Kannappan, PL.

Czech digital mathematics library

On a multiplace functional equation related to information measures and functions

Institute of Mathematics AS CR, v. v. i.

KybernetikaPalaniappan KannappanOn a multiplace functional equation related to information measures and functionsKybernetika, Vol. 19 (1983), No. 2, 110--120Persistent URL: http://dml.cz/dmlcz/124981Terms of use:© Institute of Information Theory and Automation AS CR, 1983Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This paper has been digitized, optimized for electronic delivery and stamped withdigital signature within the project DML-CZ: The Czech Digital Mathematics Libraryhttp://project.dml.czK Y B E R N E T I K A - V O L U M E 19 (1983), N U M B E R 2 ON A MULTIPLACE FUNCTIONAL EQUATION RELATED TO INFORMATION MEASURES AND FUNCTIONS PL. KANNAPPAN* This paper deals with a functional equation which is related to measures of information such as Shannon's entropy, inaccuracy, divergence, entropy of degree /fete, and information functions. The general solutions of the functional equation are determined, without the assumption of any regularity condition on the functions involved. 1. INTRODUCTION Real valued functions satisfying the functional equation (i.i) /(*) + (i - » / / ( — ; ) - I w + c1 - y/f(r^) are called information functions of degree /?, which is related to the Shannon's entropy when/? = 1, otherwise to the entropy of degree ft (+ 0) [2, 4]. In [5,7,2] a generaliza-tion of (1.1), (1.2) F(x) + (1 - xf G (j-L^j = H(y) + (l - yfK (~^j for x, y e [0, 1[ with x + y e [0, 1] was considered, giving details of its connection with directed divergence of type /?, inaccuracy of type (S etc. The purpose of this paper is to consider in generalization of these and other results, a multiplace functional equation (1.3) fix, y) + (i - xf g (-JL- , ^ - _ ) = h(u, v) + (i- Uf k (-2L- , - 2 - ) \1 - x 1.— y) \l — u 1 — v) * Work partially supported by an NSERC of Canada grant. 110 for x, u £ [0, l [ with x + u e [0, 1], y, v, y + v e ]0, l [" and to find the general solution of (1.3) by simple methods using familiar functional equations such as Pexider equations e tc , without any further assumptions (regularity conditions) on the functions, by relating (1.3) to (1.2). 2. SOLUTION OF (1.3) Theorem. Let j , h: [0, l [ x ]0, l[" -> R reals, g, k: [0, l ] x ]0, l[" -> R satisfy the functional equation (1.3) for x, u e [0, 1 [ with x + u e [0, 1], y, v, y + v e ]0, 1 [", fi e R. Then the general solution is given by (2.1) f(x, y) = x l(x) + (1 - x) /(l - x) + x Ms(y) + (1 - x) M_(l - y) + + C_x + C2 , _?(„, y) = u /(«) + (1 - u) 1(1 -u) + u M,(y) + (1 - u) M_(l - y) + + C3M + C5 - C2 + C4 - C3 , h(x, y) = x l(x) + (1 - x) 1(1 -x) + xMt(y) + (1 - x) M_(l - y) + + C4X + C5 , k(u, y) = u l(u) + (1 - «) /(l - u) + u Ms(y) + (l - u) M_(l - y) + + (C_ - C5 + C2 - C4 + C3) U + C4 - C3 , for x 6 [0,1[, u e [0, l ] , y e ]0, 1 [", if p = 1; (2.2) ,Y, ... _ fM i( i - >•) + M 20') + «i • x el°> if ^ X ' J J J M _ ( l - y ) + M_(y) + a2) x = 0 fM.