This paper focuses on a novel method for the stability of fractional partial differential equations with a power fractional derivative. By some estimates of convex functions, we extend the Lyapunov functionals for ODE systems to their fractional-order counterparts with and without the Laplacian operator term. Accordingly, it is shown that the stability conditions in these systems can be derived from their integer-order equivalents. The efficiency of the proposed method is demonstrated through some applications.
This paper investigates fixed point results for fuzzy mappings in the framework of fuzzy quasi-metric spaces by employing an altering distance control function. The obtained results are established in more general settings, including left K-complete fuzzy quasi-metric spaces as well as G-bicomplete fuzzy quasi-metric spaces. We derive results for both single-valued and multivalued mappings. Furthermore, an application to the Divide and Conquer algorithm is presented. Several well-known fixed point theorems are recovered as special cases of the results proved herein.
A common approach in the theory of generalized metric spaces, particularly in spaces which lack symmetry, is to pass to a symmetrized function. In the literature on quasi-partial b-metric spaces, it has been repeatedly assumed without proof that the standard symmetrization d_q(x,y)=q(x,y)+q(y,x)-q(x,x)-q(y,y) is always a b-metric. We first provide a definitive counterexample to disprove this general claim. Next, from an elementary estimate derived from the axiom (QPb4), we obtain a convenient sufficient criterion: if a self-distance modulus 𝔐(q) is finite, then d_q is a b-metric with explicit constant K^*(q)=s+(s-1)𝔐(q). We then show by example that this condition is not necessary. Finally, once d_q is known to be a b-metric, we explain how fixed point results from complete b-metric spaces may be transferred back to the quasi-partial b-metric setting.
This work introduces a novel modified Halpern-type proximal point algorithm, designed for a finite family of k-demimetric mappings, resolvents of monotone operators, and resolvents of mixed equilibrium problems. In the context of Hadamard spaces, we establish the convergence of the perturbed algorithm to a common zero of the finite family of mixed equilibrium problems and monotone operators, as well as to a common fixed point of the finite family of k-demimetric mappings. Furthermore, we provide a numerical example to illustrate the effectiveness of the proposed algorithm.
In this article, we explore the existence and controllability results for a class of ψ-Hilfer fractional Sobolev-type stochastic differential systems with infinite delay. Sufficient conditions for controllability results are obtained by using the notion of the measure of noncompactness, stochastic theory, fractional calculus, and the Mönch fixed point theorem. A key feature of our work is the use of the ψ-Hilfer fractional derivative (FD), which provides a unified framework by generalizing several well-known fractional operators, including the Hilfer, Caputo, and Riemann–Liouville(R-L) derivatives. This flexibility makes our approach particularly effective for capturing memory effects and improving the modeling accuracy of real-world dynamical systems. Finally, we present a concrete example to demonstrate the applicability of our theoretical findings.
This paper introduces the novel concept of perturb supra metric spaces, which provides a mathematical framework to account for potential errors in distance measurements. We establish several significant fixed point theorems in this generalized setting, including extensions of the Banach’s contraction principle, Kannan type contractions, Chatterjea type contractions, and Reich type contractions. The theoretical results are substantiated with comprehensive examples and applied to analyze a fractional order epidemiological model of Foot and Mouth Disease (FMD) using the Atangana-Baleanu- Caputo derivative operator. Our work demonstrates the robustness of fixed point(FP) theory under measurement perturbations and offers new tools for analyzing nonlinear problems in generalized metric spaces(ms).
We establish finite-step probabilistic upper bounds for the contraction ratios represented by ρ _k = Δ _k+1/Δ _k arising in iterated Pearson row–row correlation dynamics. Let (P_k)_k≥ 0 denote the sequence generated by the Pearson correlation update, with increments Δ _k := P_k+1-P_k_F, ratios ρ _k := Δ _k+1/Δ _k (Δ _k>0), and normalized step size δ _k := Δ _k/n. Although Δ _k→ 0 along convergent trajectories, finite-step ratios may exceed unity, a phenomenon not captured by local linearization analyses. For fixed matrix dimension n and under the probability measure ℙ induced by random initialization of P_0 with independent and identically distributed uniform [-1,1] entries, we construct explicit state-dependent bounds B_p:ℝ_+→ℝ_+ in the post-transient regime k≥ 2. These bounds are piecewise-constant functions B^q_p(δ ) obtained as empirical conditional p-quantiles of logρ _k given δ _k under logarithmic binning. Deterministic enlargements are introduced via uniform multiplicative adjustments, yielding pointwise larger families while preserving the learned δ-dependence. Independent validation confirms that the constructed bounds satisfy ℙ(ρ≤ B_p(δ )) ≥ p with empirical coverage matching nominal levels across n∈ [3,2000]. Analysis of the baseline empirical 0.95-quantile bound shows ℙ(ρ≤ 1 |δ≤ 0.03) ≥ 0.95 for all tested dimensions, and ℙ(ρ≤ 1.7) ≥ 0.95 for 21 of 22 dimensions. These results provide the first finite-step probabilistic control for this nonlinear normalization map.
