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BULL. AUSTRAL. MATH. SOC. VOL. 24 ( 1 9 8 1 ) , 46G99, 46A05 69-77. MEAN VALUE THEOREMS AND A TAYLOR THEOREM FOR VECTOR VALUED FUNCTIONS RUDOLF W B O R N ? Two mean value theorems and a Taylor theorem for functions with values in a locally convex topological vector space are proved without the use of the Hahn-Banach extension theorem. 1. Introduction It is well known that the classical mean value theorem does not hold for vector valued functions. For example, if fix) = e , x € [a, b] = [0, 2TT] then the equation (1) fib) - f(a) = f'(c)(b-a) cannot hold for any e € (0, 2TT) . [f(b) = f{a) , |/'(c)| = 1 for every c € (a, b) .) In the mathematical literature there appear two kinds of valid generalization of the mean value theorem for vector valued functions. (a) The equation (l) is replaced by an inequality involving norms or seminorms. For complex valued function this goes back to Darboux [3] who proved: if f : [a, b] -*• C has a continuous derivative then there exists a c i (a, b) such that (2) \f(b)-f{a)\ 5 \f'(c)\(b-a) . For linear normed spaces this generalization without the assumption of continuity of /' is due to Aziz and Diaz [7], the sign |*| in inequality (2) in this case must be interpreted as a norm. For further Received 17 February 1981. This research was partly supported by a grant from Deutscher Akademischer Austauschdienst. The author wishes to express his gratitude to W.F. Eberlein for making available his paper [S]. 69 Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 12 Jul 2017 at 22:24:09, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700007449 70 Rudolf Vyborny development see [2], [4], and for history of the mean value theorem 1101. (Also concerning generalizations (b) to follow.) (b) The equation (l) is replaced by the statement that hi lies in the closed convex hull of the range of the derivative. sequel, the convex hull of a bar. A shall be denoted by co A In the and a closure by Hence nb (3) l~/aia) € ^ { / ' U ) ; a < t <b) . This generalization for complex valued functions goes back to Weierstrass [72, p. 58], for linear normed spaces is due to Wazewski [16]. There is a nice example by Eberlein [S] showing that the bar cannot be omitted in (3) in case of an infinite dimensional space. See also [17]. McLeod [73] proved that the bar can be omitted if the space is finite dimensional. For locally convex spaces see [6] and [73]. There is another feature to these generalizations, namely, the occurrence of an exceptional set E c (a, b) where the function is either not required to be differentiate or the values of the derivative are not taken into account (McLeod [73] actually works with two exceptional sets). The classical mean value theorem is no longer true even if ton; E is a single- however, in both generalizations mentioned above some infinite sets are permitted. There is yet another feature common to both generalizations, namely, most of the proofs (an exception is, for example, [4]) employ the HahnBanach theorem. It is one of the aims of this paper to prove a Cauchy mean value theorem of Weierstrass' type for functions with values in a locally convex topological vector space. By avoiding the Hahn-Banach Theorem and hence also the axiom of choice (and by not using Solovey's axiom either) we obtain a theorem of constructive functional analysis in the sense of Garnir [7 7]. In a certain sense Weierstrass' mean value theorem is stronger than Darboux mean value theorem [S], [75]. There exist very many generalizations of the Taylor theorem to vector valued functions. Besides the integral form of the remainder (see, for example, [7], [74, p. 188]) which is most often used there also exist Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 12 Jul 2017 at 22:24:09, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700007449 Mean value theorems 71 estimates of the remainder similar to Darboux type generalizations of the Mean Value Theorem [5], [9, Theorem 33. In this paper we prove an estimate of the remainder in the spirit of Weierstrass generalization of the Mean Value Theorem. The advantage of our approach lies in the fact that we use only the mere existence of the (n+l)st derivative. 2. Mean Value Theorems LEMMA, If 6 : [a, b] •* i?+ (that is, 6 is real valued and positive), M c [a, b] , G = {(a., $.)