The main purpose of this paper is to introduce a new general-type proximal point
algorithm for finding a common element of the set of solutions of monotone inclusion
problem, the set of minimizers of a convex function, and the set of solutions of fixed
point problem with composite operators: the composition of quasi-nonexpansive and
firmly nonexpansive mappings in real Hilbert spaces. We prove that the sequence xn
which is generated by the proposed iterative algorithm converges strongly to a
common element of the three sets above without commuting assumption on the
mappings. Finally, applications of our theorems to find a common solution of some
nonlinear problems, namely, composite minimization problems, convex optimization
problems, and fixed point problems, are given to validate our new findings.