Contractive Research Articles

Existence and uniqueness of the solution to initial and inverse problems for integro-differential heat equations with fractional load

This work is devoted to the unique solvability of the direct and inverse problems for a multidimensional heat equation with a fractional load in Holder spaces. In the problem under consideration, the loaded term is in the form of a fractional integral operator for the time variable. We prove the existence and uniqueness of the solution to these problems by the contraction mapping theorem and the theory of integral equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/64/abstr.html

On the Generalized (p,q)-ϕ-Calculus with Respect to Another Function

In the present paper, we generalized some of the operators defined in (p,q)-calculus with respect to another function. More precisely, the generalized (p,q)-ϕ-derivatives and (p,q)-ϕ-integrals were introduced with respect to the strictly increasing function ϕ with the help of different orders of the q-shifting, p-shifting, and (q/p)-shifting operators. Then, after proving some related properties, and as an application, we considered a generalized (p,q)-ϕ-difference problem and studied the existence property for its unique solutions with the help of the Banach contraction mapping principle.

Convergence of Graph-Based Fixed Point Results with Application to Fredholm Integral Equation

In this manuscript, we present a novel concept termed graphical Θc-Kannan contraction within the context of graphically controlled metric-type spaces. Unlike traditional Kannan contraction, this novel concept presents a modified method of contraction mapping. We discuss the significance and the existence of fixed point results within the framework of this novel contraction. To strengthen the credibility of our theoretical remarks, we provide a comparison example demonstrating the efficiency of our suggested framework. Our study not only broadens the theoretical foundations inside graphically controlled metric-type spaces by introducing and examining visual Θc-Kannan contraction, but it also demonstrates the practical significance of our innovations through significant examples. Furthermore, applying our findings to second-order differential equations by constructing integral equations into the domain of Fredholm sheds light on the broader implications of our research in the field of mathematical analysis and contributes to the advancement of this field.

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.

Creating new contractive mappings to obtain fixed points and data-dependence results under auxiliary functions

This manuscript is concerned with obtaining results for fixed points that arise from new contractive mappings on controlled metric spaces. These mappings are a mixture of Wardowski’s contractions with both multivalued, nonlinear mappings and auxiliary functions. It is also proved that the obtained fixed-point outcomes are well-posed. Additionally, a data-dependence result for fixed points is given. To aid with understanding, several illustrative examples are also provided. Numerous findings that are currently in the literature are specific instances of the findings that were made.

Fixed Point Results of Fuzzy Multivalued Graphic Contractive Mappings in Generalized Parametric Metric Spaces

Fixed Point Results of Fuzzy Multivalued Graphic Contractive Mappings in Generalized Parametric Metric Spaces

Model-free adaptive consensus design for a class of unknown heterogeneous nonlinear multi-agent systems with packet dropouts

This paper studies the consensus problem for a class of unknown heterogeneous nonlinear multi-agent systems via a network with random packet dropouts. Based on the dynamic linearization technique, novel model-free adaptive consensus protocols with the data compensation mechanism are designed for both leaderless and leader-following cases. The advantage of this approach is that only neighborhood input and output data of the agents are required in the protocol design. For the stability analysis, a new Squeeze Theorem based method is developed to derive the theoretic results instead of the traditional contraction mapping principle used in model-free adaptive control. It is shown that the consensus can be achieved for both leaderless and leader-following cases if the communication topology is strongly connected. Finally, numerical simulations verifying the correctness of the theoretical results are given.

