On superadditive rates of convergence

M ODÉLISATION MATHÉMATIQUE ET ANALYSE NUMÉRIQUE F LORIAN A. P OTRA On superadditive rates of convergence Modélisation mathématique et analyse numérique, tome 19, no 4 (1985), p. 671-685 <http://www.numdam.org/item?id=M2AN_1985__19_4_671_0> © AFCET, 1985, tous droits réservés. L’accès aux archives de la revue « Modélisation mathématique et analyse numérique » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ MATH€MAT1CALM0OEUJHGANONUMERlCALAHAlYStS MODÉUSATION MATHÉMATIQUE ET AJULYSE NUMERIQUE (vol. 19, n° 4, 1985, p. 671 à 685) ON SUPERADDITIVE RATES OF CONVERGENCE (*) by Florian A. POTRA (*) Communicated by Françoise CHATELIN Abstract. — In the present paper we prove that a superadditive function is a rate of convergence, in the sense ofV, Ptâk, if and only ifits itérâtes are pointwise convergent to zero. Moreover if the function is continuons from the right then this is equivalent to the inequality w(t) < t. These results are generalized for rates of convergence of several variables. Resumé. — Dans ce travail nous démontrons qu'une fonction superadditive est un taux de convergence, dans le sens de V. Ptâk, si et seulement si ses itérées convergent ponctuellement vers zéro. De plus, si la fonction est continue à droite, alors ce f ait est équivalent à l'inégalité w(t) < t. Ces résultats sont généralisés aux taux de convergence de plusieurs variables. 1. INTRODUCTION The method of nondiscrete induction was introduced in 1966 by V. Ptâk [16] in connection with some quantitative refînements of the closed graph theorem. Since then the method has successfully been applied to the study of various itérative constructions in analysis and numerical analysis. The results were published in a series of papers ([1, 2,4-28, 30]) and they were recently collected in a book [15]. Many of the theorems obtained by using this method are sharp in the sense that neither the convergence conditions nor the error estimâtes can be improved in the class of problems considered. This is explained by the fact that in the application of the method of nondiscrete induction the classical way of measuring convergence is replaced by a more refined one which makes it possible to obtain estimâtes sharp not only asymptotically but throughout the whole process. A rate of convergence is defined as a function, not as a number. (*) Received in December 1984. l1) Department of Mathematics, University of Iowa, Iowa City, Iowa 52242, U.S.A. M 2 AN Modélisation mathématique et Analyse numérique 0399-0516/85/04/671/15/S 3,50» Mathematical Modelling and Numerical Analysis © AFCET-Gauthier-Villars 672 F. A. POTRA DÉFINITION 1 . 1 : Let T dénote either the positive semiaxis ]0, oo[ or a half open interval oftheform ]0, b]. Afunction w : T -> T is called a rate of convergence on T if f »=o t GT win\t) < oo , (1) where win) dénotes the rcth iterate ofw in the sensé ofthe usual function compo- sition (Le., wm(t) = u w(n+1)(t) = w{w{n)(t% n = 0, 1, 2, ...)- D By defining the rate of convergence of an itérative procedure as a function we can retain more information at each step and obtain a sharp fùiai estimate. This departure from tradition in measuring convergence is justified by V. Ptak in a paper suggestively entitled « What should be a rate of convergence ? » [25]. In the same paper he stresses the importance of the rates of convergence which satisfy a functional inequality of the form 5o w ^ Wos (2) where s(t) dénotes the sum of the series (1). This inequality is not implied by Définition 1.1 and V. Ptak shows that it is satisfied at least by convex rates of convergence. It is easy to prove that (2) holds for superadditive rates of convergence continuous from the left (see Proposition 3.2). Every convex rate of convergence is superadditive but the converse is not true. For example the rate of convergence of Newton's