Abstract
Combining the implicit midpoint method and the splitting method, we present a new iterative algorithm with errors to solve the problems of finding zeros of the sum of m-accretive operators and μ-inversely strongly accretive operators in a real q-uniformly smooth and uniformly convex Banach space. We obtain some strong convergence theorems, which demonstrate the relationship between the zero of the sum of m-accretive operator and μ-inversely strongly accretive operator and the solution of one kind variational inequality. Moreover, the applications of the main results on the nonlinear problems with Neumann boundaries and Signorini boundaries are demonstrated.
Highlights
Introduction and preliminaries LetE be a real Banach space with norm · and let E∗ denote the dual space of E
In, Marino and Xu presented the following iterative algorithm in the frame of Hilbert spaces in [ ], which sets up the relationship between fixed point of a nonexpansive mapping and the solution of one kind variational inequality x ∈ C, xn+ = αnγ f + (I – αnA)Txn, n ≥, ( )
We infer that each cluster point of {xt} is equal to p, which is the unique solution of the variational inequality ( )
Summary
In , Marino and Xu presented the following iterative algorithm in the frame of Hilbert spaces in [ ], which sets up the relationship between fixed point of a nonexpansive mapping and the solution of one kind variational inequality x ∈ C, xn+ = αnγ f (xn) + (I – αnA)Txn, n ≥ , where f is a contraction, A is a strongly positive linear bounded operator, and T is nonexpansive. Lemma (see [ ]) Let E be a real uniformly convex Banach space, C be a nonempty, closed, and convex subset of E and T : C → E be a nonexpansive mapping such that Fix(T) = ∅, I – T is demiclosed at zero. Lemma (see [ ]) Let E be a real q-uniformly smooth Banach space with constant Kq. A be a μ-inversely strongly accretive operator
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
