Abstract
In this paper, we introduce and study a class of new Picard-Mann iterative methods with mixed errors for common fixed points of two different nonexpansive and contraction operators. We also give convergence and stability analysis of the new Picard-Mann iterative approximation and propose numerical examples to show that the new Picard-Mann iteration converges more effectively than the Picard iterative process, Mann iterative process, Picard-Mann iterative process due to Khan and other related iterative processes. Furthermore, as an application, we explore iterative approximation of solutions for an elliptic boundary value problem in Hilbert spaces by using the new Picard-Mann iterative methods with mixed errors for contraction operators.
Highlights
In order to find a weak solution of the following elliptic boundary value problem: ⎧⎨– u = f (x, u), x ∈, ⎩u(x) = 0, x∈∂, (1.1)where ⊂ Rn is a bounded domain, f : × R → R is a Carathéodory function, Ayadi et al [1] proved a new global minimization theorem in Hilbert spaces by using the notion of a nonexpansive potential operator
Multidimensional dynamical systems are frequently formulated by partial differential equations, which generally depend on space and time, i.e., parabolic or evolutionary type equations, and are treated with emphasis on various real-world applications inmechanics of solids and fluids, electrical devices, engineering, chemistry, biology, etc
Two numerical examples to verify effectiveness of the new Picard-Mann iteration are presented, and a new iterative approximation of solutions for an elliptic boundary value problem in Hilbert spaces is investigated by using the new Picard-Mann iterative methods with mixed errors for nonexpansive operators, which are different from the method proposed in [1]
Summary
In order to find a weak solution of the following elliptic boundary value problem (so-called Dirichlet problem):. Following on the works of Khan [18], by using an up-to-date method for approximating common fixed points of countable families of nonlinear operators, Deng [19] introduced a modified Picard-Mann hybrid iterative algorithm for a sequence of nonexpansive mappings and established strong convergence and weak convergence of the iterative sequence generated by the modified hybrid iterative algorithm in a convex Banach space. Motivated and inspired by the above works, we aim in this paper to introduce and study a class of new Picard-Mann iterative methods with mixed errors for common fixed points of two different nonexpansive and contraction operators. Two numerical examples to verify effectiveness of the new Picard-Mann iteration are presented, and a new iterative approximation of solutions for an elliptic boundary value problem in Hilbert spaces is investigated by using the new Picard-Mann iterative methods with mixed errors for nonexpansive operators, which are different from the method proposed in [1].
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