ON A FUNCTIONAL EQUATION ASSOCIATED WITH THE TRAPEZOIDAL RULE - II

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1 Bulletin of the Institute of Mathematics Academia Sinica New Series) Vol ), No., pp. 9-3 ON A FUNCTIONAL EQUATION ASSOCIATED WITH THE TRAPEZOIDAL RULE - II BY PRASANNA K. SAHOO Astract In this paper, we find the solution f, f, f 3, f 4, f 5, g : R R of f y) g x) = y x) [f x) + f 3 sx + ty) +f 4 tx + sy) + f 5y)] for all real numers x and y. Here s and t are any two a priori chosen real parameters. This functional equation is a generalization of a functional equation that arises in connection with the trapezoidal rule for the numerical evaluation of definite integrals and is a generalization of a functional equation studied in [0].. Introduction Let R e the set of all real numers. The trapezoidal rule is an elementary numerical method for evaluating a definite integral a ft)dt. The method consists of partitioning the interval [a, ] into suintervals of equal lengths and then interpolating the graph of f over each suinterval with a linear function. If a = x o < x < x < < x n = is a partition of [a,] into n suintervals, each of length a n, then a ft)dt a n [fx o) + fx ) + + fx n ) + fx n )]. This approximation formula is called the trapezoidal rule. It is well known that the error ound for trapezoidal rule approximation is a ft)dt a n [fx o) + fx ) + + fx n ) + fx n )] K a)3 n Received April 9, 006 and in revised form Feruary 6, 008. AMS Suject Classification: Primary 39B. Key words and phrases: Additive map, functional equation, trapezoidal rule. 9

2 0 PRASANNA K. SAHOO [March where K = sup { f ) x) x [a,] }. It is easy to note from this inequality that if f is two times continuously differentiale and f ) x) = 0, then a ft)dt = a n [fx o) + fx ) + + fx n ) + fx n )]. This is oviously true if n = 3 and it reduces to a ft)dt = a 6 [fx o ) + fx ) + fx ) + fx 3 )]. Letting a = x, = y, x = x+y 3 and x = x+y 3 in the aove formula, we otain y x ft)dt = y x 6 [ fx) + f x + y 3 ) + f x + y 3 ) ] + fy). ) This integral equation ) holds for all x,y R if f is a polynomial of degree at most one. However, it is not ovious that if ) holds for all x,y R, then the only solution f is the polynomial of degree one. The integral equation ) leads to the functional equation gy) gx) = y x 6 [ fx) + f x + y 3 ) + f x + y 3 ) ] + fy) where g is an antiderivative of f. The aove equation is a special case of the functional equation f y) g x) = y x)[f x) + f 3 sx + ty) + f 4 tx + sy) + f 5 y)] 3) where s,t are two real a priori chosen parameters, and f,f,f 3,f 4,f 5,g : R R are unknown functions. It should e noted that if we consider n = in the approximation formula, then the functional equation )[ ) ] y x x + y gy) gx) = fx) + f + fy) 4 arrises analogously and it is a special case of gy) gx) = y x) [φx) + ψy) + hsx + ty)]. )

3 009] THE TRAPEZOIDAL RULE This functional equation was treated y Kannappan, Riedel and Sahoo [6] also see [9]) without any regularity conditions. Interested reader should see [-9, -] for related functional equations whose solutions are polynomials. The present paper is a continuation of the author s works in [0]. In this paper, our goal is to determine the general solution of the functional equation 3) assuming the unknown functions g,f,f,f 5 : R R to e twice differentiale and f 3,f 4 : R R to e four-time differentiale.. Some Auxiliary Results The following result from [0] will e instrumental in solving the functional equation 3). Lemma. Let s and t e any two a priori chosen real parameters. Suppose g : R R and f : R R are twice differentiale and k : R R is four time differentiale. The functions f,g,h,k : R R satisfy the functional equation 3), that is gy) hx) = y x) [fx) + k sx + ty) + k tx + sy) + fy)] for all x,y R if and only if hx) = gx) and ax + x + c if s = 0 = t ax + x + c if s = 0, t 0 ax + x + c if s 0, t = 0 3ax 4 + x 3 + cx + d + β)x + α if s = t 0 ax 3 + cx + β d)x + α if s = t 0 3 a i ist[s i + t i ]x i+ gx) = i= + [ i +s i +t i )a i ]x i+ + 0 x+c 0 if s t, s t) st i=0 5 a i ist[s i + t i ]x i+ i= + [ i +s i +t i )a i ]x i+ + 0 x+c 0 if s t, s t) =st i=0

