Differentiability on semilinear spaces
The aim of this short note is to introduce a natural differentiability concept between semilinear spaces that may preserve the basic properties of the Gâteaux and Hukuhara differentials and may be efficiently used in applications on stochastic manifolds, set-valued and fuzzy differential equations, etc. More precisely, if \(S_1, S_2\) are topological semilinear spaces, \(A\subset S_1\) is open then a continuous mapping \(f(\cdot):A\to S_2\) is said to be differentiable at \(x_0\in A\) if there exist the neighborhoods \(U\subset A\) of \(x_0\) and \(I\subset\mathbb R\) of \(0\) and the continuous mappings \(df(\cdot,\cdot):U_0\times S_1\to S_2\), \(\varphi(\cdot,\cdot,\cdot):I\times U_0\times S_1\to S_2\) such that \(df(x,\cdot), \;x\in U_0\) are semilinear and satisfy: \[ \begin{gathered} f(x+t.y)=f(x)+t.df(x,y)+t.\varphi(t,x,y), \\ \lim_{t\to 0+}\varphi(t,x,y)=0, \;t\in I, \;x\in U_0, \;y\in S_1.\end{gathered} \] Several simple propositions containing the basic properties of the new differential, \(df(\cdot,\cdot),\) are proved.
scientific article; zbMATH DE number 5597870