Search Results (482)

This paper proposes and analyzes a modified version of the Hănțilă Method (HM) for solving electrical circuits with nonlinear inductive elements. The method replaces the nonlinear inductor with a generator comprising a nonlinear source and a linear impedance. The value of the nonlinear source is determined by defining a Picard–Banach fixed-point sequence that converges to the solution. This approach transfers the nonlinearity from the inductance to the generator’s source. The resulting circuit consists of linear and nonlinear sources and only linear components. It is solved in the harmonic domain using classical theorems and algorithms. A comparative analysis is performed on an RL circuit (a transformer primary winding at no-load). Both accuracy and computational effort are evaluated. The proposed iterative method is compared against steady-state time-domain transient analysis and frequency-domain approaches (Harmonic Balance Method) implemented in commercial software. This method is particularly suitable for analyzing circuits with nonlinear inductive components, such as iron-core coils and equivalent circuits for electrical machines. Full article

This work addresses the Ulam–Hyers stability of mild solutions of fractional noninstantaneous evolution nonlinear equations subject to noninstantaneous impulses in the Banach space through an approach based on the measure of noncompactness. Mild solutions are formulated via operators generated by a closed linear operator and a probability density function. Existence results are derived by applying the fixed point theorem for k-set contractive operators. Furthermore, Ulam–Hyers stability is established under certain hypotheses, and an illustrative example is provided to validate the derived results. Full article

This investigation is concerned with a time-fractional diffusion equation governed by the Hilfer fractional derivative of order $1<{\rho }_1<2$, subject to homogeneous Neumann boundary data and generalized initial conditions of the Hilfer type. This paper addresses three interconnected problems: a direct problem and two inverse source problems (ISPs). In the first ISP, the objective is to recover an unknown space-dependent source function from measurements taken at a specified final time. In the second ISP, the goal is to determine an unknown time-dependent coefficient through an integral-type over-specification condition. By employing eigenfunction expansions in conjunction with the LT technique, we derive explicit series representations of the solutions in terms of the Mittag-Leffler function. Rigorous existence and uniqueness results for classical solutions are established for all three problems. The second ISP is reformulated as a Volterra integral equation, whose unique solvability is demonstrated via the Banach fixed point theorem. Both ISPs are shown to be ill-posed in the Hadamard sense, indicating instability with respect to data perturbations. Numerical experiments are also presented to validate the theoretical findings and to illustrate the performance of the proposed reconstruction methods. As limiting cases, the formulations corresponding to the Riemann–Liouville and Caputo fractional derivatives are recovered, illustrating the generality of the proposed framework. Full article

This paper investigates an infinite-horizon discounted stochastic maintenance optimization problem within the framework of dynamic programming. The system degradation is modeled by a discrete-time stochastic process affected by maintenance actions and random disturbances, while the objective is to minimize the expected discounted maintenance and degradation costs. The analysis is carried out on the Banach space of continuous functions equipped with the supremum norm. It is proved that the associated Bellman operator is well defined, maps the function space into itself, and is a contraction with contraction modulus equal to the discount factor. Consequently, the existence and uniqueness of the optimal value function follow from the Banach Fixed Point Theorem, and the convergence of value iteration is established. In addition, rigorous a priori and a posteriori error estimates are derived, providing theoretical stopping criteria for numerical computation. The theoretical results are complemented by numerical experiments illustrating the optimal stationary maintenance policy, the stability of the computed solution under grid refinement, and the influence of the discount factor on both the optimal policy and the convergence rate of value iteration. Full article

This paper establishes the existence, uniqueness, and Ulam–Hyers stability of solutions for a novel class of coupled sequential hybrid Hilfer fractional q-Duffing equations. By integrating Dhage’s hybrid structure with generalized Hilfer q-operators, we extend recent results on fractional quantum systems. Existence is proven via Dhage’s fixed point theorem in Banach algebras, while uniqueness follows from the Banach contraction principle with explicit verification of operator invariance. All auxiliary functions satisfy rigorous continuity, boundedness, and Lipschitz conditions, and the solution representation is derived with complete calculation of q-integration constants. The theoretical findings are rigorously validated through a detailed numerical example that explicitly verifies all contraction and stability constants. Full article