} is a family (finite or infinite) of disjoint open intervals, [ex., 3.] c [a, b] and if la, b] c U (£-6U), C+6(5)) u U (a., 6.) then there exists a partition (P) of [a, b] , (P) a = xQ < xx < x2 < ... < x = b with the following property: for every j , j = 0, 1, ...,rc-1, either [x., x. ,) is one of the intervals \ (a., 6.) € G or there is a J and ,.x.6(Z.) . The latter intervals will be referred to as Proof. If G ^-intervals. is empty an indirect proof based on bisecting and on the nested interval theorem is easy. [a, b] The general case can be reduced to the case of finitely many intervals for which G is empty. This is done by applying Borel's covering theorem to the system of open intervals consisting of intervals of G and of intervals , C €M. THEOREM 1. If (i) X is a real locally convex topological vector space, (ii) f : [a, b] -*• X is continuous at a and b from the right and left respectively, Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 12 Jul 2017 at 22:24:09, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700007449 72 Rudo I f Vyborny (Hi) g : [a, b] -*• R and (iv) E is strictly increasing, continuous at b , from the right and from the left, respectively, is a subset of {a, b) and differentiable on on a f {a, b) - E ; and g are moreover g' is positive (a, b) - E , { f'(t) 1 J ,),(; t € (a, b)-Ef c C , g \ t) ) (vi) for every neighbourhood of zero U there exists a family of disjoint open intervals (a., 8.) such that (5) EC and for any finite (6) U (o i set of &) , i's , I [fiaj-ffcift (vii) €U , 0 €C , then f(b)-f(a) g(b)-g(a) Proof. Denote q - neighbourhood of zero. exists an element and iv.CQi—~ anii , 6 C ' 1st V be a convex, symmetric The theorem will be proved if we show that there c €C such that q-c € V . There exist positive 6 (a) <5(£>) such that (7a) /(*) - f(a) i2lb and fix) -f{b) (Tb) for every a 5 x 5 a+&(a) fi and b-5(b) 2 x 2 b , respectively. £ € (a, b)-E there exists a positive Further, for 6(5) such that g'U) * * ' for 0 < \x-E,\ 5 6(£) . Let (a., 8.) be the system of intervals from Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 12 Jul 2017 at 22:24:09, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700007449 Mean value theorems (vi). 73 Let (P) be the partition from Lemma for M = (a, b) - E . If [x •, x. ) is a ^-interval then it follows from (8) using strict 3 3 -*monotonicity of g and convexity of V that (9) _/_ I „i-. i ~ _ r j c i ^ u " • t We denote by ]T the sum which is extended over those (x., x. ) 0 J+L ^-interval and in J ~* is a the remaining indices. ( i f t h e r e i s no We define j for which the summation is understood over X. = 0 TT-T ?—\ — , c = ) X. —, tr \ g(b)-g(a) ^ j ff'UJ ^ - i n t e r v a l in (P) we define o = 0 ). Since 0 < A. < 1 , I* £ X. 5 1 we have, by using (vii.) , that by using Cui/' with c € C . Now we estimate y = -V , (7) and (9)- Hence we established q - c; q-o € V and the theorem is proved. I. The hypothesis REMARK 0 €C cannot be omitted without affecting the validity of Theorem 1. For a counterexample see [6, Remark 0, pp. 295-296]. However, if the hypothesis (v) of Theorem 1 is replaced by the following stronger hypothesis, (viS) E is a subset of {a, b) such that, for every and every neighbourhood of zero disjoint open intervals e > 0 U , there is a family of (a., 3.) , [a., $•] c [a, b] such that (5)j (6) hold and moreover do) then the hypothesis (vii) can be omitted. THEOREM More precisely, we have 2. If hypotheses (i)-(v) and (viS) hold then Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 12 Jul 2017 at 22:24:09, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700007449 74 RudoIf Vyborny Proof. We keep the notation from the proof of Theorem 1, (a., $.) If now satisfy (viS) rather than (vi). We choose a positive £ < ite(fc)-0(a)} , {g{bf^g(a)) « * V • particular there are a=y'x., have ^-intervals and d = i- e . Then Tt folios that o = £ X. e Is satisfying j ' A^. > ± , in , /.." \ . Define d € coK'|| ; ? € (a, b)-E> . Finally, we q - d = q - — q + — iq-c) d V + — V c 3V . This completes the proof. REMARK 2. (a) and f Some examples when hypothesis (viS) is satisfied are: g continuous throughout [a, b] and ff is at most countable; (b) / and g are absolutely continuous and E is of measure zero; (c) / of (d) f is absolutely continuous with respect to g and E E is of is ^-measure zero; is a-Holder continuous, g{x) = x , and a-Hausdorff measure zero. REMARK 3. One can obtain a mean value theorem for a mapping F from one locally convex topological vector space to another by using either Theorem 1 or Theorem 2 on the function f of a real variable defined by ftt) = F[a+t[b-a)) . 