Fixed Point Results for Almost Contraction Mappings in Fuzzy Metric Space

في بعض المسائل الرياضية والحاسوبية والاقتصادية والنمذجة، يكون وجود حل لمشكلة نظرية أو مشكلة في العالم الحقيقي مرادفًا لوجود نقطة صامدة (Fp) لدالة مناسبة. وبالتالي، فأن نظرية النقطة الصامدة Fp تلعب دورًا أساسيًا في مجموعة واسعة من السياقات الرياضية والعلمية. تعتبر نظرية النقطة الصامدة Fp في حد ذاتها مزيجًا مذهلاً مكون من التحليل (التحليل الصرف والتحليل التطبيقي)، والهندسة، والطوبولوجيا. لقد أظهرت السنوات الأخيرة أن نظرية النقطة الصامدة Fp هي أداة قوية ومفيدة للغاية في دراسة الحالات غير الخطية. تهتم نظريات النقطة الصامدة Fp بدالة f من المجموعة X الى المجموعة X نفسها والتي في ظل ظروف معينة، فأنها تسمح بوجود نقطة صامدة Fp ، بعبارة اخرى انه بمعنى لكل نقطة x موجودة في المجموعة X (x∈X) بحيث أن f(x)=x . يقدم هذا العمل ويثبت نظرية النقطة الصامدة Fp لأنواع مختلفة من دوال الانكماش في الفضاء المتري الضبابي (FM-space) والتي تسمى بدالة الانكماش القريبة ودالة الانكماش الضعيفة القريبة ( Ψ̃ ,Φ̃). في البداية، تم التذكير بمفهوم الفضاء المتري الضبابي(FM-space) وبعض المصطلحات المستخدمة في الإطار الضبابي. ثم بعد ذلك تم اعطاء مفهوم دالة المحاكاة. يتم استخدام مفهوم دالة المحاكاة هذا لتقديم تعريف دالة الانكماش Z̃ القريبة في إطار الفضاء المتري الضبابي. بالإضافة إلى ذلك، لقد تم استخدام هذا المفهوم(دالة الانكماش Z̃ القريبة) لبرهان وجود النقطة الصامدة ووحدانية النقطة الصامدة لهذا النوع من الدوال.ثم بعد ذلك تم تقديم فكرة دالة الانكماش الضعيفة القريبة ( Ψ̃ ,Φ̃) في إطار الفضاء المتري الضبابي(FM-space) ، بالإضافة إلى تقديم نظرية النقطة الصامدة Fp لهذا النوع من من الدوال. وفي نهاية البحث تم تقديم بعض الأمثلة لدعم النتائج.

An analytic approach for nonlinear collisional fragmentation model arising in bubble column

The phenomenon of coagulation and breakage of particles plays a pivotal role in diverse fields. It aids in tracking the development of aerosols and granules in the pharmaceutical sector, coagulation or breakage of droplets in chemical engineering, understanding blood clotting mechanisms in biology, and facilitating cheese production through the action of enzymes within the dairy industry. A significant portion of research in this direction concentrates on coagulation or linear breakage processes. In the case of linear case, bubble particles break down due to inherent stresses or specific conditions of the breakage event. However, in many practical situations, particle division is primarily due to forces exerted during collisions between particles, necessitating an approach that accounts for nonlinear collisional breakage. Despite its critical role in a wide array of engineering and physical operations, the study of this nonlinear fragmentation phenomenon has not been extensively pursued. This article introduces an innovative semi-analytical method that leverages the beyond linear use of equation superposition function to address the nonlinear integro-partial differential model of collisional breakage population balance. This approach is versatile, allowing for the resolution of both linear/nonlinear equations while sidestepping the complexities associated with discretization of domain. To assess the precision of this method, we conduct a thorough convergence analysis. This process utilizes the principle of contractive mapping in the Banach space, a globally recognized strategy for verifying convergence. We explore a variety of kernel parameters associated with collisional kernels, alongside breakage and initial distribution functions, to derive novel iterative solutions. Comparing our findings with those obtained through the finite volume method regarding number density functions and their integral moments, we demonstrate the reliability and accuracy of our approach. The consistency and correctness of our method are further validated by depicting the errors between the exact and approximated solutions in graphical and tabular formats.

On the (k,φ)-Hilfer Langevin fractional coupled system having multi point boundary conditions and fractional integrals

In this manuscript, we investigate the (k,φ)-Hilfer Langevin fractional coupled system having multi point boundary conditions and fractional integrals. This study is crucial as it addresses the complexities of (k,φ)- Hilfer Langevin coupled system with multi point conditions, which have applications in many scientific and engineering fields. Unlike previous research, our work introduces novel techniques for analyzing these system, leading to more accurate and comprehensive results. The major findings include new existence and uniqueness theorems, along with improved stability conditions, which enhance the understanding and potential applications of these systems. We demonstrate the existence, uniqueness and different kinds of Ulam stability for our suggested model. We prove the uniqueness of the solution by utilizing Banach contraction mapping principle. The existence of solution is studied by using Krasnoselskii's fixed point theorem. We also explore the different types of Ulam stability under the specific conditions. We presented an example at the end to support our main results.

