On Fichera's existence principle in functional analysis (Q2712813)

For a metric space \(( X,d) \) denote by \(\operatorname {Lip}X\) the space of all functions \(F:X\rightarrow\mathbb{R}\) which are Lipschitz on \(X\), i.e., such that NEWLINE\[NEWLINE\|F\|_{\operatorname {Lip}X}:=\sup\left\{ \frac{|F( x) -F( y)|}{d( x,y)}:x,y\in X,\;x\neq y\right\} <\infty. NEWLINE\]NEWLINE Let \(H\) be an arbitrary set and \(( H_{1},d_{1}),( H_{2},d_{2}) \) metric spaces. If \(T_{1}:H\rightarrow H_{1}\) and \(T_{2}:H\rightarrow H_{2}\) are two mappings and \(F\in \operatorname {Lip}H_{1}\), then one considers the functional equation NEWLINE\[NEWLINEU( T_{2}v) =F( T_{1}v), \qquad \forall v\in H,\tag{1}NEWLINE\]NEWLINE where \(U\in \operatorname {Lip}H_{2}\) is the unknown function. Main result: The equation (1) has a solution \(U\in \operatorname {Lip}H_{2}\) with the property \(\|U\|_{\operatorname {Lip}H_{2}}\leq k\cdot\|F\|_{\operatorname {Lip}H_{1}}\) (with given \(k>0\)) if and only if \(d_{1}( T_{1} u,T_{1}v) \leq kd_{2}( T_{2}u,T_{2}vt)\), for all \(u,v\in H\). This general result is applied to prove the solvability of other problems.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00046].