method MO = . rC , (3) is superadditive on U+ but it is convex only on the interval [0, a^jT\ (see définitions in Sections 2.1 and 2.7). These facts demonstrate that the class of superadditive rates of convergence deserves special attention. In the present paper we in tend to show that for superadditive functions w : T -* T condition (1) is equivalent to the simpler condition lim w{n)(t) = 0, te T. M-*OD (4) Moreover, if w : T -» T is superadditive and continuous from the right then (1) is equivalent to the inequality w(t)<t9 teT. (5) M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis ON SUPERADDITIVE RATES OF CONVERGENCE 673 The above-mentioned results will be proved in Section 3 in the more gênerai context of^-dimensional rates of convergence. The notion of a /7-dimensional rate of convergence (or rate of convergence of type (p. 1)) was first introduced in [7] in connection with the study of Régula Falsi and it represents a natural generalization of the notion given in Définition 1.1 (see also [9] and [15]). 2. CONVEXITY AND SUPERADDITIVITY We have mentioned in the introduction that convex rates of convergence are superadditive. This is a conséquence of the fact that a convex function ƒ : R+ -> R which vanishes at zero is superadditive. This fact holds for functions of several variables as well, with a proper generalization of the notion of convexity. We will consider the notion of S-convexity, which was introduced (in a more gênerai context) by J. W. Schmidt and H. Leonhardt [29], and will compare it with some more popular generalizations of the notion of a convex function, in order to give the reader a better understanding of its significance. In what follows we will consider the Euclidean space Up endowed with the natural (component-wise) partial ordering « < » (Le., for any w = (wl5 ..., wp), v = (vu ..., vp)fromRp the relation u ^ v is equivalent to ut ^ vifori=l,2,...,p). The set R?. is called the non-negative cône of Rp. Two éléments x, y of Up are called comparable if either x ^ y or y ^ x holds. We will identify the space (IRP)* of ail linear functionals defined on Up with the space Up itself. This motivâtes also the notation uv for the scalar product between the vectors u and v. We will dénote by el3 e2,..., ep the standard basis of Up and by e the vector (1, 1,..., 1). Obviously e = e1 + e2 + *•• + ep. DÉFINITION 2 . 1 : Let D be a convex subset of Up. A function ƒ : D -> R is called : à) convex, if (6) f(Xx + (1 - X) y) ^ Xf(x) + (1 - X) f (y) for ail X e [0, 1] and ail x,yeD; b) order convex, if (6) holds for ail X e [0, 1] and ail comparable x, yfrom D ; c) S-convex, if there is a mapping 5/(., .):A>= {(x,y)eDx D;x¥=y}^Up, 5f(x,y)(x-y)=f(x)-f(y), for ail (x,y)eA, 8/(x, y) ^ 6f(u, v), for ail (x, y), (u, v)eA vol. 19, n° 4,1985 with x ^ u, y ^ v . (7) D (8) 674 F. A. POTRA For p = 1 the above defîned notions coincide. If p > 1 convexity implies order-convexity, but the reverse is not true. A similar statement holds if we replace convexity by S-convexity. 2.2 : Let D be a convex subset ofUp and let ƒ : D -> U be an S-convex function. Then ƒ is also order-convex. Proof : Consider two points x, y e D with x < y and let X be a number between 0 and 1. Dénote v = Xx + (1 - X) y. We have obviously x ^ v ^ y, v — x = (1 — X) (y — x), y — v = X(y — x) so that we can write : PROPOSITION Xf(x) + (1 - X) f (y) - f(v) = X(f(x) - f(v)) + (1-X) (f(y) - f(v)) = Xbf{x, v) (x - v) + (1 - X) &f{y, v) (y - v) x)^O. D We will see in what follows that the reverse of the above proposition is not true. Before that let us give a useful characterization of S-convexity : 2 . 