Vol. 33 No. 3 (2026): In Progress

Open Access

Articles

Article ID: 4409

On numerical radius inequalities via McCarthy inequality

by Muhammad Fazeel Anwar, Saira Iqbal, Muhammad Saeed Akram

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This paper aims to establish new inequalities for the numerical radius of bounded linear operators defined on a complex Hilbert space. By employing a generalized form of the McCarthy inequality, we derive several upper bounds for the numerical radius of a single operator as well as for expressions involving sums and products of operators. The obtained results extend and refine a number of existing inequalities in the literature. In particular, many known numerical radius inequalities are recovered as special cases of our results, thereby providing a unified framework for their analysis. The refinement is based on the Akkouchi-Ighachane version of the Hölder-McCarthy inequality, which allows the usual McCarthy term to be replaced by a smaller parameter-dependent expression before taking the supremum. This gives a common refinement mechanism for estimates involving a single operator, Cartesian decompositions, finite sums, block matrices and the Euclidean operator radius. We also state explicitly the parameter choices which recover the earlier inequalities and include simple finite-dimensional comparisons showing that, for suitable non-normal matrices, the refined bounds may be strictly sharper than the corresponding classical estimates. These comparisons confirm that the refinement provides a usable quantitative improvement, rather than only a formal parameter extension of existing bounds in concrete cases.

Open Access

Article

Article ID: 4390

Not loss of stability but steady-state shift: A systems biology explanation for elevated fasting blood glucose in type 2 diabetes

by Guanyu Wang

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Elevated fasting blood glucose is a hallmark clinical feature of prediabetes and type 2 diabetes, reflecting underlying pathological alterations in the body’s glucose-insulin regulatory system. This study employs a validated mathematical model of the glucose-insulin negative feedback loop to investigate the fundamental mechanisms of this elevation from a systems biology perspective. We specifically analyzed whether hyperglycemia arises from decreased system stability or an upward shift in the steady-state set-point. Our findings demonstrate that the intrinsic stability of the glucose-insulin regulatory system remains largely unchanged from healthy states through early-stage diabetes. Contrary to the hypothesis of stability decay, the essence of elevated fasting glucose is an upward shift of the steady-state level driven primarily by hepatic insulin resistance, which increases basal hepatic glucose output. While system stability is preserved during early progression, the dynamic coupling between glucose and insulin weakens; low-frequency oscillations inherent to the healthy system gradually diminish and eventually disappear as insulin resistance intensifies, signaling a decoupling of dynamics before stability loss. Significant weakening of system stability, characterized by a markedly reduced convergence rate following perturbation, occurs only in late-stage diabetes accompanied by severe pancreatic damage and compromised insulin secretory capacity. These results redefine the pathophysiological understanding of fasting hyperglycemia, suggesting that early therapeutic strategies should target recalibrating the glucose set point and hepatic sensitivity rather than bolstering system stability. This research provides valuable theoretical insights into the pathogenesis of diabetes and highlights potential dynamic biomarkers for monitoring disease progression.

Open Access

Article

Article ID: 4327

Channeling paths identification and mitigation by using integrated profile control and flooding based on meshless connection element method

by Wei Yong, Zhijie Wei, Jian Zhang, Wensheng Zhou, Yuyang Liu, Wentao Zhan, Wei Liu, Yixin Zhang, Jiaxu Mei

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Addressing the challenge of quantitatively identifying deep thief zones in mature oilfields during the high water-cut stage, this study proposes a robust quantitative characterization model for thief channels based on the non-Euclidean, meshless Connection Element Method (CEM) to directly guide in-depth fluid diversion and integrated profile control and flooding treatments through automated flow path tracking rooted in a directed-graph depth-first search algorithm. To systematically capture the complex subsurface topological network, a comprehensive multi-parameter connectivity parameter system was constructed by integrating key dynamic indicators, including connection conductivity, connection volume, and path splitting coefficients. By dynamically coupling these parameters with the field-wide Lorentz coefficient, a dimensionless channeling factor was defined to establish a rigorous four-level quantitative standard—ranging from extreme channeling to matrix seepage—thereby successfully advancing preferential pathway evaluation from qualitative inference to spatial grading and precise localization. Quantitative validation against conventional commercial grid-based compositional simulators demonstrates the superior fidelity and performance forecasting efficiency of the proposed method: the CEM framework achieves an exceptionally accurate water-cut prediction Root Mean Square Error (RMSE) of approximately 3.8%. Crucially, by abstracting continuous domains into streamlined networks, it drastically compresses structural degrees of freedom, successfully accelerating the operational execution runtime from 7.3 s to a mere 1.6 s. Ultimately, this work provides a computationally highly efficient, physically sound novel approach for the reliable mapping and graded quantification of deep dominant channeling pathways in mature heterogeneous oilfields.

