Abstract
We define generalized additive set-valued functional equations, which are related with the following generalized additive functional equations: f( x 1 +⋯+ x l )=(l−1)f( x 1 + ⋯ + x l − 1 l − 1 )+f( x l ), f ( x 1 + ⋯ + x l − 1 l − 1 + x l ) + f ( x 1 + ⋯ + x l − 2 + x l l − 1 + x l − 1 ) + ⋯ + f ( x 2 + ⋯ + x l l − 1 + x 1 ) = 2 [ f ( x 1 ) + f ( x 2 ) + ⋯ + f ( x l ) ] for a fixed integer l with l>1, and they prove the Hyers-Ulam stability of the generalized additive set-valued functional equations by using the fixed point method.MSC: 39B52, 54C60, 91B44.
Highlights
1 Introduction and preliminaries After the pioneering papers were written by Aumann [ ] and Debreu [ ], set-valued functions in Banach spaces have been developed in the last decades
We can refer to the papers by Arrow and Debreu [ ], McKenzie [ ], the monographs by Hindenbrand [ ], Aubin and Frankowska [ ], Castaing and Valadier [ ], Klein and Thompson [ ] and the survey by Hess [ ]
The theory of set-valued functions has been much related with the control theory and the mathematical economics
Summary
Introduction and preliminariesAfter the pioneering papers were written by Aumann [ ] and Debreu [ ], set-valued functions in Banach spaces have been developed in the last decades. = 2 f (x1) + f (x2) + · · · + f (xl) for a fixed integer l with l > 1, and they prove the Hyers-Ulam stability of the generalized additive set-valued functional equations by using the fixed point method. Let f : → (Ccb(Y ), h) be a set-valued function from a complete finite measure space ( , , ν) into Ccb(Y ).
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