Demiclosedness and weak convergence of supper hybrid mappings in Banach spaces

We introduce and study a new class of mapping in Banach Spaces, termed (α , β,γ) - supper hybrid mappings, which generalize the well – known ( α , β ) - generalized hybrid mappings. This extended framework encompasses a broader spectrum of nonlinear of nonlinear operators and allows for refined control via an additional parameter γ ≥ 0. We establish several foundational properties of supper hybrid mappings, including quasi – nonexpansivenes and the demiclosedness principle at zero. Furthermore, we prove a nonlinear ergodic theorem of Baillon’s type in Hilbert spaces for supper hybrid mappings, demonstrated weak convergence of the Cesàro means to a fixed point. Our approach leverages metric projections and techniques inspired by Takahashi, thereby extending classical fixed point theory to this new operator class.