Open Access

Article

Article ID: 4591

Stability and controllability for fractional damped differential equations with ψ-Hilfer derivatives

by Junjie Xie, Xiaoyue Han, Zhenbin Fan

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This study investigates the existence, uniqueness, finite-time stability and controllability for a class of nonlinear ψ-Hilfer fractional differential equations with damping terms, and the main mathematical contributions are achieved through several techniques including parameter selection, reasonable application of weighting methods and transformations. Specifically, by leveraging the generalized Laplace transform method and weighted norm technique, the explicit representation of solutions is derived. The existence and uniqueness of solutions are established via the generalized Banach contraction principle and Schauder fixed point theorem. Then finite-time stability criteria are obtained via inequality estimates and fractional Gronwall inequalities, each providing complementary insights into the behavior of systems within a bounded time interval. Furthermore, we establish controllability criteria for linear and nonlinear systems, the latter relies on the successive approximation method. Finally, several numerical examples are presented to validate our findings. These examples illustrate the necessity of the two existence theorems, since their respective assumptions are not always simultaneously satisfied in practical applications. In particular, we examined the magnitude of the error generated when the controllability conditions for the nonlinear system are not satisfied. These results contribute to the broader understanding of fractional damped systems governed by ψ-Hilfer derivatives and offer practical tools for analyzing stability and controllability in complex damped dynamical systems.

Open Access

Article

Article ID: 4549

On a class of bivalued hybrid cyclic and non-cyclic contractive self-mappings on unions of possibly disjoint subsets in metric spaces

by Manuel De la Sen, Asier Ibeas, Hasanen A. Hammad

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

This paper presents and analyzes a class of self-mappings in metric spaces which are endowed of mixed characteristics between that of contractive cyclic self-mappings on union of subsets and that of contractive mappings within the individual subsets. Stability analysis of dynamic systems can be focused on by formalizing the relationship between cyclic contractive mappings over non-necessarily intersecting subsets and the stability criteria for switched systems with cyclic dynamics. The case of disjoint set structures necessitates the application of best proximity theory. Therefore, this manuscript introduces the mentioned class of hybrid cyclic/non-cyclic bivalued contractive self-mappings which can operate iteration-by-iteration in either a cyclic operation mode (the selected image lies in the next adjacent subset) or in an intra mode operation (the selected image lies in the current subset). Unlike cyclic operators, these proposed self-mappings allow dynamic switches to either the current subset or the next adjacent one in a cyclic disposal, governed by an iteration-dependent image selection sequence. This “modus-operandi” is possible since one of the images of each point at each iteration is activated for the cyclic operation mode while the other one is activated for an intra mode mode within some subset. Under the key assumptions that the subsets are closed, at least one best proximity set is a singleton, and the subset itself is boundedly compact, new boundedness and convergence theorems are established. It is emphasized how cyclic contractive properties are useful in the context of asymptotic stability of hybrid systems, validating the theoretical framework through illustrative examples.

Open Access

Review

Article ID: 4514

Frontier advances in solving fractional differential equations via physics-informed neural networks: A comprehensive review

by Yi Tian, Xiuqing Peng, Ji-huan He

Advances in Differential Equations and Control Processes, Vol.33, No.3, 2026;

Fractional differential equations (FDEs) serve as indispensable mathematical instruments for modeling complex dynamical systems characterized by inherent memory retention, hereditary evolution, and anomalous diffusion—behaviors that elude accurate description by conventional integer-order differential models. Traditional numerical algorithms for FDEs are plagued by substantial mesh discretization overhead, prohibitive computational costs, and poor compatibility with various inverse parameter identification tasks. As a groundbreaking computational framework, physics-informed neural networks (PINNs) synergistically combine data-driven fitting capabilities with rigorous physical governing laws, offering transformative solutions to general differential equations. Cutting-edge advancements in fractional PINNs (fPINNs) are systematically synthesized in this comprehensive review. Physical motivations for adopting FDEs are elaborated, the fundamental architecture of vanilla PINNs is delineated, and core obstacles originating from the non-local integral characteristics of fractional derivative operators are dissected to reveal critical restrictions on direct application of standard PINNs to FDEs. All mainstream fPINN approaches are taxonomically grouped into three distinct technical branches, with focused mathematical derivations and in-depth discussions dedicated to fractional-order inverse identification. A homotopy-augmented simultaneous training scheme is formulated to mitigate unstable joint optimization of fractional orders and neural network weights, enabling a smooth transition of optimization targets from simplistic integer-order dynamics toward complex fractional non-local responses. Relative advantages and disadvantages of simultaneous and two-stage sequential identification workflows are thoroughly contrasted, and clear, actionable criteria are delivered to facilitate selection of appropriate identification pipelines for diverse fractional modeling scenarios. Systematic theoretical references and operable technical guidance for fractional modeling and scientific machine learning are provided throughout the review