The unique solution of some operator equations via fractional differential equations

In this paper we consider the existence and uniqueness of positive solutions to the following operator equation in an ordered Banach space $E$$$A(x,x)+B(x,x)=x,~x\in P,$$where $P$ is a cone in $E$. We study an application for fractional differential equations.

Solution of system of the Fredholm integral equation of the first kind

The article is devoted to the study of a system of two inhomogeneous Fredholm integral equations of the first kind with two required functions depending on one variable. Integral equations describe the restoration of a blurred image, production costs, etc. Fredholm integral equations with one desired function have been considered in many works, but relatively few works have been devoted to systems of such equations. The questions of stability for the solution of systems and the construction of a regularizing system of equations were investigated, but the solution was not constructed in an explicit form. In this paper, the kernels depend on two variables. The case is considered: in the kernels and inhomogeneities, the variables are separated in the equations; these functions are decomposed on the basis of two functions on the interval of integration. Examples of basic functions are given. A condition is determined under which the system has a unique solution in the chosen basis, formulated as a theorem. The solution is found in the form of an expansion in this basis. To illustrate the results obtained, an example is considered

Transformation digitale et achat public : procédures dématérialisées, cas de l’Université Hassan II – Casablanca –

La commande publique représente une part importante des dépenses publiques : entre 15% et 17% du PIB du Maroc. La dématérialisation de la procédure des marchés publics se présente comme seule et unique solution devant les entités publiques pour renforcer leur efficacité en termes d’engagements et d’émissions. La dématérialisation de la procédure des achats des Universités va permettre d’améliorer leur efficacité et à participer à l’atteinte des objectifs de bonne gouvernance dans la gestion de la chose publique. Dans le cadre de cette recherche, nous allons adopter une méthode d’étude qualitative basée sur un entretien en profondeur. Cette étude aura comme objectif d’étudier la fonction des achats au sein de l’Université Hassan II – Casablanca et de proposer une démarche qui intègre la digitalisation comme étant une variable importante du développement de cette fonction. L’administration dudit guide au niveau de 19 établissements, nous a permis de collecter les réponses de 13 établissements, soit un taux de couverture de la population cible de 68,42%.

Pareto-based Dynamic Difficulty Adjustment of a Competitive Exergame for Arm Rehabilitation (Preprint)

BACKGROUND Lack of motivation is a major hindrance to frequent and intense exercise which is critical to rehabilitating people with arm disabilities due to old age, neurological disorders or stroke. Recently, the use of interpersonal exergames has been associated with increased motivation and exercise intensity in arm rehabilitation and is becoming a common trend. However, the Dynamic Difficulty Adjustment (DDA) of such games is still an open issue because unlike the traditional DDA frameworks where game intensity is simply adapted to suit the players' performance, the aim of DDA for exergames is to optimize the conflicting objectives namely of intensity and performance. Objective: To design a dedicated DDA for rehabilitation exergames that optimize the conflicting objectives of intensity and performance by generating a set of feasible trade-off solutions. Based on the rehabilitative needs, the tradeoff worth information of each solution is to be used to select a unique solution. OBJECTIVE To design a dedicated DDA for rehabilitation exergames that optimizes the conflicting objectives of intensity and performance by generating a set of feasible trade-off solutions. Based on the rehabilitative needs, the tradeoff worth information of each solution is to be used to select a unique solution. METHODS We designed a Pareto-based DDA for a competitive exergame that optimizes the two conflicting objectives. Using a set of feasible solutions generated during the first episode of the game, the proposed Pareto-based DDA is used to modify the parameters of the game. Optimizing conflicting objectives generally results in a set of trade-off solutions called Pareto optimal set instead of a single solution. Therefore, the DDA is capable of selecting a single solution from the optimal Pareto based on the trade-off worth information of each solution in the optimal Pareto set. RESULTS Results: Experimental results with 12 unimpaired participants show the capability of the proposed Pareto-based DDA to online adjust the game parameters effectively based on a trade-off between the intensity and motivation. CONCLUSIONS Since rehabilitation outcomes rely on both intensity and motivation, unlike traditional DDA approaches, the capability of Pareto-based DDA to provide trade-off solutions between conflicting objectives of intensity and motivation is very promising to rehabilitation outcomes. However, multi-session investigation over a period of time needs to be carried out to examine if they influence rehabilitation outcomes positively. CLINICALTRIAL This work is not a clinical trial. Although humans participated in this study, they participate in the evaluation of a single-session of a rehabilitation exergame rather than a comprehensive rehabilitation intervention with no health outcomes investigated.

Wardowski’s Contraction and Fixed Point Technique for Solving Systems of Functional and Integral Equations

In this manuscript, some tripled fixed point results are presented in the framework of complete metric spaces. Furthermore, Wardowski’s contraction was mainly applied to discuss some theoretical results with and without a directed graph under suitable assertions. Moreover, some consequences and supportive examples are derived to strengthen the main results. In the last part of the paper, the obtained theoretical results are used to find a unique solution to a system of functional and integral equations.

STATIONARY DISTRIBUTION AND EXTINCTION OF STOCHASTIC CORONAVIRUS (COVID-19) EPIDEMIC MODEL

The aim of this paper is to model corona-virus (COVID-19) taking into account random perturbations. The suggested model is composed of four different classes i.e. the susceptible population, the smart lockdown class, the infectious population, and the recovered population. We investigate the proposed problem for the derivation of at least one unique solution in the positive feasible region of nonlocal solution. For one stationary ergodic distribution, the necessary result of existence is developed by applying the Lyapunov function and the condition for the extinction of the disease is also established. The obtained results show that the effect of Brownian motion and noise terms on the transmission of the epidemic is very high. If the noise is large the infection may decrease or vanish. For validation of our obtained scheme, the results for all the classes of the problem have been simulated numerically.

On a Conjecture for the One-Dimensional Perturbed Gelfand Problem for the Combustion Theory

We investigate the well-known one-dimensional perturbed Gelfand boundary value problem and approximate the values of α0,λ* and λ* such that this problem has a unique solution when 0<α<α0 and λ>0, and has three solutions when α>α0 and λ*<λ<λ*. The solutions of this problem are always even functions due to its symmetric boundary values and autonomous characteristics. We use numerical computation to show that 4.0686722336<α0<4.0686722344. This result improves the existing result for α0≈4.069 and increases the accuracy of α0 to 10−8. We developed an algorithm that reduces errors and increases efficiency in our computation. The interval of λ for this problem to have three solutions for given values of α is also computed with accuracy up to 10−14.