O przybliżonych rozwiązaniach nieliniowego równania funkcyjneg

Głowacki, Eugeniusz

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Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: On approximate solutions of a non-linear functional equation   Author: Eugeniusz Głowacki  Citation style: Głowacki Eugeniusz. (1972). On approximate solutions of a non-linear functional equation. "Prace Naukowe Uniwersytetu Śląskiego w Katowicach. Prace Matematyczne" (Nr 2 (1972), s. 13-17)  PRACE NAU K O W E  U N IW ERSYTETU  ŚLĄSKIEG O  W  K A TO W IC A C H  N R  I IPRACE M ATEM ATYCZN E  II, 1972EUGENIUSZ G ŁO W A C K IO n approximate solutions of a non-linear functional equationIn paper [1] we have considered approximate solutions of some linear functional equations. The role of the approximate solution was played by the n-th term of a suitable functional sequence whose limit was the exact solution. A  similar method may be applied to more general equa­tions.In the present paper we construct approximate solutions, with a preassigned accuracy, for continuous solutions of the functional equation( 1) <p(x) =  h {x ,< p [f (x ) ]} ,where cp(x) is the required real-valued function of a real variable. The functions h{x, y) and f (x ) are given real-valued functions of two, respec­tively one, real variable.In the sequel we shall assume that the function f (x ) fulfils the following condition:(I ) f (x ) is continuous and stricly increasing • in an interval [f, a), moreover i  <  f(x ) <  x  for x  e(£, a).(In other words, f  e  ft? ([£, a]); cf. [2]).Let fn(x ) denote the n-th  iterate of the function f(x ): f° (x ) =  x, fn+1 (x) =  f [ f n(x )], n =  0, 1 , 2, . . .Under hypothesis (I) we have /(|) =  -, and the iterates f n(x ) are defined, continuous and strictly increasing in [f, a); furthermore, for every fixed xe(£ , a), the sequence {/ " (x )} is stricly decreasing and lim f n(x ) =  | (cf. [2]).Let i) be a real number such that <2) 7i(f, y) =  y.Let us fix  arbitrarily a c, 0 <  c <  a — £.W e shall make the following assumptions concerning the function h(x, y ):(II) The function h(x, y ) is continuous in the rectangleD =  [ ! , f  +  c] X [17 —  d,y +  d], d > 0 ,moreover, h(D ) d  [t)—d, t] +  d].(III) There exist constants O < 0 < 1 ,  0 <  3 ^  c, 0 <  a ^  d such that(3) Ih(x,y) —  h(x, ÿ)| <  0  |y —  y\ for (x, y ),  (x , ÿ ) e  [I, I  +  S] X [y —  a, y +  a]and(4) |h(x, rj) -  h(£, y)\ <  (1 -  0)® for z e [ f , f  +  3].(IV ) The function h(x, y) fulfils in the whole rectangle D the Lipschitz condition with a constant L  >  0, i. e.(5) \h(x,y) —  h(x,ÿ )| <  L  \y —  y\ for (x,y), {x,ÿ) e  D.As has been proved in [2], under hypotheses (I), (II), and (III) equation( 1 ) has exactly one solution tp which is continuous in [|, f  +  c] and fulfils the condition tp(£) =  rj. This solution is given by the formula(6) cp (x) =  lim <pn (x),n->oowhere(7) cpn +1 (x) =  h {x , 9?„[/(xO]}, n =  0, 1, 2,. . .,and 9?0(x) is an arbitrary continuous function defined in [|, £ +  c], taking values in [y—d, y +  d] and fulfilling the condition(8) ■ <Pa {£) =  y.The functions <pn(x ) defined by (7) may be regarded as approximate solutions of equation (1 ).Let us fix  arbitrarily e £> 0. W e are going to find the index n in such a manner that function g?