Search Results (909)

In this paper, based on the Schauder fixed point theorem, the (generalized) Darbo fixed point theorem, and the (generalized) Banach contraction mapping principle, we study the mixed-type fractional impulse evolution equation with non-local and delay terms, and obtain the existence and uniqueness theorems under whether the operator is compact or not. The order of the derivative in this paper is $0<\alpha <1$, this fractional order introduces a series of problems concerning compactness, continuity, and convergence. We overcome these problems using methods such as H$\ddot{o}$lder inequality and Minkowski inequality. Moreover, under the condition of the non-compact measure, the non-negative constant is extended to an unbounded Lebesgue-integrable function. In addition, when obtaining the uniqueness of the solution through the (generalized) Banach contraction mapping principle, the non-negative constant L in the Lipschitz condition is extended to an unbounded Lebesgue integrable function. Finally, a case study is conducted to demonstrate the validity of the theoretical results. Full article

We introduce $(\mathrm{\Omega },\mathrm{\Pi })$-contractions in complete S-metric spaces, where $\mathrm{\Omega },\mathrm{\Pi }:(0,\infty )\to \mathbb{R}$ satisfy $\mathrm{\Pi }\left(t\right)<\mathrm{\Omega }\left(t\right)$. Under natural comparison conditions, we prove asymptotic regularity and S-Cauchy Picard iterates. With either a closed-graph condition or ${\displaystyle \underset{t\to 0^{+}}{\lim \sup }\mathrm{\Pi }\left(t\right)<\underset{t\to {\epsilon }^{+}}{\lim \inf }\mathrm{\Omega }\left(t\right)}$ for every $\epsilon >0$, we establish unique fixed points and strong convergence from any initial point. Our results generalize the Banach contraction principle to S-metric spaces and subsume recent theorems on asymptotically regular mappings and implicit contractions. An application solves a nonlinear boundary value problem for diffusion between parallel walls. Full article

In this paper, we begin with a brief overview of quasi-contraction mappings in metric spaces and their extensions to b-metric spaces and strong b-metric spaces. We then work in the framework of strong b-metric spaces and establish new fixed-point theorems for $\kappa$-multi-valued quasi-contraction mappings. Our results extend and unify several existing fixed-point theorems in the literature. Finally, we present an application to the existence of solutions for a nonlinear integral equation, illustrating the applicability of the obtained results. Full article

This paper investigates the output synchronization of fractional-order cyber-physical systems (FOCPSs) under communication constraints. To address limited bandwidth and high transmission costs, an event-triggered encoding-decoding sampled-data iterative learning control (ET-EDSDILC) protocol is proposed. The control law integrates a quantized sampling framework with an encoding–decoding mechanism to reconstruct control signals and address communication constraints. Furthermore, an event-triggered mechanism based on error energy attenuation (EEA) is developed to adjust communication frequency by monitoring error trends, thereby reducing unnecessary data transmissions. By applying fractional-order calculus and the contraction mapping principle, sufficient conditions for output synchronization are derived. Numerical simulations show that the proposed ET-EDSDILC framework reduces communication overhead and data redundancy while maintaining tracking performance, offering a solution for FOCPSs under communication constraints. Full article

In this paper, we introduce a new class of contractions, called dual connected-image contractions, for a pair of self-mappings on a metric space endowed with a directed graph. This concept extends the notion of connected-image contractions by incorporating the interaction between two mappings through the graph structure. By employing a class of auxiliary functions, we establish existence and uniqueness results for common fixed points under this generalized contractive framework. The proposed approach not only unifies and extends several known results in the literature but also provides greater flexibility in handling nonlinear problems. As an application, the theoretical results are applied to a class of nonlinear fractional differential equations with nonlocal integral boundary conditions, where the existence of solutions is established by reformulating the problems as equivalent integral equations and applying the developed common fixed point framework. Full article