4 PRASANNA K. SAHOO [March ax + 4η0) if s = 0 = t ax + ηtx) if s = 0, t 0 ax + ηsx) if s 0, t = 0 ax 3 + x + c d)x + β if s = t 0 fx) = 3ax + cx + β if s = t 0 3 [ a i ists i +t i ) s i +t i ) ] x i + x+ 0 if s t,s t) st i= 5 [ a i ists i +t i ) s i +t i ) ] x i + x+ 0 if s t,s t) =st i= ηx) if s = 0 = t ηx) if s = 0, t 0 ηx) if s 0, t = 0 a x ) 3 x ) 4 δ x s + d 4 if s = t 0 kx) = a x ) s d k x) if s = t 0 3 a i x i if s t, s t) st i=0 5 a i x i i=0 if s t, s t) = st where η : R R is an aritrary function, and a i i = 0,,,...,5), i i = 0,),a,, c, d, c 0, α, β, δ are aritrary real constants. Lemma. Let s and t e any two a priori chosen real parameters. Suppose φ : R R is twice differentiale. The functions φ,ψ : R R satisfy the functional equation y x) [ψx) + φsx + ty) φtx + sy) ψy)] = 0 4) for all x,y R if and only if ψx) = ωtx) + α + β if s = 0, t 0 ωsx) + α + β if s 0, t = 0 α if s = t ax + α if s = t 0 at s )x + t s)x + α if s t

5 009] THE TRAPEZOIDAL RULE 3 φx) = ωx) if s = 0, t 0 ωx) if s 0, t = 0 ωx) if s = t a x s + φ x) if s = t 0 ax + x + c if s t where ω : R R is an aritrary function, and a,,c,α,β are aritrary real constants. Proof. From 4) we have ψx) + φsx + ty) = φtx + sy) + ψy) 5) for all x,y R with x y. It is easy to see that 5) also holds in the case x = y. Letting y = 0 in 5), we otain ψx) = φtx) φsx) + α 6) where α is a constant given y α = ψ0). Letting 6) into 5), we see that φsx + ty) φsx) φty) = φsy + tx) φtx) φsy) 7) for all x,y R with x y. Now we consider several cases. Case. Suppose s = 0 and t 0. Then from 6), we have ψx) = φtx) + β + α 8) where the constant β is given y β = φ0). In this case, letting this ψx) in 8) into 5), we see that ψx) is a solution for any aritrary function φx). Case. Suppose s 0 and t = 0. This is case symmetric to Case and hence we have ψx) = φsx) + β + α and φx) = ωx) 9) where the constant β is given y β = φ0) and ωx) is an aritrary function.

6 4 PRASANNA K. SAHOO [March Case 3. Suppose s = t. Then from 6), we have ψx) = α 0) where the constant α is given y α = ψ0). In this case, letting this ψx) in 9) into 5), we see that ψx) is a solution for any aritrary function φx). Case 4. Suppose s = t 0. Then from 6), we have ψx) = φ sx) φsx) + α ) where the constant α is given y α = ψ0). From 7), we have φsx y)) φ sx y)) = φsx) φ sx) φsy) φ sy)) ) for all x,y R with x y. Defining Ax) = φsx) φ sx) 3) for all x R, we have from 3) Ax y) = Ax) Ay) 4) for all x,y R with x y. Letting x = 0 in 4), we otain A y) = Ay). 5) Replacing y y y in 4) and using 5) we have Ax + y) = Ax) A y) = Ax) + Ay) 6) for all x,y R. Hence A : R R is an additive function. Since φ is differentiale, A : R R is also differentiale and hence Ax) = ax 7) where a is an aritrary constant. From 3) and 7) we have φx) = a x + φ x) 8) s

