Fixed Point Theorem Research Articles

Intuitionistic Fuzzy Partial Metric Spaces

Fuzzy logic is a theory that is used as an alternative to classical structures in both application and algebraic fields. The fixed point theorem is a theorem widely used in mathematics, especially in metric spaces and partial metric spaces. The fixed point theorem is used on classical metric structures, but it is also widely used on fuzzy metric spaces, fuzzy partial metric spaces and intuitionistic fuzzy metric spaces. In this paper, intuitionistic fuzzy partial metric spaces are defined, their basic properties and examples are obtained. For it, open ball, convergent sequence, and Cauchy sequence are defined and their basic properties are introduced. Furthermore, the relations between intuitionistic fuzzy partial metric spaces, classical metric spaces, fuzzy metric spaces, fuzzy partial metric spaces, and intuitionistic fuzzy metric spaces are analyzed. As a result of this investigation, it is shown that from each classical metric, classical partial metric, and intuitionistic fuzzy metric, an intuitionistic fuzzy partial metric can be obtained. Moreover, it is achieved that an intuitionistic fuzzy metric is also an intuitionistic fuzzy partial metric space. Thus, a new structure is given by transferring the partial metric structure to intuitionistic fuzzy metric spaces.

The strong waves for a non-local dispersal epidemic model with latent stage and multiple parallel infectious stages

Considering the phenomenon of multiple parallel infection and latent stages in the spatial spread of the disease, a non-local dispersal epidemic model, which describes such a mechanism of disease transmission, is proposed. Our attention is directed toward the existence of strong traveling wave solutions connecting disease-free equilibrium and endemic equilibrium using the fixed point theorem, the method of upper–lower solutions and some limit techniques. When the basic reproduction number [Formula: see text], a unique critical wave speed [Formula: see text] is defined, and then when wave speed [Formula: see text], the existence of the traveling wave solution is established. In particular, the existence of strong wave solutions at wave speed [Formula: see text] is also discussed. Furthermore, when [Formula: see text] and [Formula: see text] or [Formula: see text] and [Formula: see text], the non-existence of traveling wave solution also is established. Finally, we present the numerical simulations and investigate the effect of the critical wave speed and non-local diffusion on disease propagation.

Multi-bump solutions for a Schrödinger equation with prescribed L2-norm via a fixed point approach

This paper focuses on investigating the existence of multi-bump solutions for the following non-homogeneous Schrödinger equation with prescribed [Formula: see text]-norm in [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are viewed as a potential functions, [Formula: see text] belongs to the dual of our working space and the [Formula: see text] is considered a continuous function that satisfies exponential critical growth when [Formula: see text] and subcritical growth when [Formula: see text]. We consider that function [Formula: see text] has a potential well [Formula: see text] consisting of [Formula: see text] isolated connected components. Instead of using variational methods, we adopt a modified fixed point theorem to investigate the problem. Specifically, for [Formula: see text] sufficiently large, this approach establishes the existence of at least [Formula: see text] multi-bump positive solutions for the above equation.

Two fixed point results on b-metric space

In this paper, we present two fixed point theorems on $b$-metric spaces. First, we obtain a general result, which includes many fixed point theorems on $b$-metric space in the literature as a special case, by using implicit relation technique, Then, we define the concept of nearly Lipschitzian mapping in $b$-metric spaces, and we obtain a fixed point theorem for such mappings.

Multivalued Probabilistic ω-Contraction in b-Menger Spaces with Aapplication

In this article, we extend the concept of multivalued $\omega$-contraction mappings within the framework of b-Menger spaces. We introduce a novel fixed point theorem specific to these mappings, significantly broadening the existing body of knowledge in this area. Our primary result directly leads to the derivation of an equivalent fixed point theorem for fuzzy b-metric spaces, showcasing the versatility and applicability of our findings in more generalized and complex metric structures. Furthermore, we provide a concrete application of the principal theorem in the context of ordinary b-metric spaces, demonstrating the practical implications and effectiveness of our theoretical advancements. This research contributes to the deeper understanding and potential applications of fixed point theory in various scientific and engineering domains, where probabilistic and fuzzy structures play a critical role.

Solutions to nonlinear elliptic problems with nonhomogeneous operators and mixed nonlocal boundary conditions

We investigate the existence, multiplicity and nonexistence of positive solutions to nonlinear (singular) elliptic problems involving nonhomogeneous operators and mixed nonlocal boundary conditions based on the behaviors of the nonlinear term near \(0\) and \(\infty\). In particular, we discuss the existence of at least three positive solutions to the mixed nonlocal boundary problems, which is new finding even for the problems involving homogeneous operators. The novelty of this study lies in constructing completely continuous operators related to nonlinear elliptic problems involving complicated boundary conditions. We emphasize that only one fixed point theorem is used to obtain the existence and multiplicity results, despite generalizingand extending most of the problems in previous literature. For more information see https://ejde.math.txstate.edu/Volumes/2025/66/abstr.html

Upper and lower solution method for control of second-order Kolmogorov type systems

In this paper, an upper and lower solution method for the control of second-order Kolmogorov systems is introduced. Two iterative algorithms, one exact and one approximate, are proposed and their convergence is studied. The technique is based on Perov's fixed point theorem, matrices convergent to zero, and the use of Bielecki's norm.