In this article, we first prove the existence of a class of vector-valued and multivariate α-fractal functions on a hype-rectangle of ℝ^q. After that, we show that the set of functions with any possible fixed dimension in the continuous function space is dense with respect to uniform norm. Building on this, we also study dimension preserving approximation in this general setting. We then introduce and study some constrained approximation aspects based on dimension-preserving notion. Further, we prove some approximation theoretic results (such as Weierstrass approximation theorem) of the constructed vector-valued and multivariate α-fractal functions on a hype-rectangle of ℝ^q. In particular, we show the existence of a dense subset of α-fractal functions in the continuous functions space.
In this study, we introduce a new generalization of metric spaces, called Perturbed Parametric Metric Spaces (PPMS). This framework extends the classical metric space by incorporating perturbation functions and the presence of a non-negative parameter τ in its distance function, rather than a standard two-variable metric d(μ ,ν ) , providing a more flexible and robust treatment of distance measurement that reflects underlying variations and imperfections in complex systems. Specifically, the measurement of distance between two points is subject to errors, often arising from factors such as instrumental inaccuracies or environmental influences. We develop the foundational properties and initiate some topological notions of PPMS, provide illustrative examples, and establish several fixed-point theorems within this setting with application to fixed-circle problem. These results demonstrate the applicability of PPMS in nonlinear analysis and show how it unifies and generalizes various existing metric-type spaces. Our approach opens new perspectives for future research in functional analysis.
A contractive iterated function system (IFS) defined on a complete metric space X possesses a unique compact attractor F⊂ X. Obtaining the transformations of F is generally nontrivial, due to the nature of construction of F. In this work, we review a method based on barycentric coordinates for constructing affine transformations of F (translation, scaling, reflection, rotation, shear, and their compositions) and develop its stronger theoretical framework. This approach yields an exact solution to a specific inverse problem: given the affine image w(F) as a target set, we derive an explicit formula for the IFS whose fixed point is exactly w(F). Taking into account the definition of the Hutchinson operator, we also obtain the corresponding Hutchinson operator, whose fixed point is also w(F). Pseudocode and examples are included to facilitate implementation and verification.
This paper develops a unified and extended framework for contraction-type mappings in metric spaces by introducing extended unified interpolative Ćirić–Reich–Rus type (α ,β ,F)-contractions together with their r-order counterparts. Within this general setting, rigorous existence and uniqueness results for fixed points of self-mappings are established. The proposed approach not only subsumes numerous classical and contemporary contraction principles as particular cases, but also elucidates the structural relationships among them. The scope and effectiveness of the new framework are further demonstrated through applications to nonlinear integral equations, confirming its relevance and potential for broader analytical investigations.
This paper proposes an advanced group decision-making model based on Type-2 fermatean fuzzy sets (T2FFS), incorporating newly developed Hamming and Euclidean distance metrics. The proposed framework addresses the limitations of conventional fuzzy set approaches by improving and handling uncertainty and vagueness in expert evaluations. The distance metrics can effectively and accurately represent the inherent uncertainty and hesitation levels of decision makers when assessing alternatives, thereby avoiding one of the main limitations of existing type-1 and type-2 distance metrics in fuzzy environments. To demonstrate the model’s practical relevance, it is applied to a real-world problem involving selecting electric vehicles (EVs) for sustainable transportation. A comparative analysis with existing methodologies demonstrates the enhanced accuracy and reliability of the proposed technique.
In this paper, anti-periodic boundary value problems for Caputo fractional differential equations involving the p-Laplacian operator and a singular nonlinearity of the form t^-γ are studied. Using tools from functional analysis together with Schaefer fixed point theorem, a global existence result for the considered problem is obtained. In order to apply the fixed point argument, we first establish the equivalence between the fractional differential problem and a corresponding Volterra integral equation. The singular term plays a crucial role throughout the analysis and requires additional estimates. An illustrative example is provided to demonstrate the applicability of the main theorem.
