Stability of a Mixed Type Additive, Quadratic, Cubic and Quartic Functional Equation

M. Eshaghi-Gordji (), S. Kaboli-Gharetapeh (), M.S. Moslehian () and S. Zolfaghari ()
Additional contact information
M. Eshaghi-Gordji: Semnan University
S. Kaboli-Gharetapeh: Payame Noor University of Mashhad
M.S. Moslehian: Ferdowsi University of Mashhad
S. Zolfaghari: Semnan University

Chapter Chapter 6 in Nonlinear Analysis and Variational Problems, 2010, pp 65-80 from Springer

Abstract: Abstract We find the general solution of the functional equation $$\begin{array}{l} D_f {\rm{(}}x,y{\rm{)}}\,\,{\rm{: = }}f{\rm{(}}x + {\rm{2}}y{\rm{)}} + f{\rm{(}}x - {\rm{2}}y{\rm{)}} - {\rm{4[}}f{\rm{(}}x + y{\rm{)}} - f{\rm{(}}x - y{\rm{)]}} - f{\rm{(4}}y{\rm{)}} + {\rm{4}}f{\rm{(3}}y{\rm{)}} \\ - {\rm{6}}f{\rm{(2}}y{\rm{)}} + {\rm{4}}f{\rm{(}}y{\rm{)}} + {\rm{6}}f{\rm{(}}x{\rm{) }} = {\rm{ 0}}{\rm{.}} \\ \end{array}$$ in the context of linear spaces. We prove that if a mapping f from a linear space X into a Banach space Y satisfies f(0)=0 and $$\|D_f(x,y)\|\leq\epsilon \quad (x,y\in X),$$ where ε > 0, then there exist a unique additive mapping $$A:X\to Y,$$ a unique quadratic mapping $$Q_1:X\to Y,$$ a unique cubic mapping $$C:X\to Y$$ and a unique quartic mapping $$Q_2:X\to Y$$ such that $$\|f(x)-A(x)-Q_1(x)-C(x)-Q_2(x)\|\leq\frac{1087 \epsilon}{140}\quad \forall x\in X.$$

Keywords: Banach Space; Functional Equation; Positive Real Number; Real Vector Space; Ulam Stability (search for similar items in EconPapers)
Date: 2010
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-1-4419-0158-3_6

Ordering information: This item can be ordered from
http://www.springer.com/9781441901583

DOI: 10.1007/978-1-4419-0158-3_6

Access Statistics for this chapter

More chapters in Springer Optimization and Its Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().