A Modified Algorithms for New Krasnoselskii's Type for Strongly Monotone and Lipschitz Mappings

Abstract

Let E be a 2 uniformly smooth and convex real Banach space and let a mapping A: E → E^∗ be lipschitz and strongly monotone such that A⁻¹(0)≠∅. For an arbitrary ({x₁}, {y₁})∈E, we define the sequences {xn} and {yn} by
y_n = x_n − θ_nJ⁻¹(Ax_n), n≥1
x_n+1 = y_n − λ_nJ⁻¹(Ay_n), n≥1
where λ_n and θ_n are positive real number and J is the duality mapping of E. Letting (λ_n, θ_n)∈(0, 1), then x_n and y_n converges strongly to ρ^∗, a unique solution of the equation Ax = 0. We also applied our algorithm in convex minimization and also proved the convergence of it in L_p, l_p or W^m,p. At the end we proposed the algorithm of it in L_p(Ω) and its inverse L^q(Ω).