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The manuscript introduces the concept of a double-composed metric-like space, which is an extension of the notion of a double-composed metric space. In this new space, the self-distance is not necessarily zero, but if the distance metric equals zero, it must be for identical points of distance. Furthermore, this paper presents several results related to this novel concept in the literature, with a particular focus on Hardy–Rogers type contractions. It establishes fixed point theorems accompanied by some illustrative examples that elucidate the findings. Finally, this research provides an application to nonlinear integral equation to substantiate our theorems.

The multi-term time-fractional order diffusion-wave equation (MT-TFDWE) is an important mathematical model for processes exhibiting anomalous diffusion and wave propagation with memory effects. This article develops a robust numerical technique based on the Chebyshev collocation method (CCM) coupled with the Laplace transform (LT) to solve the time-fractional diffusion-wave equation. The CCM is utilized to discretize the spatial domain, which ensures remarkable accuracy and excellent efficiency in capturing the variations of spatial solutions. The LT is used to handle the time-fractional derivative, which converts the problem into an algebraic equation in a simple form. However, while using the LT, the main difficulty arises in calculating its inverse. In many situations, the analytical inversion of LT becomes a cumbersome job. Therefore, the numerical techniques are then used to obtain the time domain solution from the frequency domain solution. Various numerical inverse Laplace transform methods (NILTMs) have been developed by the researchers. In this work, we use the contour integration method (CIM), which is capable of handling complex inversion tasks efficiently. This hybrid technique provides a powerful tool for the numerical solution of the time-fractional diffusion-wave equation. The accuracy and efficiency of the proposed technique are validated through four test problems.

Abstract In this article, we introduce the structure of a new extended cone b-metric-like space over a Banach algebra. In this generalized space, we define the notion of generalized Reich-type mappings and prove a fixed point result. We also prove a fixed point result using α ∗ − ψ $\alpha^{\ast}-\psi$ multivalued contraction and provide some of its consequences. Finally, we furnish with applications to establish the validity of our results.

Abstract In the present article, we have introduced the notions of γ-admissibility for the pair of q-ROF set-valued maps and admissible hybrid q-ROF Z $\mathcal{Z}$ -contraction. Notions introduced in the article generalizes the existing concepts in fuzzy literature. Common fixed point result for a pair of γ-admissible q-ROF mappings in b-metric spaces utilizing the introduced contraction is presented. A nontrivial example to support the obtained results is also included. As an application, we have discussed the existence of solution of system of non-linear n-th order differential inclusions with non-local and integral boundary conditions.

Abstract In this paper, we introduce the notion of unified interpolative contractions of the Reich–Rus–Ćirić type and give some results about the fixed points for these mappings in the framework of relational metric spaces. We present examples where the results of some previous research are not relevant. Also, we apply our results to solving problems related to nonlinear matrix equations, emphasizing their practical importance.

Abstract The survey of the available literature shows that a lot of important invariant point problems of Banach and Heilpern types have been examined in both metric and quasimetric spaces. However, a handful of the existing results employed the recent approaches of interpolative contractions. Therefore, based on the new idea of interpolation techniques in fixed point theory, this article studies new notions of L-fuzzy contractions and investigates conditions for the existence of L-fuzzy fixed points for such mappings. On the fact that fixed points of point-to-point mappings satisfying interpolative-type contraction are not always unique, whence making the concepts more fitted for invariant point results of crisp set-valued maps, new multi-valued analogues of the key findings put forward in this work are derived. Comparative illustrations, which indicate the preeminence of the results presented herein, are constructed. From application viewpoint, one of the theorems so obtained is employed to introduce new solvability conditions of Fredholm-type integral inclusions.

In this paper, we introduce the concept of elliptic-valued b-metric spaces, extending the notions of elliptic-valued metric spaces and complex-valued metric spaces. We present several fixed-point results that involve rational and product terms within this novel space framework. To support our main findings, we offer numerical examples. Additionally, we demonstrate an application of Urysohn integral equations.

Abstract The current paper introduces a novel generalization of cone metric spaces called type I and type II composed cone metric spaces. Therefore, examples of a type I and type II composed cone metric space, which is not a cone metric space, are given. We establish some results of fixed point precisely about Hardy–Rogers type contraction on C2CMS and provide examples. Finally, we present an application of our results and how our results solve the Fredholm integral equation of generalizing several existing and unique fixed point theorems.