(l - y) + M3(y) - M3(l - y) + fo_ - a_ + b2 , x e ]0, l [ a(x, y) - JM_(1 - y) + M3(y) - M3(l - y) + a3 , x = 0 [M_(l - v) + M3(y) - M3(l - y) + a5, x = 1 /7.x v . = fM i ( l - >') + M2(l - y) + M3(y) - M3(l - y) + 6. , x e ]0 , l [ I ' ^ |M_(1 - y) + M2(l - .v) + M3(y) - M3(l - y) + a_ + a3- b2 , x = 0 [Af_(l - y) - M3(l - j,) + M2(y) + b2 , x e ] 0 , 1 [ /c(x, y) = J M _ ( 1 - y) - M3(l - j ) + M2(y) + a2 - ax-b2, x = 0 (M_(l - y) - M3(l - y) + M2(y) + a, + a4 - bl , x = 1 f o r y e ] 0 , l[",if/i = 0; (2.3) f(x, y) = - „ ( x ) + fllx2 + a2(l - x)2 + a3, xe [0, 1[ a(x, y) = 4 X ) + fl4*2 + «s(l - ^')2 - a2 > xe[0, l] h(x, y) = - t /(x) + a4x2 + a6(l - x)2 + a3 , x e [0, 1[ k(x, y) = - _ ( x ) + a_x2 + «5(1 - x)2 - a6 , x e [0, 1] 111 forj>e]0, l[",if/J = 2; and (2.4) f(x, y) = at* + a2(l ~ xf + a3 , g(x, y) = hx" + b2(\ - xf - a2 , h(x, y) = hx" + b3(l - xf + a3 , k(x, y) = OiX11 + b2(l - xf - b3 , for any y e ]0, l[", if /? + 0 ,1 , 2, where /, M ; and d are arbitrary solutions of (2.5) l(uv) =*/(«)+ l(v)(u,ve]0, oo[) with (2.6) 0 . / ( 0 ) : = 0 , (2.7) M;(ut;) = Mf(«) + M,{t>) («, v e ]0, 1["), (2.8) d(x + y) = <*) + d(j) rf(x^) = x d(y) + y d(x) x, y e R (that is d is a real derivation) and C;, ah b( are arbitrary constants. Remark 1. The convention 0 . 1(0) := 0 is used only to represent/, g, h, k as given in (2.1) and is not used in the proof at all. We also follow the convention 0" = 0 (P * 0), 1" = 1. The following lemma will be useful in the sequel and elsewhere. Lemma. If M^w) = M2(l - u) + C, where Mx and M 2 are solutions of (2.7), then Mx = M2 = 0 and C = 0. That is, 1, Mt(u) and M2(l - u) aie linearly independent. Proof. By (2.7), M±(uv) = Mx(u) + M,(v) = M2(l - u) + M2(l - v) + 2C = = M2(l - « - v + uvj + 1C = Mi(u + v - uv) + C That is, Mx(uvj(u + v - uv)) = C for u, u e ]0, l[". For fixed u e ]0, 1[", as v varies over ]0,1[", uvj(u + v — uv) varies over ]0 ,«[ . As u is arbitrary, it follows that M,(() = C, for t e ]0, l[". Thus Mx = 0, C = 0. Hence M2 = 0. This proves the lemma. • Proof of the Theorem. Case 1. Let us first treat the case when fi = T. For fixed u, >> e ]0 , 1[", (1.3) becomes a special case of (1.2) (ft = 1) treated in [5, 2], with F(x)=f(x,y), G(x) = ( x , r £ - V tf(x) = % , , ) , x(x) = / c ^ , r ^ - ) 112 so that F, G, H, K are given by (2.9) F(x) = m(x) + dxx + d2 G(x) = m(x) + d3x + ds - d2 + d4 - d3 , x e ]0, 1 [ H(x) = m(x) + d4x + ds K(x) = m(x) + (d. - ds + d2 - d4 + d3)x + d4 - d3 where m is a symmetric (m(x) = m(\ — x)) solution of m(x) + (1 - x) m LlL-\ = = m(y) + (1 - y) m (~^\ , for x, y, x + y e]0,1[ . Then from [9, 10], it follows that m(x) = x l(x) + (1 - x) l(x), x e ]0 ,1[ where / is an arbitrary solution of (2.5). Either by [2] or by the examination of F, G, H, K at the boundary points, the solution (2.9) can be extended to 0 ,1 , that is valid at the appropriate boundary points too. Thus from (2.9) and the form of m(x), with the convention (2.6) we have (2.10) f(x, y) = x l(x) + (1 - x) 1(1 - x) + dt(y, v)x + d2(y, v), g U, - — - ) = u l(u) + (1 - «) 1(1 -u)+ d3(y, v)u + + (ds - d2 + d4- d3) (y, v), h(x, v) = x l(x) + (1 - x) 1(1 - x) + d4(y, v) x + ds(y, v) k (u, -^--\ = u /(H) + (1 - u) 1(1 -u) + + (d, - ds + d2- d4 + d3) (y, v)u + (d4 - d3) (y, v) for x e [0,1[, u e [0, l ] , y, v, y + v e ]0, 1 [". Here / would be a function of v and y. Now we determine the d{ and /. From the forms of/and h in (2.10) we can conclude that / is independent of y and v, and that (2.11) dt(y, v) = a function of y alone = dx(y) say; similarly d2(y, v) = d2(y) say; d4(y, v) = d4(v) say; ds(y, v) = ds(v) say; and from the forms of g and k in (2.10) that, (2-12) d3(y,v)=d3(^j, 113 (2.13) (d5 ~d2 + dA- d3) (y, v) = A UlL-\ , (2.14) (dt -d5 + d2-d4 + d3) (y, v) = B (-^~j , (2.15) (d4 - d3)(y, v) = C(-^A , for y, v, y + ve]0, l [" . Now (2.11), (2+4) and (2.15) give, (d, + d2)(y) - d5(v) = (B + C)(yj(l - v)), which is the Pexider equation, (dy + d2)(rs) = d5(l - s) + (B + C)(r), (r = yl(l—v) e]0,1[", s — 1 - v 6 ]0, l["). The general solution is given by [1] as, (2.16) (dy +d2)(y) = L1(y) + al + a2, d5(y) = Ly(i -y) + ax, (B + C) (y) = L,(y) + a2 , y e ]0. 1[», with arbitrary constants a1; a2 and Lt an arbitrary solution of (2.7). Similarly, (2.11), (2.12) and (2.13) imply the Pexider equation (d4 + d5) (v) = d2(y) + (d3 + A). . (vl(l - y)) so that (2.17) (d4 + d5(y) = L2(y) + 6. + b2, d2(y) = L2(l - y) + b2 , (d3 + A) (y) = L2(y) + b2 , for y 6 ]0, 1 [", where by, b2 are arbitrary constants and L2 is a solution of (2.7). From (2.16), (2.17) and (2.15), we get, (2+8) dt(y) = Ly(y) - L2(l - y) + a, + a2 - by , (2.19) d4(y) = L2(y) - Ly(l - v) + by + b2 - a, . Now, (2.15J can be written as, (2.20) dA(v) = d(^L\ + c( y 1 - y) \l - v which by (2.19) and by the substitution v\(i - y) = t, yj(l - v) = r e ] 0 , l[", thus v = t(i — r)/(l — rt), 1 — v = 1 — t/(l - rt), reduces to the Pexider equation \d3(t) - L2(t) + Ly(i - t)] + (C(r) - L2(l - r) -by-b2 + ay) = = Ly(i - tr) - L2(l - tr) Now, the use of the lemma gives Lx = L2 and (2.21) d3(y) ~ Ly(y) ~ Ly(l - y) + C3 , where C3 is a constant. Now the functions du d2, d3, d4, ds, are given by (2.18), (2.16), (2.21), (2.19) and 114 (2A6) respectively. Thus the d;'s are of the form (2.22) d,(y) = L,(y) - L,(l - y) + Cu d2(y) = L.(l - v) + C2 , d3(y) = M y ) - L2(I - y) + C3. y e ]o, i[-rf4(y) = L ^ ) - L^ l - y) + C4, d5(y) - L.(l - y) + C5 , where E! is a solution of (2.7) and C,'s are constants. Indeed j , g, h, k given by (2A0) with d,'s given by (2.22) satisfy the functional equation (1.3) when /? = 1. Thus j , g, h, k are of the form given in (2.1). Case 2. Let us now consider the case when /? — 0. As in Case 1, for fixed t>, y, e e ]0 , 1[" (1.3) can be reduced to a special case of (1.2) (ft = 0) [5, 3], so that E, G, H, K are given by E(x) = m(x) + a , G(x) = m(x) + b , H(x) = m(x) + c , K(x) = m(x) + d, for x e ]0, 1[, where E(x) = j(x, y), G(x) = g(x, vj(\ - y)), H(x) = h(x, v), K(x) = = k(x, y/(l — v)) and m, a symmetric solution of m(x) + m (—— ) = m(u) + m ( — ) , x, u,x + u e ]0 , 1[ . \1 - xj \ l - uj By letting T(p, q) = m(p) + 2m(q), it is easy to show (as in [6]) that Tis symmetric, that is, m is a constant, so that E(x) = C1; G(x) = C2 , H(x) = C3 and K(x) = C4 on ]0, l [ . The use of [3] or the examination of E, G, H, K satisfying (1.3) at the boundary ,, N f C3 + a - C2 , x = 0 *(*-o)-R *.ju[ x = 0 fC, x = 0 xe]0, 1[ k(x, —̂ — j = ic . + C2 - C3, xe]0, l[ x = 1 V 1 ~ V [Ci + b_ c3, x = 1 where a, ft, C, C; are functions of y, v e ]0, l[". From the forms of/, a, ft, k it is easy to see that Ct is a function of y only, say C!