 1, 2,..., k, be ll the points of the division d such tht f(ξ pi ) [g(γ i ) g(β i )] γi β i f dg. Then the system {([β i, γ i ], σ i ), i = 1, 2,..., k}, where σ i = ξ pi, fulfils (2.3.38) nd (2.3.39) nd hence we cn use Lemm 2.3.17 to prove tht the inequlity k f(ξ p i ) [g(γ i ) g(β i )] i=1 γi β i f dg < ε 2 is true. Similrly, if ω i = t qi nd θ i = t qi 1, i = 1, 2,..., r re ll points of the division d such tht f(ξ qi ) [g(ω i ) g(θ i )] then the inequlity r f(ξ q i ) [g(ω i ) g(θ i )] i=1 ωi θ i f dg, ωi θ i f dg < ε 2 follows from Lemm 2.3.17, s well. Summrizing, we conclude tht m j f(ξ j) [g(t j ) g(t j 1 )] f dg t j 1 j=1 = + k f(ξ p i ) [g(γ i ) g(β i )] i=1 r f(ξ q i ) [g(ω i ) g(θ i )] i=1 This completes the proof. < ε 2 + ε 2 = ε. γi β i ωi θ i f dg f dg Theorem 2.3.19. (Substitution) Let h : [, b] R be bounded on [, b] nd let f, g : [, b] R be such tht the integrl integrl h(t) f(t) d[g(t)] f dg exists. Then the
29 exists if nd only if the integrl [ h(t) d ] f(s) d[g(s)] exists, nd in this cse the reltion (2.3.37) is true. Proof. Let h(t) C < on [, b]. Let us ssume tht the integrl h(t)f(t) d[g(t)] exists nd let ε > be given. There exists guge δ 1 on [, b] such tht m h(ξ k )f(ξ j ) [g(t j ) g(t j 1 )] h(t)f(t) d[g(t)] < ε 2 j=1 is stisfied for ny δ 1 fine prtition (d, ξ) of [, b]. By Lemm 2.3.18 there exists guge δ on [, b] such tht δ(t) δ 1 (t) on [, b] nd m j f(ξ j) [g(t j ) g(t j 1 )] f dg < ε t j 1 2C j=1 is true for ny δ fine prtition (d, ξ) of [, b]. Let us denote k(t) = f dg for t [, b]. Then for ny δ fine prtition D = (d, ξ) of [, b] we hve S D(h k) h(t)f(t) d[g(t)] m j m = h(ξ j ) f dg h(ξ j )f(ξ j ) [g(t j ) g(t j 1 )] t j 1 j=1 + j=1 m h(ξ j )f(ξ j ) [g(t j ) g(t j 1 )] j=1 h(t)f(t) d[g(t)] m [ j ] h(ξ j ) f dg f(ξ j ) [g(t j ) g(t j 1 )] j=1 t j 1 m + h(ξ j ) f(ξ j ) [g(t j ) g(t j 1 )] h f dg < ε. j=1 This implies the existence of the integrl h dk nd the reltion (2.3.37). The second impliction cn be proved in n nlogous wy.
3 The convergence result 2.3.1 enbles us to extend the known theorems on the chnge of the integrtion order in iterted integrls d c [ ] g(t) d h(t, s) d[f(s)], ( d c ) g(t) d t [h(t, s)] d[f(s)], (2.3.41) where < c < d < nd h is of strongly bounded vrition on [c, d] [, b] (cf. 1.2.5). Theorem 2.3.2. (Unsymmetric Fubini Theorem) Let h : [c, d] [, b] R be such tht v (h) + vr b h(c,.) + vrd ch(., ) <. Then for ny f BV[, b] nd ny g G(c, d) both the integrls (2.3.41) exist nd d [ ] ( d ) g(t) d h(t, s) d[f(s)] = g(t) d t [h(t, s)] d[f(s)]. (2.3.42) c Proof. Let us notice tht by [45, Theorem I.6.2] our ssertion is true if g is lso supposed to be of bounded vrition. In the generl cse of g G[, b] there exists sequence {g n } n=1 S[, b] such tht lim n g g n =. Then, since the function v(t) = c h(t, s) d[f(s)] is of bounded vrition on [c, d] (cf. the first prt of the proof of [45, Theorem I.6.2]), the integrl on the left-hnd side of (2.3.42) exists nd by Corollry 2.3.1 nd [45, Theorem I.6.2] we hve d c Let us denote [ ] g(t) d h(t, s) d[f(s)] = lim n ( d = lim n w n (t) = d c c d c [ g n (t) d ] h(t, s) d[f(s)] ) g n (t) d t [h(t, s)] d[f(s)]. (2.3.43) g n (t) d t [h(t, s)] for s [, b] nd n N. Then w n BV[, b] for ny n N (cf. [45, Theorem I.6.18]) nd by [45, Theorem I.4.17] mentioned here in Remrk 2.3.5 we obtin lim w n(s) = n d c g n (t) d t [h(t, s)] := w(s) on [, b].
