Fixed point and quadratic \(p\)-functional inequalities in non-Archimedean Banach spaces (Q2792521)

After a brief presentation of the notion of non-Archimedean Banach spaces and a short story of the Hyers-Ulam stability of functional equations, the authors investigate the following functional equation: NEWLINE\[NEWLINE f\Big(\frac{x+y+z}{2}\Big)+f\Big(\frac{x-y-z}{2}\Big)+f\Big(\frac{y-x-z}{2}\Big)+f\Big(\frac{z-x-y}{2}\Big)=f(x)+f(y)+f(z) NEWLINE\]NEWLINE with \(f:X \to Y\), \(X\) a non-Archimedean vector space, \(Y\) a non-Archimedean Banach space, proving that its solutions are quadratic functions.NEWLINENEWLINEFurthermore, they consider the functional inequality NEWLINE\[NEWLINE\left\| f\Big(\frac{x+y+z}{2}\Big)+f\Big(\frac{x-y-z}{2}\Big)+f\Big(\frac{y-x-z}{2}\Big)+f\Big(\frac{z-x-y}{2}\Big)-f(x)-f(y)-f(z)\right\| \leqNEWLINE\]NEWLINE NEWLINE\[NEWLINE\| \rho(f(x+y+z)+f(x-y-z)+f(y-x-z)+f(z-x-y)-4f(x)-4f(y)-4f(z)) \| + \varphi(x,y,z),NEWLINE\]NEWLINE where \(\rho\) is a fixed non-Archimedean number with \(|\rho|<1/|4|\). Assuming \(f\) to be even, \(\varphi:X^3 \to [0,\infty)\) such that \(\varphi(0,0,0)=0\) and NEWLINE\[NEWLINE \varphi\Big(\frac{x}{2},\frac{y}{2},\frac{z}{2}\Big) \leq \frac{L}{|4|}\varphi(x,y,z), NEWLINE\]NEWLINE they prove that there exists a unique quadratic function \(Q\) such that NEWLINE\[NEWLINE \| f(x)-Q(x)\| \leq \frac{1}{1-L}\varphi(x,0,0). NEWLINE\]NEWLINENEWLINENEWLINEThe proof of this stability theorem is based on a fixed point theorem.NEWLINENEWLINESimilar results under different conditions are then presented.