On the existence of Q–Monopole–Ball soliton with unit magnetic charge

This manuscript investigates the controllability of systems involving the Prabhakar fractional derivative under prescribed control. First, the controllability of the linear system is studied using the Grammian matrix technique. For the nonlinear system, the existence and uniqueness of solutions are established through the application of the Schauder fixed-point theorem, and its controllability is subsequently analysed. To illustrate the practical applicability and effectiveness of the theoretical results, numerical examples are provided and discussed in detail.

This paper addresses the existence of mild solutions and the approximate controllability of a class of higher-order Hilfer fractional semi-linear neutral stochastic differential equations with non-instantaneous impulses in Hilbert spaces. The system is driven by both fractional Brownian motion and Poisson jumps, thereby capturing long-range dependence as well as random discontinuities. By combining techniques from fractional calculus, stochastic analysis, and operator theory, we establish sufficient conditions for the existence of mild solutions. The analysis is carried out through the construction of suitable solution operator families and the application of Sadovskii’s fixed point theorem in an appropriate phase space framework. In addition, we investigate the controllability properties of the system and derive criteria ensuring approximate controllability of the underlying fractional neutral dynamics. The proposed approach relies on the structural properties of the higher-order Hilfer fractional derivative, estimates for stochastic integrals with respect to fractional Brownian motion, and compactness arguments adapted to non-instantaneous impulsive effects. The inclusion of Poisson jumps and neutral terms introduces significant analytical difficulties, which are overcome using refined resolvent operator techniques and fractional power estimates. An illustrative example is presented to demonstrate the applicability of the theoretical results. The results obtained generalize and unify several recent developments in the theory of fractional stochastic systems and provide a flexible framework for analyzing controlled dynamical models with memory, randomness, and impulsive behavior.

Predicting clinically significant drug-drug interactions (DDIs) continues to be an unresolved challenge in contemporary pharmacovigilance, primarily due to the inadequacy of current computational frameworks in addressing the nonlinear, multi-scale characteristics of simultaneous drug metabolism. This paper presents the Quantum Graph-Differential (QGD) model an exact mathematical framework that combines quantum-inspired graph theory with a set of interconnected fractional differential equations to describe and forecast pairwise drug-drug interactions (DDIs). The principal component of our construction is the quantum interaction graph [Formula: see text], wherein the vertex set represents distinct drug molecules as quantum states within a finite-dimensional Hilbert space, and the complex-valued edge weights are obtained from the overlap of shared metabolic pathways and transporter affinity profiles.A Schrödinger-type equation on [Formula: see text] governs drug-drug coupling, and the graph Hamiltonian [Formula: see text] is constructed from a novel fractional quantum graph Laplacian [Formula: see text], [Formula: see text]. A hybrid quantum-classical dynamical model is created by coupling the time evolution of the interaction wavefunction [Formula: see text] to a compartmental pharmacokinetic/pharmacodynamic (PK/PD) ordinary differential equation system. Using Banach fixed-point and semigroup theory, we prove existence, uniqueness, and long-time asymptotic stability of solutions. Using the QGD framework on a selected dataset of 7,428 clinically confirmed DDI pairs from DrugBank v5.1, TWOSIDES, and FAERS, our model outperforms five established baselines by 1.5-13.9 percentage points in AUC, with an average precision of 0.948 and an AUC of 0.962. Quantum edge weighting alone explains a 3.7% relative F1 gain over unweighted graph methods, according to ablation experiments. These results show that quantifiable, interpretable improvements in DDI prediction can be obtained by incorporating quantum mechanical concepts into graph-differential frameworks.

The problem of relative exact controllability for a class of Riemann–Liouville fractional stochastic delay differential systems is considered. First of all, necessary and sufficient conditions for the relative controllability of linear systems are proved. Next, under the premise of several reasonable assumptions, we are able to prove the relative exact controllability of the corresponding linear stochastic system. Subsequently, we innovatively combine Rothe's fixed point theorem with Itô's isometry, taking them as core analytical tools to overcome the research difficulties in the relative exact controllability of nonlinear stochastic systems. Finally, the effectiveness of the proposed method is clearly demonstrated through a specific example.

Abstract In this paper, we investigate the Ulam-Hyers-Rassias stability of non-local problems for differential equations with Caputo fractional derivatives involving deviating arguments on unbounded intervals. By applying the Banach fixed-point theorem, we establish conditions for the existence and uniqueness of mild solutions within the space of continuous functions endowed with the Bielecki norm. To illustrate the validity of theoretical results and the applicability of Ulam-Hyers-Rassias stability in estimation, concrete examples are provided.