In this publication, our objective was to introduce and establish the concepts of $ \kappa _{G_{m}} $-contraction and generalized $ (\alpha, \kappa _{G_{m}}) $-contraction in complete $ G_{m} $-metric spaces, which led to the discovery of novel fixed points, coincidence points, and common fixed points. Additionally, we demonstrated the usefulness of our main results by applying it to the investigation of the integral equation. Also, we presenting a noteworthy example demonstrating the practicality of our primary hypothesis.

Abstract A hybrid system interacts with the discrete and continuous dynamics of a physical dynamical system. The notion of a hybrid system gives embedded control systems a great advantage. The Langevin differential equation can accurately depict many physical phenomena and help researchers effectively represent anomalous diffusion. This paper considers a fractional hybrid Langevin differential equation, including the ψ-Caputo fractional operator. Furthermore, some novel boundaries selected are considered to be a problem. We used the Schauder and Banach fixed-point theorems to prove the existence and uniqueness of solutions to the considered problem. Additionally, the Ulam-Hyer stability is evaluated. Finally, we present a representative example to verify the theoretical outcomes of our findings.

This paper examines the analysis of some rational-type contractions within the context of extended b-metric spaces, establishes a theoretical foundation for rational-type contractions and demonstrates their application in Volterra integral inclusions and Urysohn integral equations. Some examples are provided for the authenticity of the findings.

The area related to fractional order partial differential equations (FOPDEs) has also attracted attention. Therefore, we focuss our attention to study nonlinear Benney equations of using power law type Caputo derivative of fractional order. To compute a series-type solution for the problem under consideration, we combine the Adomian approach with the Laplace transform. Our findings are supported by graphical representations and numerical interpretations, illustrating the applicability and insights gained from this approach in solving complex real-world problems.

In this paper, we present fixed point theorem for mappings defined on b-metric spaces, which satisfies extended quasi-contractive inequality with nonlinear comparison function. Our result generalizes and improves several recent results from fixed point theory.

Neutrosophic sets provide greater versatility in dealing with a variety of uncertainties, including independent, partially independent, and entirely dependent scenarios, which q-ROF soft sets cannot handle. Indeterminacy, on the other hand, is ignored completely or partially by q-ROF soft sets. To address this issue, this study offers a unique novel concept as known as q-RONSS, which combines neutrosophic set with q-ROF soft set. This technique addresses vagueness using a set of truth, indeterminacy, and false membership degrees associated with the parametrization tool, with the condition that the sum of the qth power of the truth, indeterminacy, and false membership degrees be less than or equal to one. In addition, this study outlines operational laws for the suggested structure. The main purpose this article is to define some averaging and geometric operators based on the q-rung orthopair neutrosophic soft set. Furthermore, this article provides a step-by-step method and a mathematical model for the suggested techniques. To solve a MADM issue, this research article proposes a numerical example of people selection for a specific position in a real estate business based on a variety of criteria. Finally, to demonstrate the proposed model's superiority and authenticity, this article performs several analyses, including sensitivity analysis, to address the reliability and influence of various parameter ''q'' values on the alternatives and the ultimate ranking outcomes using the averaging and geometric operators. A comparison of the proposed operators to current operators demonstrates the validity of the proposed structure. Furthermore, a comparison of the proposed structure to current theories demonstrates its superiority by overcoming their limits and offering a more flexible and adaptable framework. Finally, this study reviews the findings and consequences of our research.

In this manuscript, utilizing interpolative contractions with fractional forms, some unique fixed-point results were studied in the context of extended $ b $-metric spaces. For the validity of the presented results some examples are given. In the last section an existence theorem is provided to study the existence of a solution for the Fredholm integral equation.

In this paper, we presented Tauberian type results that intricately link the quasi-asymptotic behavior of both even and odd distributions to the corresponding asymptotic properties of their fractional Fourier cosine and sine transforms. We also obtained a structural theorem of Abelian type for the quasi-asymptotic boundedness of even (resp. odd) distributions with respect to their fractional Fourier cosine transform (FrFCT) (resp. fractional Fourier sine transform (FrFST)). In both cases, we quantified the scaling asymptotic properties of distributions by asymptotic comparisons with Karamata regularly varying functions.