Common fixed point theorems of integral type in intuitionistic fuzzy metric space using control function with applications

In this paper, first, we investigate the existence and uniqueness of common fixed point for two pairs of weakly compatible maps which form a new type of integral-type condition via CLR property in the context of intuitionistic fuzzy metric space. The given results not only associate but also establish various existing results in the literature. Second, we present application of common fixed point theorem for the existence and uniqueness of solutions for functional equations occurring in dynamic programming of multistage decision processes. We also give an illustrative example which yields the main result.

Some fixed point results on controlled rectangular partial metric spaces with a graph structure

In this work, we define a controlled rectangular partial metric space, which is a generalization of a controlled rectangular metric space and a controlled metric space. Furthermore, we derive some fixed point results with suitable conditions in this setting. As an application, we give a fixed point theorem for graphic contractions on a controlled rectangular partial metric space via a graph structure.

Tuning Mechanism and Parameter Optimization of a Dynamic Vibration Absorber with Inerter and Negative Stiffness Under Delayed FOPID

The dynamic vibration absorber (DVA) based on delayed fractional PID (DFOPID) can achieve a more superior vibration suppression effect. However, the strong nonlinear characteristics of the system and the computational burden resulting from its high dimensionality make solving and optimizing more challenging. This paper presents a coupled model of DFOPID and DVA, exploring its parameter tuning mechanism and optimization problem. First, using the averaging method and Lyapunov stability theory, the amplitude-frequency equation and the stability condition of the steady-state solution of the primary system are derived. Numerical simulations validate the accuracy of the analytical result. Next, based on the mechanics of vibration, the approximate expressions of the controller under different differential conditions are calculated, and their equivalent action mechanisms are analyzed. Finally, by minimizing the maximum amplitude of the primary system as the objective function, the Particle Swarm Optimization (PSO) algorithm is applied to optimize the parameters of the passive DVA and the DVA models controlled by PID, FOPID, and DFOPID, successfully addressing the parameter optimization challenges posed by traditional fixed-point theory. The vibration reduction performance is compared across different loading environments. The results demonstrate that the model presented in this paper performs the best, exhibiting excellent vibration suppression and robustness.

APPLICATION OF A FRACTAL FRACTIONAL DERIVATIVE WITH A POWER-LAW KERNEL TO THE GLUCOSE–INSULIN INTERACTION SYSTEM BASED ON NEWTON’S INTERPOLATION POLYNOMIALS

This paper introduces novel numerical schemes based on Newton’s interpolation polynomial for solving a fractal-fractional glucose–insulin regulatory system modeled using the Lotka–Volterra framework. The model incorporates fractal-fractional derivatives with a power-law kernel, extending classical glucose–insulin dynamics. Fixed-point theory establishes solutions’ existence and uniqueness under the Caputo fractal-fractional operator. Hyers–Ulams stability is investigated. By using linear controllers, the Caputo fractal-fractional-order glucose–insulin system can be controlled to equilibrium. This control system has the potential to improve the management of diabetes by providing precise regulation of blood glucose levels. Numerical examples illustrate the effectiveness of the proposed method. To solve the system, the Atangana–Seda numerical scheme, incorporating Newton’s interpolation polynomials, is applied. Simulated results are compared with analytical findings, confirming the accuracy and applicability of the approach to real-world biological systems.

A New Class of (α,η,(Q,h),L)-Contractions in Triple Controlled Metric-Type Spaces with Application to Polynomial Sine-Type Equations

This paper introduces a novel class of generalized contractions, termed (α,η,(Q,h),L)-contraction mapping, within the context of triple controlled metric-type spaces, extending the framework of fixed point theory in controlled structures. The proposed mapping is defined using α-admissible and η-subadmissible functions, in conjunction with a control pair (Q,h) of upper class of type I, and incorporates Wardowski’s function L-contraction condition. Under suitable hypotheses, we establish both the existence and uniqueness of fixed points for this class of mappings. Several corollaries are derived as special cases of the main result. Moreover, we provide a nontrivial application by analyzing the solvability of a nonlinear equation involving powers of the sine function, thereby illustrating the utility of the developed theory.

Elegant Rational Contractive Conditions with Applications to Implicit Functional Integral Equations

This paper aims to introduce several new classes of contraction mappings inspired by convex and rational contraction mappings. We establish the existence and uniqueness of fixed points for each newly proposed contraction mapping in metric spaces. To validate our theoretical findings, we provide several illustrative examples that demonstrate cases where well-known fixed point results, such as the Banach contraction principle, the Kannan fixed point theorem, the Chatterjea fixed point theorem, the Jaggi fixed point theorem, and the Istratescu fixed point theorem, are not applicable. As an application, we employ our theoretical fixed point results to investigate the existence and uniqueness of solutions to nonlinear implicit integral equations.

