Search Results (132)

Non-unique fixed-point theorems play a pivotal role in the mathematical modeling to solve certain typical equations, which admit more than one solution. In such situations, traditional outcomes fail due to uniqueness of fixed points. The primary aim of the present article is to investigate a non-unique fixed-point theorem in the framework of a metric space endowed with a local class of transitive binary relations. To obtain our main objective, we introduce a new nonlinear contraction-inequality that subsumes the ideas involved in four noted contraction conditions, namely: almost contraction, Boyd–Wong contraction, Pant contraction and relational contraction. We also establish the corresponding uniqueness theorem for the proposed contraction under some additional hypotheses. Several examples are furnished to illustrate the legitimacy of our newly proved results. In particular, we deduce a fixed-point theorem for almost Boyd–Wong contractions in the setting of abstract metric space. Our results generalize, enhance, expand, consolidate and develop a number of known results existing in the literature. The practical relevance of the theoretical findings is demonstrated by applying to study the existence and uniqueness of solution of a specific periodic boundary value problem. Full article

We unify a known technique used for fixed points and coupled, tripled and N-tupled fixed points for weak monotone maps, i.e., maps that exhibit monotone properties for each of their variables. We weaken the classical contractive condition in partially ordered metric spaces by requiring it to hold only for a sequence of successive iterations, generated by the considered map, provided that it is a monotone one. We show that some known results are a direct consequence of the main result. The introduced technique shows that the partial order in the constructed Cartesian space is induced by both the partial order in the considered metric space and by the monotone properties of the investigated maps. We illustrate the main result, which is applied to solve a nonlinear matrix equation, following key ideas from Berzig, Duan & Samet. We present an illustrative example. We comment that a similar approach can be used to solve systems of nonlinear matrix equations. Full article

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

This paper establishes the well-posedness of Cauchy-type problems with non-symmetric initial conditions for nonlinear implicit Hilfer fractional differential equations of general fractional orders in weighted function spaces. Using fixed-point techniques, we first prove the existence of solutions via Schaefer’s fixed-point theorem. The uniqueness and Ulam–Hyers stability are then derived using Banach’s contraction principle. By introducing a novel singular-kernel Gronwall inequality, we extend the analysis to Ulam–Hyers–Rassias stability and continuous dependence on initial data. The theoretical framework is unified for general fractional orders and validated through examples, demonstrating its applicability to implicit systems with memory effects. Key contributions include weighted-space analysis and stability criteria for this class of equations. Full article

This paper is concerned with the investigation of a nonlocal sequential multistrip boundary-value problem for fractional differential inclusions, involving $(k_1,{\psi }_1)$-Hilfer and $(k_2,{\psi }_2)$-Caputo fractional derivative operators, and $(k_2,{\psi }_2)$- Riemann–Liouville fractional integral operators. The problem considered in the present study is of a more general nature as the $(k_1,{\psi }_1)$-Hilfer fractional derivative operator specializes to several other fractional derivative operators by fixing the values of the function ${\psi }_1$ and the parameter $\beta$. Also the $(k_2,{\psi }_2)$-Riemann–Liouville fractional integral operator appearing in the multistrip boundary conditions is a generalized form of the ${\psi }_2$-Riemann–Liouville, $k_2$-Riemann–Liouville, and the usual Riemann–Liouville fractional integral operators (see the details in the paragraph after the formulation of the problem. Our study includes both convex and non-convex valued maps. In the upper semicontinuous case, we prove four existence results with the aid of the Leray–Schauder nonlinear alternative for multivalued maps, Mertelli’s fixed-point theorem, the nonlinear alternative for contractive maps, and Krasnoselskii’s multivalued fixed-point theorem when the multivalued map is convex-valued and $L^1$-Carathéodory. The lower semicontinuous case is discussed by making use of the nonlinear alternative of the Leray–Schauder type for single-valued maps together with Bressan and Colombo’s selection theorem for lower semicontinuous maps with decomposable values. Our final result for the Lipschitz case relies on the Covitz–Nadler fixed-point theorem for contractive multivalued maps. Examples are offered for illustrating the results presented in this study. Full article

The study of fractional differential equations is gaining increasing significance due to their wide-ranging applications across various fields. Different methods, including fixed-point theory, variational approaches, and the lower and upper solutions method, are employed to analyze the existence and uniqueness of solutions to fractional differential equations. This paper investigates the existence and uniqueness of solutions to a class of nonlinear fractional differential equations involving mixed Caputo–Riemann fractional derivatives with integral initial conditions, set within a Banach space. Sufficient conditions are provided for the existence and uniqueness of solutions based on the problem’s parameters. The results are derived by constructing the Green’s function for the initial value problem. Schauder’s fixed-point theorem is used to prove existence, while Banach’s contraction mapping principle ensures uniqueness. Finally, an example is given to demonstrate the practical application of the results. Full article