3 : Let D be a convex subset of Up. A function if and only if PROPOSITION is S-convex i ( ƒ (x + sed - ƒ (x)) < \ (f(y + ted - f (y)) f:D->U (9) for ail i = 1, 2,..., p, x, y e D, s, t e U\{ 0 } satisfying x ^ y and x + set ^ y + tei. (10) Proof : Suppose (10) is satisfïed, If ƒ is S-convex we can write - ( ƒ (x + set) - ƒ (x)) = - 5/(x + sei9 x) se( s s = 5/(x + seu x) et < 8f(y + te;, y) et which proves the necessity of the condition stated in the proposition. Then, taking x = y in (9), it is easy to see that the foliowing limits exist : t>0 t->0 r<0 M 2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis ON SUPERADDITIVE RATES OF CONVERGENCE 675 + Dénote D£/(x) = (Df f(x) + Dt f(x))/2. For any two points x = {xl9 x 2 ,..., xp), y — CVIJ y il •-) JP) from W define a vector ô/(x, y) e [RP whose zth component is given by [ / ( /( ) )] if xt± yt\ and by D J ( l 5 ..., x £ _ ls x (> if x£ = yt. It is easy to check that conditions (7) and (8) are satisfied for the mapping ô/(., .) defined as above. The proof is complete. D In order to clarify the relation between the notions introduced in Définition 2.1, let us suppose that the function ƒ is twice G-differentiable. Then ƒ is convex if and only if ƒ "(x) hh ^ 0 for ail h e Up , xeD . (11) Also, ƒ is order convex if and only if f"(x) hh>0 for ail h e Up+ , xeD. (12) For a proof of the above statements see, for example, [3]. Using Proposition 2.3, we can easiiy prove that ƒ is S-convex if and oniy if f"(x)hk^0 for ail h,keUp+, xeD. (13) Condition (11) means that the matrix /"(x) is semipositive defmite while condition (13) means that the matrix /"(x) has all the entries nonnegative. This observation shows us that there are many functions which are convex but not S-convex as well as functions which are S-convex but not convex. As an example from the first category let us take ƒ : U2 -> R , f ( x u x2) = x 2 x - x x x 2 + x \ (14) and as an example from the second category f:U2^U, f(xl9x2) = xl + 4xiX 2 + x i . (15) It is known that convex functions are continuous (for a proof, see [3]). For order-convex functions (and even for S-convex functions) we have only continuity from the left and from the right in the sensé of the following définition : vol. 19,110 4, 1985 676 F. A. POTRA DÉFINITION 2 . 4 : A function f:D<^Mp^>Mis called continuons from the right (resp. left) at xe D if for any e > 0 there is a 5 > 0 such that \f(x)-f(y)\<e whenever y e D and x ^ y ^ x + 8e (resp. x — he < y ^ x). D More precisely we have the following. 2 . 5 : Let D be an open convex subset of Up and letf:D^Ube an order convex function. Then ƒ is continuons from the left andfrom the right at each point of D. THEOREM The above theorem can be proved by adapting the proof of Theorem 3.4.2 of [3] and using the following lemma. LEMMA 2 . 6 : Let Up be the hyper cube Up = \ x = (xl9 ..., xp) GUP; max ^ i <j< and let u[p\ u{2p\ ..., u{$ be its vertices. Iff:Up-*U any xe Up we have /(x)< is order convex then for max f(uip)). Proof : Our lemma is trivially verified for p = 1. Suppose it holds for a given p ^ 1. By reordering the vertices of Up+1 we may suppose that the last coordinate of (16) is equal to — 1 while the last coordinate of U (p+i) jjip+x) u{p+l) 2P+1> W 2 P + 2 ' ' " 'U2P+1 (\T\ V1 ') equals 1. Let us dénote by Up the convex envelope of the points (16) and by Up the convex envelope of the points (17). From the induction hypothesis we have max / ( 4 P + 1 ) ) , x'eU'p and M"+1)), max x"eu;. 1 Now any x = (x l5 x 2 , ..., x p , xp+1) e Up+1 can be written as x - Xx' + (1 - X) x" , Xe [0, 1] M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis ON SUPERADDITIVE RATES OF CONVERGENCE 677 with x' = (x 1( ..., xp, - 1) e U'p, x" = (xu ..., xp, 1) e £/;. We have clearly x' < x" so that using the order convexity of ƒ we deduce that f(x) < Xf (x') + (1 - X) /(x") < max 1 ^fc^2P The proof is complete. • We note that the above lemma does not hold if we replace Up by some other convex polyhedron. Indeed if we consider the function ƒ given by (15) and the square of vertices (1, 0), (0, 1), (— 1, 0), (0, - 1) we have /(l,0)=/(0,l)=/(-l,0)=/(0,-l) = l