Fixed Point Theorem Research Articles

A noncompact Schauder fixed point theorem in random normed modules and its applications

A noncompact Schauder fixed point theorem in random normed modules and its applications

Fixed Point Results for Fuzzy Enriched Contraction in Fuzzy Banach Spaces with Applications to Fractals and Dynamic Market Equillibrium

We introduce fuzzy enriched contraction, which extends the classical notion of fuzzy Banach contraction and encompasses specific fuzzy non-expansive mappings. Our investigation establishes both the presence and uniqueness of fixed points considering this broad category of operators using a Krasnoselskij iterative scheme for their approximation. We also show the graphical representation of fuzzy enriched contraction and analyze its graph for different values of beta. The implications of these findings extend to significant results within fuzzy fixed-point theory, enriching the understanding of iterative processes in fuzzy metric spaces. To demonstrate the versatility of our innovative concepts and the associated fixed-point theorems, we provide illustrative examples that showcase their applicability across diverse domains, including the generation of fractals. This demonstrates the relevance of fuzzy enriched contraction to iterated function systems, enabling the study of fractal structures under various contractive conditions. Additionally, we explore practical applications of fuzzy enriched contraction in dynamic market equilibrium, offering new insights into stability and convergence in economic models. Through this unified framework, we open new avenues for both theoretical advancements and real world applications in fuzzy systems.

On the singular-hybrid type of the Langevin fractional differential equation with a numerical approach

The nonlinear fractional Langevin equation is an important equation used in various areas of physics and mathematics. In this article, we focus on the singular-hybrid case of the generalized Langevin equations and prove the existence of a solution using fixed point theory. We also provide numerical and graphical simulations to demonstrate the convergence and existence of a solution for this singular problem. Finally, we present numerical examples in the form of tables and graphs to examine the accuracy of the method.

Granular fuzzy calculus on time scales and its applications to fuzzy dynamic equations

This paper introduces the foundational theory of fuzzy calculus on time scales, utilizing granular arithmetic operations between fuzzy intervals. These operations are developed based on the concept of the horizontal membership function (HMF), which is applied in multidimensional fuzzy arithmetic (MFA). Furthermore, the paper explores the existence of a unique solution and the continuous dependence of the solution to fuzzy dynamic equations on initial data, employing the Banach fixed-point theorem under a new metric for fuzzy functions in time scales involving the generalized exponential function. Finally, to highlight the practical significance of these results and their potential applications, the paper presents mathematical models relevant to nuclear physics and biology.

Some fixed point theorems in generalized parametric metric spaces and applications to ordinary differential equations

Abstract The objective of this work is the construction of `Boyd-Wong fixed point theorem’ in the setting of generalized parametric metric space and discussion its application on existence criteria of solutions to a second order initial value problem. Also an analogue of `Banach type fixed point theorem’ of generalized parametric metric space is proved and its application on the existence of a solution of first order periodic boundary value differential equation is examined.

A fixed point theorem in T-partial b-metric spaces and an application in dynamic programming

Here, a fixed point theorem is proved in a new class of spaces called [Formula: see text]-partial [Formula: see text]-metric spaces without using the compactness. This class of spaces is a new generalization of [Formula: see text]-metric spaces such that the generated topology is a Hausdorff topology. Moreover, an application in dynamic programming is given to illustrate the usability of the obtained results.

Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation

Abstract In this paper, we present the existence of global mild solutions for the Hilfer fractional neutral evolution equations (HFNEEs), regardless of whether the semigroups are compact or noncompact. We achieve our main results by utilizing the generalization Ascoli-Arzelà theorem, Krasnoselskii’s fixed point theorem, Laplace transform, and measures of noncompactness. To demonstrate the feasibility of our method, we provide an example.

The discrete fractional Karamata theorem and its applications

Abstract We establish a fractional extension of the discrete Karamata integration theorem for two types of fractional sum operators. This result, along with other properties of regularly varying sequences and the tools such as comparison theorems and a fixed point theorem in FK-space, is then used to study asymptotic properties of solutions to a nonlinear fractional difference equation. We also establish and utilize a fractional extension of the Stolz–Cesáro theorem, i.e., of the discrete l’Hospital rule. Both, the discrete fractional Karamata theorem and the discrete fractional l’Hospital rule, are believed to find applications in a broader context within the discrete fractional calculus.

Study of a non-linear Volterra integro-differential equation with a non-linear unknown source term

In this paper, we propose a new class of integro-differential nonlinear Volterra equations with a non-linear unknown source term. Our study focuses on both theoretical analysis and numerical approximation of solutions to these equations. In the theoretical analysis partition, we define the specific assumptions on the integral kernel's characteristics, on the source term function and also on the derivative of the solutions of equations, under which the Krasnoselskii's fixed point theorem can be applied and in order to ensures the existence and uniqueness of solutions to the proposed integro-differential equations. In the numerical approximation partition, we use a well-known numerical technique for solving integral equations called Nyström method. This method allows us to get an approximate solutions of the proposed equations. Furthermore, we provide some illustrative examples and according to the numerical method described in this paper we approach them, where the comparison results between the exact and approximate solutions of these examples confirm the effectiveness of the numerical Nyström method for approaching the solutions of this class of integro-differential equations.