Herein, a concept referred to as graphical fuzzy b-metric spaces is introduced, which generalizes the framework of fuzzy b-metric with the aid of graphical structures. Certain topological aspects corresponding to these spaces are explored, and related fixed point findings are established. These findings extend previously established results in the literature. Instances together with an application concerning equations of motion are included to showcase the feasibility of the established findings.
In this paper, we introduce a novel class of mappings, referred to as noncyclic generalized θ-contractions. By employing the geometric concept of WUC property in metric spaces, we establish new existence and convergence theorems for the fixed points associated with these mappings. The results presented herein generalize and improve several existing fixed point theorems related to generalized φ-contractions. Furthermore, we address the issue of error estimation and derive both a priori and a posteriori error bounds for the fixed points obtained via the Picard iterative process applied to a noncyclic generalized θ-contraction mapping defined on a uniformly convex Banach space. A distinctive feature of our analysis lies in avoiding the use of geometric progression techniques. Consequently, the resulting error estimates hold unconditionally in uniformly convex Banach spaces, thereby removing the need for any restrictive power-type condition on the modulus of convexity. We then present a comprehensive example to illustrate and validate the applicability and robustness of the main theoretical results. Finally, we apply the existence and convergence results for optimal pairs of fixed points to a system of differential equations.
The aim of this paper is to investigate the approximate controllability of fractional stochastic differential equations involving the Hilfer derivative of order 1<μ <2 and type ν∈ [0,1]. The analysis is carried out within the framework of fractional calculus, where the existence of mild solutions is established by employing properties of multivalued maps together with fixed-point methods. Initially, we study the approximate controllability of the considered stochastic system, and subsequently, an illustrative application is provided to demonstrate the effectiveness of the proposed approach and validate the theoretical results. Unlike existing studies that mainly address deterministic systems or stochastic models with Caputo-type derivatives, this work considers Hilfer-type stochastic evolution equations with multivalued control operators. By combining measurable selection techniques, Mainardi/Wright kernel representations, and a β-regularized controllability operator, we obtain new mean-square approximate controllability results not covered in the current literature.
Using the concept of the degree of nondensifiability in Banach algebras, we establish fixed point theorems for the product of two operators that are contractions with respect to this measure. Furthermore, we present a Krasnosel’skii-type fixed point theorem for the sum of two operators in Banach algebras.
Pneumonia presents a persistent and significant global health challenge, with its complex transmission dynamics exacerbated by the prevalence of asymptomatic carriers and diverse clinical presentations. This study addresses the limitation of traditional epidemiological models by incorporating the impact of public awareness on disease dynamics. We propose and analyze a novel six-compartment mathematical model using a system of ordinary differential equations. The model stratifies the population into susceptible (S), Exposed E(t), unaware infected ( I_u ), aware infected ( I_a ), treated (T), and recovered (R) compartments. We rigorously establish the positivity and boundedness of solutions and determine the stability of both the disease-free and endemic equilibriums. A comprehensive sensitivity analysis of the basic reproduction number ( R_0 ) is performed to identify the most influential parameters affecting disease spread. Our analysis confirms the stability of the disease-free equilibrium when R_0 < 1 and the existence and stability of the endemic equilibrium when R_0 > 1 . The sensitivity analysis reveals that increased public awareness and enhanced treatment rates are the most critical parameters for reducing disease transmission. Numerical simulations validate these findings, demonstrating a significant decrease in disease prevalence with higher awareness and treatment rates. The findings underscore the vital role of public health campaigns in mitigating pneumonia outbreaks. We show that integrating behavioral factors like public awareness into epidemiological models provides a more accurate representation of disease dynamics and highlights the necessity of proactive, data-driven interventions for effective and long-term disease control.
The present paper is devoted to discussing a class of nonlinear Caputo-type fractional integro-differential equations with two-point type boundary value conditions. We investigate the existence and uniqueness of the solutions by virtue of the classical Leray-Schauder alternative principle and the Banach contraction principle. Furthermore, by means of a novel Gronwall-type inequality, we prove the Hyers-Ulam stability of boundary value problems of multi-term Caputo fractional differential equations. Finally, some numerical examples are given to illustrate the results.
