Fenyő, I.

Pap, E.

English

Periodica Polytechnica (Budapest University of Technology and Economics)

ON THE FUNCTIONAL EQUATION f(x 1 )+f(x2) = f(Y1) +f(Y2) By 1. FENYO and E. PAP* Department of Mathematics. Technical University Budapest Received February 17. 1981 1. In the theory of partial differential equations, namely at the homoge-neous wave equation appears a functional equation. A necessary and sufficient condition as it is well known - that a function f(x, y) of the class C2(R2) should be a solution of the above-mentioned differential equation in R2, is that for all possible rectangles whose sides are segments of characteristics of the homogeneous wave equation with label the vertices V; (i = 1,2.3,4) VI and V4 are denoted as opposite vertices - satisfy the functional equation f( VI) + + f(V4 )= f(V2 )+ f(V3 )· Our aim is to investigate a functional equation closely connected with the preceeding functional equation, but on a more general algebraic structure, namely on a semi-group and free of geometrical interpretation. Roughly spoken, we investigate a functional equation in which the sum of images are equal if the sum of arguments are equal. Theorem 2 states, that if this is true for two independent arguments, then it remains true for an arbitrary number of arguments. Such a functional equation is closely connected with the Cauchy and Jensen functional equations (Theorem 1) as well as with the Ryff ([5J) functional equation. We give also the most general solution of the investigated functional equation on the reals, as wel! as, the general solution of it if the unknown is not a function but a distribution. Also a method is given to solve the corresponding inhomogeneous functional equation. 2. Let (X. + ) be a commutative semi-groups with root functions j'll: X->X for all llEN, i.e.(see [4J) for all x, yE X. i'n(X) + i'Il(Y) = i'n(x + y) ni'n(X) = x Let A be a subset of X with the property that Xi E A (i = 1,2 .... , n) implies 1',,(x 1 + ... +xll)EA for all liEN. Let 0: +) be a semi-group (we shall use the * Institute of Mathematics. University of Novi Sad. Yugoslavia 116 I. FESYO- E. PAP additive notation also m the noncommutative case). We consider the functional equation I(x l ) + I(x 2 ) = I(YI)+ I(Y2) (1) for I: X -+ Y on restricted domain, for all (x I' X 2), (y I' Y2 ) E A x A such that XI +X2 =YI+Y2' THEOREM 1. If the function I: X -+ Y satisfies the equation (1), then I satisfies (2) for alh 1- X2' ... , x 11 E A and all 11 E N. The left hand side of (2) is independent of the order of summation. PROOF. We prove the theorem by induction on n. (2) is obviously true for n = 1. Let us suppose its validity for all 11 ~ m - 1. For 11 = m we take X 2 = x 3 = ... =X"'. Since rl=~' (x l +(m-1)x 7 )EA - m . -Y2 = j'",((m -1)x 1 +x2 ) E A and then by (1 ) we obtain Hence I(x 1 )+ -1 )j(x2 )= f(x 1 )+ I(x 2 l+ Im -2)f(x2 )= = f(j'n,(.'( 1 + (m -1 )x 2 ) + I()'m((m -l)x 1 + x 2 ) + (m - 2)f(x2 ). Now we apply the equality (2) for 11 = m 1 on the last m-I factors in the preceding relation. In this way we obtain (3) = (m -1 )I(f'", 1 C'm({m -l)x 1 + x 2 ) + (nl- 2)x2)). We shall prove that ;'", 1(i'm((m-l)x l +x2)+(m-2)x1 )=j'm(x 1+(m-l)x2) (4) OS THE FLACTIOSAL EQL'ATIOS 117 holds. Let us start from the left hand side + i'm- 1 (j'm(m(m -1 )X2» . Since i'm 0 i'm - 1 = i'm 1 0 I'm (see [4J), we can write L = j'm(/'m -1 ((m -1 )X1) + i'm -1 (X2 + (m 2 - 2m)X2) = and so we obtain the relation (4). By (4) and (3) we have (5) Now we consider arbitrary elements Xj E A (i = 1, 2 .... ,11). By (2) for n = m-I and (5) we have this is exactly the relation (2) for n = m. By the commutativity of the operation + in X, we have where p( . ) denotes any permutation of the numbers 1. 