On Type I Blowup of Some Nonlinear Heat Equations With a Potential

In this paper, we are concerned with the following initial-boundary value problem{ut=Δu+Q(|x|)|u|p−1u,x∈BR,t>0u(x,t)=0,x∈∂BR,t>0u(x,0)=u0(x),x∈BR, where p≥ps:=N+2N−2, u0∈L∞(BR), and Q(r)∈C1([0,R]), 0<C_≤Q(r)≤C‾<∞,Q′(r)≤0. We extend the asymptotic behavior results, which is well-known when Q is constant according to Matano-Merle (cf. [26]), for the blow-up solutions. More precisely, we show that when ps≤p<p⁎, the blowup of radial solution to this problem is always of Type I, and hence partially generalize the conclusions in [26] for Q≡1. This extension is nontrivial due to the appearance of Q. The quasi-monotonicity formula established by the third author and Cheng in [8] allows us to use an energy method to get a priori estimates on the rescaled solutions. The contraction mapping principle shows the existence of singular stationary solutions to an associated elliptic equation with a potential. In the end, the properties of zero number for solutions lead to the nonexistence of type II singularity for the problem.

Disquisition on convergence, stability, and data dependence for a new fast iterative process

This paper introduces a novel fast iterative process designed for approximating fixed points of contraction and weak contraction mappings. The study presents strong convergence results for this newly proposed iterative process, and proving its efficiency. Analytical and numerical evidences are provided to establish that the proposed iterative method converges more rapidly than several existing processes. Furthermore, stability results and dependence analysis are presented for the newly developed iterative process, enhancing its practical applicability and robustness.

The Robin Problems in the Coupled System of Wave Equations on a Half-Line

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-value problems related to the coupled system of wave equations, utilizing the Unified Transform Method in conjunction with the Hadamard norm while considering the influence of external forces. Furthermore, we demonstrate that replacing the external force with a nonlinear term alters the iteration map defined by the unified transform solutions, making it a contraction map in a suitable solution space. By employing the contraction mapping theorem, we establish the existence of a unique solution. Finally, we show that the data-to-solution map is locally Lipschitz continuous, thus confirming the local well-posedness of the coupled system of wave equations under consideration.

Common Tripled Fixed Point Theorem on M- Fuzzy Metric Space for Occasionally Weakly Compatible Mappings

The fixed point theorems, which are primarily existential in nature, serve as a fundamental topological toolkit for the qualitative analysis of solutions to both linear and nonlinear equations in various branches of mathematics. Many authors have extended and generalized these results in different ways, particularly in the context of fuzzy metric spaces and fuzzy mappings. Numerous researchers have also proved common fixed point theorems under the condition of compatible mappings for fizzy metric spaces. Coupled common fixed point theorems for fuzzy metric spaces with the condition of weakly compatible mappings were attempted to be proved by many authors. Tripled fixed points have emerged as a significant area of research within fixed point theory. Berinde and Borcut introduced the concept of a tripled fixed point for nonlinear mappings in partially ordered metric spaces. They also established a common fixed point theorem for contractive type mappings in M-fuzzy metric spaces. Later, other authors extended these results for common tripled fixed point theorems in fuzzy metric spaces. In this paper we introduce a new technique for proving some new common tripled fixed point theorems for Occasionally Weakly Compatible Mappings in M-fuzzy metric spaces, a method which is not previously utilized by authors in this field. Additionally, we provide illustrative example to support our findings, which represent an improvement over recent results found in the literature.

(α, F)-Geraghty-type generalized F-contractions on non-Archimedean fuzzy metric-unlike spaces

Abstract In this study, we generalize fuzzy metric-like, non-Archimedean fuzzy metric-like, and all the variants of fuzzy metric spaces. We propose the idea of fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike, respectively. We also propose the idea of ( α , F ) \left(\alpha ,F) -Geraghty-type generalized F F -contraction mappings utilizing fuzzy metric-unlike and non-Archimedean fuzzy metric-unlike spaces. We investigate the presence of unique fixed points using the recently introduced contraction mappings. In order to complement our study, we consider an application to dynamic market equilibrium.

