Abstract
We construct an implicit sequence suitable for the approximation of solutions ofK-positive definite operator equations in real Banach spaces. Furthermore, implicit error estimate is obtained and the convergence is shown to be faster in comparsion to the explicit error estimate obtained by Osilike and Udomene (2001).
Highlights
Let E be a real Banach space and let J denote the normalized duality mapping from E to 2E∗ defined byJ (x) = {f∗ ∈ E∗ : ⟨x, f∗⟩ = ‖x‖2, f∗ = ‖x‖}, (1)where E∗ denotes the dual space of E and ⟨⋅, ⋅⟩ denotes the generalized duality pairing
Since fixed point problems and solving nonlinear equations based on implicit iterative processes have been considered by many authors
It is our purpose in this paper to introduce implicit scheme which converges strongly to the solution of the Kpd operator equation Ax = f in a separable Banach space
Summary
Converges strongly to the solution of the equation Ax = f. If x∗ denotes the solution of the equation Ax = f, For convergence results of this scheme and related iterative schemes, see, for example, [9,10,11,12,13,14,15]. Since fixed point problems and solving (or approximating) nonlinear equations based on implicit iterative processes have been considered by many authors (see, e.g., [17,18,19,20,21]).
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