We study a Caputo fractional three-point boundary value problem of order $\alpha \in (c-1,c]$ and establish three interrelated contributions, all resting on a single refined pointwise ${\mathbf{L}}^2$ estimate for the associated Green function $\mathbf{Gr}(t,s)$. First, we derive a tighter upper bound for ${\sup }_{0<t<1}{\int }_0^1{\mathbf{Gr}}^2(t,s)ds$ by retaining a sign-definite negative mixed term that the classical ${\mathbf{L}}^1$-based analysis of Shivanian discards. The resulting admissible Lipschitz constant ${\kappa }_B$ is explicit in $\alpha$, a, b, c and exceeds Shivanian’s constant ${\kappa }_S$ under an explicit algebraic condition; a closed-form refinement ${\kappa }_B^{\ast }\ge {\kappa }_B$ follows by maximising the pointwise bound in closed form. On Shivanian’s benchmark, the admissible constant rises from $14.646$ to $23.220$ and then to $32.165$. Second, the same contraction constant yields an explicit Ulam–Hyers stability theorem for this problem. While a stability estimate already follows from the classical ${\mathbf{L}}^1$ condition, the refined constant both enlarges the range of admissible Lipschitz constants for which stability is certified and yields a strictly smaller stability constant. Third, we establish quantitative continuous-dependence bounds with respect to the nonlinearity h and the boundary parameter a, and characterize the deterioration of the contraction-based boundary-parameter estimate as the problem approaches resonance. For a fixed Lipschitz constant, this estimate becomes singular as the perturbed contraction factor approaches one and ceases to apply once the contraction condition fails. A more accurate analysis of the Green function thus simultaneously sharpens solvability conditions, stability estimates, and sensitivity bounds for nonlinear Caputo fractional boundary value problems. Full article

This paper investigates the well-posedness of a class of nonlinear implicit fractional differential equations involving the $\psi$-Hilfer fractional derivative of complex order $\alpha$ with $\Re \left(\alpha \right)\in (n-1,n)$ for $n\in \mathbb{N}$, under general initial conditions in weighted spaces. The implicit nature of the problem, where the highest-order derivative appears nonlinearly on both sides of the equation, presents significant analytical challenges. By transforming the fractional Cauchy problem into an equivalent Volterra integral equation, we employ fixed-point theory to establish existence via Schaefer’s fixed-point theorem and uniqueness via Banach’s fixed-point theorem under suitable Lipschitz-type conditions. A generalized Gronwall inequality with singular kernels is developed to handle the nonlocal memory effects inherent to fractional operators. We further investigate four types of Ulam stability, namely Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability, demonstrating that small perturbations in the equation yield correspondingly small changes in the solution. Continuous dependence on initial conditions is also established. The theoretical framework is applied to a physically motivated fractional nonlinear oscillator with saturating acceleration-dependent feedback, where explicit verification of the hypotheses is provided. Full article

This paper considers a class of fractional stochastic differential systems driven jointly by the Rosenblatt process and a compensated Poisson random measure. The Rosenblatt process captures non-Gaussian fluctuations with memory effects; the Poisson jumps model sudden random shocks. By employing the mild solution formulation associated with fractional resolvent operators, we establish the existence of solutions via Krasnoselskii’s fixed point theorem (KFPT) and prove uniqueness through the Banach contraction under suitable Lipschitz conditions on the drift, diffusion, and jump coefficients together with contraction assumptions. Furthermore, Ulam–Hyers stability is established, ensuring that approximate solutions remain close to exact solutions in the mean-square sense. We also establish Mittag–Leffler-type continuous dependence on initial data, demonstrating that solutions depend continuously on their initial histories in the mean-square sense. In addition, open-loop approximate trajectory realization is established through the construction of an explicit open-loop control law that steers the stochastic system along any prescribed admissible trajectory under suitable invertibility and regularity assumptions. An example validates the theoretical results, demonstrating fractional memory, Rosenblatt noise, and Poisson jumps. Full article

The primary objective of this study is to establish some new fixed point results in the framework of perturbed $\mathcal{F}$-metric spaces. Specifically, we develop fixed point theorems for three classes of contractions: Banach-type, rational-type, and interpolative contractions. To demonstrate the validity and applicability of the obtained results, some nontrivial examples are provided. Moreover, our findings yield various consequences that extend existing results in both $\mathcal{F}$-metric spaces and perturbed metric spaces. As an application of the main result, we examine the existence of solutions for a Caputo fractional initial value problem. Full article

The present article introduces and investigates a class of implicit fractional differential equations with time-varying state-dependent feedback, a setting that significantly extends existing fractional delay models. Existence and uniqueness results are established via the Banach fixed-point theorem, while sharp exponential growth estimates for the solutions are derived using a generalized Gronwall-type inequality. These results provide new analytical insights into the qualitative behavior of fractional systems with dynamically evolving feedback mechanisms. Full article