7 009] THE TRAPEZOIDAL RULE 5 for all x R, and from ) we otain Replacing a y a, we have the asserted solution ψx) = ax + α. 9) φx) = a x s + φ x) and ψx) = ax + α. Case 5. Suppose s t 0. Differentiating 7) twice, first with respect to x and then with respect to y, we otain φ sx + ty) = φ sy + tx) 0) for all x,y R. Since s t, letting u = sx + ty and v = sy + tx we see that u and v are linearly independent and 0) yields φ u) = φ v) ) for all u,v R. Hence φ u) = a, where a is a constant. Integrating we have φx) = ax + x + c ) where,c are constants of integration. Using ) in 6), we otain ψx) = at s )x + t s)x + c. 3) Letting φx) in ) and ψx) in 3) into 5), we see that φx) and ψx) satisfy the functional equation with aritrary constants a,,c. Remark. For the case s = t, the unknown function φx) could not e determined explicitly. We have found the explicit form of φx) φ x). As the referee noticed, in this case the unknown function φx) is an aritrary function satisfying φx) φ x) = a x s. One can rephrase in this way to avoid explaining ignotum per ignotum ut it is asically the same as what we have in the lemma. 3. Main Result Now we present the solution of the functional equation 3).

8 6 PRASANNA K. SAHOO [March Theorem. Let s and t e any two a priori chosen real parameters. Suppose g,f,f,f 5 : R R are twice differentiale and f 3,f 4 : R R are four time differentiale. The functions g,f,f,f 3,f 4,f 5 : R R satisfy the functional equation 3), that is f y) g x) = y x) [f x) + f 3 sx + ty) + f 4 tx + sy) + f 5 y)] for all x,y R if and only if g x) = f x) and f x) = f x) = ax + x + c) if s = 0 = t ax + x + c) if s = 0, t 0 ax + x + c) if s 0, t = 0 3ax4 + x 3 + cx + β + d)x + α), if s = t 0 ax 3 + cx + β d)x + α ) if s = t 0 3csts + t)x 4 + 4dstx 3 + [γ + es + t)]x +[γ + α]x + δ x + ε 5asts 3 + t 3 )x 6 + 4sts + t )x 5 +3csts + t)x 4 + 4dstx 3 if s t, s t) st + [γ + es + t)]x + [γ + α]x +δ x + ε if s t, s t) = st ) ax + α + 4η0) if s = 0 = t ax + ηtx) + ωtx) + α + β) if s = 0, t 0 ax + ηsx) + ωsx) + α + β) if s 0, t = 0 ax 3 + x + c δ)x + β + α ) if s = t 0 3ax + cx + β + x + e ) if s = t 0 c[3sts + t) s 3 + t 3 )]x 3 + d[4st s + t )]x + γ x + δ + [ At s )x + B t s)x + D ] if s t, s t) st a[5sts 3 + t 3 ) s 5 + t 5 )]x 5 + [4sts + t ) s 4 + t 4 )]x 4 + c[3sts + t) s 3 + t 3 )]x 3 +d[4st s + t )]x + γ x + δ + [ At s )x + B t s)x + D ] if s t, s t) = st

