Fixed Point Technique Research Articles

Solving an Integral Equation via Tricomplex-Valued Controlled Metric Spaces

In this present study, we propose the concept of tricomplex-controlled metric spaces as a generalization of both controlled metric-type spaces and tricomplex metric-type spaces. In this work, we establish fixed point results using Banach, Kannan and Fisher-type contractions supported with nontrivial examples in the setting of the proposed space. We apply the derived result to find the analytical solution of an integral equation using the fixed point technique under the same metric.

Solution of an integral equation in G-metric spaces

In this paper, we prove a fixed point result in G-metric spaces satisfying generalized contractive condition with new auxiliary functions, which generalize the result of Bhardwaj and Kumar. The motivation of this paper is to observe the solution of an integral equation using the fixed point technique in G -metric space.

A fixed point result on a DCMLS and applications to matrix equations and random integral equations

In this work, we present a fixed point result in the context of a double controlled metric-like space involving its control functions. A matrix equation has been solved via a fixed point technique. We also give some concrete and numerical examples. Finally, we ensure the existence of a solution of a random integral equation of Fredholm type.

Approximate controllability of Hilfer fractional neutral stochastic systems of the Sobolev type by using almost sectorial operators

<abstract><p>The main aim of this work is to conduct an analysis of the approximate controllability of Hilfer fractional (HF) neutral stochastic differential systems under the condition of an almost sectorial operator with delay. The theoretical ideas linked to stochastic analysis, fractional calculus and semigroup theory, along with the fixed-point technique, are utilized to establish the key results of this article. More precisely, the main theorem of this study is devoted to proving the fact that the relevant linear system is approximately controllable. Finally, to help this research be as comprehensive as possible, we provide a theoretical application and filter system.</p></abstract>

Modeling Drug Concentration in Blood through Caputo-Fabrizio and Caputo Fractional Derivatives

This study focuses on the dynamics of drug concentration in the blood. In general, the concentration level of a drug in the blood is evaluated by the mean of an ordinary and first-order differential equation. More precisely, it is solved through an initial value problem. We proposed a new modeling technique for studying drug concentration in blood dynamics. This technique is based on two fractional derivatives, namely, Caputo and Caputo-Fabrizio derivatives. We first provided comprehensive and detailed proof of the existence of at least one solution to the problem; we later proved the uniqueness of the existing solution. The proof was written using the Caputo-Fabrizio fractional derivative and some fixed-point techniques. Stability via the Ulam-Hyers (UH) technique was also investigated. The application of the proposed model on two real data sets revealed that the Caputo derivative was more suitable in this study. Indeed, for the first data set, the model based on the Caputo derivative yielded a Mean Squared Error (MSE) of 0.03095 with a corresponding best value of fractional order of derivative of 1.00360. Caputo-Fabrizio-basedderivative appeared to be the second-best method for the problem, with an MSE of 0.04324 for a corresponding best fractional derivative order of 0.43532. For the second experiment, Caputo derivative-based model still performed the best as it yielded an MSE of 0.04066, whereas the classical and the Caputo-Fabrizio methods were tied with the same MSE of 0.07299. Another interesting finding was that the MSE yielded by the Caputo-Fabrizio fractional derivative coincided with the MSE obtained from the classical approach.

Exponential stability of delayed neutral impulsive stochastic integro-differential systems perturbed by fractional Brownian motion and Poisson jumps

In this manuscript, we investigate the existence, uniqueness, and exponential stability of a delayed neutral impulsive stochastic integro-differential equation driven by fractional Brownian motion in a separable Hilbert space and Poisson jumps. The results are obtained, using the theory of resolvent operators, stochastic analysis, and a fixed-point technique. Lastly, an example is provided to show the validity of the obtained results.

