On the Semi-Local Convergence of an Ostrowski-Type Method for Solving Equations

Symmetries play a crucial role in the dynamics of physical systems. As an example, microworld and quantum physics problems are modeled on principles of symmetry. These problems are then formulated as equations defined on suitable abstract spaces. Then, these equations can be solved using iterative methods. In this article, an Ostrowski-type method for solving equations in Banach space is extended. This is achieved by finding a stricter set than before containing the iterates. The convergence analysis becomes finer. Due to the general nature of our technique, it can be utilized to enlarge the utilization of other methods. Examples finish the paper.

“The street is ours”. A comparative analysis of street trading, Covid-19 and new street geographies in Harare, Zimbabwe and Kisumu, Kenya

This collaborative and comparative paper deals with the impact of Covid-19 on the use and governance of public space and street trade in particular in two major African cities. The importance of street trading for urban food security and urban-based livelihoods is beyond dispute. Trading on the streets does, however, not occur in neutral or abstract spaces, but rather in lived-in and contested spaces, governed by what is referred to as ‘street geographies’, evoking outbreaks of violence and repression. Vendors are subjected to the politics of municipalities and the state to modernize the socio-spatial ordering of the city and the urban food economy through restructuring, regulating, and restricting street vending. Street vendors are harassed, streets are swept clean, and hygiene standards imposed. We argue here that the everyday struggle for the street has intensified since and during the Covid-19 pandemic. Mobility and the use of urban space either being restricted by the city-state or being defended and opened up by street traders, is common to the situation in Harare and Kisumu. Covid-19, we pose, redefines, and creates ‘new’ street geographies. These geographies pivot on agency and creativity employed by street trade actors while navigating the lockdown measures imposed by state actors. Traders navigate the space or room for manoeuvre they create for themselves, but this space unfolds only temporarily, opens for a few only and closes for most of the street traders who become more uncertain and vulnerable than ever before, irrespective of whether they are licensed, paying rents for vending stalls to the city, or ‘illegally’ vending on the street.

Monotone Iterative Technique for the Periodic Solutions of High-Order Delayed Differential Equations in Abstract Spaces

This paper deals with the existence of ω-periodic solutions for nth-order ordinary differential equation involving fixed delay in Banach space E. Lnu(t)=f(t,u(t),u(t−τ)),t∈R, where Lnu(t):=u(n)(t)+∑i=0n−1aiu(i)(t), ai∈R, i=0,1,⋯,n−1, are constants, f(t,x,y):R×E×E⟶E is continuous and ω-periodic with respect to t, τ>0. By applying the approach of upper and lower solutions and the monotone iterative technique, some existence and uniqueness theorems are proved under essential conditions.

Some remarks on bv(s)-metric spaces and fixed point results with an application

We compare the newly defined bv(s)-metric spaces with several other abstract spaces like metric spaces, b-metric spaces and show that some well-known results, which hold in the latter class of spaces, may not hold in bv(s)-metric spaces. Besides, we introduce the notions of sequential compactness and bounded compactness in the framework of bv(s)-metric spaces. Using these notions, we prove some fixed point results involving Nemytzki–Edelstein type mappings in this setting, from which several comparable fixed point results can be deduced. In addition to these, we find some existence and uniqueness criteria for the solution to a certain type of mixed Fredholm–Volterra integral equations.

Extending the Convergence Domain of Methods of Linear Interpolation for the Solution of Nonlinear Equations

Solving equations in abstract spaces is important since many problems from diverse disciplines require it. The solutions of these equations cannot be obtained in a form closed. That difficulty forces us to develop ever improving iterative methods. In this paper we improve the applicability of such methods. Our technique is very general and can be used to expand the applicability of other methods. We use two methods of linear interpolation namely the Secant as well as the Kurchatov method. The investigation of Kurchatov’s method is done under rather strict conditions. In this work, using the majorant principle of Kantorovich and our new idea of the restricted convergence domain, we present an improved semilocal convergence of these methods. We determine the quadratical order of convergence of the Kurchatov method and order 1 + 5 2 for the Secant method. We find improved a priori and a posteriori estimations of the method’s error.

Existence Results of Mild Solutions for the Fractional Stochastic Evolution Equations of Sobolev Type

In this paper, by utilizing the resolvent operator theory, the stochastic analysis method and Picard type iterative technique, we first investigate the existence as well as the uniqueness of mild solutions for a class of α ∈ ( 1 , 2 ) -order Riemann–Liouville fractional stochastic evolution equations of Sobolev type in abstract spaces. Then the symmetrical technique is used to deal with the α ∈ ( 1 , 2 ) -order Caputo fractional stochastic evolution equations of Sobolev type in abstract spaces. Two examples are given as applications to the obtained results.