9 009] THE TRAPEZOIDAL RULE 7 f 3 x) = f 4 x) = f 5 x) = a x 4 s ηx) + ωx)) if s = 0 = t ηx) + ωx)) if s = 0, t 0 ηx) + ωx)) if s 0, t = 0 ) 3 + x ) δ x d 4 + ωx) if s = t 0 a x ) s d x s) f4 x) if s = t 0, cx 3 + dx + ex + α + Ax + B x + C) if s t, s t) st ax 5 + x 4 + cx 3 + dx + ex + α + Ax + B x + C) if s t, s t) = st a x 4 s ηx) ωx)) if s = 0 = t ηx) ωx)) if s = 0, t 0 ηx) ωx)) if s 0, t = 0 ) 3 + x ) δ x d 4 ωx) if s = t 0 a x ) s d + x s) f3 x) if s = t 0 cx 3 + dx + ex + α Ax + B x + C) if s t, s t) st ax 5 + x 4 + cx 3 + dx + ex + α Ax + B x + C) c[3sts + t) s 3 + t 3 )]x 3 if s t, s t) = st ) ax α + 4η0) if s = 0 = t ax + ηtx) ωtx) α β) if s = 0, t 0 ax + ηsx) ωsx) α β) if s 0, t = 0 ax 3 + x + c δ)x + β α ) if s = t 0 3ax + cx + β x e ) if s = t 0 +d[4st s + t )]x + γ x + δ [ At s )x + B t s)x + D ] if s t, s t) st a[5sts 3 + t 3 ) s 5 + t 5 )]x 5 + [4sts + t ) s 4 + t 4 )]x 4 + c[3sts + t) s 3 + t 3 )]x 3 +d[4st s + t )]x + γ x + δ [ At s )x + B t s)x + D ] if s t, s t) = st where η,ω : R R are aritrary functions and A,B,C,D,a,,c,d,e,α,β,γ, δ, ε are aritrary real constants.

10 8 PRASANNA K. SAHOO [March Proof. Letting x = y in 3) we see that f x) = g x) for all x R. Hence 3) yields f y) f x) = y x)[f x) + f 3 sx + ty) + f 4 tx + sy) + f 5 y)] 4) for all x,y R. Interchanging x with y in the functional equation 4), we otain f y) f x) = y x)[f y) + f 3 sy + tx) + f 4 ty + sx) + f 5 x)] 5) for all x,y R. Adding 4) and 5), we get gy) gx) = y x)[fx) + k sx + ty) + k tx + sy) + fy)] 6) where fx) = f x) + f 5 x) kx) = [f 3x) + f 4 x)] gx) = f x). 7) Similarly, sutracting 5) from 4), we get for all x,y R, where y x) [ψx) + φsx + ty) φtx + sy) ψy)] = 0 8) { ψx) = f x) f 5 x) φx) = f 3 x) f 4 x). 9) Now we consider several cases. Case. Suppose s = 0 = t. Then from 7), 9), Lemma and Lemma, we otain f x) = ax + x + c f x) + f 5 x) = ax + 4η0) f 3 x) + f 4 x) = ηx) f x) f 5 x) = α f 3 x) f 4 x) = ωx), 30)

11 009] THE TRAPEZOIDAL RULE 9 where η,ω : R R are aritrary functions and a,,c,α are aritrary constants. Hence from 30) we have f x) = ax + x + c) ) f x) = ax + α + 4η0) f 3 x) = ηx) + ωx)) f 4 x) = ηx) ωx)) ) f 5 x) = ax α + 4η0). 3) Case. Suppose s = 0 and t 0. Then from 7), 9), Lemma and Lemma, we otain f x) = ax + x + c f x) + f 5 x) = ax + ηtx) f 3 x) + f 4 x) = ηx) f x) f 5 x) = ωtx) + α + β f 3 x) f 4 x) = ωx), 3) where η,ω : R R are aritrary functions and a,,c,α,β are aritrary constants. Hence from 3), we get f x) = ax + x + c) f x) = ax + ηtx) + ωtx) + α + β) f 3 x) = ηx) + ωx)) f 4 x) = ηx) ωx)) f 5 x) = ax + ηtx) ωtx) α β). 33) Case 3. Suppose s 0 and t = 0. Then from 7), 9), Lemma and Lemma, we otain f x) = ax + x + c f x) + f 5 x) = ax + ηsx) f 3 x) + f 4 x) = ηx) f x) f 5 x) = ωsx) + α + β f 3 x) f 4 x) = ωx), 34) where η,ω : R R are aritrary functions and a,,c,α,β are aritrary

