Search Results (926)

This paper studies local and global existence, uniqueness, and Lipschitz continuous dependence of mild solutions for a semilinear Hadamard fractional integro-differential equation in a Banach space ${\displaystyle X}$, with nonlocal initial conditions, a state-dependent Volterra kernel embedded in the nonlinearity, and impulsive effects at finitely many fixed times, where ${\displaystyle -A}$ generates a compact ${\displaystyle C_0}$-semigroup on ${\displaystyle X}$. The mild-solution formula is derived from first principles via ${\displaystyle \tau =lnt}$ and the Riemann–Liouville variation of parameters formula. Local existence follows from Schauder’s fixed-point theorem, global existence from a Hadamard–Gronwall inequality with an explicit blow-up alternative, uniqueness from a logarithmic power-weight contraction with an explicit condition on the Lipschitz constants, and continuous dependence with an explicit stability constant ${\displaystyle C_{\mathrm{dep}}}$; two concrete examples verify all hypotheses. Beyond this compact-semigroup theory, we prove two further results: an existence theorem under a Kuratowski measure-of-noncompactness condition via a Hadamard-adapted Mönch fixed-point argument, removing the compactness assumption on ${\displaystyle {\left\{T\left(t\right)\right\}}_{t\ge 1}}$ altogether, and an Ulam Hyers Rassias stability theorem whose constant reuses the same contraction weight ${\displaystyle {\lambda }^{*}}$ from the uniqueness theorem. Both extend recent noncompact-semigroup and Ulam-stability results in the literature to the present nonlocal, impulsive, state-dependent-kernel setting. Full article

We construct a resolvent operator framework for multi-term fractional Cauchy problems with damping in Banach spaces, where the fractional derivatives are of the Caputo–Fabrizio (CF) type. Using Laplace transform techniques, we obtain necessary and sufficient conditions for the generation of the associated resolvent families and establish the existence and uniqueness of generalized solutions. Since the CF derivative vanishes at the origin, the classical initial condition $u\left(0^{+}\right)=x$ is unattainable when $Ax\ne 0$; the resolvent constructed here gives rise to a generalized solution with $u\left(0^{+}\right)=T\left(0\right)x\ne x$. We make this initial jump precise, show that it vanishes at the rate $O(1-\alpha )$ as $\alpha \to 1^{-}$, and indicate when it is physically meaningful. The abstract theory is applied to a biharmonic plate model with structural damping, producing explicit analytical solutions and a quantitative bound on the jump. Full article

We consider impulsive stochastic delay differential equations with state-dependent impulse times, where the k-th impulse occurs at a time $\tau$ satisfying $\tau ={\tau }_k\left(u\left({\tau }^{-}\right)\right)$. In generalized Lipschitz and growth conditions for coefficients, establishing existence and uniqueness is conditional on the successive approximations sharing a common finite impulse-time structure. The proof of our main result is based on the successive approximation scheme along with Itô’s isometry and Bihari’s inequality. We prove that the approximating sequence is bounded and convergent in a suitable Banach space, which gives a unique local solution. Full article

This paper develops a Banach space-valued approximation framework based on symmetrized neural network (SNN) operators generated by a deformation-dependent sigmoidal activation function. Symmetry is introduced directly at the activation level through a reciprocal-deformation mechanism, yielding a positive, even, normalized, and localized density kernel satisfying the partition of unity. The resulting construction provides normalized compact-interval and whole-line quasi-interpolation operators for Banach space-valued functions. Quantitative pointwise and uniform convergence estimates are established through the first modulus of continuity and are extended to higher-order and Caputo–Bochner fractional approximation. Numerical diagnostics support the theoretical kernel properties, and fractional approximation experiments compare the SNN and classical NN operators under common computational conditions. A controlled blind-prediction experiment on a synthetic monthly temperature-like series uses a strict fit–validation–test protocol and a parameter-matched operator comparison, with seasonal ARIMA and MLP models as external baselines. Across five independent realizations, the SNN attains the best mean predictive performance, with $R^2=0.9500$, NMAE $=0.0452$, and NRMSE $=0.0570$. A vector-valued experiment in $Y={\mathbb{R}}^2$ further illustrates the non-scalar applicability of the Banach space framework. In addition, the normalized SNN kernel weights provide an intrinsic node-level interpretation mechanism without requiring an external post hoc explainability method. 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 paper develops a unified asymptotic theory for inverse-probability-weighted conditional U-statistics of arbitrary fixed order in the presence of missing-at-random responses and infinite-dimensional functional covariates. The target is a conditional higher-order functional generated by a measurable response kernel and evaluated locally on a separable Banach space. Localization is formulated through delta sequences, providing a common framework for kernel, partition, regressogram, orthogonal series, and related smoothing procedures without recourse to finite-dimensional density arguments. For bounded kernels, we establish uniform almost-complete convergence over pseudo-compact functional domains and obtain a sharp decomposition into deterministic localization bias and stochastic fluctuation. The latter is governed by the localized-kernel variance, the envelope of the delta sequence, the metric complexity of the indexing domain, and the small-ball concentration of the functional covariate. Unbounded kernels are treated under explicit weighted moment, truncation, and summability conditions. The feasible theory quantifies the additional perturbation induced by estimating the propensity score and identifies conditions under which this first-stage uncertainty is asymptotically negligible. Pointwise distributional theory is derived through a denominator linearization combined with the Hoeffding decomposition of the centered localized kernel. The Gaussian limit is driven by the first projection, while the higher-order canonical components are shown to be negligible under explicit local-mass, moment, and noncancellation assumptions. This yields oracle-equivalent feasible inference, a consistent first-projection variance estimator, and asymptotically valid studentized confidence intervals. A finite-grid adaptive comparison principle is also developed for data-driven resolution selection. The scope of the theory is illustrated through conditional rank functionals, discrimination with incomplete labels, metric-learning criteria, and functional prediction. Synthetic and semi-synthetic studies based on functional classification, phoneme log-periodograms, and growth trajectories document the finite-sample interaction between covariate-dependent label observation, local information loss, propensity estimation, and inverse-weighting variance. Full article