Fixed-Point Theorems in Vector-Valued Fuzzy Metric Spaces

This paper presents a comprehensive study on fixed-point theorems within the framework of vector-valued fuzzy metric spaces, extending classical notions of both metric and fuzzy metric spaces. In this generalized setting, the metric function takes values in a partially ordered Banach space, providing a richer structure for analyzing convergence in multi-dimensional and uncertain environments. We establish sufficient conditions for the existence of fixed points for self-mappings in these spaces. The theoretical results are supported by illustrative examples and are positioned to contribute meaningfully to both pure and applied mathematical disciplines, particularly in systems involving fuzzy logic, uncertainty modeling, and vector-valued analysis.

Equilibria for abstract economies with convex or acyclic valued correspondences

Abstract Using collectively fixed point theorems, we establish new equilibrium results for abstract economies where the constraint and preference correspondences are Kakutani or acyclic maps. First we present a variety of new collectively fixed point theorems based on a generalized Schauder type theorem. This then enables us to establish new equilibrium results for abstract economies where, in particular, we obtain new general results for acyclic valued constraint and preference correspondences.

A second-order difference scheme for time-fractional Burgers-Fisher equation with initial singularity

In this paper, we investigate a numerical method for time-fractional Burgers-Fisher equation with initial singularity. Initially, the Caputo fractional derivative and the spatial derivative are discretized by the L 2 - 1 σ formula on a time-graded mesh and the central difference formula, respectively. We also employ the Newton linearization method to handle the nonlinear term. Subsequently, we analyze the existence of the numerical solution using the Browder fixed point theorem. Based on the cut-off function technique, the stability and convergence of the scheme are obtained by the energy method. The theoretical results indicate that the proposed scheme can achieve second-order accuracy in both time and space with an appropriate choice of graded mesh. Ultimately, numerical experiments are conducted to demonstrate the feasibility and efficiency of our method.

Mathematical modelling of child mortality with Caputo–Fabrizio fractional derivative

In this article, we proposed a fractional-order mathematical model of Child mortality. We analyzed the existence of a unique solution for our model using the fixed point theory and Picard–Lindelöf technique. We propose a Caputo operator for modeling child mortality in a given population of 1000 susceptible under five children. Our stability analysis was based on the fixed point theory, which was used to prove that our Picard iteration was stable. Using the Julia software and some real world values for our parameters, we numerically simulated the system through graphs. Our findings were that, reducing child mortality rates alone is insufficient to significantly improve survival rates for children under five. To make a real impact, a holistic approach is necessary, including access to healthcare, proper nutrition, vaccination programs, hygiene practices, clean water sources and comprehensive public health campaigns can greatly enhance the survival rates of children under five.

New Fixed Point Theorem on Extended b-Metric Space Using a Class of Contraction Mappings

Abstract The main purpose of this paper is to investigate a fixed point for a class of contractive maps over a complete extended b-metric space. In this article, we used the Bianchini-type contraction and BK-type contractive mapping to find new results in the direction of a complete extended b-metric space. To illustrate our results, we give several examples.

Optimal control of conformable fractional neutral stochastic integrodifferential systems with infinite delay

The main concern of the manuscript deals with the optimal control problem of conformable fractional neutral stochastic integrodifferential systems (CFNSIDEs) with infinite delay. Initially, we investigate the existence of mild solutions for the CFNSIDEs using stochastic analysis techniques and Banach fixed point theorem. In the later part, we establish the existence of mild solutions of the CFNSIDEs with infinite delay. Furthermore, the existence of optimal control of the corresponding Lagrange optimal control problem is investigated. An example is provided to illustrate the applications of obtained results. Numerical simulation for the optimization of the control for studied problem justifies the analytical results.

Hyperbolic Stefan problem with nonlocal boundary conditions

In this paper, we consider problems with unknown boundaries for hyperbolic equations and systems with free boundaries with two independent variables. The boundary conditions for such equations in the linear or quasilinear cases are given in nonlocal (non-separable and integral) form. The hyperbolic Stefan and Darboux-Stefan problems (the line of initial conditions degenerates to a point) are considered. There are proved the existence and uniqueness theorems of generalized solution, which are continuous solutions of equivalent systems of the second kind Volterra integral equations.The method of characteristics based on a combination of the Banach fixed point theorem allows us to obtain global generalized solutions in terms of the time variable in the case of linear hyperbolic equations with free boundaries and local solutions for quasilinear equations.Nonlocal (non-separable and integral) conditions require additional solvability conditions that are not present in the case of generally accepted boundary conditions for hyperbolic equations and systems. The paper provides examples indicating the significance of the conditions for the solvability of the corresponding problems. The corresponding solutions may have discontinuities along the characteristics of the hyperbolic equations. This additionally requires setting the conditions for matching the initial data of the problems at the corner points of the considered domains.This paper extends the results on the problems with nonlocal conditions for hyperbolic equations and systems to the case of hyperbolic equations with free boundaries.