(y), similarly C3, a, C2, b and C are of the form C3(v), a(vj({ - y)), C2(vj(l - >•)), 6(»/(l - y)) and c(y/(l - o)) and (2.24) (c3 + c - c2) (y, v) = A(y) (say); (C3 + a - c2) (y, v) = B(v) (say) ( c , + c2 - c3) (y, p) = E ( j ^ ; ) (say) a n d (Ci + b ~ C3) (y, v) = = « ( -^) ( ! W ) . 115 points reveals that (2.23) .., , \c3 + c 1^-Ҷс. - C2 , X = 0 x є ] 0 , 1[ / „ \ ía' л   . fl í x ł - \ C x є ] 0 , l [ k(. x = 1 V вvi-y)-\ï; From the third equation in (2.24) and the foims of Cx, C%, C3 given above we get (2.25) Cj(v) + C2 (—^A = C3(v) + E ( ~ A , fO" 3», v,y + ve~\0, 1[". Now we prove by induction on n, that (2.26) Cx(y) = M.(l - v) + M20) + ax C2(y) = M.(l - j>) + M30) - M3(l - y) + bx ~ ax + b2 C3(y) = Mj(l - v) + M2(l - v) + M3(>>) - M3(l ~ y) + bx, for y e ]0 , l [ " , where M ; are solutions (2.7), ay, bx, b2 arbitrary constants. For n = 1, the result follows by [10]. Let the result (2.26) hold for (n - 1). Now to prove (2.26) for n. Let y = (/, s), v = (r, w) with /, r, / + r 6 ] 0 , l [ " ~ \ s, w, s + we]0, 1[, so that (2.25) can be rewritten as c . M + ca(------. - ^ - ) = C3(»,w) + E(-J~, -L-). \ 1 - / 1 - s / \ 1 - r 1 - wj For fixed s, w e ]0,1 [, by induction hypothesis, from the above operation we obtain (2.27) C,(/, s) = m.(l - f) + m2(f) + ax C2(t, -^—J = ™x(l - /) - m3(l - f) + m3(f) + bx - ax + c4 C3(/, w) = m.(l - /) + m2(l - /) - m3(l — t) + f7/3(/) + /jj E (/, — — ) = mi(l - t) - m3(l - /) + m2(t) + C4, V 1 - w) for f e ]0 , l [ " ~ \ s,w,s + w e ] 0 , 1[ where mt, solutions of a(fr) = a(f) + a(r) for /, r e ]0, 1["_ 1 and constants a1; bx, C4 are functions of s and w. From the forms of Cx, C2, C3 and E given above and the lemma, we conclude that, (2.28) mx(l - t, s, w) = nx(l - t, s) (say), m2(t, s, w) = n2(t, s), ax(s,w) = ch(s). (2.29) (mx - m3) (1 - f, s, w) = n ( l - f, - — - j , m3(t, s, w) = n3(f, •—- j , (fe, - «! + c4) (s, w) = e ( j - ^ J 116 (2.30) m3(t, s, w) = v3(t, w), bx(s, w) = bx(w) (2.31) m2(t, s, w) = v2 ft, -~~\ , cA(s, w) = c4 (~A • From the forms m2 given by (2.28) and (2.31) it is easy to see that m2(t, s, w) is independent of s and w, that is, m2 is a function of t only say m2(t). Similarly m3(t, s, w) occurring in (2.29) and (2.30) is a function of t only say m3(t). From this it follows that mt(l — t, s, w) occurring in (2.28) and (2.29) is a function of tonly say mi(l - t). Further, from (2.28) to (2.31), we get w . ) _ . l W + Ҷ_i-)-.(-i-) for w, 5, w + s 6 ]0, l [ . Then by [10], it follows that (2.32) b,(w) = /j(l - w) + l2(w) + a2 c4(w) = 1,(1 - w) + l3(w) - /3(i - w) + b2- a2 + b3 a,(w) = h(l - w) + Z2(l - w) - /3(1 - w) + l3(w) + b2 , for w e ] 0 , l [ , where Z; are solutions of (2.5) and a2, b2, b3 are constants, From (2.27) and (2.32), it follows that Cv C2, C3 indeed are of the form given by (2.26). Similarly, the first, the second and the fourth equations in (2.24) and the forms of Cj's given by (2.26) yield c-') + c (r^.)"