Abstract
We prove the Hyers–Ulam stability of the functional equation *f(a1x1+a2x2,b1y1+b2y2)=C1f(x1,y1)+C2f(x1,y2)+C3f(x2,y1)+C4f(x2,y2)\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned}&f(a_1x_1+a_2x_2,b_1y_1+b_2y_2)=C_{1}f(x_1,y_1)\\nonumber \\\\ \\nonumber \\\\&\\quad +C_{2}f(x_1,y_2)+C_{3}f(x_2,y_1)+C_{4}f(x_2,y_2) \\end{aligned}$$\\end{document}in the class of functions from a real or complex linear space into a Banach space over the same field. We also study, using the fixed point method, the generalized stability of (*) in the same class of functions. Our results generalize some known outcomes.
Highlights
Problem of studying the stability of functional equations has begun with a question posed by S
Cieplinski starting with a bilinear mapping, i.e., linear in each of its arguments, considered the following functional equation
Firstly knowing already the form of solutions of (1) we prove its Hyers–Ulam stability, in the cases excluded in [10]
Summary
Problem of studying the stability of functional equations has begun with a question posed by S. Ulam (see, e.g., [17]) and an answer given by D.H. Hyers [13]. Since a number of papers investigating the so called Hyers– Ulam stability have appeared. The results concern various generalizations of the problem and these kind of research have their origins in the papers by T. Aoki [1], D.G. Bourgin [7], Th. Rassias [16], P. Let X and Y be linear spaces over the same field F ∈ {R, C}, a1, a2, b1, b2 ∈ F \ {0}, C1, C2, C3, C4 ∈ F and f : X2 → Y. Cieplinski starting with a bilinear mapping, i.e., linear in each of its arguments, considered the following functional equation
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