The main concern of this work is to study the sufficient conditions for existence, uniqueness, and approximate controllability results for the nonlinear Caputo conformable fractional neutral-type delayed integro-differential system with nonlocal conditions in a Hilbert space. Since, the conformable derivative retains several fundamental properties of classical calculus including mean value theorem, Rolle's theorem, product, quotient, and linearity rules. This distinguishes it from traditional fractional derivatives such as Riemann-Liouville, Caputo, and Hilfer. Therefore, the conformable derivative is simpler and faster but ignores history while the Caputo conformable fractional derivative offers a balance capturing some memory with easier calculations. Firstly, the proposed system is reformulated into an equivalent fixed point problem implementing the Riemann-Liouville conformable fractional integral operator. The Schauder fixed point theorem is used to derive the existence of mild solution. The Banach contraction principle is then applied to show the uniqueness of mild solution. The main tools applied to derive the results are theory of fractional calculus, semigroup of bounded linear operators, and fixed point theorems. Further, the approximate controllability result for the proposed system is established under the consideration that the corresponding linear system is approximate controllable. An illustrative example is presented to demonstrate the applicability of theoretical results.

This paper studies a nonlinear class of fractional difference problems depending on parameter-dependent summation constraints. An explicit formula for the Green's function and its positivity and bounds for a specific interval of parameters are found. By use of the Guo–Krasnoselskii fixed point theorem with an appropriate cone, we obtain the existence criteria for multiple distinct positive solutions. This is achieved through a ‘trapping mechanism’ based on a series of alternating growth conditions of the nonlinearity. In addition, the use of the obtained results is justified through an illustrative example.

Abstract This paper investigates the existence of traveling wave solutions for diffusive two-species Lotka–Volterra systems with delays in both the reaction and diffusion terms under partial monotonicity assumptions. The model incorporates small-memory effects in the homogeneous diffusion term, representing a modification of the random-walk interpretation underlying Fick’s law. We extend the partial (cross) monotone iteration method to systems satisfying a partial quasi-monotone condition through the construction of appropriate upper and lower solutions. Convergence of the iteration is established using Schauder’s fixed point theorem.

Abstract This study investigates the existence and uniqueness of solutions for a coupled system of Langevin-type fractional differential equations featuring generalized Ψ-Caputo derivatives in arbitrary Banach spaces. While prior research has predominantly focused on finite-dimensional or specific function spaces, this work extends the framework to infinite-dimensional settings, offering a more versatile analytical approach. Uniqueness is established using Banach’s fixed-point theorem under Lipschitz-type conditions, while existence is proven via Monch’s fixed-point principle combined with the measure of noncompactness—a powerful tool for infinite-dimensional problems. Demonstrative examples, including cases in the space of null sequences, validate the theoretical framework. The findings enhance the study of fractional coupled systems, introducing a flexible and comprehensive approach that integrates diverse fractional operators and strengthens foundational results.

This note revisits Theorem 3.2 of Crettez et al. (2025), which establishes the existence of a pure-strategy Nash equilibrium in convex and compact games with possibly discontinuous payoffs under an F–π–ϖ weak robust better-reply correspondence property. Instead of the original fixed-point construction, the proof here proceeds via a single Caristi–Khamsi fixed point theorem for set-valued maps, under a mild additional lower semicontinuity assumption on the associated deviation–profit potential. The key step is to associate with each strategy profile a deviation–profit potential that measures the maximal unilateral gain from weakly robust deviations and to show that this potential decreases along the F–π–ϖ weak robust betterreply correspondence. The resulting Caristi inequality yields a fixed point of the induced better-reply map, which is a Nash equilibrium. The argument provides a structural reinterpretation of the result of Crettez et al. (2025), highlighting the potential-theoretic nature of the robustness condition and separating the economic structure of profitable deviations from the topological fixed-point machinery.

