A new iteration method is introduced to find fixed points of Lipschitzian pseudo-contractive maps in Hilbert spaces. The paper shows that if T is a Lipschitzian pseudo-contractive map on a compact convex subset E of a Hilbert space, and x₁ is any point in E, then the sequence defined by x_{n+1} = α_n T[β_n T x_n + (1 - β_n)x_n] + (1 - α_n)x_n converges strongly to a fixed point of T, provided that the sequences {α_n} and {β_n} satisfy certain conditions: 0 ≤ α_n ≤ β_n ≤ 1 for all n, lim_{n→∞} β_n = 0, and ∑_{n=1}^∞ α_n β_n = ∞. As a particular case, α_n = β_n = n^{-1/2} is used. The proof involves using properties of pseudo-contractive maps and the Lipschitz condition. It shows that the sequence {x_n} converges strongly to a fixed point of T. The key steps involve deriving inequalities that show the sequence {x_n} is contractive and bounded, leading to the conclusion that it converges to a fixed point. The paper also references previous work on mean value iterations for finding fixed points of strictly pseudo-contractive maps. The author thanks Professors H. Fujita and T. Kawata for their suggestions.A new iteration method is introduced to find fixed points of Lipschitzian pseudo-contractive maps in Hilbert spaces. The paper shows that if T is a Lipschitzian pseudo-contractive map on a compact convex subset E of a Hilbert space, and x₁ is any point in E, then the sequence defined by x_{n+1} = α_n T[β_n T x_n + (1 - β_n)x_n] + (1 - α_n)x_n converges strongly to a fixed point of T, provided that the sequences {α_n} and {β_n} satisfy certain conditions: 0 ≤ α_n ≤ β_n ≤ 1 for all n, lim_{n→∞} β_n = 0, and ∑_{n=1}^∞ α_n β_n = ∞. As a particular case, α_n = β_n = n^{-1/2} is used. The proof involves using properties of pseudo-contractive maps and the Lipschitz condition. It shows that the sequence {x_n} converges strongly to a fixed point of T. The key steps involve deriving inequalities that show the sequence {x_n} is contractive and bounded, leading to the conclusion that it converges to a fixed point. The paper also references previous work on mean value iterations for finding fixed points of strictly pseudo-contractive maps. The author thanks Professors H. Fujita and T. Kawata for their suggestions.