31 As w n (s) w(s) g n g ( vr d ch(., s) ) g n g ( v (h) + vr d ch(., ) ) for ny s [, b] (cf. [45, Lemm I.6.6]), we hve lim w n w =. n It mens tht w G[, b] nd by Theorem 2.3.8 the integrl w(s) d[f(s)] = exists s well. Since obviously lim n ( d c ( d ) g n (t) d t [h(t, s)] d[f(s)] = lim n = c w(s) d[f(s)] = the reltion (2.3.42) follows from (2.3). ) g(t) d t [h(t, s)] d[f(s)] ( d c w n (s) d[f(s)] ) g(t) d t [h(t, s)] d[f(s)], 2.4. Liner bounded functionls on the spce of left-continuous regulted functions By Theorem 2.3.8 the expression Φ η (x) = q x() + p dx (2.4.1) is defined for ny x G[, b] nd ny η = (p, q) BV[, b] R. Moreover, for ny η BV[, b] R, the reltion (2.4.1) defines liner bounded functionl on G L [, b]. Proposition 2.3.3 immeditely implies Lemm 2.4.1. Let η = (p, q) BV[, b] R. Then Φ η (χ [,b] ) = q, (2.4.2) Φ η (χ (τ,b] ) = p(τ) for ll τ [, b), Φ η (χ [b] ) = p(b). Corollry 2.4.2. Let η = (p, q) BV[, b] R nd Φ η (x) = for ll x S[, b] which re left-continuous on (, b). Then p(t) on [, b] nd q =.
32 Lemm 2.4.3. Let x G[, b] nd η = (p, q) BV[, b] R. Then Φ η (x) = x() if p on [, b] nd q = 1, (2.4.3) Φ η (x) = x(b) if p 1 on [, b] nd q = 1, Φ η (x) = x(τ ) if p = χ [,τ) on [, b], τ (, b] nd q = 1, Φ η (x) = x(τ+) if p = χ [,τ] on [, b], τ [, b) nd q = 1. Proof follows from Proposition 2.3.1. Corollry 2.4.4. Let x G[, b] nd Φ η (x) = for ll η = (p, q) BV[, b] R. Then x() = x(+) = x(τ ) = x(τ+) = x(b ) = x(b) (2.4.4) holds for ny τ (, b). In prticulr, if x G L [, b] (x is left-continuous on (, b)) nd Φ η (x) = for ll η = (p, q) BV[, b] R, then x(t) on [, b]. Remrk 2.4.5. The spce BV[, b] R is supposed to be equipped with the usul norm ( η BV R = q + p BV for η = (p, q) BV[, b] R). Obviously, BV[, b] R is Bnch spce with respect to this norm. Proposition 2.4.6. The spces G L [, b] nd BV[, b] R form dul pir with respect to the biliner form i.e. nd x G L [, b], η BV[, b] R Φ η (x), (2.4.5) Φ η (x) = for ll x G L [, b] = η = BV[, b] R Φ η (x) = for ll η BV[, b] R = x = G L [, b]. Proof follows from Corollries 2.4.2 nd 2.4.4. On the other hnd, we hve Lemm 2.4.7. Let Φ be liner bounded functionl on G L [, b] nd let p(t) = { Φ(χ(t,b] ) if t [, b), Φ(χ [b] ) if t = b. (2.4.6) Then p BV[, b] nd where p() + p(b) + vr b p 2 Φ, (2.4.7) Φ = sup Φ(x). x G L[,b], x 1