and ƒ ( - 1 / 2 , - 1 / 2 ) = 1 + 1/2 . Let us state now two very simple results to be used later : DÉFINITION 2.7 : A function f: D c Up -> IR is called superadditive if f(x + y) > f(x) + f(y) for all x, y e D with x + yeD . DÉFINITION 2.8 : A function f :D c Up -• U is called isotone if f(x) < fiy) whenever x, y e D and x < y. PROPOSITION 2.9 : Let T dénote either the whole positive axis ]0, oo[ or a half open interval of the form JÜ, b\ and set where 0 IJ the origin of Up. If f : Do -> U is S-convex and /(O) = 0 then f is superadditive. Proof : f(x +y)- f(x) - f(y) - f(x + y) - f{x) - (f(y) - /(O)) = PROPOSITION 2.10 : If f : D <= Up ^> M is superadditive and nonnegative (Le., f(D) c R + ) then f is isotone, Proof : If x, y e D and x < y then fiy) =f(x + y-x) >f(x) +f(y-x)^o. • It is interesting to remark that Proposition 2.9 does not hold if we replace S-convexity by convexity. For example the function f given by (14) is convex but if we take x = (1, 9) and y = (9, 1) then we have /(x+j>)=/(10, 10) = 100 and ƒ(x) + f{y) = 2(81 - 9 + 1) = 146. vol. 19, n° 4,1985 678 F. A. POTRA 3. p-DEMENSIONAL SUPERADDITIVE RATES OF CONVERGENCE Let T dénote as before either the set of all positive real numbers or a half open interval of the form ]0, b]. Let w b e a mapping of the cartesian product Tp into T and let us consider the « itérâtes » w{n) of w given for each (tut2,...,tp)eTp t = by the following récurrent scheme : w ( 0 ) (0 = tp, w<"+ *>(*) = w(n>(£2,.., tp, w(t)), n = 0, 1, 2, .... (18) 3.1 : A mapping w : Tp -> T with the above itération law is called a p-dimensional rate of convergence {or rate of convergence of type (/?, 1)) on T if DÉFINITION f w{n\t) < oo, teTp. «=o (19) Itisconvenienttoattachtothemappingw : Tp -• Tamapping ~w :TP -> Tp defined for every t - (tl912, ...s tp) e Tp by w(t) = (t29.., tp9 w(t)). It is easily seen that if we dénote by w{n) the itérâtes of w in the sensé of the usual composition of functions (i.e., w(0)(t) = t, vftn+l){t) = w(win)(t))) then from (18) it follows that . w{n+1\t) = w(w{n)(t)) = win\w(t)), w™(t) = (w*-p+ l\t\ n = 0, 1, 2, ..., .., w(«>(0), (20) n = p - 1 , ^ /i + 1,.... (200 Let us dénote by s{t) the sum of the series (19) and let us consider the mappings st : Tp -> R, f = 1, 2, ...,/> given by st{t) = s{i) + P Ë t,, t = (tu t2,... tp) e T p . Finally let us define the mapping s : Tp -> Tp, J(t) = ( Sl (0, s2(0, », M0) • With the above notation we have : 3.2 : If the p-dimensional rate of convergence w :TP -> T is superadditive and continuons from the left then PROPOSITION s(w(t)) ^ w(J(t)), for all teTp. (21) M 2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis ON SUPERADDITIVE RATES OF CONVERGENCE 679 Proof : First, let us observe that Using the superadditivity of w and (20) we M/"X can write \k=0 Finally, letting n to tend to infinity in the above inequality, we obtain (21). • The inequality (21) was first proved by V. Ptak [25] for unidimensional convex rates of convergence. The heuristic motivation for the importance of such an inequality given in the above-mentioned paper can be generalized to the /7-dimensional case as follows : Suppose { xn}n^0 is a séquence of points belonging to a complete metric space (X, d) such that >Xn+l) < M4(xn-p> X n-p+l)> -94(xn_l9 X„)) where w is an isotone /?-dimensional rate of convergence. It is easy to see that under this assumption the séquence {xn}n^0 has to be convergent. Moreover if we dénote x* = lim xn, r0 = (d(xOi x j , . . . , d(xp_l9 xpj) «-•oo then we have the estimâtes If the séquence { xn } n ^ 0 is obtained via an itérative procedure then at a certain stage the distances d(xn, x n+1 ) are known for, say, ail n < N while the distances d{xm x*) are generally not known. It is then important to note that if (21) holds then the estimâtes en wilî satisfy a relation similar to the relation satisfied by the distances d{xn, x„ +1 ). Indeed we have : 0))) - w(en_p+l,..., O . In what follows we will show that for superadditive functions w :TP -> T, condition (19), which is generally very difficult to verify, can be replaced by much simpler ones. vol. 19, n° 4,1985 680 F. A. POTRA 3 . 