In this paper, we introduce a novel class of quadruple controlled metric-type spaces formulated in a four-dimensional setting. Within this framework, we extend the notion of a $\Theta$-contraction mapping to accommodate such spaces. By employing these concepts, we establish several new fixed-point theorems that significantly generalize and enhance the existing results in the literature. The validity and effectiveness of the proposed approach are demonstrated through carefully constructed illustrative examples within the defined space and for the derived fixed-point results. Finally, an application to a boundary value problem is presented, highlighting the potential of the developed theory. Furthermore, a coupled four-component thermo-chemical system was examined as an additional application, demonstrating the broader applicability of the proposed framework. Full article

In the digital twin (DT) network, effective edge data processing is essential to meet the real-time requirements of DT models. However, edge servers (ESs) are self-interested and have limited computation resources. The virtual content operator (VCO) cannot observe their true computing capabilities, leading to participation reluctance and information asymmetry. To address these challenges, this paper proposes a contract-learning integration method for computing incentive and data offloading. A two-dimensional computation-reward contract incentive mechanism is designed to motivate ESs to provide computation resources for data pre-processing, where both continuous and discrete distributions of ES types are considered. Then, ESs upload the processed results to the VCO for DT model mapping, synchronization, and final construction. Based on the individual rationality and incentive compatibility constraints, the optimal incentive reward and computing resource allocation strategies are analytically derived to maximize the VCO’s utility. Then, based on the signed contracts, a multi-agent double deep Q-network algorithm is developed to jointly optimize the binary data offloading decision, transmission bandwidth, and transmission power for the minimal system delay. The algorithm learns adaptive strategies in the dynamic network environment and mitigates Q-value overestimation. Numerical results demonstrate that the proposed method improves system performance in terms of computing incentive and data offloading. Full article

High penetration of distributed photovoltaic (PV) generation has intensified voltage violations and stochastic voltage fluctuations in distribution networks, while existing voltage–var control methods still have limitations in terms of communication dependence, scalability, and edge deployment. To address these issues, this paper proposes a stability-constrained voltage–var self-optimizing control method for distribution networks based on the Bandit-Guided Online Self-Tuning Group Relative Policy Optimization (BOST-GRPO) algorithm. First, based on the LinDistFlow linearized power-flow model, a communication-free, decentralized, and locally observable reinforcement learning control environment is constructed, enabling each node to independently generate reactive power regulation commands using only local voltage measurements. Second, a contraction-mapping-based stability constraint is embedded into the policy output layer, theoretically guaranteeing the local exponential convergence of nodal voltage deviations around the equilibrium point and reducing the risk of voltage instability caused by overly aggressive policy actions. Meanwhile, device capacity constraints are incorporated into the policy output through a tanh-based action mapping, ensuring the physical feasibility of control commands. On this basis, BOST-GRPO realizes the online self-tuning of key hyperparameters within a single training process through a Bandit-guided mechanism, thereby avoiding the repeated training overhead caused by traditional offline hyperparameter tuning. Simulation results on the IEEE 33-bus system show that the proposed method outperforms benchmark reinforcement learning algorithms in final test cost, voltage deviation suppression, steady-state error, and regulation speed. Further tests under sensitivity matrix mismatch, different initial voltage disturbance intensities, and the extended IEEE 69-bus system demonstrate that the proposed method achieves good robustness and scalability. Full article