^ ) + c ,(rh)"6-so that COO = Af ,(1 - y) - M 3 ( l - v) + M2(y) + as , a(y) = M,(l - y) + M3(y) - M3(l - y) + a3 , b(y) = M,(l - y) + M3(y) - M,(l - y) + aA , ye ]0, l[» . With these values of Cu C2, C3, a, b, C the functions j , g, h, k given in (2.23) satisfy (1.3) for /? = 0, provided they are of the form given by (2.2). Case 3. Let us treat the case /? = 2. For fixed v, y e ]0, l[", (1.3) can be rewritten as a special case of (1.2) (/? = 2) with F(x) = j(x, y), G(x) = g(x, vftl - y)), H(x) = h(x, v), K(x) = k(x, y/(l - «)), so that from [5], we see that F(x) = n(l - x) + a,x2 + a2(l - x)2 + a3 , G(x) = n(x) + btx2 + b2(i - xf + b3, H(x) = n(x) + C,x2 + C3(l - xf + C3 , K(x) = n(l - x) + clxx2 + cl2(\ - xf + cl3, xe ]0, l [ 117 where at, bt, c;, di are constants and n is a solution of „(i -x) + (i - ^ " ( 7 ^ ) = »(0 + (i - «r»(- - — ) • From [2, 11], it follows that n(x) = d(x) + a(x2 + (1 - x)2 - 1), for x e ]0, l [ where rf is a derivation, that is a solution of (2.8). These F, G, H, K with n given above satisfy (1.2) for fi = 2, provided / ( * , >>) = F(x) = -d(x) + atx2 + a2(l - x)2 + a3, , g (u, - ^ - ) = G(M) = d(u) + Ctu2 + 6.(1 - «)2 - «2 , h(x, v) = H(x) = d(x) + C,x2 + C2(l - x)2 + a3 , k (it, '——) = K(u) = -d(u) + axii2 + 6,(1 - u)2 - C2 , x, u e]0, l [ where d satisfying (2.8), at, bi, C; constants are functions of y and v. Examination at the boundary points with the help of the equation (1.3) shows that (also refer to [2]), the above forms hold for x e [0, l [ , u e [0, 1]. From the forms of/ and h it is easy to see that d is independent of y and v. Noting that \,x2, (1 — x)2 are linearly independent, au occurring in / and k shows that ax(y, v) is a function of y and v/(l — y), that is a± is a constant. Similarly, we can show that a2, a3, Cu C2 and b1 are constants. That i s / , g, h, k are of the form (2.3) for x e [0, 1[, u e [0, 1], y e ]0, l[". Case 4. Finally, let us consider fi any real number, other than 0, 1 and 2. As before, for fixed y, ve~\0, l[", (1.3) takes the form (1.2) with F(x) = f(x, y), . G(x) = g(x, vj(l - y)), H(x) = h(x, v), K(x) = k(x, y/(l - v)), so that, from [5], it follows that f(x, y) = F(x) = ay + a2(i - xf + a3 , g (u, ~J—\ = G(u) = a y + b2(i - uf - a2 , h(x, v) = H(x) = bi*? + Cj(l - xf + a3 , k (u, -?—\ = K(u) = ay + b2(\ -uf - C\, for x e [0, l [ , u e [0,1] , y,v, y + ve ]0, 1[" where the constants ah bt, Cj are func-tions of y and v. The argument showing that ax, a2, a3, bu b2, C3 are constants is analogous 118 to that given in Case 3 (notice the fact that 1, x11, (1 - xf (/? 4= 0, 1, 2) are linearly independent). Thus, in this casej, g, h, k are of the from given by (2.4). This completes the proof of the theorem. • Remark. 