3 : Let w : Tp -> T be a superadditive function. Then w is a p-dimensional rate of convergence on T if and only if THEOREM lim w(n)(0 = 0 teTp, for ail (22) «-•QO Proof : The necessity of condition (22) is obvious. In order to prove the sufïîciency we note fîrst that (22) implies W&ç,..,o<ç, ^r; (23) Indeed if we suppose that there is an r] G T such that w(r\, T|, ..., r|) > r| then because of the isotony of w (see Proposition 2.10) we have w(n)(r), r|,..., r|) ^ rj which contradicts (22). Now, let us fîx a point teTp and a positive number e > 0. Let us dénote a — e — w(e, e,..., e), From (22) it foliows that there is an integer N = N(t, e) such that w(n)(0 < a/p for any n 7* N . (24) We will prove by induction with respect to k that win)(t) + w(n+1)(t) + - + witt+k)(t) ^ 8 for any n > N . (25) For k = 0, 1, 2, ...,p — 1 this follows immediately from (24). Suppose (25) holds for a given k ^ p — 1. Using the superadditivity of w and (20') we have : win)(t) + win+1)(t) + - + win+k+1\t) < wn(t) + •» + w ( " + p - 1 } (0 + w{W+p-1(t) + •" + wn+k{t)) < a + 1^(8, 8, ..., 8) = 8 . Thus (25) holds for ail n ^ N and ail k = 0, 1, 2,... Hence the series (19) is convergent. The proof is complete. • In the proof of the above theorem we have seen that if w : Tp -> T is an isotone function then (22) implies (23). In what follows we show that for right continuous functions the converse is also true. 3.4 : Let w :TP -> T be isotone and continuous from the right, Then condition (22) is equivalent to PROPOSITION W (Ç,Ç,..,o<Ç, ^ G T . (23) Proof : We only have to prove that (23) implies (22). Consider a vector t = (tl912, ..., tp) G Tp and set r| = max { tu t2,..., tp }. The séquence M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis ON SUPERADDITIVE RATES OF CONVERGENCE 681 an = win)(r\, T|s ..., r|) is a nonincreasing séquence of positive numbers so that it is convergent. Set a = lim an and suppose a > 0. Then from the right continuity of w we have a = lim an+1 n-*oo n->oo = lim w(a B _ p + 1 , Û B _ P + 2 Ï - , <O = w(o, a, ..., a) «->oo which contradicts (23). Hence lim an = 0. From the isotony of H> it foliows H — • oo that w{n)(t) ^ an so that (22) is satisfïed. D Now from Theorem 3.3 and Proposition 3.4 we can immediately deduce the main resuit of our paper : 3.5 : If w : Tp -> T is superadditive and continuous from the right then conditions (19), (22) and (23) are equivalent. D COROLLARY In particular, for S-convex functions we have : 3.6 : Let w : Tp -> T be an S-convex function. Then w is a p-dimensional rate of convergence on T if and only ifone of the conditions (22) or (23) is satisfïed. COROLLARY Proof : By virtue of 2.2, 2.5, 2.6 and 3.5 we only have to prove that if w : Tp -> T is S-convex and if there is a séquence an e Tp such that aB+i < a» > n = 09 1, 2, ... ; lim an = 0 , lim W(ÛB) = 0 thenbytakingvt>(0) = O,wextendsto an S-convex function on D 0 = Tp u { 0 } . Let us fix a point xeTp and consider the séquence { Sw(x, a j }„ ^ 0 . According to 2.1, c) we have 8w(x» an+1) < 5w(x, a„), B) n = 0, 1, 2, ... (x - fln) = w(x) - w(an). (26) (27) From (26) it folio ws that all the séquences { 5w(x, an) et }„^ 0 (/ = 1, 2, ...,/>) are nonincreasing. Hence d{ = lim 5w(x, an) e{ exists although it might be n-*co P equal to — oo. From (27) it folio ws that £ d{{xe^ = w(x) > 0 and taking into account the fact that xet > 0 for i = 1, 2,...,/? it folio ws that df > — oo for i = 1, 2, ...,/?. Hence lim 8w(x, an) exists and we take by définition bw(x, 0) = lim 5w(x, a„) . vol. 19, n° 4, 1985 682 F. A. POTRA From (27) it follows that 8w(x, 0) (x - 