Classical fixed-point theorems for Rakotch, Edelstein, and Bianchini contractions require the contractive condition to hold for the mapping itself at every iteration, which severely limits their applicability to many real-world problems. In this paper, we break this limitation by shifting the contractive requirement to the p-th iterate of the mapping. We introduce three novel classes of p-Rakotch, p-Edelstein, and p-Bianchini contractions and prove that each guarantees the existence of a unique fixed point and global convergence of the Picard sequence from any initial point, under appropriate metric space assumptions (completeness for Rakotch and Bianchini; compactness with continuity for Edelstein). A key feature of our approach is that the original mapping T need not satisfy any contractive condition; only its p-th iterate $T^p$ needs to. This allows us to handle mappings where classical theorems simply do not apply. To validate our theoretical findings, we provide explicit numerical examples for $p=3,4,5$. More importantly, we demonstrate the practical power of our results through six diverse applications: ordinary differential equations with large coefficients; planar discrete dynamical systems; nonlinear Hammerstein integral equations; Caputo fractional differential equations with large linear terms; fractional equations exploiting the smoothing property; and implicit fractional differential equations. In each application, the classical contractive condition fails, yet our p-iterate approach succeeds. When $p=1$, all three theorems reduce to their classical counterparts, confirming that our framework is a natural and faithful generalization. Full article

Spinal cord injury (SCI) is frequently accompanied by abnormal muscle tone and spasticity, which impair voluntary motor control, mobility, and quality of life. Although classically defined as velocity-dependent hyperreflexia, tone abnormalities after SCI encompass a broader spectrum, including sustained muscle activation, co-contraction, clonus, and non–velocity-dependent resistance to movement. These manifestations arise from distributed changes across spinal and supraspinal motor systems. At the segmental level, SCI induces maladaptive plasticity involving motoneurons, interneurons, sensory afferents, and muscle, including dysregulated persistent inward currents, altered inhibitory neurotransmission, afferent hyperexcitability, synaptic reorganization, and structural muscle remodeling. In parallel, supraspinal adaptations—including cortical motor map reorganization, reduced intracortical inhibition, corticospinal–reticulospinal imbalance, loss of monoaminergic modulation, and altered brainstem and cerebellar regulation—further amplify spinal circuit gain and impair inhibitory control of tone. Current pharmacologic treatments largely suppress symptoms without addressing these underlying circuit changes, while invasive neuromodulatory strategies are limited by surgical risk or state-dependent effects. This review synthesizes emerging insights into the multilevel mechanisms regulating abnormal tone after SCI and examines neuromodulatory approaches targeting spinal and supraspinal networks. Particular attention is given to transcutaneous spinal cord stimulation (TcSCS), a non-invasive method capable of modulating segmental reflex circuits and descending control pathways. Advances in transcriptomic and epigenetic profiling may further enable mechanism-based therapies and biomarker-guided strategies for treating spasticity. Full article

In this study, the influence of thermal loading in a supersonic flight environment on the mechanical stiffness of elastic structures and the corresponding aeroelastic stability limits is investigated analytically. Recognizing that elevated temperatures inherently alter constituent elastic properties, a temperature-dependent continuous elasticity framework is incorporated directly into the governing differential operators of the structural domain. The macro-mechanical behavior of representative panel- and wing-type elements is modeled utilizing the Euler–Bernoulli beam formulation, while high-speed supersonic aerodynamic effects are represented through linearized first-order piston theory. The continuous spatial displacement fields are discretized by means of a modal expansion, and the coupled aeroelastic system is subsequently transformed into a finite set of dynamic state-space equations using the Ritz–Galerkin truncation method. The numerical and analytical outputs demonstrate that aerothermal softening not only induces continuous erosion in the material stiffness but also directly modulates the aeroelastic pole trajectories, thereby prematurely contracting the safe supersonic flight envelope. The primary novelty of the proposed framework lies in the derivation of explicit analytical expressions that directly map temperature-dependent stiffness variations onto supersonic aeroelastic instability boundaries. Because this approach is formulated in a generalized analytical form, it can be applied across diverse material systems, geometric profiles, and thermal conditions with reduced computational overhead compared to full fluid–structure interaction solvers, thereby providing a theoretical basis for preliminary stability assessment of supersonic aerospace configurations operating under high-temperature conditions. Full article

