A Krasnoselskii–Ishikawa Iterative Algorithm for Monotone Reich Contractions in Partially Ordered Banach Spaces with an Application

Iterative algorithms have been utilized for the computation of approximate solutions of stationary and evolutionary problems associated with differential equations. The aim of this article is to introduce concepts of monotone Reich and Chatterjea nonexpansive mappings on partially ordered Banach spaces. We describe sufficient conditions for the existence of an approximate fixed-point sequence (AFPS) and prove certain fixed-point results using the Krasnoselskii–Ishikawa iterative algorithm. Moreover, we present some interesting examples to highlight the superiority of our results. Lastly, we provide both weak and strong convergence results for such mappings and consider an application of our results to prove the existence of a solution to an initial value problem.

Existence of positive S-asymptotically periodic solutions of the fractional evolution equations in ordered Banach spaces

In this paper, we discuss the asymptotically periodic problem for the abstract fractional evolution equation under order conditions and growth conditions. Without assuming the existence of upper and lower solutions, some new results on the existence of the positive S-asymptotically ω-periodic mild solutions are obtained by using monotone iterative method and fixed point theorem. It is worth noting that Lipschitz condition is no longer needed, which makes our results more widely applicable.

Stability criteria for positive linear discrete-time systems

We prove new characterisations of exponential stability for positive linear discrete-time systems in ordered Banach spaces, in terms of small-gain conditions. Such conditions have played an important role in the finite-dimensional systems theory, but are relatively unexplored in the infinite-dimensional setting, yet. Our results are applicable to discrete-time systems in ordered Banach spaces that have a normal and generating positive cone. Moreover, we show that our stability criteria can be considerably simplified if the cone has non-empty interior or if the operator under consideration is quasi-compact. To place our results into context we include an overview of known stability criteria for linear (and not necessarily positive) operators and provide full proofs for several folklore characterizations from this domain.

Random Fixed Point Theorems and Applications to Random First-Order Vector-Valued Differential Equations

In this paper, we establish several random fixed point theorems for random operators satisfying some iterative condition w.r.t. a measure of noncompactness. We also discuss the case of monotone random operators in ordered Banach spaces. Our results extend several earlier works, including Itoh’s random fixed point theorem. As an application, we discuss the existence of random solutions to a class of random first-order vector-valued ordinary differential equations with lack of compactness.

Geometric Characterization of Injective Banach Lattices

This is a continuation of the authors’ previous study of the geometric characterizations of the preduals of injective Banach lattices. We seek the properties of the unit ball of a Banach space which make the space isometric or isomorphic to an injective Banach lattice. The study bases on the Boolean valued transfer principle for injective Banach lattices. The latter states that each such lattice serves as an interpretation of an AL-space in an appropriate Boolean valued model of set theory. External identification of the internal Boolean valued properties of the corresponding AL-spaces yields a characterization of injective Banach lattices among Banach spaces and ordered Banach spaces. We also describe the structure of the dual space and present some dual characterization of injective Banach lattices.