A presentation for the semigroup of order-preserving full contraction mappings of a finite chain

For n ∈ Z + , let X n = { 1 , … , n } be a finite chain and T n be the full transformation semigroup on Xn . If O C T n is the set of all order-preserving and full contraction mappings on Xn , then it is well-known that O C T n is a submonoid of T n . In this paper we give a monoid presentation of O C T n .

Super Metric Space and Fixed Point Results

Banach contraction principle is the beginning of the Metric fixed point theory. This principle gives existence and uniqueness of fixed points and methods for obtaining approximate fixed points. It is the basic tool of finding fixed points of all contraction type maps. It has a constructive proof which makes the theorem worthy because it yields an algorithm for computing a fixed point. Banach fixed point result has been extended by various authors in many directions either by weakening the conditions of contraction mapping or by changing the abstract structure. Several generalizations and extensions of metric spaces have been introduced. Among these, the prominent extensions are b-metric space, fuzzy metric space, partial metric space and a lot more of their combinations. In particular, a new structure namely Super metric space is introduced. In the present paper, we generalize and extend the fixed point results of fixed point theory in literature in the framework of super metric space.

Positive Solutions to a System of Coupled Hadamard Fractional Boundary Value Problems

We explore the existence, uniqueness, and multiplicity of positive solutions to a system of Hadamard fractional differential equations that contain fractional integral terms. Defined on a finite interval, this system is subject to general coupled nonlocal boundary conditions encompassing Riemann–Stieltjes integrals and Hadamard fractional derivatives. To establish the main results, we employ several fixed-point theorems, namely the Banach contraction mapping principle, the Schauder fixed-point theorem, the Leggett–Williams fixed-point theorem, and the Guo–Krasnosel’skii fixed-point theorem.

Extension of Meir-Keeler-Khan (ψ − α) Type Contraction in Partial Metric Space

In numerous scientific and engineering domains, fractional-order derivatives and integral operators are frequently used to represent many complex phenomena. They also have numerous practical applications in the area of fixed point iteration. In this article, we introduce the notion of generalized Meir-Keeler-Khan-Rational type (ψ−α)-contraction mapping and propose fixed point results in partial metric spaces. Our proposed results extend, unify, and generalize existing findings in the literature. In regards to applicability, we provide evidence for the existence of a solution for the fractional-order differential operator. In addition, the solution of the integral equation and its uniqueness are also discussed. Finally, we conclude that our results are superior and generalized as compared to the existing ones.

Fixed points of regular set-valued mappings in quasi-metric spaces

Metric fixed point theory is becoming increasingly significant across various fields, including data science and iterative methods for solving optimization problems. This paper aims to introduce new fixed point theorems for set-valued mappings under novel regularity conditions, such as orbital regularity and orbital pseudo-Lipschitzness. Instead of traditional metric spaces, we adopt the framework of quasi-metric spaces, motivated by the need to address problems in spaces that are not necessarily metric, such as function spaces of homogeneous type. We also explore the stability of the set of fixed points under variations of the set-valued mapping. Additionally, we provide estimates for the distances from a given point to the set of fixed points and between two sets of fixed points. Building on these findings, we extend the discussion to similar problems involving fixed, coincidence, and cyclic/double fixed points within this framework. Our results generalize recent findings from the literature, including those in Ait Mansour M, Bahraoui MA, El Bekkali A. [Metric regularity and Lyusternik-Graves theorem via approximate fixed points of set-valued maps in noncomplete metric spaces. Set-Valued Var Anal. 2022;30(1):233–256. doi: 10.1007/s11228-020-00553-1], Dontchev AL, Rockafellar RT. [Implicit functions and solution mappings. a view from variational analysis. Dordrecht: Springer; 2009. Springer Monographs in Mathematics], Ioffe AD. [Variational analysis of regular mappings. Springer, Cham; 2017. Springer Monographs in Mathematics; theory and applications. doi: 10.1007/978-3-319-64277-2], Lim TC. [On fixed point stability for set-valued contractive mappings with applications to generalized differential equations. J Math Anal Appl. 1985;110(2):436–441. doi: 10.1016/0022-247X(85)90306-3] and Tron NH. [Coincidence and fixed points of set-valued mappings via regularity in metric spaces. Set-Valued Var Anal. 2023;31(2):22. Paper No. 17. doi: 10.1007/s11228-023-00680-5].