Some fixed point results for enriched nonexpansive type mappings in Banach spaces

[EN] In this paper, we introduce two new classes of nonlinear mappings and present some new existence and convergence theorems for these mappings in Banach spaces. More precisely, we employ the Krasnosel'skii iterative method to obtain fixed points of Suzuki-enriched nonexpansive mappings under different conditions. Moreover, we approximate the fixed point of enriched-quasinonexpansive mappings via Ishikawa iterative method. The first author acknowledges the support from the GES 4.0 fellowship, University of Johannesburg, South Africa.Shukla, R.; Pant, R. (2022). Some fixed point results for enriched nonexpansive type mappings in Banach spaces. Applied General Topology. 23(1):31-43. https://doi.org/10.4995/agt.2022.16165314323

The fixed point property in direct sums and modulus R(a,X)

We show that the direct sum of Banach spaces $X_{1},..., X_{r}$ with a strictly monotone norm has the weak fixed point property for nonexpansive mappings whenever $M(X_{i})>1$ for each $i=1,...,r$. In particular, $(X_{1} \oplus ... \oplus X_{r})_{\psi}$ enjoys the fixed point property if Banach spaces $X_{i}$ are uniformly nonsquare. This combined with the earlier results gives a definitive answer for r=2: the direct sum of uniformly nonsquare spaces $X_{1}, X_{2}$ with any monotone norm has FPP. Our results are extended for asymptotically nonexpansive mappings in the intermediate sense.Comment: 12 page

Geometry of Banach spaces and some fixed point theorems

The aim of this thesis is to study the geometry of Banach spaces, the existence of fixed points and the convergence of iterative sequences of certain mappings in Banach spaces. -- We introduce some of the basic definitions and give a brief survey of some well-known results on fixed points for different mappings. -- We also introduce and discuss different classifications of Banach spaces. A few results, similar to those of uniformly convex Banach spaces, have been given for weakly uniformly convex and weakly* uniformly convex Banach spaces. -- Finally considering more general mappings, of types Diaz and Metcalf [46], Dotson [48], Kirk [87], some new results and various generalizations have been given on the asymptotic regularity and the convergence of the iterative sequences in Banach spaces. We end with mentioning some of the applications of fixed point theory in brief

Existence Results for Some Damped Second-Order Volterra Integro-Differential Equations

In this paper we make a subtle use of operator theory techniques and the well-known Schauder fixed-point principle to establish the existence of pseudo-almost automorphic solutions to some second-order damped integro-differential equations with pseudo-almost automorphic coefficients. In order to illustrate our main results, we will study the existence of pseudo-almost automorphic solutions to a structurally damped plate-like boundary value problem.Comment: 20 pages. arXiv admin note: substantial text overlap with arXiv:1402.563

Mann-Type Viscosity Approximation Methods for Multivalued Variational Inclusions with Finitely Many Variational Inequality Constraints in Banach Spaces

We introduce Mann-type viscosity approximation methods for finding solutions of a multivalued variational inclusion (MVVI) which are also common ones of finitely many variational inequality problems and common fixed points of a countable family of nonexpansive mappings in real smooth Banach spaces. Here the Mann-type viscosity approximation methods are based on the Mann iteration method and viscosity approximation method. We consider and analyze Mann-type viscosity iterative algorithms not only in the setting of uniformly convex and 2-uniformly smooth Banach space but also in a uniformly convex Banach space having a uniformly Gáteaux differentiable norm. Under suitable assumptions, we derive some strong convergence theorems. In addition, we also give some applications of these theorems; for instance, we prove strong convergence theorems for finding a common fixed point of a finite family of strictly pseudocontractive mappings and a countable family of nonexpansive mappings in uniformly convex and 2-uniformly smooth Banach spaces. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature

Best proximity pair results for relatively nonexpansive mappings in geodesic spaces

Given $A$ and $B$ two nonempty subsets in a metric space, a mapping $T : A \cup B \rightarrow A \cup B$ is relatively nonexpansive if $d(Tx,Ty) \leq d(x,y) \text{for every} x\in A, y\in B.$ A best proximity point for such a mapping is a point $x \in A \cup B$ such that $d(x,Tx)=\text{dist}(A,B)$. In this work, we extend the results given in [A.A. Eldred, W.A. Kirk, P. Veeramani, Proximal normal structure and relatively nonexpansive mappings, Studia Math., 171 (2005), 283-293] for relatively nonexpansive mappings in Banach spaces to more general metric spaces. Namely, we give existence results of best proximity points for cyclic and noncyclic relatively nonexpansive mappings in the context of Busemann convex reflexive metric spaces. Moreover, particular results are proved in the setting of CAT(0) and uniformly convex geodesic spaces. Finally, we show that proximal normal structure is a sufficient but not necessary condition for the existence in $A \times B$ of a pair of best proximity points