Mathematical Evaluation and Dynamic Transmissions of a Tumor Growth Model Using a Generalized Singular and Non-Local Kernel

Currently, fractional calculus plays a critical role in improving control techniques, analyzing disease transmission dynamics, and solving several other real-world problems. This research investigates the time-fractional tumor growth model using an innovative approach. The new modified fractional derivative operator employs a singular and non-local kernel, based on Atangana and Baleanu's concepts with the Caputo derivative. The tumor growth model used the newly modified fractional operator, which provided numerical simulation. With the introduction of this new operator, we provide significant analysis for the tumor growth epidemic model. We have proven the uniqueness and stability conditions of the model by utilizing Banach's fixed point theory and the Picard successive approximation method. Using the Laplace-Adomian decomposition method (LADM), we found the numerical solution to the Modified Atangana-Baleanu-Caputo derivative model. We have verified the convergence analysis of the suggested scheme. We ultimately utilize the suggested method to obtain numeric outcomes and simulations for the tumor growth model. The study investigates the effect of multiple biological variables on the transmission of tumor growth dynamics.

Analysis of multi-term arbitrary order implicit differential equations with variable type delay

Due to their capacity to simulate intricate dynamic systems containing memory effects and non-local interactions, fractional differential equations have attracted a great deal of attention lately. This study examines multi-term fractional differential equations with variable type delay with the goal of illuminating their complex dynamics and analytical characteristics. The introduction to fractional calculus and the justification for its use in many scientific and technical domains sets the stage for the remainder of the essay. It describes the importance of including variable type delay in differential equations and then applying it to model more sophisticated and realistic behaviours of real-world phenomena. The research study then presents the mathematical formulation of variable type delay and multi-term fractional differential equations. The system’s novelty stems from its unique combination of variable delay, generalized multi terms fractional differential operators (n and m), and integral implicit parameters, and studying the stability of the the newly formulated system as compared to the work in the existing literature. While the variable type delay is introduced as a function of time to describe instances where the delay is not constant, the fractional order derivatives are generated using the Caputo approach. The existence, uniqueness, and stability of solutions are the main topics of the theoretical analysis of the suggested differential equations. In order to establish important mathematical features, the inquiry makes use of spectral techniques, and fixed-point theorems. The study finishes by summarizing the major discoveries and outlining potential future research avenues in this developing field. It highlights the potential contribution of multi-term fractional differential equations with variable type delay to improving the control and design of complex systems. Overall, this study adds to the growing body of knowledge in the field of fractional calculus and provides insightful information about the investigation of multi-term fractional differential equations with variable type delay, making it pertinent for academics and practitioners from a variety of fields.

On the relation of Kannan contraction and Banach contraction

Abstract Kannan contraction is broadly investigated topic in Metric Fixed Point Theory due to its importance in omitting the continuity presumption. Of a great significance is also its role in characterizing completeness of a metric space through existence and uniqueness of a fixed point of arbitrary Kannan contraction in the observed setting. The concept of Kannan contraction has been adapted, extended and transferred to various types of spaces including cone metric spaces, quasi metric spaces, b-metric spaces, partial metric spaces, among others. The main aim of this article is to prove that for any Kannan contraction T on a complete metric space (X, d) there exists another metric on the set X in relation to which T is a Banach contraction while the completeness is preserved. In that way, all results on Kannan contraction may be derived as corollaries of the Banach contraction principle. The converse also holds since, by altering the metric, Banach contraction becomes a Kannan contraction. The obtained theoretical results are substantiated with adequate examples.

A study on the solvability of fractional integral equation in a Banach algebra via Petryshyn's fixed point theorem

This study focuses on the nonlinear fractional functional integral equation (FFIE) concerning the Riemann-Liouville operator. In certain weaker conditions, the authors demonstrate that the FFIE has a solution, which is defined within the Banach algebra 0 $ ]]> C [ 0 , a ] , a > 0 . Our analysis relies on the Petryshyn's fixed point theorem and the notion of measure of non-compactness (MNC). In addition, our results include numerous authors' work under less restrictive conditions. Furthermore, we provide an illustrative example of fractional functional integral equations to support our proven results.

A φ-Contractivity Fixed Point Theory and Associated φ-Invariant Self-Similar Sets

In this paper, we apply a generalized variant of the concept of fixed point theory due to contraction mappings on metric spaces to construct a general class of iterated function systems relative to the so-called φ-contraction mappings on a metric space. In particular, we give a general framework to the Hutchinson method of constructing self-similar sets as fixed points of suitable mappings issued from the φ-contractions on the metric space. The results may open a new axis in the generalization of self-similar sets and associated self-similar functions. Moreover, our results may be extended to general metric spaces with suitable assumptions. The theoretical results are applied for the computation of the fractal dimension of a concrete example of the new self-similar sets.

