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We discuss the set-valued dynamics related to the theory of functional equations. We look for selections of convex set-valued functions satisfying set-valued Euler-Lagrange inclusions. We improve and extend upon some of the results in [13, 20], but under weaker assumptions. Some applications of our results are also provided.

In this paper, using the Brzdek’s fixed point theorem [9, Theorem 1] in non-Archimedean (2,β)-Banach spaces, we prove some stability and hyperstability results for an p-th root functional equation $$f\left( {\left( {\sum\limits_{i = 1}^k {a_i x_i^p } } \right)^{1/p} } \right) = \sum\limits_{i = 1}^k {a_i f\left( {x_i } \right)} ,$$ where p ∈ {1, ⋯, 5}, a1, ⋯, ak are fixed nonzero reals when p ∊ {1, 3, 5} and are fixed positive reals when p ∈ {2, 4}.

We present results on approximate solutions to the biadditive equation f ( x + y , z - w ) + f ( x - y , z + w ) = 2 f ( x , z ) - 2 f ( y , w ) on a restricted domain. The proof is based on a quite recent fixed point theorem in some function spaces. Our main results state that, under some weak natural assumptions, functions satisfying the equation approximately (in some sense) must be actually solutions to it. In this way we obtain inequalities characterizing biadditive mappings and inner product spaces. Our outcomes are connected with the well known issues of Ulam stability and hyperstability.