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In this paper, the explicit multistep, explicit multistep-SP and implicit multistep iterative sequences are introduced in the context of modular function spaces and proven to converge to the fixed point of a multivalued map T such that PρT, an associate multivalued map, is a ρ-contractive-like mapping.

The concepts of relative ρ-stability and weak ρ-stability are introduced, and conditions in which these multistep iterations are relatively ρ-stable, weakly ρ-stable and ρ-stable are established for the newly introduced strong ρ-quasi-contractive-like class of maps.

Noor type, Ishikawa type and Mann type iterative sequences are deduced as corollaries in this study.

The results obtained in this work are complementary to those proved in normed and metric spaces in the literature.

The purpose of this paper is to extend the recent results of Okeke et al. (2018) to the class of multivalued ρ-quasi-contractive mappings in modular function spaces. We approximate fixed points of this class of nonlinear multivalued mappings in modular function spaces. Moreover, we extend the concepts of T-stability, almost T-stability and summably almost T-stability to modular function spaces and give some results.

The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. The authors apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend generalize compliment and include several known results as special cases.

The results of this paper are theoretical and analytical in nature.

The authors prove the existence and uniqueness of fixed point of mappings satisfying Geraghty-type contractions in the setting of preordered modular G-metric spaces. apply the results in solving nonlinear Volterra-Fredholm-type integral equations. The results extend, generalize, compliment and include several known results as special cases.

The results are theoretical and analytical.

The results were applied to solving nonlinear integral equations.

The results has several social applications.

The results of this paper are new.

This paper aims to prove some fixed-point theorems for a general class of mappings in modular G-metric spaces. The results of this paper generalize and extend several known results to modular G-metric spaces, including the results of Mutlu et al. [1]. Furthermore, the authors produce an example to demonstrate the applicability of the results.

The results of this paper are theoretical and analytical in nature.

The authors established some fixed-point theorems for a general class of mappings in modular G-metric spaces. The results generalize and extend several known results to modular G-metric spaces, including the results of Mutlu et al. [1]. An example was constructed to demonstrate the applicability of the results.

Analytical and theoretical results.

The results of this paper can be applied in science and engineering.

The results of this paper is applicable in certain social sciences.

The results of this paper are new and will open up new areas of research in mathematical sciences.

Eddy currents are investigated in a ferromagnetic bar exposed to a transverse magnetic field. Such an open boundary field problem is analysed applying the Galerkin finite element method coupled with a separation of variables. A steady state is considered, introducing time periodic conditions. The resultant system of nonlinear equations is solved using an iterative procedure based on Brouwer's fixed‐point theorem referred to the nonlinear material reluctivity. Numerical results are presented for a massive conductor made of cast steel and cast iron. The eddy‐current distribution and characteristics of power losses are illustrated in a graphic form.

In this paper, we use the notion of cyclic representation of a nonempty set with respect to a pair of mappings to obtain coincidence points and common fixed points for a pair of self-mappings satisfying some generalized contraction- type conditions involving a control function in partial metric spaces. Moreover, we provide some examples to analyze and illustrate our main results.

Theoretical method.

We establish some coincidence points and common fixed point results in partial metric spaces.

Results of this study are new and interesting.

In this paper, Picard–S hybrid iterative process is defined, which is a hybrid of Picard and S-iterative process. This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid and Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.

This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings.

Showed the fastest convergence of this new iteration and then other iteration defined in this paper. The author finds the solution of delay differential equation using this hybrid iteration. For new iteration, the author also proved a theorem for nonexpansive mapping.

This new iteration converges faster than all of Picard, Krasnoselskii, Mann, Ishikawa, S-iteration, Picard–Mann hybrid, Picard–Krasnoselskii hybrid, Picard–Ishikawa hybrid iterative processes for contraction mappings and to find the solution of delay differential equation, using this hybrid iteration also proved some results for Picard–S hybrid iterative process for nonexpansive mappings.

This paper is concerned with the existence of mild solutions for a class of fractional semilinear integro-differential equations having non-instantaneous impulses. The result is obtained by using noncompact semigroup theory and fixed point theorem. The obtained result is illustrated by an example at the end.

The non‐well‐founded properties of the empty set provide existence to abstract topological spaces, in which intersections of subspaces with non‐equal dimensions give topologically closed structures. This let space‐time emerge as an ordered sequence of sections upon a defined combination rule, in which interactivity between observable structures is allowed and conditions for conscious perception phenomena are fulfilled. Conditions for evolution and rules for optimal evolution of ecosystems also infer as corollaries. A physical universe and life constitute one single self‐organized and self managed system where life appears as the physical‐like realization of conditions of functionality of the embedding mathematical spaces. The system includes self‐ethical and moral guidelines which should inspire human behavior on Planet Earth.

The purpose of this paper is to describe a finite element formulation to approximate thermally coupled flows using both the Boussinesq and the low Mach number models with particular emphasis on the numerical implementation of the algorithm developed.

The formulation, that allows us to consider convection dominated problems using equal order interpolation for all the valuables of the problem, is based on the subgrid scale concept. The full Newton linearization strategy gives rise to monolithic treatment of the coupling of variables whereas some fixed point schemes permit the segregated treatment of velocity‐pressure and temperature. A relaxation scheme based on the Armijo rule has also been developed.

A full Newtown linearization turns out to be very efficient for steady‐state problems and very robust when it is combined with a line search strategy. A segregated treatment of velocity‐pressure and temperature happens to be more appropriate for transient problems.

A fractional step scheme, splitting also momentum and continuity equations, could be further analysed.

The results presented in the paper are useful to decide the solution strategy for a given problem.

The numerical implementation of a stabilized finite element approximation of thermally coupled flows is described. The implementation algorithm is developed considering several possibilities for the solution of the discrete nonlinear problem.