Solvability of a system of an operator equation and finitely many equality constraints

Abstract

The existence of solutions for the problem
\[

\mathcal{T}u=u, \ \alpha_j(u)=\theta_\mathcal{E},\;j=1,2,\dots,r

\]

is proved, by considering \'Ciri\'c-Joti\'c contraction condition,
where $\mathcal{T},\alpha_j: \mathcal{E}\to \mathcal{E}$ $(j=1,2,
\cdots,r)$ are mappings and $(\mathcal{E},\|\cdot \|)$ is a Banach
space. A set of sufficient conditions on $\mathcal{T},\alpha_j$ is
used which ensure the existence of, possibly, non-unique solution
to the underlying system. A common fixed point result is derived
from the obtained result and some illustrative examples are given
in order to justify the established results. An application to nonlinear matrix equations is also
presented