„(x) should d iffer from the exact solution rp (x) in the whole interval [f, £ +  c] by less than s.Let po(x) be an arbitrary continuous function in [£, £ +  c] which takes values in [y — d, y +  d], fulfils condition (8) and is such that(9) |ç?0(x) —  y\ <  a for x  e [f,  £ +  3].It follows from (9), (3) and (4) that for x  e  [£, £ +  3] we havei9?1 (x) — |h(x, ??o[/(^)]) —  h(x, 77)1 +  |h(x, y) —  h{£, y)\ <^  0 l<M;f(x)] —  y\ +  (1 — 0)o <  a, and, as is easily proved by induction,(10) |pn(x) —  j? K  0 f or x  e [f, f  +  3], n =  0, 1, 2 , . . .Let N  be the least non-negative integer such that(1 1 ) f N( i  +  c ) e [ ł , £  +  31.Hence it follows by hypothesis (I) that(12) f n(x) e  [i, £ +  ô] for n ^ N  and x e [ U  +  c],Let us write(13) K  =  max | çq(x) —  <p0(x)j,[£, £ +  à](14) a =  logft Q) +  N.'  6 0 L n kWe have the followingTHEOREM. Under hypotheses (I), (II), ( I I I )  and (IV ) for every n >  a and every x  e  [I, £ +  c], we haveI Vn ( x ) ~ y (x ) I <  E, where <pn (x ) and <p (x ) are defined by (6) and (7), and <po(x) fu lfils ine­quality (9).P r o o f .  By (5), (3), (10), (12) and (13) we get the estimation of \<Pn+i(x) ~~ <Pn(x)|, valid for n ^  N  and x  e  [f, £ +  c]I<Pn +  l W  —  <Pn(x)\  <  L N  \cpn -JV +  1 tfN(x )] —  <Pn -N  [,fN(x )]| <<  L N 0 n_N \<p,mx)\ —  po[f»(x)]j <  ( —  ) &nK.0For the difference 9?n(x) —  <p(x) we get hence for n ^  IV and x  e  [I, £ +  c]■ V / LI<pn(x ) —  cp(x)I <  ^  |Ç5S + !(x ) —  Ç9s(x)| <  ( ^ l  • K  • 0 s =s =  nN-  | ' i f  ■ K  ®”Q l  1 —  0For n >  a, where a is defined by (14). and for every x  e  (I, I  +  cjI <Pn (x ) — <P (x ) I <  £which was to be proved.I f  ô =  c, a =  d, then for a we have the simpler formula.  *(1 —  0 ) a -  l0g 0  k  'It results from the estimation obtained in the proof that the convergenceof the sequence (<pn(x ) }  is the more rapid the smaller is the num­ber 0, but if  we reduce 0  then N  increases and a need not decrease,This may be seen from the followingEXAM PLE. W e consider the functional equation{x +  2) cp<p{x) =  --------------------------  for x e [0,l].(x +  2) q> "t* 1We have hereh(x, y ) =  — , f (x )  =  f  , î  =  0, c =  1,(x +  2) y +  1 2We take y =  —  and d =  —  . Then D =  [0,1] X [0,1] and2 2|h (x, y ) —h (x, y)| <  3 |y — ÿ| for (x, y), (x, ÿ ) e  D  .For an arbitrarily chosen 0  with —  <  0  <  1 inequalities (3) and (4)2are fulfilled for4(1 — 0 ) ( 2 0  — 1) 20 —  1à — -----------------------a -----------------------.202 —  0  +  1 40I f  we take v>o{x) =  —  , then2K  =  max |9?i(x) —  ??o(a;)| =  (1 —  0 ) «[0, Ô]and for a we have the formulaa =  logg. — —  +  N  ® 3NoXSince f n(x ) = ----  . N  is the least non-negative integer such that2 n2 * > —  .<5Let us take e =  10-3. The exact determination of the least number a is too complicated, but trying various values we can find that the3possible best estimation we have for 0  =  — . Then we have51 4 < a <  15.Therefore for the equation consideredtaste) —  y>te)l <  10-3 for xe[0,ij.[1] E. G ł o w a c k i :  On approximate solutions o f a linear functional equation. Prace Naukowe Uniwersytetu Śląskiego N r 2, Prace Mat. 1 (1969) p. 41— 51.[2] M. K  u c z m a: Functional equations in a single variable. Monografie matema­tyczne 46, Warszawa 1968.EUGENIUSZ G ŁO W A C K IO PR Z Y B L IŻ O N Y C H  R O Z W IĄ Z A N IA C H  N IELIN IO W EG O  R Ó W N A N IAFUNKCYJNEGOS t r e s z c z e n i eW  pracy rozważane są przybliżenia ciągłego rozwiązania równania funkcyjne­go cp(x) =  h {x ,ip [f(x )]}, gdzie cp jest funkcją szukaną. Podobnie jak w  pracy [1] jako rozwiązanie przybliżone przyjm uje się n-ty wyraz odpowiedniego ciągu funk­cyjnego, którego granicą jest dokładne rozwiązanie.Celem pracy jest dobór liczby rzeczywistej a w  ten sposób, by n-te przybli­żenie (pn (x ) rozwiązania różniło się od dokładnego rozwiązania o z góry zadaną wielkość w  góry zadanym przedziale, gdy tylko n >  a.Praca zawiera 1 twierdzenie podające sposób doboru liczby a, przy czym sposób ten jest również zilustrowany na odpowiednim przykładzie.Oddano do Redakcji 27. 5. 1970 r.2 Prace matematyczne U

On approximate solutions of a non-linear functional equation

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On approximate solutions of a non-linear functional equation

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