The goal of this paper was to improve some known results of fixed points by using $ w $-distances and properties of locally symmetric $ \mathcal{H} $-transitivity of binary relations. Also, we gave the application of the obtained results for finding the solution of nonlinear matrix equations. Finally, we gave a numerical example to demonstrate the applicability of our results.

This paper presents the stability theorem for the $ T $-Picard iteration scheme and establishes the existence and uniqueness theorem for fixed points concerning $ T $-mean nonexpansive mappings within $ b $-metric-like spaces. The outcome of our fixed point theorem substantiated the existence and uniqueness of solutions to the Fredholm-Hammerstein integral equations defined on time scales. Additionally, we provided two numerical examples from distinct time scales to support our findings empirically.

This paper delves into fixed point findings within a complete partially ordered $ b $-metric space, focusing on mappings that adhere to weakly contractive conditions in the presence of essential topological characteristics. These findings represent modifications of established results and further extend analogous outcomes in the existing literature. The conclusions are substantiated by illustrative examples that strengthen the conclusion of the paper.

Abstract The analysis of stability in functional equations (FEs) within neutrosophic normed spaces is a significant challenge due to the inherent uncertainties and complexities involved. This paper proposes a novel approach to address this challenge, offering a comprehensive framework for investigating stability properties in such contexts. Neutrosophic normed spaces are a generalization of traditional normed spaces that incorporate neutrosophic logic. By providing a systematic methodology for addressing stability concerns in neutrosophic normed spaces, our approach facilitates enhanced understanding and control of complex systems characterized by indeterminacy and uncertainty. The primary focus of this research is to propose a novel class of Euler-Lagrange additive FE and investigate its Ulam-Hyers stability in neutrosophic normed spaces. Direct and fixed point techniques are utilized to achieve the required results.

Abstract This paper introduces an innovative inertial simultaneous cyclic iterative algorithm designed to address a range of mathematical problems within the realm of split equality variational inequalities. Specifically, the algorithm accommodates finite families of split equality variational inequality problems, infinite families of split equality variational inclusion problems, and multiple-sets split equality fixed point problems involving demicontractive operators in infinite-dimensional Hilbert spaces. The algorithm integrates well-established methods, including the cyclic method, the inertial method, the viscosity approximation method, and the projection method. We establish the strong convergence of this proposed algorithm, demonstrating its applicability in various scenarios and unifying disparate findings from existing literature. Additionally, a numerical example is presented to validate the primary convergence theorem.

We discuss the existence of a fixed point for a self mapping and its uniqueness satisfying (ϕ˙,η˙)-generalized contractive condition including altering distance functions of rational terms in an ordered b-metric space. It is also discussed whether the two self-maps under the same contraction condition can be coincident and coupled coincident. The results are backed up by a dearth of numerical examples and application to nonlinear quadratic integral equation.

This work presents an accurate and efficient method, for solving a two dimensional time-fractional Oldroyd-B fluid model. The proposed method couples the Laplace transform (LT) with a radial basis functions based local meshless method (LRBFM). The suggested numerical scheme first uses the LT which transform the given equation to an elliptic equation in LT space, and then it utilizes the LRBFM to solve transformed equation in LT space, and then the solution is converted back into the time domain via the improved Talbot's scheme. The local meshless methods are widely recognized for scattered data interpolation and for solving PDEs in complex shaped domains. The adaptability, simplicity, and ease of use are features that have led to the popularity of local meshless methods. The local meshless methods are easy and straightforward, they only requires to solve linear system of equations. The main objective of using the LT is to avoid the computation of costly convolution integral in time-fractional derivative and the effect of time stepping on accuracy and stability of numerical solution. The stability and the convergence of the proposed numerical scheme are discussed. Further, the Ulam-Hyers (UH) stability of the proposed model is discussed. The accuracy and efficiency of the suggested numerical approach have been demonstrated using numerical experiments on five different domains with regular nodes distribution.