2, .... n. Hence by (2) we have This completes the proof. REMARK 1. It is evident that every solution of the Cauchy functional equation is also a solution of the equation (1). If (:r; + ) is a commutative 118 semi-group with the root function 6n for alln E N, then by (2) every solution of the equation satisfies also the generalized Jensen-equation for all XiEA (i=1, 2, ... , n) and all nEN. REMARK 2. Any homogeneous real solution of (l) satisfies the Ryff-equation ([5J) af(ax)+ bf(bx+a)=bf(bx)+ af(ax + b). 3. THEOREM 2. If f: X --> }' is a solution of (1 ) for which the condition f(2x) = 2f(x) (6 ) holds for all x E A. then f satisfies (7) forallxi'YkEA (i=1,2, .... n;k=1.2, ... ,m)andalln,mENsuchthat Xl + ... +X,,=YI + ... +Ym holds. PROOF. It is obvious that in the case n = m Theorem 2 follows (without using (6)) from the Theorem 1. Let now be n ~ m. Then we have for arbitrary elements Xi' Yk (i=1, 2 .... , n; k=l, 2, ... , m) fulfilling the condition by Theorem 1 f(x l )+··· +f(xn)=nf(i'n(.z)) f(YI) + ... + f(Ym) = mf(i'm(Z)). We shall now prove (7) in the equivalent form nf(x) = mf(j'm(nx)) for )J~m, where x=}'n(z), Let be first n=m+ 1. By (6) and Theorem 1 we have (m+ l)f(x)= (m-l)f(x)+ f(2x)=mf(Ym((m+ l)x)) (8 ) OS THE FCSCTlOSAL EQCATlOS 119 SInce fm((m+ 1)x)=j'm((m-l)x+2x). (8) is proved for this particular case. Let us now suppose that (8) is true for n fulfilliIlg the inequalities m+ 1 ~n~m+k-1. Hence (m+k)f(x)= f(x)+ (m+k-l)f(x)= f(x)+mf(l'm((m+k 1 ). For n=m+ 1 we have by (8) (m+k)f(x)=mf(j'm((m+k)x)) . i.e. the relation (8) is valid also for 11 = m + k. This the of Theorem 2. 4.1t seems to be interesting to look for the most general solmion j: R--+ under the condition t·, ) r( ) J'( \, 1'( ) , (x 1 + j X 2 =.l Yl J T .l) 2 Introducing the new variables x = Xl' Y =.r l' .: = x + X 2 = Y + .1'1' we can write instead of our functional equation f(X)- f(y)+ f(.: x) f(z For X=Z we have f(x- y)- f(x)+ f(y)=c. where c is a constant. This is (in general) an inhomogeneous equation. Its general solution is the sum of a partial solution solution of the corresponding homogeneous equation. Obviously f (x) = c (x E R) is a partial solution of (10) which also satisfies (9). The corresponding homogeneous functional equation. is the f(x - ,v) = fix) - f(y). from this if x = 0 11) f( - y)= - fly) 120 as (11) implies f (0) = O. We see at once that the most general solution of (11) is an odd additive function which also satisfies (9). THEOREM 3. The most general solution of (9) has the form f(x)=c+l1(x) (x E RJ, where h is an arbitrary odd additive function and c an arbitrary real constant. 