Relation-Theoretic Fixed Point Results for Block-Structured Nonlinear Operators in Hilbert Spaces

Relation-theoretic fixed point theory extends classical contraction principles by allowing contractive conditions to be imposed only on pairs of elements that satisfy a prescribed binary relation. This framework has proved useful for studying nonlinear operators that do not satisfy global contraction conditions in metric or normed spaces.

Let H be a Hilbert space admitting a finite orthogonal decomposition H = H₁ ⊕ H₂ ⊕ ··· ⊕ H_k. Motivated by this structure, a binary relation R is introduced on H by defining xRy whenever the difference x − y belongs to one of the component subspaces H_i. This relation reflects the block structure of the space and restricts attention to pairs of elements that differ along a single orthogonal direction.

Nonlinear mappings T : H \to H that preserve this relation and satisfy a Banach-type contraction condition on related pairs, ‖Tx − Ty‖ ≤ α‖x − y‖, with 0 < α < 1, for all x, y with xRy, are considered. Under suitable relational admissibility conditions, the convergence behaviour of the associated Picard iteration defined by xₙ₊₁ = T(xₙ) is analyzed. It is shown that the generated sequence converges strongly to a fixed point of the operator.

The results illustrate how orthogonal decompositions of Hilbert spaces naturally induce relational structures that support relation-theoretic fixed point arguments for nonlinear operators whose behaviour may not be contractive in the global sense.