This paper focuses on using efficient Haar wavelet collocation method to nonlinear parabolic differential equations in order to solve steady state pore-scale creeping fluid flow in porous medium, substituting finite Haar series for space derivatives, incontrast forward difference approach is used to estimate the time derivative for porous media equation. These equations has vast number of application in engineering and various other fields such as, in petroleum engineering, there are often situations where fluids are near critical conditions. As oil pressure reduces below the bubble point when gas forms liquid condensations, or when the surface tension become very low i.e, transition from miscibility to immiscibility. To model gas production by pressure decline requires an accurate description of physical mechanism involved in the appearance of the gas bubbles. The observed stability with theoretical convergence assertion, the conservation law of mass and energy are stated. This supports the convergence of the preferred paradigm. A convergent approximation to the porous media equation is obtained by suggested approach. By calculating the maximum error norm and the experimental rate of convergence for various problems, the method efficiency and dependability are demonstrated. The method is accurate, easily applicable, and efficient.

This article employs the Laplace transform approach to solve the Bagley–Torvik equation including Caputo’s fractional derivative. Laplace transform is a powerful method for enabling solving integer and non-integer order ODEs and PDEs in engineering and science. Using the Laplace transform for solving integer and non-integer order ODEs and PDEs, however, sometimes leads to solutions in the Laplace domain that are difficult to convert back into the time domain using analytical techniques. Therefore, the best option to convert the Laplace domain solution back into real domain is to use numerical inversion techniques. But it is important to remember that the inverse Laplace transform is not a well-posed problem. There is not a single, general approach that can address every challenge in this matter. Numerous algorithms are devised for numerically inverting the Laplace transform. In this paper, we have evaluated three numerical inverse Laplace transform techniques. These methods include the Fourier series method, the Gauss–Hermite quadrature method, and Zakian’s method. Different test problems are used to evaluate these inversion methods. The effectiveness and efficiency of these methods are demonstrated through graphical representations and tabular data. The comparison of the obtained results with the results of other methods available in literature led us to conclude that the proposed methods exhibit rapid convergence with optimal accuracy without any time instability.

In this work, a family of hybrid contractions, termed Jaggi-type hybrid (ℵ-τ)-contractive mapping is proposed in metric space equipped with a graph and new conditions under which the mapping is a Picard operator are studied. The novel ideas proposed in this manuscript are exemplified to display the validity of the presented results and to show how they differ from the existing ones. Additionally, some corollaries which reduce our proposed notion to some recently announced concepts in the existing findings are indicated and examined. Finally, we study Ulam-type stability for the fixed point equations with hybrid contractions.

This study introduces a novel neural network approach integrating q-spherical fuzzy rough Frank aggregation operators, aiming to enhance AI systems’ resilience to uncertain and imprecise data in military transport systems. Three new operators are developed: q-spherical fuzzy rough Frank weighted averaging (q-SFRFWA), q-spherical fuzzy rough Frank ordered weighted averaging (q-SFRFOWA), and q-spherical fuzzy rough Frank hybrid weighted averaging (q-SFRFHWA), tailored to handle complex decision-making scenarios. We demonstrate their efficacy in multiple attribute decision-making using q-spherical fuzzy rough data, providing valuable insights and expanding the knowledge base in this domain. Through numerical examples, we illustrate the practical application of these operators, validating their effectiveness and relevance in real-world settings. Comparative and sensitivity analyses further corroborate the superiority of our proposed approach over existing methods. This research offers a robust decision-making framework equipped to manage intricate and unreliable data, promising significant advancements in military transport systems and beyond.

Abstract In this article, we analyze the boundedness for the fractional bilinear Hardy operators on variable exponent weighted Morrey–Herz spaces M K ˙ q , p ( ⋅ ) α ( ⋅ ) , λ ( w ) ${M\dot{K}^{\alpha (\cdot ),\lambda}_{q,p(\cdot)}(w)}$ . Similar estimates are obtained for their commutators when the symbol functions belong to BMO space with variable exponents.

Abstract In this paper, we propose the notion of extended neutrosophic rectangular metric space and prove some fixed point results under contraction mapping. Finally, as an application of the obtained results, we prove the existence and uniqueness of the Caputo fractional differential equation.

Linear correlated fuzzy differential equations (LCFDEs) are a valuable approach to handling physical problems, optimizations problems, linear programming problems etc. with uncertainty. But, LCFDEs employed on spaces with symmetric basic fuzzy numbers often exhibit multiple solutions due to the extension process. This abundance of solutions poses challenges in the existing literature's solution methods for LCFDEs. These limitations have led to reduced applicability of LCFDEs in dealing with such types of problems. Therefore, in the current study, we focus on establishing existence and uniqueness results for LCFDEs. Moreover, we will discuss solutions in the canonical form of LCFDEs in the space of symmetric basic fuzzy number which is currently absent in the literature. To enhance the practicality of our work, we provide examples and plots to illustrate our findings.