3. Taylor's Theorem Let a+x € Z j X and Y be locally convex topological spaces, a € X 3 f : X -* Y , n + 1 times Gateaux differentiable at all points of the segment [a, a+x] . g'(t) > 0 0 < t < 1 . Then for Let further g : [0, l] ->• R be continuous and Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 12 Jul 2017 at 22:24:09, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700007449 Mean v a l u e (11) theorems 75 f{a*x) - I i=0 -, 0 < t < 1 n! In particular, for n g(t) = -(l-t) (,.s l) t . , LrJn+l) (12) f(a+X) - II /(l>)(fl()(«)t) «{ ™{L REMARK f 4. For convenience of exposition we assumed the existence of (a+tx) for all t d [0, l] . One can, however, weaken this hypothesis and allow an exceptional set Proof. \ZT)\ ]> 0 < t < l} . E , similarly as in paragraph 2. Let us define Hit) = fia+tx) + ^f^ f'(a+tx)(x) + .. f^n+1\a+tx) An easy c a l c u l a t i o n shows H'(t) = ^^— the mean value theorem to H, g and the interval ) [x^1] . Applying [0, l] , we obtain (ll). The relation (12) is then immediate. References [1] A.K. Aziz and J.B. Diaz, "On a neam-value theorem of the differential calculus of vector-valued functions, and uniqueness theorems for ordinary differential equations in a linear-normed space", Contrib. Differential Equations 1 (l°63), 251-269. [2] A.K. Aziz, J.B. Diaz and W. Mlak, "On a mean value theorem for vector-valued functions, with applications to uniqueness theorems for 'right-hand-derivative' equations", J. Math. Anal. Appl. 16 (1966), 302-307. [3] G. Darboux, "Sur les developpments en serie des fonctions d'une seule variable", Liouville J. (3) 11 (1876), 291-312. Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 12 Jul 2017 at 22:24:09, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700007449 76 [4] RudoIf Vyborny J.B. Diaz and R. Vyborny, "On mean value theorems for strongly continuous vector valued functions", Contrib. Differential Equations 3 (196*1), 107-118. [5] Joaquin B. Diaz and R. Vyborny, "A fractional mean value theorem, and a Taylor theorem, for strongly continuous vector valued functions", Czechoslovak Math. J. 15 (90) (1965), 299-301*. [6] J.B. Diaz and R. Vyborny, "Generalized mean value theorems of the differential calculus", J. Austral. Math. Soa. Ser. A 20 (1975), 290-300. [7] J. Dieudonne, Foundations of modern analysis (Pure and Applied Mathematics, 10. [S] Academic Press, New York and London, i960). W.F. Eberlein, "The mean value theorem for vector valued functions", preprint. [9] T.M. Flett, "Mean value theorems for vector-valued functions", Tohoku Math. J. 24 (1972), lltl-151. [70] T.M. Flett, "Some historical notes and speculations concerning the mean value theorems of the differential calculus", Bull. Inst. Math. Appl. 10 (197*0, no. 3, 66-72. [77] H.G. Garnir, "Solovay's axiom and functional analysis", Functional analysis and its applications, 189-20*4 (Proc. Internat. Conf., Madras, 1973- Lecture Notes in Mathematics, 399. Springer- Verlag, Berlin, Heidelberg, New York, 197*0. [7 2] Ch. Hermite, Cours de la Faculte des Sciences de Paris sur les integrales definies, la thSorie des fonctions d'une variable imaginaire, et les fonctions elliptiques, *t ed., entierement refondue (Redige par M. Andoyer. [7 3] Hermann, Paris, 1891). Robert M. McLeod, "Mean value theorems for vector valued functions", Proc. Edinburgh Math. Soc. (2) 14 (196**-1965), 197-209. Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 12 Jul 2017 at 22:24:09, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700007449 Mean value theorems 77 [74] M.Z. Nashed, "Differentiability and related properties of nonlinear operators: some aspects of the role of differentials in non- linear functional analysis", Nonlinear functional analysis and applications, 103-309 (Proc. Advanced Seminar, Mathematics Research Center, University of Wisconsin, Madison, 1970. Publication No. 26 of the Mathematics Research Center, The University of Wisconsin. Academic Press, New York, London, 1971). [75] Jacek Szarski, Differential inequalities (Monografie Matematiyczne, 43. PWN - Polish Scientific Publishers, Warszawa, 1965). [76] T. Wazewski, "Une generalisation des theoremes sur les accroissements finis au cas des espaces de Banach et application a. la generalisation du theoreme de l'Hopital", Ann. Soc. Polon. Math. 24 (1951), no. 2, 132-ll»7 (195*0. [7 7] Sadayuki Yamamuro, Differential calculus in topological linear spaces Lecture Notes in Mathematics, 374. Springer-Verlag, Berlin, Heidelberg, New York, 197*1). Department of Mathematics, University of Queensland, St Lucia, Queensland 4067, Australia. Downloaded from https:/www.cambridge.org/core. IP address: 88.99.165.207, on 12 Jul 2017 at 22:24:09, subject to the Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0004972700007449