0) = 5w(x, 0) x = w{x) = w(x) - w{0). We have to prove that 8w(x} 0) ^ 5w(u, v) for ail u,vsD0 If v with x ^ u . = 0 we have Sw(x, 0) = lim 8w(x, an) < lim 8w(w, an) = 8w(w, 0). If f # 0 then there is an integer iV such that an ^ v for n ^ N. Hence 8w(x, 0) = lim ôw(x, a„) In a similar way by defming 8w(0, x) = lim 8w(an, x) we can prove that 8w(0, x) (0 - x) = w(Q) - w(x) and 8w(0, x) ^ 8(w, v) for ail u,v e Do with x ^ u. The proof is complete. D From the proof of the above corollary it follows that any S-convex rate of convergence w : Tp -> T is superadditive. The reciprocal is clearly not true. It is interesting to note that the rate of convergence of Newton's method (3) is convex only on the interval [0, a s/ï] but it is superadditive on U+. The proof of this fact is elementary but not very straightforward and it will be given in the following. 4. APPENDIX The rate of convergence of Newton's method (3) has been intensely studied (see [12, 15, 20 and 21]). For this function the series (1) is convergent for any t e U+ and its sum has the simple explicit expression : s(t) = t - a + Explicit expressions for the finite sums sn(i) = t + w{t) have also been found (see [20] or [15]). These explicit expressions were used to obtain sharp and elegant a priori and a posteriori estimâtes for Newton's M2 AN Modélisation mathématique et Analyse numérique Mathematical Modelling and Numerical Analysis ON SUPERADDITIVE RATES OF CONVERGENCE 683 method for solving nonlinear operator équations in Banach spaces (see [23, 12 and 15]). The second derivative of w is of the form „, x (2 a2 - t1) a2 which shows that w is convex on [0, a yfï\ and concave on [a ^/ï, + oo[. In what follows we will show that w is superadditive on [0, + oo[. We have to prove that [a2 _|_ (x _|_ y ) 2 ! 1 ' 2 (a -f X ) ' (<2 + V ) If a = O this is trivially satisfied. If a > O then by dividing both sides of the inequality by a and denoting u = x/a, v = J/Ö this reduces to ^ [1 + (u + i;) 2 ] 1 ' 2 " (1 + i/2)1'2 " (1 + » 2 ) 1/2 Let us fix v and consider the left hand side of the above inequality as a function of w, say f(ti). We have obviously /(O) = 0. We will prove that ƒ (w) > 0 for u > 0 by showing that : 1) There is an M > 0 such that ƒ (w) > 0 for ail u ^ M. 2) The derivative of ƒ has a unique root M* > 0 such that f'(ü) > 0 for u < w* and ƒ'(M) < 0 for M > w*. Indeed if 1) and 2) hold then for any M < M* we have ƒ (w) > ƒ (0) = 0 while if u > u* then by taking z > max { w, M } we get f(u) > ƒ (z) > 0. In order to prove statement 1) we first perform the change of variable e = l/w and obtain a function of 8, We have (1 + zv)2 (1 + 82)1/2 - [e 2 + (1 + sv)2f/2 = (1 + st;)2 + 82[(1 + 8z;)2 + 1] sv(Z + m sv(Z + m ( l + £v)2 ( 1 + g2)1/2 + so that we can write g(e) = sv[a(e) (1 + t;2)1/2 - i(e) v] vol. 19, n°4, 1985 684 F. A. POTRA where lim a(e) = lim b(e) = 1. This shows that g(e) > 0 for s sufficiently small, which complètes the proof of statement 1). If we dénote t(2 + t2) then the derivative of ƒ can be written as ƒ'(M) - *(w + y) - *(«) . It is easy to prove that for any given v there is a unique w > 0 (more precisely we have [(^/S - 1)/2] 1/2 < w < ^/2) such that h(u + u) = A(M). (Intuitively this can be easily realized by drawing the graph of h.) Hence the équation f'{u) = Ohas a unique positive solution M*. We have obviously ƒ'(0) = h(v) > 0 and from the mean value theorem it follows that there is a w > yJ2 such that which complètes the proof of statement 2). REFERENCES [1] I. KOKËC, On aprobiem of V. Ptâk, Cas. pro pest. mat. 103 (1978), 365-379. [2] J. KJUZKOVÂ, P. VRBOVÂ, A remark on a factorization theorem, Comm. Math. Univ. Carol (CMUC) 15 (1974), 611-614. [3] J. M. ORTEGA, W. C RHEINBOLDT, Itérative solution of nonlinear équations in several variables, Academie Press, London, 1970. [4] H. 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