5. We can easily get also the general solution of the inhomogeneous functional equation f(x- y)- I(x)+ I(y)=d(x, y) (x·YER) (12) if any exists. Here d is a given bounded function. First let us calculate a partial solution. e.g. a bounded one. (12) implies I( -x)+ f(x)=d(O, xJ+ nO) (xER). (13 ) Now we substitute in (12) y=2x. Regarding (13) we get f(2x)-2f(xJ=d(x. 2x)-d(0, x)- I(O). Putting y = 3x into (12) and apply (13) with 2x instead of x we get f(3x)-3 ) d(x. 2x) +d(x. 3xj -d(O. x) - d(O. 2x) -2f(O). By induction finx) )=d{x. 2x)+ . .. +d(x.llx)-d(O, x)- ... --d(x,(11-1 )x) - (11-1 )f(O). From this follows as I(O) =d(x. x) nllX) -f(x)= [d(x.x)-d(O.x)]+ ... +[d(X.IIX)-d(O.l1x)] + d(O.l1x) 11 11 11 f (0). OS THE FCSCTlOSAL EQl ATlO.\ 121 By the assumption that f and d are bounded lim f(llx)/Il=O Tl- X lim d(O. ilx)/n 0 Tl- :r~ for every x. therefore [d(x.x) d(O,xl]+ ... +[d(x,ilx)-d(O,llx)] fix) - ./(0) = lim . (14) n- x This means, if (12) has a solution at all, then the unique bounded solution has the form ( Denoting by g an odd additive f + g is the most general solution of (12). If we consider now the corresponding inhomogeneous functional equation I(Xl- f(y)+ f(::-x)-f(::-y)=p(x.y.::) (IS 1 and put:: x. then this overgoes into f y)-I{x)+j f )- y.xJ. (16) Letbed(x,y) c y,x){c= Obviously (10) is not equivalent to \vould be interesting and useful to find conditions under 04 is a solution of (10), one of (15 J. 6. Looking at the relation between the wave equation and the functional equation (9) mentioned in the introduction. it seems to be of some to find the' genera! solution of (9 J. if f is not a function but a . the distributional differential operator D=D, to every term of (9 J, we get 1. I =0. where LO, 1) is the dual operator of f. (l(1. -l)cp)it) = J <;'I)(x. x -I) dx. 122 cp is an arbitrary Schwartz-testfunction (see [6J, especially p. 48-49). But then D2f = 0 and therefore the most general solution of (9) is the distribution generated by the function ax + b (a and b are arbitrary constants )*. Summary The paper investigates the functional equation f(x,)+ f(x 2 )= f(yd+ f(Y2) under the condition x, + X2 = .1', + Y2 on a semigroup. Also the general solution of it is given on the reals as well the general solution if the unknown is a distribution. A method is proposed which provides the solution of the corresponding inhomogeneous functional equation. References I. ACZEL. J.: Lectures on functional equations and their applications. Academic Press. New York. (1966). 2. FE:-;Yo. 1.: Remark on a paper of J. A. Baker. Aequationes Math. 8. (1972) 103. 3. KliCZ~IA. M.: Functional equations on restricted domains. Aequationes Math. 18. (1978) 1. 4. PAP. E.: n-convex functions on a semigroup with root function. Zbornik PMF u Novom Sadu. 6. (1976) 7. 5. R YFF. J. V.: The functional equation aj(ax) + bj(bx+a) =N(bx) +aj(ax+b). Bull. Amer. Math. Soc. 82. (1976) 325. 6. FE:-;Yo.1. (E).: Sur les equations distributionnelles. In: Functional equations and Inequalities. CIME. Ill. ciclo. Ed. Cremonese. Rorna (1971) 43. Prof. Dr Istvan FENYO Prof. Dr. Endre PAP H -1521 Budapest Institute of Mathematics. University of Novi Sad 21 000 Novi Sad. Yugoslavia * The part of the paper of the second author has been sponsored by SIZ za NOUCNI rad SRBIJE over Mathematical Institute in Belgrade. 

ON THE FUNCTIONAL EQUATION f(x1 )+f(x2) = f(Y1) +f(Y2)

https://pp.bme.hu/ee/article/download/4772/3877

ON THE FUNCTIONAL EQUATION f(x1 )+f(x2) = f(Y1) +f(Y2)

Abstract

Similar works

Full text

Available Versions

Periodica Polytechnica (Budapest University of Technology and Economics)