Proof is nlogous to tht of prt c (i) of [15, Theorem 5.1]. Indeed, for n rbitrry division {t, t 1,..., t m } of [, b] we hve sup p() c + p(b) c m+1 + c j 1,c j R m [p(t j ) p(t j 1 )] c j j=1 = sup m 1 Φ(c χ (,b] + c m+1 χ [b] c j χ (tj 1,t j] + c m χ (tm 1,b)) c j 1,c j R sup Φ(h) = 2 Φ. h 2,h G L[,b] In prticulr, for c = sgn p(), c m+1 = sgn p(b) nd c j = sgn(p(t j ) p(t j 1 )), j = 1, 2,..., m, we get p() + p(b) + j=1 m p(t j ) p(t j 1 ) 2 Φ, j=1 nd the inequlity (2.4.7) immeditely follows. Using the ides from the proof of [15, Theorem 5.1] we my now prove the following representtion theorem. Theorem 2.4.8. Φ is liner bounded functionl on G L [, b] (Φ G L(, b)) if nd only if there is n η = (p, q) BV[, b] R such tht Φ(x) = Φ η (x) ( := q x() + The mpping is n isomorphism. 33 p dx ) for ny x G L [, b]. (2.4.8) Ξ : η BV[, b] R Φ η G (, b) L Proof. Let liner bounded functionl Φ on G L [, b] be given nd let us put q = Φ(χ [,b] ) nd p(t) = { Φ(χ(t,b] ) if t [, b), Φ(χ [b] ) if t = b. (2.4.9) Then Lemm 2.3.6 implies η = (p, q) BV[, b] R nd by Lemm 2.4.1 we hve Φ(χ [,b] ) = Φ η (χ [,b] ), Φ(χ [b] ) = Φ η (χ [b] ) Φ(χ (t,b] ) = Φ η (χ (t,b] ) for ll t [, b). Since ll functions from S[, b] G L [, b] obviously re finite liner combintions of the functions χ [,b], χ (τ,b], τ [, b), χ [b], nd
34 it follows tht Φ(x) = Φ η (x) is true for ny x S[, b] G L [, b]. Now, the density of S[, b] G L [, b] in G L [, b] implies tht Φ(x) = Φ η (x) for ll x G L [, b]. This completes the proof of the first ssertion of the theorem. Lemm 2.3.6 yields tht Φ η (x) ( p() + p(b) + vr b p + q ) x is true for ny x G L [, b] nd, consequently, Φ η p() + p(b) + vr b p + q 2 ( p BV + q ) = 2 η BV R. On the other hnd, ccording to Lemm 2.4.7 we hve p BV ( p() + p(b) + vr b p) 2 Φ. Furthermore, in virtue of (2.4.9) we hve q Φ nd hence It mens tht η BV R = p BV + q 2 Φ. 1 2 Φ η BV R 3 Φ nd this completes the proof of the theorem. 2.5. Liner bounded functionls on the spce of regulr regulted functions Recll tht the subspce of G[, b] consisting of ll functions regulted on [, b] nd such tht f(t) = 1 [f(t ) + f(t+)] for ll t (, b) 2 is denoted by G reg [, b] nd the functions belonging to G reg [, b] re usully sid to be regulr on (, b). In this section we shll show tht liner bounded functionls on G reg [, b] my be represented in the form (2.4.8), s well. To this im the following lemms will be helpful. Lemm 2.5.1. A function f : [, b] R is finite step function on [, b] which is regulr on (, b) (f S[, b] G reg [, b]) if nd only if there