12 0 PRASANNA K. SAHOO [March constants. Hence from 34), we get f x) = ax + x + c) f x) = ax + ηsx) + ωsx) + α + β) f 3 x) = ηx) + ωx)) f 4 x) = ηx) ωx)) f 5 x) = ax + ηsx) ωsx) α β). 35) Case 4. Suppose s = t 0. Then from 7), 9), Lemma and Lemma, we otain f x) = 3ax 4 + x 3 + cx + β + d)x + α f x) + f 5 x) = ax 3 + x + c δ)x + β f 3 x) + f 4 x) = a x ) 3 x ) δ x d ) 4 f x) f 5 x) = α f 3 x) f 4 x) = ωx), 36) where ω : R R is an aritrary function and a,,c,d,α,β,δ are aritrary constants. Hence from 36), we get f x) = 3ax4 + x 3 + cx + β + d)x + α) f x) = ax 3 + x + c δ)x + β + α ) f 3 x) = a x ) 3 x ) δ x d 4 + ωx) f 4 x) = a x ) 3 x ) δ x 4 s + d 4 ωx) f 5 x) = ax 3 + x + c δ)x + β α ). 37) Case 5. Suppose s = t 0. Then from 7), 9), Lemma and Lemma, we otain f x) = ax 3 + cx + β d)x + α f x) + f 5 x) = 3ax + cx + β f 3 x) + f 4 x) + f 3 x) + f 4 x) = a x s) d f x) f 5 x) = x + e f 3 x) f 4 x) f 3 x) + f 4 x) = x s, 38)

13 009] THE TRAPEZOIDAL RULE where a,, c, d, e, α, β are aritrary constants. Hence from 38), we get f x) = ax 3 + cx + β d)x + α ) f x) = 3ax + cx + β + x + e ) f 3 x) = a x s) d x s) f4 x) f 4 x) = a x s) d + x s) f3 x) f 5 x) = 3ax + cx + β x e ). 39) Case 6. Suppose s t 0 and s t) st. Then from 7), 9), Lemma and Lemma, we otain f x) = { 3csts + t)x 4 + 4dstx 3 + [γ + es + t)]x +[γ + α]x + δ x + ε } f x) + f 5 x) = c[3sts + t) s 3 + t 3 )]x 3 +d[4st s + t )]x + γ x + δ f 3 x) + f 4 x) = cx 3 + dx + ex + α) f x) f 5 x) = At s )x + B t s)x + D f 3 x) f 4 x) = Ax + B x + C, 40) where c,d,e,a,b,c,d,α,β,γ,δ, ε are aritrary constants. Hence from 40), we get f x) = 3csts + t)x 4 + 4dstx 3 + [γ + es + t)]x +[γ + α]x + δ x + ε f x) = c[3sts + t) s 3 + t 3 )]x 3 + d[4st s + t )]x +γ x + δ + [ At s )x + B t s)x + D ] f 3 x) = cx 3 + dx + ex + α + Ax + B x + C) f 4 x) = cx 3 + dx + ex + α Ax + B x + C) f 5 x) = c[3sts + t) s 3 + t 3 )]x 3 + d[4st s + t )]x +γ x + δ [ At s )x + B t s)x + D ]. 4) Case 7. Suppose s t 0 and s t) = st. Then from 7), 9),