A study on the existence of numerical and analytical solutions for fractional integrodifferential equations in Hilfer type with simulation

<abstract><p>Previous studies have shown that fractional derivative operators have become an integral part of modeling natural and physical phenomena. During the progress and evolution of these operators, it has become clear to researchers that each of these operators has special capacities for investigating phenomena in engineering sciences, physics, biological mathematics, etc. Fixed point theory and its famous contractions have always served as useful tools in these studies. In this regard, in this work, we considered the Hilfer-type fractional operator to study the proposed integrodifferential equation. We have used the capabilities of measure theory and fixed point techniques to provide the required space to guarantee the existence of the solution. The Schauder and Arzela-Ascoli theorems play a fundamental role in the existence of solutions. Finally, we provided two examples with some graphical and numerical simulation to make our results more objective.</p></abstract>

Interpretation on nonlocal neutral functional differential equations with delay

<abstract><p>This work deals with the existence and continuous dependence of an integral solution for neutral integro-differential equations with a nonlocal condition. This result is established by using an integrated resolvent operator under conditions of Lipschitz continuity and uniqueness via the Banach fixed point technique. We also study the existence of a strict solution on reflexive and general Banach spaces. In the last section, an example is provided related to this theory.</p></abstract>

A non-linear fractional neutral dynamic equations: existence and stability results on time scales

<abstract><p>The outcomes of a nonlinear fractional neutral dynamic equation with initial conditions on time scales are examined in this work using the Riemann-Liouville nabla ($ \nabla $) derivative. The existence, uniqueness, and stability results for the solution are examined by using standard fixed point techniques. For the result illustration, an example is given along with the graph using MATLAB.</p></abstract>

Applying fixed point techniques for obtaining a positive definite solution to nonlinear matrix equations

<abstract><p>In this manuscript, the concept of rational-type multivalued $ F- $contraction mappings is investigated. In addition, some nice fixed point results are obtained using this concept in the setting of $ MM- $spaces and ordered $ MM- $spaces. Our findings extend, unify, and generalize a large body of work along the same lines. Moreover, to support and strengthen our results, non-trivial and extensive examples are presented. Ultimately, the theoretical results are involved in obtaining a positive, definite solution to nonlinear matrix equations as an application.</p></abstract>

A computational method for investigating a quantum integrodifferential inclusion with simulations and heatmaps

<abstract><p>We aim to investigate an integro-differential inclusion using a novel computational approach in this research. The use of quantum calculus, and consequently the creation of discrete space, allows the computer and computational algorithms to solve our desired problem. Furthermore, to guarantee the existence of the solution, we use the endpoint property based on fixed point methods, which is one of the most recent techniques in fixed point theory. The above will show the novelty of our work, because most researchers use classical fixed point techniques in continuous space. Moreover, the sensitivity of the parameters involved in controlling the existence of the solution can be recognized from the heatmaps. For a better understanding of the issue and validation of the results, we presented numerical algorithms, tables and some figures in our examples that are presented at the end of the work.</p></abstract>

Discussion on iterative process of nonlocal controllability exploration for Hilfer neutral impulsive fractional integro-differential equation

<abstract><p>This manuscript primarily focuses on the nonlocal controllability results of Hilfer neutral impulsive fractional integro-differential equations of order $ 0\leq w\leq1 $ and $ 0 < g < 1 $ in a Banach space. The outcomes are derived from the strongly continuous operator, Wright function, linear operator, and bounded operator. First, we explore the existence and uniqueness of the results of the mild solution of Hilfer's neutral impulsive fractional integro-differential equations using Schauder's fixed point theorem and an iterative process. In order to determine nonlocal controllability, the Banach fixed point technique is used. We employed some specific numerical computations and applications to examine the effectiveness of the results.</p></abstract>

ON FRACTAL-FRACTIONAL WATERBORNE DISEASE MODEL: A STUDY ON THEORETICAL AND NUMERICAL ASPECTS OF SOLUTIONS VIA SIMULATIONS

Waterborne diseases are illnesses caused by pathogenic bacteria that spread through water and have a negative influence on human health. Due to the involvement of most countries in this vital issue, accurate analysis of mathematical models of such diseases is one of the first priorities of researchers. In this regard, in this paper, we turn to a waterborne disease model for solution’s existence, HU-stability, and computational analysis. We transform the model to an analogous fractal-fractional integral form and study its qualitative analysis using an iterative convergent sequence and fixed-point technique to see whether there is a solution. We use Lagrange’s interpolation to construct numerical algorithms for the fractal-fractional waterborne disease model in terms of computations. The approach is then put to the test in a case study, yielding some interesting outcomes.