35 re rel numbers α 1, α 2,..., α m nd division d = {t, t 1,..., t m } of [, b] such tht N f(t) = α j h j (t) on [, b], where j= h = 1, h 1 = χ (,b], h j = 1 2 χ [t j] + χ (tj,b] for j = 2, 3,..., m 1 nd h m = χ [b]. Proof. Obviously function f : [, b] R belongs to S[, b] G reg [, b] if nd only if there re rel numbers c, c 1,..., c N+1 nd division d = {t, t 1,..., t N } of [, b] such tht i.e. f(t) = c if t =, c j if t (t j 1, t j ) for some j = 1, 2,..., m, c j+c j+1 2, if t = t j for some j = 1, 2,..., m 1, c m+1 if t = b. f(t) = c χ [] (t) + + 1 2 ( m 1 m c j χ (tj 1,t j)(t) j=1 ) (c j + c j+1 ) χ [tj](t) j=1 + c m+1 χ [b] (t) for t [, b]. (2.5.1) It is esy to verify tht the right-hnd side of (2.5.1) my be rerrnged s f = c χ [,b] c χ (,b] + 1 2 m 1 j=1 m j=1 c j χ [tj] c m χ [b] + 1 2 m 1 c j χ (tj 1,b] c j χ (tj,b] m 1 j=1 j=1 c j+1 χ [tj] + c m+1 χ [b] m 1 m 1 = c χ [,b] c χ (,b] + c j+1 χ (tj,b] c j χ (tj,b] 1 2 m 1 j=1 c j χ [tj] + 1 2 j= m 1 j=1 j=1 c j+1 χ [tj] + c m+1 χ [b] c m χ [b] m 1 = c χ [,b] + (c 1 c ) χ (,b] + (c j+1 c j ) ( χ (tj,b] + 1 2 χ ) [t j] + (c m+1 c m ) χ [b], j=1
36 wherefrom the ssertion of the lemm immeditely follows. Lemm 2.5.2. The set S[, b] G reg [, b] is dense in G reg [, b]. Proof. Let x G reg [, b] nd ε > be given. Since cl(s[, b]) = G[, b], there exists ξ S[, b] such tht x(t) ξ(t) < ε is true for ny t [, b]. Consequently, we hve x(t ) ξ(t ) < ε nd x(s+) ξ(s+) < ε (2.5.2) for t [, b), s (, b]. Let us put ξ (t) = 1 2 ξ() if t =, ( ) ξ(t+) + ξ(t ) if t (, b), ξ(b) if t = b. Obviously ξ (t ) = ξ(t ) nd ξ (s+) = ξ(s+) for ll t (, b] nd s [, b), respectively. In prticulr, ξ (t) = ξ(t) for ny point t of continuity of ξ. It follows tht ξ S[, b] G reg [, b]. Furthermore, in virtue of (2.5.2) we hve for ny t (, b) x(t) ξ (t) = 1 2 [x(t ) ξ(t )] + [x(t+) ξ(t+)] < ε, wherefrom the ssertion of the lemm immeditely follows. Lemm 2.5.3. Let Φ be n rbitrry liner bounded functionl on G reg [, b]. Let us define Φ(χ (,b] ), if t =, p(t) = Φ( 1 2 χ [t] + χ (t.b] ), if t (, b), (2.5.3) Φ(χ [b] ), if t = b. Then (i.e. p BV[, b]). vr b p Φ = sup Φ(x) x G reg[,b], x 1 Proof. Let d = {t, t 1,..., t m } be n rbitrry division of [, b] nd let α j R, j = 1, 2,..., m, be such tht α j 1 for ll j = 1, 2,..., m. Then m j=1 α j [p(t j ) p(t j 1 )] = α 1 [ Φ( 1 2 χ [t 1] + χ (t1,b]) Φ(χ (,b] ) ] (2.5.4)
37 m 1 + j=2 α j [ Φ( 1 2 χ [t j] + χ (tj,b]) Φ( 1 2 χ [t j 1] + χ (tj 1,b]) ] + α m [ Φ(χ[b] ) Φ( 1 2 χ [t m 1] + χ (tm 1,b]) ] = Φ(h), where [1 h = α 1 2 χ ] [ [t 1] + χ (t1,b] χ (,b] + αm χ[b] 1 2 χ ] [t m 1] χ (tm 1,b] m 1 [1 + α j 2 χ [t j] + χ (tj,b] 1 2 χ ] [t j 1] χ (tj 1,b] j=2 [1 = α 1 2 χ ] [1 [t 1] χ (,t1] αm 2 χ ] [t m 1] + χ (tm 1,b) m 1 [1 + α j 2 χ [t j] χ (tj 1,t j] 1 2 χ ] [t j 1] j=2 [1 = α 1 2 χ ] [1 [t 1] + χ (,t1) αm 2 χ ] [t m 1] + χ (tm 1,b) 1 m 1 α j χ [tj] 1 m 1 m 1 α j χ [tj 1] α j χ (tj 1,t 2 2 j) = 1 2 m 1 = j=1 m 1 j=1 j=2 α j χ [tj] 1 2 α j + α j+1 2 = α 1 χ (,t1) m 1 j=2 j=2 χ [tj] j=2 j=2 m m α j χ [tj 1] α j χ (tj 1,t j) j=1 m α j χ (tj 1,t j) j=1 (α j + α j+1 2 χ [tj] + α j χ (tj 1,t j)) αm χ (tm 1,b). It is esy to see tht h S[, b] G reg [, b] nd h(t) 1 for ll t [, b]. Consequently, by (2.5.4), we hve tht sup m α j 1,j=1,2,...,m j=1 α j [p(t j ) p(t j 1 )] sup Φ(x) x G reg[,b], x 1 is true for ny division d = {t, t 1,..., t m } of [, b]. In prticulr, choosing we get j=1 α j = sgn[p(t j ) p(t j 1 )] for j = 1, 2,..., m, m p(t j ) p(t j 1 ) sup Φ(x) < x G reg[,b], x 1 nd this yields vr b p Φ <.
38 Lemm 2.5.4. Let Φ be n rbitrry liner bounded functionl on G reg [, b] nd let η = (p, q) BV[, b] R be given by (2.5.3) nd q = Φ(χ [,b] ). Let us define Φ η (x) = q x() + Then Φ η is liner bounded functionl on G[, b], nd p dx for x G[, b]. (2.5.5) Φ η (x) = Φ(x) for ll x G reg [, b] (2.5.6) sup Φ η (x) q + 2 ( p() + vr b p). (2.5.7) x G[,b], x 1 Proof. By Theorem 2.3.8, Φ η (x) is defined nd Φ η (x) ( q + p() + p(b) + vr b p) x for ll x G[, b]. (2.5.8) It mens tht Φ η is liner bounded functionl on G[, b] nd the inequlity (2.5.7) is true. It is esy to verify tht the reltion (2.5.6) holds for ny function h from the set { χ [,b], χ (,b], 1 } 2 χ [τ] + χ (τ,b], χ [b] ; τ (, b). According to Lemms 2.5.1 nd 2.5.2 this implies tht (2.5.6) holds for ll x G reg [, b]. Lemm 2.5.5. Let η = (p, q) BV[, b] R. Then Φ η (x) = for ll x S[, b] G reg [, b] if nd only if q = nd p(t) on [, b]. Proof. Let η = (p, q) BV[, b] R nd let Φ η (x) = for ll x S[, b] G reg [, b]. Then Φ(χ [,b] ) = q =. Furthermore, by Proposition 2.3.3 we hve nd Φ η (χ (,b] ) = p() =, Φ η ( 1 2 χ [τ] + χ (τ,b] ) = p(τ) = for τ (, b) Φ η (χ [b] ) = p(b) =. By Lemm 2.5.1 this completes the proof. Remrk 2.5.6. Let us notice tht if x G reg [, b], then Φ η (x) = for ll η = (p, q) BV[, b] R if nd only if x(t) on [, b]. In fct, let x G[, b] nd let Φ η (x) = for ll η = (p, q) BV[, b] R. Then by Corollry 2.4.4 we hve x() = x(+) = x(t ) = x(t+) = x(b ) = x(b) = for ll t (, b). In prticulr, if x G reg [, b], then x(t) = for ny t [, b].
39 Theorem 2.5.7. A mpping Φ : G reg [, b] R is liner bounded functionl on G reg [, b] (Φ G reg[, b]) if nd only if there is n η = (p, q) BV[, b] R such tht Φ = Φ η, where Φ η is given by (2.5.5). The mpping Ξ : η BV[, b] R Φ η G [, b] genertes n isomorphism between reg BV[, b] R nd G reg[, b]. Proof. By Lemms 2.5.4 nd 2.5.5 nd by the inequlity (2.5.7) the mpping Ξ is bounded liner one-to-one mpping of BV[, b] R onto G [, b]. Consequently, by the Bounded Inverse Theorem, the mpping Ξ 1 is bounded, reg s well.