We study positive–negative guarded systems of language equations over a fixed finite alphabet. The ambient space is the complete ultrametric space of all formal languages equipped with a length-based distance, where two languages are close whenever they agree on all words up to a sufficiently large length. The systems considered here contain both positive recursive dependencies and negative dependencies expressed through language complements. To handle this mixed structure, we introduce a suitable product order on pairs of languages and prove that the associated system operator has the weak monotone property. We show that the complement is an isometry for the length-based ultrametric and establish a signed wrapping estimate for guarded positive and negative language terms. These estimates lead to an ordered contraction principle for comparable pairs. As a consequence, the canonical lower and upper Picard iterations converge to the same limit, which is the unique fixed pair of the system. We also derive an explicit convergence rate and a finite-depth certification result: after a prescribed number of iterations, the approximants agree with the fixed-point semantics on all words below a given length. Additional symmetry assumptions are shown to force the unique fixed pair to be diagonal, reducing the system to a single language equation. Finally, we discuss an application to trace-based policies for tool-using AI agents. In this interpretation, finite executions of an agent are represented as words over an alphabet of observable tool-events, and the two components of the fixed point provide a stable semantics for policy-defined admissible and risky trace classes. The resulting framework gives a mathematically certified method for finite-depth analysis of recursive trace-based policies based on ultrametric fixed-point techniques. Full article

In this paper, we propose a new three-step iterative scheme to approximate fixed points of contraction operators in Banach spaces. Under standard Lipschitz conditions, we establish the existence, uniqueness, and strong convergence of the iterative sequence. The convergence rate and data dependence of the method are also investigated. A comparative analysis with Noor, Picard–S, Abbas–Nazir, SP, and NIP iterative methods is presented. As an application, the proposed scheme is employed to solve a fractional viscoelastic model involving a Caputo derivative of order $0<\alpha <1$, which is reformulated as a Volterra integral equation. The numerical results, including error analysis and graphical illustrations, demonstrate that the proposed method achieves faster convergence and a higher accuracy. Full article

Urban ethnic enclaves are historically layered cultural landscapes whose public signage encodes community vitality, power relations, and cultural identity in ways that conventional land-use inventories cannot capture. Addressing the absence of scalable, longitudinal computational methods for monitoring such at-risk landscapes, this study develops a reproducible digital-mapping pipeline that operationalises linguistic-landscape analysis as a cultural-heritage monitoring tool for heritage-sensitive land-use planning. Taking Daerim-dong—Seoul’s primary Joseonjok (Korean Chinese) enclave—as a case, we process 38,640 Kakao Map Road View images across 17 annual cross-sections (2008–2024). The pipeline integrates four methodological components: a bounded Spatial Weighting Correction that adjusts for uneven historical coverage; zero-shot semantic sign-function classification using the Qwen2-7B-Instruct model; an exploratory Difference-in-Differences design probing the 2016–2017 THAAD geopolitical disruption; and a Boundary Permeability Ratio (BPR) for tracking enclave edge dynamics. The results document a three-phase trajectory—rapid bilingual expansion (2008–2016), stabilisation (2016–2019), and a COVID-period contraction (2019–2024)—and show that raw sign-count metrics can systematically overstate minority-language decline during economic crises once crisis-period signage is isolated. The BPR is presented as a candidate leading indicator of enclave contraction whose operational thresholds remain to be calibrated through multi-enclave validation. As a methodological proof-of-concept, the study illustrates how computational street-view analysis can support cultural-landscape governance, offering urban planners and heritage managers an actionable, transparent baseline for monitoring at-risk multicultural urban landscapes. Full article

In this work, we develop two new classes of rational contraction mappings along with the corresponding fixed point theorems for these types of contractions in suprametric spaces. Furthermore, we use the obtained results to investigate two nonlinear systems, namely, a fractional chaotic financial system and a nonlinear fractional differential equation under integral boundary conditions. Both these nonlinear problems are transformed into fixed point problems in appropriate suprametric spaces, thereby demonstrating the applicability of the developed rational contraction results to nonlinear systems with memory effects. Full article