Abstract The objective of this work is to investigate the perturbed Gerdjikov–Ivanov (GI) equation along spatio-temporal dispersion which explains the dynamics of soliton dispersion and evolution of propagation distance in optical fibers, photonic crystal fibers (PCF), and metamaterials. The algorithms, namely hyperbolic extended function method and generalized Kudryashov’s method, are constructed to obtain the new soliton solutions. The dark, bright, periodic, and singular solitons are derived of the considered equation with the appropriate choice of parameters. These results are novel, confirm the stability of optical solitons, and have not been studied earlier. The explanation of evaluated results is given by sketching the various graphs in 3D, contour and 2D plots by using Maple 18. Graphical simulations divulge that varying the wave velocity affects the dynamical behaviors of the model. In summary, this research adds to our knowledge on how the perturbed GI equation with spatio-temporal dispersion behaves. The obtained soliton solutions and the methods offer computational tools for further analysis in this field. This work represents an advancement in our understanding of soliton dynamics and their applications in photonic systems.

The computational study of fixed-point problems in distance spaces is an active and important research area. The purpose of this paper is to construct a new iterative scheme in the setting of Banach space for approximating solutions of fixed-point problems. We first prove the strong convergence of the scheme for a general class of contractions under some appropriate assumptions on the domain and a parameter involved in our scheme. We then study the qualitative aspects of our scheme, such as the stability and order of convergence for the scheme. Some nonlinear problems are then considered and solved numerically by our new iterative scheme. The numerical simulations and graphical visualizations prove the high accuracy and stability of the new fixed-point scheme. Eventually, we solve a 2D nonlinear Volterra Integral Equation (VIE) via the application of our main outcome. Our results improve many related results in fixed-point iteration theory.

The research article in hand provides a new mechanism that deals with the investigation of the triangular analytical fuzzy solutions (TAFS) of the two-dimensional fuzzy fractional order wave equation (2-D FFWE) through the Hukuhara conformable fractional derivative (HCFD) along with the concept of [gH] and [gH−p] differentiability. The mechanism consists of a fuzzy traveling wave method coupled with additive operating splitting (AOS). The procedure for the aforesaid mechanism starts with the extension of the HCFD to the fuzzy conformable fractional derivative (FCFD). Furthermore, some properties of FCFD that play a vital role in this study like, ([i−gH],[ii−gH],[i−p],[ii−p])-differentiability, switching point, fuzzy chain rule, and traveling wave method are discussed in detail. Further, fuzzy traveling wave method coupled with the procedure of the additive operating splitting (AOS) method is adopted to investigate the triangular analytical fuzzy solution of the Two-dimensional fuzzy wave equation (2-D FWE). Finally, some examples are provided that support our arguments.

Abstract In this article, we present an algorithm for computing analytical solutions of linear fuzzy advection-diffusion equations and one-dimensional fuzzy heat equations involving an external source. The fuzzy problems can be solved by using the natural transform and Adomian decomposition method. The results obtained through the natural Adomian decomposition method are calculated in a series form that converges rapidly to the exact solution. To enhance the practicality of our work, we provide examples to illustrate our findings.

In this paper, we will introduce the idea of grand variable weighted Herz spaces $ {{\dot{K} ^{\alpha(\cdot), \epsilon), \theta}_{ q(\cdot)}(\tau)}} $ in which $ \alpha $ is also a variable. Our main purpose in this paper is to prove the boundedness of Hardy operators on grand variable weighted Herz spaces.

This article introduces the concept of generalized (ffF,b,ϕ˘) contraction in the context of b-metric spaces by utilizing the idea of F contraction introduced by Dariusz Wardowski. The main findings of the research focus on the existence of best proximity points for multi-valued (ffF,b,ϕ˘) contractions in partially ordered b-metric spaces. The article provides examples to illustrate the main results and demonstrates the existence of solutions to a second-order differential equation and a fractional differential equation using the established theorems. Additionally, several corollaries are presented to show that the results generalize many existing fixed-point and best proximity point theorems.