2. From [8], it follows that M satisfying (2.7) is of the form M(u) = £ m{u) , u = (uy, u2, ..., u„) e ]0 , l[" i = l where m, is a solution of (2.5). Further, if/, a, ft, A: are measurable in each variable, then /, M1, M2, M3, d occuring in (2.1), (2.2) and (2.3) are given by l(x) = a log x, Mj(u) = £ al7 logw, j = 1,2,3 and d = 0. The solutions (2.4) are regular even t = i though no regularity condition was supposed. Remark 3. Some remarks on the solutions (2.2), (2.3) and (2.4) are in order. The solution (2.2) is independent of x because of the domain of definition of x. Whereas the domain of definition of x is [0, l ] which includes the end points, the domain of definition of y is ]0, l[", which excludes the end points. If the domain of definitions of x and y were ]0, l [ and ]0, l[", then indeed the solutions/, g, h and k would depend upon x and y (refer (2.6)). The solutions (2.3) and (2.4) are indepen-dent of y, this is simply because the multiplication factors of the second and fourth terms in (1.3) depend only on the variables in the first place. In order to obtain the dependency of the solution in the second variable y also, the equation to be con-sidered would be f(x,y) + (l-xY(l-y)y9(^rx, r ^ ) = = h(u, v) + (l- uY (1 - v)y k (j±- , - ^ j for x, y e [0, l [ with x + u e [0, l ] , y, v, y + v e ]0, 1[" where (l — v)y means (1 — V'I)'1 (1 — y2)y2... (1 — y„)y", yi e ]0, l[,yteR which would be dealt elsewhere. For example, if j = g = h = k in the above equation, a solution would bej(.x, y) = = ax**/ + b(l - xf (1 - y)y - b, for suitable p, y (refer [5, 7]). ACKNOWLEDGEMENT I wish to thank Professor J. Aczel for his help during the preparation of this paper. (Received February 15. 1982.) 119 R E F E R E N C E S [1] J. Aczel: On a generalization of the functional equation of Pexider. Publ. Inst. Math. (Beograd) 4, {18) (1964), 77 -80 . [2] J. Aczel: Notes on generalized information functions. Aequationes Math. 22 (1981), 97—107. [3] J. Aczel: Information functions of degree (0, P). Utilitas Math. 18 (1980), 15 — 26. [4] Z. Daroczy: Generalized information functions. Inform, and Control 16 (1970), 36—51. [5] PL. Kannappan: Notes on generalized information function. Tohoku Math. J. 30 (1978), 2 5 1 - 2 5 5 . [6] PL. Kannappan and P. N. Rathie: On a functional equation connected with Shannon's entropy. Funkcial. Ekvac. 14 (1971), 153 — 159. [7] PL. Kannappan and P. N. Rathie: On generalized information function. Tohoku Math. J. 27 (1975), 207-212. [8] M. Kuczma: Note on additive functions of several variables. Uniw. Slaski w Katowicach Prace Nauk. - Prace Mat. 21 (1972), 4 9 - 5 1 . [9] Gy. Maksa and C. T. Ng: The fundamental equation of information on open domains (to appear in Publ. Math. Debrecen). [10] Gy. Maksa: Remark 25. Proc. of the 18 lh International Symposium On Functional Equa-tions, Univ. of Waterloo, 1980, 42—43. [11] Gy. Maska: The general solution of a functional equation related to the mixed theory of information (to appear in Aequationes Math.). Prof. Dr. PL. Kannappan, Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario. Canada N2L3G1. 120 

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http://dml.cz/bitstream/handle/10338.dmlcz/124981/Kybernetika_19-1983-2_2.pdf

On a multiplace functional equation related to information measures and functions

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Institute of Mathematics AS CR, v. v. i.