Chpter 3 Initil Vlue Problems for Liner Generlized Differentil Equtions 3.1. Introduction This chpter dels with the initil vlue problem for the liner homogeneous generlized differentil eqution x(t) x() d[a(s)] x(s) =, t [, 1], x() = x, (3.1.1) where A BV n n nd x R n re given nd solutions re functions x : [, 1] R n with bounded vrition on [, 1] (x BV n ). The bsic properties of the Perron-Stieltjes integrl with respect to sclr regulted functions were described in Chpter 2. The extension of these results to vector or mtrix vlued functions is obvious nd hence for the bsic fcts concerning integrls we shll refer to the corresponding ssertions from Chpter 2. Let P k L n n 1 for k N {} nd let X k AC n n be the corresponding fundmentl mtrices, i.e. X k (t) = I + P k (s) X k (s) ds on [, 1] for k N {}. The following two ssertions re representtive exmples of theorems on the continuous dependence of solutions of liner ordinry differentil equtions on prmeter. 4
41 Theorem 3.1.1. If then lim k P k (s) P (s) ds =, lim X k(t) = X (t) uniformly on [, 1]. k Theorem 3.1.2. (Kurzweil & Vorel, [22]) Let there exist m L 1 such tht nd let Then P k (t) m(t).e. on [, 1] for ll k N (3.1.2) lim k P k (s) ds = P (s) ds uniformly on [, 1]. lim X k(t) = X (t) uniformly on [, 1]. k Remrk 3.1.3. For t [, 1] nd k N {} denote A k (t) = P k (s) ds. Then the ssumptions of Theorem 3.1.2 cn be reformulted for A k s follows: A k AC n n for ll k N {}, sup A k L1 <, k N lim A k(t) = A (t) uniformly on [, 1]. k Besides, the ssumption (3.1.2) mens tht there exists nondecresing function h AC such tht A k (t 2 ) A k (t 1 ) h (t 2 ) h (t 1 ) for ll t 1, t 2 [, 1]. In fct, we cn put h (t) = m(s)ds on [, 1]. 3.2. A survey of known results The following bsic existence result for the initil vlue problem (3.1.1) my be found e.g. in [45] (cf. Theorem III.1.4) or in [41] (cf. Theorem 6.13).
42 Theorem 3.2.1. Let A BV n n be such tht det[i A(t)] for ll t (, 1]. (3.2.1) Then there exists unique X BV n n such tht X(t) = I + d[a(s)] X(s) on [, 1]. (3.2.2) Definition 3.2.2. For given A BV n n, the n n-mtrix vlued function X BV n n fulfilling (3.2.2) is clled the fundmentl mtrix corresponding to A. When restricted to the liner cse, Theorem 8.8 from [41], which describes the dependence of solutions of generlized differentil equtions on prmeter, reds s follows. Theorem 3.2.3. Let A BV n n stisfy (3.2.1) nd let X be the corresponding fundmentl mtrix. Let A k BV n n, k N, nd sclr nondecresing nd left-continuous on (, 1] functions h k, k N {}, be given such tht h is continuous on [, 1] nd lim A k(t) = A (t) on [, 1], (3.2.3) k A k (t 2 ) A k (t 1 ) h k (t 2 ) h k (t 1 ) for ll t 1, t 2 [, 1] nd k N {}, (3.2.4) lim sup[h k (t 2 ) h k (t 1 )] h (t 2 ) h (t 1 ) k whenever t 1 t 2 1. (3.2.5) Then for ny k N sufficiently lrge the fundmentl mtrix X k corresponding to A k exists nd lim X k(t) = X (t) uniformly on [, 1]. k Lemm 3.2.4. Under the ssumptions of Theorem 3.2.3 we hve nd sup vra k < (3.2.6) k N lim [A k(t) A k ()] = A (t) A () uniformly on [, 1]. (3.2.7) k Proof. 1 i) By (3.2.5) there is k N such tht h k (1) h k () h (1) h () + 1 for ll k k. 1 The uthor is indebted to Ivo Vrkoč for his suggestions which led to considerble simplifiction of this proof.