In this paper, we present four categories of covering-based intuitionistic hesitant fuzzy rough set (CIHFRS) models using intuitionistic hesitant fuzzy β-neighborhoods (IHF β-neighborhoods) and intuitionistic hesitant fuzzy complementary β-neighborhoods (IHFC β-neighborhoods. Through theoretical analysis of covering-based IHFRS models, we propose the intuitionistic hesitant fuzzy TOPSIS (IHF-TOPSIS) technique for order of preference by similarity to an ideal solution, addressing multicriteria decision-making (MCDM) challenges concerning the assessment of IHF data. A compelling example aptly showcases the suggested approach. Furthermore, we address MCDM problems regarding the assessment of IHF information based on CIHFRS models. Through comparison and analysis, it is evident that addressing MCDM problems by assessing IHF data using CIHFRS models proves more effective than utilizing intuitionistic fuzzy data with CIFRS models or hesitant fuzzy information with CHFRS models. IHFS emerges as a unique and superior tool for addressing real-world challenges. Additionally, covering-based rough sets (CRSs) have been successfully applied to decision problems due to their robust capability in handling unclear data. In this study, by combining CRSs with IHFS, four classes of CIFRS versions are established using IHF β-neighborhoods and IHFC β-neighborhoods. A corresponding approximation axiomatic system is developed for each. The roughness and precision degrees of CBIHFRS models are specifically talked about. The relationship among these four types of IHFRS versions and existing related versions is presented based on theoretical investigations. A method for MCDM problems through IHF information, namely, IHF-TOPSIS, is introduced to further demonstrate its effectiveness and applicability. By conducting a comparative study, the effectiveness of the suggested approach is evaluated.

Fractional order differential equations often possess inherent symmetries that play a crucial role in governing their dynamics in a variety of scientific fields. In this work, we consider numerical solutions for fractional-order linear delay differential equations. The numerical solution is obtained via the Laplace transform technique. The quadrature approximation of the Bromwich integral provides the foundation for several commonly employed strategies for inverting the Laplace transform. The key factor for quadrature approximation is the contour deformation, and numerous contours have been proposed. However, the highly convergent trapezoidal rule has always been the most common quadrature rule. In this work, the Gauss–Hermite quadrature rule is used as a substitute for the trapezoidal rule. Plotting figures of absolute error and comparing results to other methods from the literature illustrate how effectively the suggested approach works. Functional analysis was used to examine the existence of the solution and the Ulam–Hyers (UH) stability of the considered equation.

In this paper, the existence of mild solution and controllability of fractional semilinear mixed Volterra–Fredholm integro differential equations of order 1

The goal of this study is to develop a novel iterative algorithm for approximating the solutions of the monotone inclusion problem and fixed point problem of a finite family of demimetric mappings in the context of a real Hilbert space. The proposed algorithm is based on the inertial extrapolation step strategy and combines forward-backward and Tseng's methods. We introduce a demimetric operator with respect to $ M $-norm, where $ M $ is a linear, self-adjoint, positive and bounded operator. The algorithm also includes a new step for solving the fixed point problem of demimetric operators with respect to the $ M $-norm. We study the strong convergence behavior of our algorithm. Furthermore, we demonstrate the numerical efficiency of our algorithm with the help of an example. The result given in this paper extends and generalizes various existing results in the literature.

The Fourth Industrial Revolution, also known as Industry 4.0, is attracting a significant amount of attention because it has the potential to revolutionize a variety of industries by developing a production system that is fully automated and digitally integrated. The implementation of this transformation, however, calls for a significant investment of resources and may present difficulties in the process of adapting existing technology to new endeavors. Researchers have proposed integrating the Analytic Hierarchy Process (AHP) with extensions of fuzzy rough sets, such as the three-dimensional q-spherical fuzzy rough set (q-SFRS), which is effective in handling uncertainty and quantifying expert judgments, to prioritize projects related to Industry 4.0. This would allow the projects to be ranked in order of importance. In this article, a novel framework is presented that combines AHP with q-SFRS. To calculate aggregated values, the new framework uses a new formula called the q-spherical fuzzy rough arithmetic mean, when applied to a problem involving the selection of a project with five criteria for evaluation and four possible alternatives, the suggested framework produces results that are robust and competitive in comparison to those produced by other multi-criteria decision-making approaches.