Magnetic loss in grain-oriented Fe–Si sheets under different harmonic excitation and high indication

This paper presents an improved loss calculation method that takes into account the effects of harmonics and skinning effects, thus extending the loss prediction of soft magnetic materials to a larger frequency and induction range. First, the first-order and second-order slew curves are concurrently fitted into a nonlinear Preisach model to determine the hysteresis loss. Then, integrating the effects of high frequency harmonics and high induction, the fixed-point technique based on the H–B magnetic intrinsic relationship is introduced into the finite element calculation in order to calculate the flux density distribution of the silicon steel sheet more accurately. Then, the classical loss at high flux density is further calculated more accurately. Finally, losses for soft magnetic materials in the 1 kHz range are computed under various harmonic excitations. This method is compared to the traditional method and the experimental results, and this method is closer to the experimental results.

Existence Results for Nonlinear Fractional Differential Inclusions via q-ROF Fixed Point

Fractional Differential inclusions, the multivalued version of fractional differential equations, yellow play a vital role in various fields of applied sciences. In the present article, a class of q-rung orthopair fuzzy (q-ROF) set valued mappings along with q-ROF upper/lower semi-continuity have been introduced. Based on these ideas, existence theorems for a numerical solution of a distinct class of fractional differential inclusions have been achieved with the help of Schaefer type and Banach contraction fixed point theorems. A physical example is also provided to validate the hypothesis of the main results. The notion of q-rung orthopair fuzzy mappings along with the use of fixed point techniques and a new-fangled Caputo type fractional derivative are the principal novelty of this article.

A New Result Concerning Nonlocal Controllability of Hilfer Fractional Stochastic Differential Equations via almost Sectorial Operators

This manuscript mainly focused on the nonlocal controllability of Hilfer fractional stochastic differential equations via almost sectorial operators. The key ideas of the study are illustrated by using ideas from fractional calculus, the fixed point technique, and measures of noncompactness. Then, the authors establish new criteria for the mild existence of solutions and derive fundamental characteristics of the nonlocal controllability of a system. In addition, researchers offer theoretical and real-world examples to demonstrate the effectiveness and suitability of our suggested solutions.

On Nonlinear Ψ-Caputo Fractional Integro Differential Equations Involving Non-Instantaneous Conditions

We propose a solution to the symmetric nonlinear Ψ-Caputo fractional integro differential equations involving non-instantaneous impulsive boundary conditions. We investigate the existence and uniqueness of the solution for the proposed problem. Banach contraction theorem is employed to prove the uniqueness results, while Krasnoselkii’s fixed point technique is used to prove the existence results. Additionally, an example is used to explain the results. In this manner, our results represent generalized versions of some recent interesting contributions.

Atangana–Baleanu Semilinear Fractional Differential Inclusions With Infinite Delay: Existence and Approximate Controllability

Abstract The main focus of this paper is centered around approximate controllability results of Atangana–Baleanu fractional differential systems with infinite delay. Using principles and ideas from the theory of multivalued maps, fractional calculus, and Bohnenblust–Karlin fixed point techniques, the key findings are established. We begin by emphasizing the existence of mild solutions, and then demonstrate the approximate controllability of the Atangana–Baleanu fractional control system. We then apply our findings to the theory of the neutral system.

Sequential Caputo–Hadamard Fractional Differential Equations with Boundary Conditions in Banach Spaces

We present the existence of solutions for sequential Caputo–Hadamard fractional differential equations (SC-HFDE) with fractional boundary conditions (FBCs). Known fixed-point techniques are used to analyze the existence of the problem. In particular, the contraction mapping principle is used to investigate the uniqueness results. Existence results are obtained via Krasnoselkii’s theorem. An example is used to illustrate the results. In this way, our work generalizes several recent interesting results.

Hyers–Ulam stability for a class of Hadamard fractional Itô–Doob stochastic integral equations

Our goal in this work is to demonstrate the existence and uniqueness of the solution to a class of Hadamard Fractional Itô–Doob Stochastic integral equations (HFIDSIE) of order φ∈(0,1) via the fixed point technique (FPT). Hyers–Ulam stability (HUS) is investigated for HFIDSIE according to the Gronwall inequality. Two theoretical examples are provided to illustrate our results.