14 PRASANNA K. SAHOO [March Lemma and Lemma, we otain f x) = { 5asts 3 + t 3 )x 6 + 4sts + t )x 5 +3csts + t)x 4 + 4dstx 3 + [γ + es + t)]x +[γ + α]x + δ x + ε } f x) + f 5 x) = a[5sts 3 + t 3 ) s 5 + t 5 )]x 5 + [4sts + t ) s 4 + t 4 )]x 4 + c[3sts + t) s 3 + t 3 )]x 3 +d[4st s + t )]x + γ x + δ f 3 x) + f 4 x) = ax 5 + x 4 + cx 3 + dx + ex + α) f x) f 5 x) = At s )x + B t s)x + D f 3 x) f 4 x) = Ax + B x + C, 4) where a,,c,d,e,a,b,c,d,α,β,γ,δ,ε are aritrary constants. Hence from 40), we get f x) = 5asts 3 + t 3 )x 6 + 4sts + t )x 5 + 3csts + t)x 4 +4dstx 3 + [γ + es + t)]x + [γ + α]x + δ x + ε f x) = a[5sts 3 + t 3 ) s 5 + t 5 )]x 5 + [4sts + t ) s 4 + t 4 )]x 4 + c[3sts + t) s 3 + t 3 )]x 3 + d[4st s + t )]x +γ x + δ + [ At s )x + B t s)x + D ] f 3 x) = ax 5 + x 4 + cx 3 + dx + ex + α + Ax + B x + C) f 4 x) = ax 5 + x 4 + cx 3 + dx + ex + α Ax + B x + C) f 5 x) = a[5sts 3 + t 3 ) s 5 + t 5 )]x 5 + [4sts + t ) s 4 + t 4 )]x 4 + c[3sts + t) s 3 + t 3 )]x 3 + d[4st s + t )]x +γ x + δ [ At s )x + B t s)x + D ]. 43) Since there are no more cases are left, the proof of the theorem is now complete. Prolem. In Theorem, we have assumed that the functions f,f,f 5 : R R are twice differentiale and f 3,f 4 : R R are four time differentiale. The proof of Theorem heavily relies on this differentiaility assumption. Thus we pose the following prolem: Determine the general solution of the

15 009] THE TRAPEZOIDAL RULE - II 3 functional equation 3) without any regularity assumptions on the unknown functions f,f,f 3,f 4,f 5. Acknowledgments The work was partially supported y an IRI Grant from the Office of the Vice President for Research, University of Louisville. References. J. Aczel, A mean value property of the derivatives of quadratic polynomials without mean values and derivatives. Math. Mag., 58985), J. Ger, On Sahoo-Riedel equations on a real interval. Aequationes Math., 63 00), Sh. Haruki, A property of quadratic polynomials. Amer. Math. Monthly, 86979), PL. Kannappan, P. K. Sahoo, and M. S. Jacoson, A characterization of low degree polynomials. Demonstration Math., 8995), PL. Kannappan, T. Riedel and P. K. Sahoo, On a functional equation associated with Simpsons rule. Results in Mathematics, 3997), PL. Kannappan, T. Riedel and P. K. Sahoo, On a generalization of a functional equation associated with Simpson s rule. Prace Matematyczne, 5 998), T. Riedel and M. Salik, On a functional equation related to a generalization of Flett s mean value theorem. Internet. J. Math. & Math. Sci., 3000), T. Riedel and M. Salik, A different version of Flett s mean value theorem and an associated functional equation. Acta Mathematica Sinica, English Series, 0 004), P. K. Sahoo and T. R. Riedel, Mean Value Theorems and Functional Equations, World Scientific Pulishing Co., NJ, P. K. Sahoo, On a functional equation associated with the trapezoidal rule. Bull. Math. Acad. Sinica New Series), 007), L. Székelyhidi, Convolution Type Functional Equations on Topological Aelian Groups, World Scientific Pulishing Co., NJ, 99.. W. H. Wilson, On a certain general class of functional equations. Amer. J. Math., 4098), Department of Mathematics, University of Louisville, Louisville, Kentucky 409 USA. sahoo@louisville.edu