Some Results for a Class of Pantograph Integro-Fractional Stochastic Differential Equations

Symmetrical fractional differential equations have been explored through a variety of methods in recent years. In this paper, we analyze the existence and uniqueness of a class of pantograph integro-fractional stochastic differential equations (PIFSDEs) using the Banach fixed-point theorem (BFPT). Also, Gronwall inequality is used to demonstrate the Ulam–Hyers stability (UHS) of PIFSDEs. The results are illustrated by two examples.

A note on the approximate controllability of damped second-order differential inclusions with state-dependent delay

In this paper, we use Bohnenblust–Karlin's fixed point theorem and the theory of strongly continuous cosine families of bounded linear operators to study the approximate controllability of damped second-order differential inclusions with state-dependent delay in Banach spaces. In particular, we obtain a new set of sufficient conditions for the approximate controllability of damped second-order differential inclusions with state-dependent delay under the assumption that the corresponding linear system is approximately controllable. An example is provided to illustrate our main results.

A novel approach to the convergence analysis of chaotic dynamics in fractional order Chua’s attractor model employing fixed points

This article uses a fixed point method to connect with Chua’s attractor model, incorporating the Atangana–Baleanu derivative using a two-step Lagrange polynomial. We introduce novel fixed point theorems using some special contractions followed by graphical representations of the convergence behavior of the iterative process. The uniqueness and existence of fixed points are demonstrated through theoretical proofs and numerical simulations. This approach of its first kind substantiates the existence and uniqueness of the model, enriching the understanding of its chaotic dynamics. Furthermore, numerical investigations explore distinct scenarios of Chua’s model, encompassing the classic Chua’s attractor, the double scroll attractor, chaos in Chua’s circuit, and behaviors such as cycle behavior and higher parametric values. Each case is analyzed graphically, providing insights into the system’s complex dynamics and validating theoretical predictions.

Existence and uniqueness for a mixed fractional differential system with slit-strips conditions

This paper studies the existence and uniqueness of solutions for a new kind of mixed fractional differential systems with slit-strips conditions, containing Caputo-type fractional derivatives. The first result on the existence and uniqueness is based on Banach’s fixed point theorem, and the second result on the existence and uniqueness of the solution is proved by using a Schaefer-type fixed point theorem. The applicability of our primary results is finally illustrated by some examples.

A mathematical modeling of patient-derived lung cancer stem cells with fractional-order derivative

The aim of this article is to help predict the course of lung cancer patients. To make this prediction as close to reality as possible, we used data from lung cancer patients receiving treatment at Erciyes University Hospitals in Kayseri, Turkey. First, we developed a mathematical model considering the cells in the microenvironment of lung cancer tumors with the assistance of Caputo fractional derivatives. Subsequently, we identified the equilibrium points of the proposed mathematical model and examined the coexistence equilibrium point. In addition, we demonstrated the existence and uniqueness of the solutions through the fixed-point theorem. We also investigated the positivity and boundedness of the model’s solutions to show whether they are biologically meaningful. Using laboratory experimental results from cancer stem cells isolated from resected tumor tissues of lung cancer patients, we determined the most biologically realistic parameter values through the least squares curve fitting approach. Then, using these parameter values, we performed numerical simulations with the Adams-Bashforth-Moulton predictor-corrector method to validate the theoretical results. We considered different values of fractional derivatives to investigate how the model is affected by fractional derivatives. As a result, we obtained the dynamics and expectations of lung cancer and made predictions specific to individual patients. In our simulations based on the parameter values obtained from actual patient data, it has been observed that after a certain period, both tumor cells and cancer stem cells have been eliminated. Consequently, an increase in normal tissue cells and immune cells has been observed. This implies that the patient in question, and similar behaving patients, will recover and overcome cancer. The findings from this study provide insights into the dynamics and prognosis of lung cancer, opening up the possibility for more personalized and effective approaches to treatment.

Unconditional superconvergence analysis of a new energy stable nonconforming BDF2 mixed finite element method for BBM–Burgers equation

The focus of this paper is to establish a new energy stable 2-step backward differentiation formula (BDF2) fully-discrete mixed finite element method (MFEM) for the Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equation with the nonconforming rectangular EQ1rot element and zero-order Raviart–Thomas (R–T) element (EQ1rot/Q10×Q01). Based on the energy stable property, the existence and uniqueness of the numerical solution are proved by the Brouwer fixed point theorem. Subsequently, with the assistance of some typical properties of this element pair and the interpolation post-processing approach, the unconditional superclose and superconvergence results with O(h2+τ2) are obtained directly without any constraints between the spatial partition parameter h and the time step τ. It is worthy to mention that the method and analysis presented herein are very different from the time–space splitting approach utilized in the previous studies and simplify the implement. Finally, two numerical experiments are executed to validate the theoretical analysis.