This paper investigates the existence, uniqueness, and finite-time stability of solutions to a class of nonlinear systems governed by the Hadamard fractional derivative. The analysis is carried out using two fundamental tools from fixed point theory: the Krasnoselskii fixed point theorem and the Banach contraction principle. These methods provide rigorous conditions under which solutions exist and are unique. Furthermore, criteria ensuring the finite-time stability of the system are derived. To demonstrate the practicality of the theoretical results, a detailed example is presented. This paper also discusses certain assumptions and presents corollaries that naturally emerge from the main theorems. Full article

This study investigates the synergistic effects of pulsed heat load and helium plasma irradiation on the surface damage evolution of high-purity tungsten, a candidate plasma-facing material (PFM) for future fusion reactors. Using a self-developed linear plasma device, tungsten samples were exposed to controlled single-pulse heat loads (32–124 MW·m⁻²) and helium plasma fluxes (7.76 × 10²²–2.40 × 10²³ ions·m⁻²·s⁻¹). SEM and XRD analyses revealed a progressive damage mechanism involving helium bubble formation, pit collapse, coral-like nanostructure evolution, and melting-induced restructuring. These surface changes were accompanied by grain refinement, lattice contraction, and peak shifts in the (110) diffraction plane. Mechanical testing showed a flux-dependent variation in hardness, with initial hardening followed by softening due to crack propagation. Surface reflectivity significantly declined with increasing load, indicating severe optical degradation. This work demonstrates the nonlinear coupling between thermal and irradiation effects in tungsten, offering new insights into damage accumulation under realistic reactor conditions. The findings highlight the dominant role of transient heat loads in driving structural and property changes and emphasize the importance of accounting for synergistic effects in material design. These results provide essential experimental data for optimizing PFMs in divertor and first-wall applications and suggest directions for future research into cyclic loading, long-term exposure, and microstructural recovery mechanisms. Full article

In this paper, we introduce a fast iterative scheme and establish its convergence under a contractive condition. This new scheme can be viewed as an extension and generalization of existing iterative schemes such as Picard–Noor and UO iterative schemes for solving nonlinear equations. We demonstrate theoretically and numerically that the new scheme converges faster than several existing iterative schemes with the fastest known convergence rates for contractive mappings. We also analyze the stability of the new scheme and provide numerical computations to validate the analytic results. Finally, we implement the new scheme in MATLAB R2023b to simulate the dynamics of the Ebola virus disease. Full article

This paper investigates the fixed points of large enriched contractions in a convex metric space as well as in a convex G-metric space. We establish the sufficient conditions for the existence and uniqueness of fixed points for these mappings. We use the Kransnoselskij-type iterative procedure for the approximation of these fixed points in complete convex metric spaces. We demonstrate that the Kransnoselskij-type iterative approach converges to the unique fixed point associated with large enriched contractions. Our results extend and generalize classical fixed point results by introducing this novel contraction mapping. Some illustrative examples are presented to demonstrate the applicability of our theorems. In the last section, we study the existence of a solution of nonlinear equations as a practical application of our principle findings. Full article

This paper investigates the problem of ensuring the stable operation of multiple high-speed train systems under the threat of False Data Injection (FDI) attacks. Due to the wireless communication characteristics of railway networks, high-speed train systems are particularly vulnerable to FDI attacks, which can compromise the accuracy of train data and disrupt cooperative control strategies. To mitigate this risk, we propose a Distributed Model-Free Adaptive Predictive Control (DMFAPC) scheme, which is data-driven and does not rely on an accurate system model. First, by using a dynamic linearization method, we transform the nonlinear high-speed train system model into a dynamically linearized model. Then, based on the above linearized model, we design a DMFAPC control strategy that ensures bounded train velocity tracking errors even in the presence of FDI attacks. Finally, the stability of the proposed scheme is rigorously analyzed using the contraction mapping method, and simulation results demonstrate that the scheme exhibits excellent robustness and stability under attack conditions. Full article