Let $0<q<1<p<\infty$ and let $\mathbb{T}$ be an arbitrary time scale. We establish a necessary-and-sufficient two-weight criterion in the quasi-Banach range for non-negative measurable functions from $L^p({[a,\infty )}_{\mathbb{T}},{\nu }^p \Delta t)$ to $L^q({[a,\infty )}_{\mathbb{T}},{\omega }^q \Delta t)$. The criterion is the finiteness of a mixed head–tail quantity involving $U\left(t\right)={\int }_t^{\infty }\omega {\left(\tau \right)}^q \Delta \tau$ and $V\left(\sigma \left(t\right)\right)={\int }_a^{\sigma \left(t\right)}\nu {\left(\tau \right)}^{-p^{\prime }} \Delta \tau$, and explicit two-sided bounds are obtained for the optimal constant. The key technical ingredient is a gap-compatible weighted level-function construction based on interval averages over right-scattered gaps. The result recovers the continuous characterization of Sinnamon and the discrete characterization of Braverman and Stepanov, and it also yields dynamic averaging, Bennett–Copson-type, Hardy–Flett-type, and shifted quantum specializations. Full article

In this paper, we introduce and study proximal $\mathfrak{Z}$-condensing operators in strictly convex Banach spaces by combining simulation functions with measures of noncompactness. A Darbo-type best proximity point theorem is established, and several consequences corresponding to nonlinear condensing conditions are obtained. As an application, a system of nonlinear ordinary differential equations is embedded into a non-self operator problem on an enlarged product space; in this formulation, best proximity points are shown to be equivalent to classical solutions of the system. We also prove a Krasnoselskii-type best proximity point theorem for the sum of a simulation-function contraction and a compact operator and apply it to a nonlinear matrix-valued integral equation. Finally, a multiplicative best proximity point theorem is obtained in strictly convex Banach algebras and is used to study a nonlinear integral equation. The results provide a unified operator-theoretic framework for additive and multiplicative equations involving non-self mappings. 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 study addresses the analysis and numerical solution of a nonlinear time fractional diffusion problem with periodic boundary conditions. A generalized Fourier-based approach is applied to obtain a representation of the solution within a suitable Banach space, taking advantage of the periodic nature of the problem. Existence and uniqueness analysis of the solution is performed under the assumption of coordinated convexity instead of the commonly used global Lipschitz condition on the source term. Furthermore, a finite difference scheme based on the L1 approximation is proposed for the numerical solution of the problem. The consistency of the proposed method is verified through numerical experiments. In addition, the effect of fractional order on the solution behavior is investigated, and the results are supported by theoretical findings. Full article

We introduce finite-observation conditional guarded language operators on the complete length-based ultrametric space of formal languages over a finite alphabet. A finite observation map selects a guarded branch according to the membership of prescribed test words. Although each branch is Banach-contractive, branch switching may destroy global Banach contractivity. We derive depth–residual conditions yielding a max-type Kannan inequality with an explicit constant $\alpha <1/2$. Consequently, the operator has a unique fixed language, and the Picard iteration converges from every initial language. We also construct multi-branch operators that are Kannan-contractive but not Banach contractive and obtain residual error estimates, explicit convergence bounds, finite-depth certification, and eventual stabilization of the selected branch. 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 investigates a nonlinear inverse problem in a time fractional diffusion-wave equation, in which a spatially varying potential coefficient is recovered from noisy final-time measurements. A local uniqueness result for the inverse problem is first established in a finite-dimensional admissible space. To support the reconstruction, the Fréchet derivative of the forward map and its adjoint representation are derived to provide the gradient information required in the inversion procedure. A Banach space regularization scheme with a combined $L^1$ and $L^2$ penalty is then proposed to stabilize the nonlinear inverse problem, and the resulting nonsmooth minimization problem is solved by a locally linearized split Bregman iterative scheme. Numerical experiments in one and two spatial dimensions demonstrate the accuracy and stability of the proposed method for smooth, corner-type, localized, and discontinuous coefficient profiles. Full article

This work develops a progressive homotopy expansion scheme (PHES) for investigating Erdélyi–Kober (EK) fractional dynamical systems with memory effects, nonlinear interactions, and oscillatory behavior. The proposed framework is formulated in suitable Banach spaces with compact–open structures, enabling an operator-theoretic formulation for analytical and computational analysis. Existence and uniqueness of solutions are established using fixed-point arguments under suitable contraction conditions. The PHES is then constructed through recursive homotopy procedures and fractional expansion representations, and its convergence and Hyers–Ulam stability are rigorously analyzed. As an application, the proposed framework is employed to study EK fractional Brusselator oscillator networks describing memory-dependent chemical reaction dynamics. Numerical simulations for different fractional orders illustrate the influence of memory effects on oscillatory behavior, wave propagation, and collective network dynamics. A comparison with the Homotopy Perturbation Method (HPM) demonstrates excellent agreement, while the proposed PHES achieves smaller residual errors and higher numerical accuracy. These results confirm that PHES is an accurate and reliable approach for analyzing nonlinear EK fractional dynamical systems and memory-dependent oscillator networks. 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