Email this article On some equations stemming from quadrature rules
Abstract
We deal with functional equations of the type
F(y)-F(x) =(y-x)∑k=1nfk((1-λk)x+λky),
connected to quadrature rules and, in particular, we find the solutions of the following functional equation
f(x)-f(y)=(x-y)[g(x)+h(x+2y)+h(2x+y)+g(y)].
We also present a solution of the Stamate type equation
yf(x)-xf(y)=(x-y)[g(x)+h(x+2y)+h(2x+y)+g(y)].
All results are valid for functions acting on integral domains.
Keywords
Mathematics Subject Classification
References
Koclega-Kulpa, B. and T. Szostok. "On some equations connected to Hadamard inequalities." Aequationes Math. 75 (2008): 119-129.
Kuczma, M. "An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality." Warszawa-Kraków-Katowice: Panstwowe Wydawnictwo Naukowe, Uniwersytet Slaski, 1985.
Pawlikowska, I. "Solutions of two functional equations using a result of M. Sablik." Aequationes Math. 72 (2006): 177-190.
Riedel, T. and P.K. Sahoo. "Mean value theorems and functional equations." Singapore-New Jersey-Lodon-Hong Kong: World Scientific, 1998.
Sablik, M. and A. Lisak "On a problem of P.K. Sahoo." Report of meeting. The Seventh Katowice-Debrecen Winter Seminar on Functional Equations and Inequalities, January 31 - February 2, 2007. Ann. Math. Sil. 21 (2007): 70-71.
Sablik, M. "Taylor’s theorem and functional equations." Aequationes Math. 60 (2000): 258-267.
Sahoo, P.K. "On a functional equation associated with the trapezoidal rule." Report of meeting. The Forty-fourth International Symposium on Functional Equations, May 14-20, 2006. Aequationes Math. 73 (2007): 185.
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