Fixed point theorems and ergodic theorems for nonlinear mappings in Banach spaces

The purpose of this article is to study the fixed point and weak convergence problem for the new defined class of point-dependent λ-hybrid mappings relative to a Bregman distance D f in a Banach space. We at first extend the Aoyama-Iemoto-Kohsaka-Takahashi fixed point theorem for λ-hybrid mappings in Hilbert spaces in 2010 to this much wider class of nonlinear mappings in Banach spaces. Secondly, we derive an Opial-like inequality for the Bregman distance and apply it to establish a weak convergence theorem for this new class of nonlinear mappings. Some concrete examples in a Hilbert space showing that our extension is proper are also given.2010 MSC: 47H09; 47H10.

Modified block iterative algorithm for Quasi- ϕ-asymptotically nonexpansive mappings and equilibrium problem in banach spaces

Strong convergence theorems for countable families of asymptotically relatively nonexpansive mappings with applications

Stability of iterative procedures with errors for approximating common fixed points of a couple of q-contractive-like mappings in Banach spaces

Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces

Convergence of three-step iterations for asymptotically nonexpansive mappings

In this paper, we prove convergence theorems for a fixed point iterative algorithm of Krasnoselskij-Mann typeassociated to the class of enriched nonexpansive mappings in Banach spaces. The results are direct generaliza-tions of the corresponding ones in [Berinde, V.,Approximating fixed points of enriched nonexpansive mappings byKrasnoselskij iteration in Hilbert spaces, Carpathian J. Math., 35 (2019), No. 3, 293–304.], from the setting of Hilbertspaces to Banach spaces, and also of some results in [Senter, H. F. and Dotson, Jr., W. G.,Approximating fixed pointsof nonexpansive mappings, Proc. Amer. Math. Soc.,44(1974), No. 2, 375–380.], [Browder, F. E., Petryshyn, W. V., Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl., 20 (1967), 197–228.], byconsidering enriched nonexpansive mappings instead of nonexpansive mappings. Many other related resultsin literature can be obtained as particular instances of our results.

The purpose of this paper is to study the method of approximation for a zero of the sum of two maximal monotone mappings in Banach spaces and prove strong convergence of the proposed method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

<p dir="ltr">In this thesis, we study the existence of strictly positive solutions of second-order Neumann boundary value problems with nonlinearities which are allowed to take negative values via a recently established fixed point theorem for r-nowhere normal-outward maps in Banach spaces. We derive a Green's function to be used in Hammerstein integral equations. We introduce and provide proofs on linear operators which are then used in an established fixed point theorem for r-nowhere normal-outward maps in Banach spaces. We provide commentary on the application of our results to problems in physics. </p>

This paper investigates fixed points of Reich-Suzuki-type nonexpansive mappings in the context of uniformly convex Banach spaces through an M ∗ iterative method. Under some appropriate situations, some strong and weak convergence theorems are established. To support our results, a new example of Reich-Suzuki-type nonexpansive mappings is presented which exceeds the class of Suzuki-type nonexpansive mappings. The presented results extend some recently announced results of current literature.

In this paper, we prove a weak convergence theorem and a strong convergence theorem for split common fixed point problem involving a quasi-strict pseudo contractive mapping and an asymptotical nonexpansive mapping in the setting of two Banach spaces. Our results are new and seem to be the first outside Hilbert spaces.

In this paper, we prove strong convergence theorems of modified Halpern’s iteration for finding a common element of the zero point set of a maximal monotone operator and the fixed point set of a relatively nonexpansive mapping in a Banach space by using two hybrid methods. Using these results, we obtain new convergence results for resolvents of maximal monotone operators and relatively nonexpansive mappings in Banach spaces.

Iterative algorithm for generalized quasi-variational-like inclusions with fuzzy mappings in Banach spaces

On the existence of common fixed points for a pair of lipschitzian mappings in banach spaces

The fixed point property for mappings admitting a center