The goal of this paper is to establish Rothe type fixed point theorems and related general Leray-Schaduer principle for non-self compact upper semicontinuous (USC) Set-valued mappings in Hausdorff topological vector spaces (TVSs). In particular, we first establish fixed point theorems for set-valued self-mappings with local intersection property (LIP), and compact upper semicontinuous set-valued self mappings in Hausdorff -vector spaces or TVSs, then general Rothe type fixed point theorem and the general framework of Leray-Schauder principle for compact USC set-valued mappings in TVSs, and existence of maximal elements for preference mappings are developed. These new results developed in this paper do not only provide a unified positive answer to Schauder conjecture completely in TVSs, but also unify corresponding results of fixed point theory and applications in the existing literature, where p ∈ ( 0 , 1 ] . In particular, for the selfcontaining, the most fundamental results required by this paper are also provided by the Appendix enclosed to show the important role of the so-called Yuan's Sharpness Operator used.

Analytical study of Caputo–Fabrizio fractional order delayed neural network using fixed point theory

ABSTRACT We analyze a quasi‐incompressible Cahn–Hilliard–Navier–Stokes system (qCHNS) for two‐phase flows with unmatched densities. The order parameter is the volume fraction difference of the two fluids, while mass‐averaged velocity is adopted. This leads to a quasi‐incompressible model where the pressure also enters the equation of the chemical potential. We establish local existence and uniqueness of strong solutions by the Banach fixed point theorem and the maximal regularity theory.

Analysis of global behavior in a diffusive tuberculosis epidemic model structured by ages of latency and infection

This paper investigates the existence of positive solutions for a specific category of p-Laplacian tempered fractional differential equations in which the nonlinear term f contains an integral operator θ. By employing fixed point theorems for sum operators in partially ordered Banach spaces, together with Krasnosel’skii fixed point theorem, the existence of positive solutions is established. Moreover, iterative sequences are constructed to approximate the unique positive solution of the problem. Finally, three examples are presented to illustrate the main results.

We prove that a self-mapping T on a complete metric space (X,D), satisfying a generalized contractive condition involving its ρ-th iterate Tρ (ρ∈N,ρ≥2) and a uniform coefficient bound L<1, possesses a unique fixed point. The proof establishes that the Picard iteration sequence is Cauchy with the further assumption of continuity of T, and fixed point uniqueness follows directly from the contractive inequality.

In this paper, we study a class of second-order fractional boundary value problems involving Θ-Caputo derivatives of different orders. By reformulating the problem to an integral equation, we introduce an appropriate notion of a mild solution in the Θ-fractional framework. Existence results are obtained via Krasnoselskii’s fixed point theorem, while uniqueness is established using the Banach contraction principle under suitable Lipschitz-type conditions. The obtained results extend several earlier works on Caputo, Hadamard–Caputo, and Riemann–Liouville fractional derivatives. Two examples are presented to illustrate the applicability of the theoretical results.

This paper examines an m-cyclic coupled system of sequential (k,ψ)-Hilfer and (k,ψ)-Caputo fractional differential equations with boundary conditions. The nonlinearities follow a cyclic pattern: for j=1,…,m−1, fj depends on xj and xj+1 and fm depends on xm and x1, forming a closed loop of interactions. We convert the system into an equivalent integral equation and establish existence and uniqueness results using four fixed-point theorems: the Banach contraction principle, Schaefer’s theorem, Krasnosel’skiĭ’s theorem, and the Leray–Schauder alternative. A thorough Ulam–Hyers stability analysis is presented with explicit stability constants. Numerical examples illustrate the applicability of the theoretical findings.

This paper presents an innovative synthesis of generalised fixed-point theory, advanced topological degree methodologies, and high-order computational frameworks for memory-driven dynamical systems. We expand contraction principles in complete metric and ANR spaces, enhancing existence results through the Leray-Schauder degree in the context of non-compact perturbations. We develop a high-order numerical framework for fractal-fractional differential equations employing the Taylor Operational Matrix Method (TOMM) and an adaptive Adams-Bashforth-Moulton (ABM) scheme. This framework leverages global basis approximations to achieve high precision and has strong stability guarantees thanks to the Ulam-Hyers-Rassias criteria and Lyapunov-Razumikhin functionals. This framework has strong stability guarantees thanks to the Ulam-Hyers-Rassias criteria and Lyapunov-Razumikhin functionals. We incorporate sociomathematical structures, such as Kutumba-inspired familial support, into epidemic models, illustrating how topological and fractional tools collectively encapsulate memory, heterogeneity, and resilience. Applications encompass fractional epidemiology, biomedical hysteresis, and cyber-virus dynamics, demonstrating interdisciplinary effectiveness. The work connects abstract analysis with real-world complexity, giving a single method for nonlocal, socially embedded systems.