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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.

Results are given which establish a computational foundation for simplicial approximation and design centering of a convex body. A simplicial polyhedron is used to approximate the convex body and the “design center”, i.e. the point inside the body furthest in some norm from its exterior, is approximated by the point in the polyhedron furthest from its exterior. A point representation of the polyhedron is used, so that there is no necessity for computing or storing the faces of the approximation. Since in N space there can be factorially more faces than points, we are able to achieve significant efficiencies in both operation count and storage requirements, compared to previously reported methods. We give results for the 2 norm and the max norm, and demonstrate that our new method is operable in the nonconvex case, and can handle a mixed basis of faces and points as well.

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.

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.

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.

Standard macro theories have the same analytical structure as their micro counterparts. Where micro theories work with equilibrium between supply and demand for particular products, macro theories work with equilibrium applied to aggregates of products. This common approach treats the micro–macro relationship as scalable, with macro variables being aggregations over micro variables. In contrast, we pursue a systems-theoretic approach to the micro–macro relationship. This relationship is not scalable and rather entails a disjunction between micro- and macro-levels of theory. While micro phenomena are still susceptible to choice-theoretic analysis, macro phenomena are products of ecological interaction and so entail emergent phenomena. Our alternative approach treats macro theory as a form of systems theory where the behavior of the system has properties that are not reducible to properties of the individual elements within that system. Besides sketching this alternative approach, we examine some of the different insights this approach offers into such topics as unemployment and stabilization.

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.

The aim of the study is to show that merging two areas of mathematics – topology and discrete optimization – could result in a viable option to solve classical or specialized integer problems.

In the paper, discrete topology concepts are applied to propose a metaheuristic algorithm that is capable to solve binary programming problems. Particularly, some of the homotopy for paths principles are used to explore the solution space associated with four well-known NP-hard problems herein considered as follows: knapsack, set covering, bi-level single plant location with order and one-max.

Computational experimentation confirms that the proposed algorithm performs in an effective manner, and it is able to efficiently solve the sets of instances used for the benchmark. Moreover, the performance of the proposed algorithm is compared with a standard genetic algorithm (GA), a scatter search (SS) method and a memetic algorithm (MA). Acceptable results are obtained for all four implemented metaheuristics, but the path homotopy algorithm stands out.

A novel metaheuristic is proposed for the first time. It uses topology concepts to design an algorithmic framework to solve binary programming problems in an effective and efficient manner.

The purpose of this paper is to introduce the implicit midpoint rule (IMR) of nonexpansive mappings in 2- uniformly convex hyperbolic spaces and study its convergence. Strong and △-convergence theorems based on this algorithm are proved in this new setting. The results obtained hold concurrently in uniformly convex Banach spaces, CAT(0) spaces and Hilbert spaces as special cases.