This paper is about the characteristics of and a method to recognize the onset of limit cycle thermoacoustic oscillations in a gas turbine-like combustor with a premixed turbulent methane/air flame. Information on the measured time series data of the pressure and the OH* chemiluminescence is acquired and postprocessed. This is performed for a combustor with variation in two parameters: fuel/air equivalence ratio and combustor length. It is of prime importance to acknowledge the nonlinear dynamic nature of these instabilities. A method is studied to interpret thermoacoustic instability phenomena and assess quantitatively the transition of the combustor from a stable to an unstable regime. In this method, three-phase portraits are created on the basis of data retrieved from the measured acoustics and flame intensity in the laboratory-scale test combustor. In the path to limit cycle oscillation, the random distribution in the three-phase portrait contracts to an attractor. The phase portraits obtained when changing operating conditions, moving from the stable to the unstable regime and back, are analyzed. Subsequently, the attractor dimension is determined for quantitative analysis. On the basis of the trajectories from the stable to unstable and back in one run, a study is performed of the hysteresis dynamics in bifurcation diagrams. Finally, the onset of the instability is demonstrated to be recognized by the 0-1 criterion for chaos. The method was developed and demonstrated on a low-power atmospheric methane combustor with the aim to apply it subsequently on a high-power pressurized diesel combustor. Full article

The primary aim of this manuscript is to establish unique fixed point results for a class of $\Psi$-contraction operators in complete G-metric spaces. By combining and extending various fixed point theorems in the context of $\Psi$-contraction operators, we introduce a novel function, denoted as $\psi$, and explore its properties. Our work presents new theoretical results, supported by examples and applications, that enrich the study of G-metric spaces. These results not only generalize and unify a broad range of existing findings in the literature but also expand their use to boundary value problems, Fredholm-type integral equations, and nonlinear Caputo fractional differential equations. In doing so, we offer a more comprehensive understanding of fixed point theory in the G-metric space framework and broaden its scope in applied mathematics. We also offer a numerical spectral approach for solving fractional initial value problems, utilizing shifted Chebyshev polynomials to construct a semi-analytic solution that inherently satisfies the given homogeneous initial conditions. Full article

The node representation learning capability of Graph Convolutional Networks (GCNs) is fundamentally constrained by dynamic instability during feature propagation, yet existing research lacks systematic theoretical analysis of stability control mechanisms. This paper proposes a Stability-Optimized Graph Convolutional Network (SO-GCN) that enhances training stability and feature expressiveness in shallow architectures through continuous–discrete dual-domain stability constraints. By constructing continuous dynamical equations for GCNs and rigorously proving conditional stability under arbitrary parameter dimensions using nonlinear operator theory, we establish theoretical foundations. A Precision Weight Parameter Mechanism is introduced to determine critical Frobenius norm thresholds through feature contraction rates, optimized via differentiable penalty terms. Simultaneously, a Dynamic Step-size Adjustment Mechanism regulates propagation steps based on spectral properties of instantaneous Jacobian matrices and forward Euler discretization. Experimental results demonstrate SO-GCN’s superiority: 1.1–10.7% accuracy improvement on homophilic graphs (Cora/CiteSeer) and 11.22–12.09% enhancement on heterophilic graphs (Texas/Chameleon) compared to conventional GCN. Hilbert–Schmidt Independence Criterion (HSIC) analysis reveals SO-GCN’s superior inter-layer feature independence maintenance across 2–7 layers. This study establishes a novel theoretical paradigm for graph network stability analysis, with practical implications for optimizing shallow architectures in real-world applications. Full article

Soft robots with compliant bodies offer safe human–robot interaction as well as adaptability to unstructured dynamic environments. However, the nonlinear dynamics of a soft robot with infinite motion freedom pose various challenges to operation and control engineering. This research explores the motion of a pneumatic soft robot under diverse loading conditions by conducting finite element analysis (FEA) and using machine learning. The pneumatic soft robot consists of two parallel hyper-elastic tubular chambers that convert pneumatic pressure inputs into soft robot motion to mimic an elephant trunk and its motion. The body of each pneumatic chamber consists of a series of bellows to effectively facilitate the expansion, contraction, and bending of the body. The first chamber spans the entire length of the soft robot’s body, and the second chamber spans half of it. This unique asymmetric design enables the soft robot to bend and curl in various ways. Machine learning is used to establish a forward kinematic relationship between the pressure inputs and the motion responses of the soft robot using data from FEA. Accordingly, this research employs an artificial neural network that is trained on FEA data to estimate the reachable workspace of the soft robot for given pressure inputs. The trained neural network demonstrates promising estimation accuracy with an R-squared value of 0.99 and a root mean square error of 0.783. The workspaces of asymmetric double-chamber and single-chamber soft robots were compared, revealing that the double-chamber robot offers approximately 185 times more reachable workspace than the single-chamber soft robot. Full article