Existence and uniqueness of solutions in \(H_{1}(\Delta)\) of a general class of non-linear functional equations
The authors study the functional equation \[ f(z)+ \sum^m_{i=1} \alpha_i (z) f(p_iz)= g(z)+ \sum^k_{j=1} \sum^\infty_{s=3} c_s^j(z)\bigl[ f(q_jz) \bigr]^{s-1}, \;z\in\mathbb{C}, \tag{1} \] where \(m,k\) are positive integers, \(p_i,q_j\) are known complex constants and \(g(z)\), \(\alpha_i(z)\), \(c_s^j(z)\) are known functions belonging to the Banach space \(H_1(\Delta)= \{f(z)= \sum^\infty_{u=1} f_nz^{n-1}\), analytic \(\Delta\) in and \(\sum^\infty_{n=1} |f_n|<\infty\}\), where \(\Delta= \{z\in\mathbb{C}, |z|<1\}\). Using a fixed point theorem concerning holomorphic functions in abstract Banach spaces, the existence and uniqueness of a solution in \(H_1(\Delta)\) for the equation (1) is proved.
functional equations for complex functions
scientific article; zbMATH DE number 1915762