This paper investigates the existence of pseudosolutions for a class of fractional retarded dynamic equations on time scales in Banach spaces endowed with the weak topology. The proposed model combines fractional dynamics, explicit delay effects, and hybrid continuous–discrete temporal structures within a unified analytical framework. The analysis is performed by means of the $\Delta$-Henstock–Kurzweil–Pettis integral, allowing significantly weaker regularity assumptions than those required by classical integration theories. The existence result is established using the De Blasi measure of weak noncompactness together with Kubiaczyk’s fixed point theorem for weakly sequentially continuous operators. The obtained theorem extends several existing results on fractional differential equations and dynamic equations on time scales by incorporating explicit delays and generalized integration into a common framework. The obtained results provide a unified analytical framework for studying hereditary systems evolving on hybrid time domains. Full article

In this paper, we introduce a general class of quadrilateral total pairwise $\phi$-contractions and investigate fixed-point results for such mappings in metric and semimetric spaces endowed with multiple distinct metrics. We establish several fixed-point theorems for these contractions and show that the obtained results extend beyond classical metric spaces to encompass ultrametric spaces and distance spaces equipped with power-type triangle functions. Furthermore, within the framework of metric spaces, we study the existence of solutions to fractional boundary-value problems with prescribed boundary conditions by employing mappings satisfying the quadrilateral total pairwise contraction condition. Several well-known results, including the Banach contraction principle, the Boyd–Wong fixed-point theorem, and various related three-point and four-point contraction results, are recovered as particular cases of our theorems. Full article

We investigate the existence and uniqueness of solutions for two-point boundary value problems involving left and right Caputo fractional derivatives with possibly complex-valued coefficients on a finite real interval. Under suitable assumptions on the fractional order and the coefficients, we establish the existence and uniqueness of continuous weak solutions. In addition, under further $L^2$-type assumptions on the coefficients, we prove the existence of weak solutions in the Hilbert space $L^2(a,b)$. The analysis is based on an equivalent fractional integral formulation and the Banach–Caccioppoli fixed-point theorem. Full article

We study a fractional Schrödinger–Poisson system involving the spectral fractional Laplacian on a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$$(N>2s)$ subject to homogeneous Dirichlet boundary conditions. The model consists of a fractional Schrödinger equation coupled with a fractional Poisson equation through a self-consistent potential. Using the spectral Green operator associated with ${(-\Delta )}^t$, the coupled system is reduced to a single nonlocal integro-differential equation. The associated Green kernel admits a spectral representation in terms of the Dirichlet eigenpairs of the Laplacian. Under suitable assumptions on the Green kernel and Lipschitz conditions on the nonlinearities, we establish the existence of weak solutions together with local uniqueness within the contraction framework via the Banach fixed point theorem for sufficiently small coupling parameters. We further investigate the regularity of the self-consistent potential, continuous dependence on the model parameters, and a conditional convergence to the classical Schrödinger–Poisson system as $s,t\to 1^{-}$. A numerical illustration based on truncated spectral expansions is presented to demonstrate the practical implementation of the proposed framework. Unlike the predominantly variational methods available in the literature, the proposed framework combines a spectral Green-operator reduction with an operator-theoretic fixed-point analysis, providing a constructive formulation that is directly amenable to numerical implementation. Full article

This paper introduces the Contraction Mapping Optimizer (CMO), a population-based optimizer whose update rule is derived directly from the Banach fixed-point theorem rather than from a biological or physical metaphor, and for which global convergence is proved. Most metaheuristics are built on such metaphors and offer little formal insight into why their update rules converge; CMO instead makes the mechanism explicit. Each candidate solution is moved by a damped contraction map toward an attractor formed from the best solutions found so far, perturbed by a Gaussian exploration term whose amplitude vanishes over the run. Because the contraction factor is kept strictly below one by construction, the deterministic part of every update is provably a contraction, making the exploration-exploitation balance explicit and schedulable. We further prove that the full stochastic, population-based iteration converges to the global optimum, and we demonstrate CMO on a nonlinear integral equation whose solution is itself a fixed point. CMO is compared with nine established metaheuristics (GA, PSO, DE, GWO, HHO, RUN, EO, FGO and WOA) on CEC2017 (D = 30, 50, 100) and CEC2022 (D = 10, 20), on three constrained engineering design problems, and on a six-unit economic load dispatch problem, over thirty runs at a common budget. Pooled over the 111 benchmark instances, CMO attains a Friedman mean rank of 3.14, statistically inseparable from the best competitor (FGO, 2.20) and significantly ahead of the remaining seven baselines; it ranks first on both CEC2022 suites and, on the constrained applications, is second only to differential evolution. Sensitivity and ablation studies identify the vanishing exploration schedule and greedy selection as the decisive components of the design. Full article