Assume that is a mapping with satisfying.
Let be a 2-divisible Abelian group and a Banach space, and let be a mapping with satisfying the inequality.
According to the theorem of Borelli and Forti [16], we obtain the following generalization of stability theorem for the quadratic functional (1.1): let be an abelian group and a Banach space; let be a mapping with satisfying the inequality.
To prove the uniqueness, let be another generalized Euler-Lagrange type additive mapping with satisfying (2.9).
By using the random fuzzy mapping S satisfying with the corresponding function a : X → [0,1], we can define a random set-valued mapping S as follows: S : Ω × X → C B X X ), ( t, x ( t ) ) ↦ ( S t, x ( t ) ) a ( x ( t ) ), ∀ ( t, x ( t ) ) ∈ Ω × X, Where S t, x ( t ) = S ( t, x ( t ) ).
Suppose that a mapping with satisfies the inequality(3.51 for all Then there exists a unique quartic mapping satisfying (3.83).
Until 1984, Khan et al. [8] formally introduced the definition of the above family Ψ̃, and proved that any mapping F satisfying (1.4) with (psiinwidetilde{Psi}) is a Picard operator.
Theorem 2.1Let and be twoHilbert spaces, bea bounded linear operator, bean asymptotically nonexpansive mapping with the sequence satisfying,and bean asymptotically nonexpansive mapping with the sequence satisfying, and, respectively.Let,andd let the sequence be defined as follows: (2.1).
Indeed, they deal with a monotone (either order-preserving or order-reversing) mapping (mathcal F ) satisfying, with some restriction, a classical contractive condition, and are such that for some (x_{0}in mathcal{X }), either (x_{0}preceq mathcal{F }{x_{0}}) or (mathcal{F }{x_{0}}preceq x_{0}) holds.
Let be a Hilbert space, be a nonempty bounded closed convex subset of, and let be a completely continuous, uniformly -Lipschitzian and asymptotically pseudocontractive mapping with a sequence satisfying the following conditions: (i) as, (ii), where.
