We consider the equivalence of the existence of fixed points of single-valued mappings and multivalued mappings for some classes of mappings by proving some equivalence theorems for the completeness of metric spaces.

Fixed Points for Multivalued Mappings and the Metric Completeness = จุดตรึงสำหรับการส่งหลายค่าและความบริบูรณ์เชิงเมตริก / Hatairat Yingtaweesittikul

In this paper, we set up an adequate condition for the presence of a solution of the nonlinear matrix equation. To do so, we prove the existence of fixed points for multi-valued modified F-contractions in the context of complete metric spaces, which generalize, refine, and extend several existing results in the literature. An example is accompanies the obtained results to show that derived results are a proper generalization.

https://www.mdpi.com/2227-7390/8/2/212/pdf

The Existence of Solutions to Nonlinear Matrix Equations via Fixed Points of Multivalued F-Contractions

The present research paper is related to the analytical solution of fractional-order nonlinear Swift-Hohenberg equations using an efficient technique. The presented model is related to the temperature and thermal convection of fluid dynamics which can also be used to explain the formation process in liquid surfaces bounded along a horizontally well-conducting boundary. In this work Laplace Adomian decomposition method is implemented because it require small volume of calculations. Unlike the variational iteration method and Homotopy pertubation method, the suggested technique required no variational parameter and having simple calculation of fractional derivative respectively. Numerical examples verify the validity of the suggested method. It is confirmed that the present method's solutions are in close contact with the solutions of other existing methods. It is also investigated through graphs and tables that the suggested method's solutions are almost identical with different analytical methods.

The analytical analysis of nonlinear fractional-order dynamical models

D. Wardowski proved in 2012 a generalization of Banach Contraction Principle by introducing F-contractions in metric spaces. In the next ten years, a great number of researchers used Wardowski's approach, or some of its modifications, to obtain new fixed point results for single- and multivalued mappings in various kinds of spaces. In this review article, we present a survey of these investigations, including some improvements, in particular concerning conditions imposed on function F entering the contractive condition.

On F-Contractions: A Survey

Emerging Trends in the use of smart portable accessories, particularly within the context of the Internet of Things (IoT), where smart sensor devices are employed for data gathering, require advancements in energy management mechanisms. This study aims to provide an intelligent energy management mechanism for wearable/portable devices through the use of predictions, monitoring, and analysis of the performance indicators for energy harvesting, majorly focusing on the hybrid PV-wind systems. To design a robust and precise model, prediction algorithms are compared and analysed for an efficient decision support system. Levenberg–Marquardt (LM), Bayesian Regularization (BR), and Scaled Conjugate Gradient (SCG) prediction algorithms are used to develop a Shallow Neural Network (SNN) for time series prediction. The proposed SNN model uses a closed-loop NARX recurrent dynamic neural network to predict the active power and energy of a hybrid system based on the experimental data of solar irradiation, wind speed, wind direction, humidity, precipitation, ambient temperature and atmospheric pressure collected from Jan 1st 2015 to Dec 26th 2015. The historical hourly metrological data set is established using calibrated sensors deployed at Middle East Technical University (METU), NCC. The accessory considered in this study is called as Smart Umbrella System (SUS), which uses a Raspberry Pi module to fetch the weather data from the current location and store it in the cloud to be processed using SNN classified prediction algorithms. The results obtained show that using the SNN model, it is possible to obtain predictions with 0.004 error rate in a computationally efficient way within 20 s. With the experiments, we are able to observe that for the period of observation, the energy harvested is 178 Wh/d, where the system estimates energy as 176.5 Wh/d, powering the portable accessories accurately.

Use of Neural Network Based Prediction Algorithms for Powering Up Smart Portable Accessories

In the present paper, we derive a common xed point theorem for a hybrid pair of occasionally coincidentally idempotent mappings satisfying closed multi-valued F -contraction condition introduced by Wardowski (Fixed Point Theory Appl. 2012:94) via common limit range property in the frame of complete metric spaces. Also, hybrid mappings which satisfy an F -contractive condition of Hardy-Rogers type are considered. Our results improve several results from the existing literature. Two applications are presented|the proofs of existence of solutions for certain system of functional equations arising in dynamic programming, as well as for certain Volterra integral inclusion.

http://bangmod-jmcs.kmutt.ac.th/wp-content/uploads/2015/04/17gp.pdf

Common fixed point theorems for hybrid F -contractions and applications

We propose common fixed point results for two pairs of partially weakly increasing mappings in an ordered orbitally complete metric space under a generalized rational-type $\mathcal {W}$-weakly contractive condition. As an application, an existence result for certain systems of integral equations is presented.

Common fixed point theorems under generalized \mathcal {W}-weakly contractive condition in ordered orbitally complete metric spaces

In this paper, we introduce a modified implicit relation and obtain some new fixed point results for $ \sigma $-implicit type contractive conditions in relational metric-like spaces. We present some nontrivial examples to illustrative facts and compare our results with the related work. We also discuss sufficient conditions for the existence of a unique positive definite solution of the non-linear matrix equation $ \mathcal{U} = \mathcal{D} + \sum_{i = 1}^{m}\mathcal{A}_{i}^{*}\mathcal{G} \mathcal{(U)}\mathcal{A}_{i} $, where $ \mathcal{D} $ is an $ n\times n $ Hermitian positive definite matrix, $ \mathcal{A}_{1} $, $ \mathcal{A}_{2} $, $\dots$, $ \mathcal{A}_{m} $ are $ n \times n $ matrices, and $ \mathcal{G} $ is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Finally, we discuss a couple of examples, convergence and error analysis, average CPU time analysis and visualization of solution in surface plot.

Unified relational-theoretic approach in metric-like spaces with an application

In this paper, we discuss the existence of best proximity points of certain mappings via simulation functions in the frame of complete metric-like spaces. Some consequences and examples are given of the obtained results.

/pdf/best-proximity-point-results-via-simulation-functions-in-4jwz2mvrwn.pdf

Hemant Kumar Nashine

Papers

Common fixed point theorems for hybrid F -contractions and applications

Common fixed point theorems under generalized \mathcal {W}-weakly contractive condition in ordered orbitally complete metric spaces

Unified relational-theoretic approach in metric-like spaces with an application

Best Proximity Point Results via Simulation Functions in Metric-Like Spaces