Abstract Linear dynamic systems with impulsive effects are considered. For such a system we define a new impulsive exponential matrix. Necessary and sufficient conditions for exponential stability and boundedness have been established. The fundamental tool is an impulsive exponential matrix for exponential stability.

Fixed point theory can be generalized to cover multidisciplinary areas such as computer science; it can also be used for image authentication to ensure secure communication and detect any malicious modifications. In this article, we introduce the notion of complex-valued controlled rectangular metric-type spaces, where we prove fixed point theorems for self-mappings in such spaces. Furthermore, we present several examples and give two applications of our main results: solving linear systems of equations and finding a unique solution for an equation of the form $ f(x) = 0 $.

This study examines Marangoni convection in blood-based carbon nanotubes nanofluid's stagnation point flow over a time-dependent stretching surface. This study is inspired by the emerging importance of nanofluids in a variety of scientific and technical fields due to their unique and varied uses and effective thermal activities. Some examples of potential applications of these fluids include cancer treatment, magnetic refrigeration, drug delivery, and magnetic resonance imaging. Two types of nanoparticles are considered name is single wall carbon nanotube and multi wall carbon nanotube blood is takin as base fluid. Nonlinear partial differential equations are used to simulate the specified flow issue using momentum and energy conservation principles. Using a similarity transformation, the resultant is transformed into nonlinear with reduced dimensions. The relations for velocity profile and temperature distribution are calculated from the developed nonlinear ordinary differential equation by using an approximate analytical technique called the homotopy asymptotic method. Subsequently, these equations are implemented and executed within Mathematica software. The investigation focuses on significant outcomes such as momentum filed, energy filed, Skin friction coefficients, and Nusselt number. Graphs are used to interpret the effects of several factors, including the Marangoni parameter, nanoparticle volume friction, stretching parameter, slip parameter, and Prandtl number. The behavior of Nusselt's number and skin friction coefficient was also checked with the help of graphs and tables. The convergence of the solutions is checked with the help of auxiliary functions as well.

In this paper, we establish a new fixed point result for Hardy-Rogers type contractions in double controlled metric-like spaces. Our result generalizes many important theorems in the literature. We will provide an example to illustrate our results.

In this paper, we prove a common fixed-point theorem for four self-mappings with a function family on $ S_{b} $-metric spaces. In addition, we investigate some geometric properties of the fixed-point set of a given self-mapping. In this context, we obtain a fixed-disc (resp. fixed-circle), fixed-ellipse, fixed-hyperbola, fixed-Cassini curve and fixed-Apollonious circle theorems on $ S_{b} $-metric spaces.

Our aim in this paper is to prove boundedness of an intrinsic square function and higher order commutators of fractional integrals on grand Herz spaces with variable exponent $ {\dot{K} ^{a(\cdot), u), \theta}_{ s(\cdot)}(\mathbb{R}^n)} $ by applying some properties of variable exponent.

In this article, by utilizing the idea of controlled functions, we present a novel notion of triple controlled quasi rectangular metric like spaces and prove Banach fixed point principal in such spaces. A topology in such spaces and its topological properties have been discussed. The result, presented here is a new contribution to the field of fixed point theory. Examples of this new structure are given.

In this paper, we introduce quasi M-metric spaces as a generalization of M-metric spaces. We establish some fixed point results along with the examples and application of our results to integral equations and system of linear equations.

q-Rung orthopair fuzzy soft set handles the uncertainties and vagueness by membership and non-membership degree with attributes, here is no information about the neutral degree so to cover this gap and get a generalized structure, we present hybrid of picture fuzzy set and q-rung orthopair fuzzy soft set and initiate the notion of q-rung orthopair picture fuzzy soft set, which is characterized by positive, neutral and negative membership degree with attributes. The main contribution of this article is to investigate the basic operations and some averaging aggregation operators like q-rung orthopair picture fuzzy soft weighted averaging operator and q-rung orthopair picture fuzzy soft order weighted averaging operator under the environment of q-rung orthopair picture fuzzy soft set. Moreover, some fundamental properties and results of these aggregation operators are studied, and based on these proposed operators we presented a stepwise algorithm for MADM by taking the problem related to medical diagnosis under the environment of q-rung orthopair picture fuzzy soft set and finally, for the superiority we presented comparison analysis of proposed operators with existing operators.