Stability of a composite functional equation related to idempotent mappings

We deal with approximate solutions of the following composite functional equation: T(T(x)−T(y))=T(x+y)+T(x−y)−T(x)−T(y),T(T(x)−T(y))=T(x+y)+T(x−y)−T(x)−T(y), where TT is a continuous function from a Banach space to itself. An extension of Ulam’s problem, known in the literature as the Ulam–Gaˇvruta–Rassias stability problem, is answered affirmatively for this equation. As a corollary we obtain an approximate property of continuous Banach space projections.