ABSTRACT This paper is devoted to the global well‐posedness for small initial data and the large‐time behavior of solutions to the ‐dimensional () incompressible Boussinesq equations with fractional dissipation. We first establish the asymptotic stability of the system by proving that the ‐norm of the solutions decays to zero over time. Subsequently, we prove the local existence of solutions via a mollification approach and the Picard theorem, and then establish a series of a priori estimates that allow us to extend these solutions globally in time using a continuity argument. Furthermore, for initial data lying in negative Sobolev spaces, we demonstrate the global well‐posedness and propagation of regularity in these spaces. A key contribution of this work is the detailed analysis of the large‐time behavior, where we derive both upper and lower bounds for the decay rates of the solutions and their higher‐order derivatives. The fact that these bounds coincide establishes the sharpness (optimality) of the decay rates. To the best of our knowledge, this work provides the first comprehensive study on the stability and large‐time dynamics of the multi‐dimensional Boussinesq equations with general fractional dissipation. By introducing novel techniques in Fourier analysis and energy methods, we not only extend several previous results to the ‐dimensional case but also improve upon others, particularly by relaxing the restrictions on the fractional exponents and the initial data.

Abstract In this paper, we describe a computational technique to model the novel coronavirus (COVID‐19) outbreak using Caputo–Fabrizio (CF) fractional derivatives. The proposed model is formulated as an SEIHR (Susceptible–Exposed–Infected–Hospitalized–Recovered) system of nonlinear fractional differential equations. For numerical analysis, we develop a specialized scheme based on the Adams–Bashforth method (ABM) to solve the system. Our work examines how different fractional orders affect the dynamics of the COVID‐19 outbreak, demonstrating memory and disease transmission. We compare the course of the pandemic in India and Pakistan. We derive the numerical scheme through ABM for simulations to show the impact of on the COVID‐19 pandemic in both countries. This effect is more obvious in Pakistan, where the virus propagated more slowly than in India fractional order modeling captures the complex dynamics of infectious disease outbreaks and highlights the effectiveness of timely intervention measures to control the spread of COVID‐19. In addition, we study the existence and uniqueness of the model through the Picard theorem. In the last part, we present the graphical presentation of the model.

ABSTRACTIn this work, we study the time‐independent homogeneous Schrödinger equation in a bounded domain . We assume the potential is a continuously differentiable function that depends on the angular component of . A transmutation operator sending harmonic functions into solutions of the Schrödinger equation is constructed by determining its kernel along with the Goursat problem it satisfies. The existence of said kernel is proven in the sense of distributions by applying Picard's theorem. Lastly, we comment certain properties of this operator and construct a complete family of solutions for this Schrödinger equation.

ABSTRACTIn this paper, we develop some fundamental tools for fractal derivatives, such as a Leibniz‐type product rule and a chain rule. Also, a local Picard theorem for an initial value problem via fractal derivatives is obtained. Finally, the Gompertz and logistic models associated to those differential operators are considered. We also show how effective these models are compared to other well‐known (classical and conformable) models when applied to an example of real data.

In the context of normed space, Banach's fixed point theorem for mapping is studied in this paper. This idea is generalized in Banach's classical fixed-point theory. Fixed point theory explains many situations where maps provide great answers through an amazing combination of mathematical analysis. Picard- Lendell's theorem, Picard's theorem, implicit function theorem, and other results are created by other mathematicians later using this fixed-point theorem. We have come up with ideas that Banach's theorem can be used to easily deduce many well-known fixed-point theorems. Extending the Banach contraction principle to include metric space with modular spaces has been included in some recent research, the aim of study proves some properties of Banach space.

Value distribution of a pair of meromorphic functions

Let μ be a self-similar measure with compact support K. The Hausdorff dimension of K is α. The Cauchy transform of μ is denoted by F(z). For 0<β<1, we define the function F[β], which compares with the fractional derivative of F of order β. Let Φ(z)=F(1/z),|z|<1. In this paper, we prove that Φ[β] belongs to Ap for 0<p<1/(β+1), and (Φ′)[β] belongs to Ap for 1≤p<1/β≤1/(2−α), where Ap is the Bergman space. At the same time, we give a value distribution property of F, which is similar to the big Picard theorem.

Picard theorems for moduli spaces of polarized varieties

Fractional-order modelling and analysis of diabetes mellitus: Utilizing the Atangana-Baleanu Caputo (ABC) operator

Big Picard theorem for jet differentials and non-archimedean Ax-Lindemann theorem

In this paper, we study various hyperbolicity properties for a quasi-compact K\"ahler manifold $U$ which admits a complex polarized variation of Hodge structures so that each fiber of the period map is zero-dimensional. In the first part, we prove that $U$ is algebraically hyperbolic and that the generalized big Picard theorem holds for $U$. In the second part, we prove that there is a finite \'etale cover $\tilde{U}$ of $U$ from a quasi-projective manifold $\tilde{U}$ such that any projective compactification $X$ of $\tilde{U}$ is Picard hyperbolic modulo the boundary $X-\tilde{U}$, and any irreducible subvariety of $X$ not contained in $X-\tilde{U}$ is of general type. This result coarsely incorporates previous works by Nadel, Rousseau, Brunebarbe and Cadorel on the hyperbolicity of compactifications of quotients of bounded symmetric domains by torsion-free lattices.Comment: 31 pages. Final version, to appear in \'Epijournal de G\'eom\'etrie Alg\'ebrique

The concept of a Caputo fractal-fractional derivative is a new class of non-integer order derivative with a power-law kernel that has many applications in real-life scenarios. This new derivative is applied newly to model the dynamics of diabetes mellitus disease because the operator can be applied to formulate some models which describe the dynamics with memory effects. Diabetes mellitus as one of the leading diseases of our century is a type of disease that is widely observed worldwide and takes the first place in the evolution of many fatal diseases. Diabetes is tagged as a chronic, metabolic disease signalized by elevated levels of blood glucose (or blood sugar), which results over time in serious damage to the heart, blood vessels, eyes, kidneys, and nerves in the body. The present study is devoted to mathematical modeling and analysis of the diabetes mellitus model without genetic factors in the sense of fractional-fractal derivative. At first, the critical points of the diabetes mellitus model are investigated; then Picard's theorem idea is applied to investigate the existence and uniqueness of the solutions of the model under the fractional-fractal operator. The resulting discretized system of fractal-fractional differential equations is integrated in time with the MATLAB inbuilt Ode45 and Ode15s packages. A step-by-step and easy-to-adapt MATLAB algorithm is also provided for scholars to reproduce. Simulation experiments that revealed the dynamic behavior of the model for different instances of fractal-fractional parameters in the sense of the Caputo operator are displayed in the table and figures. It was observed in the numerical experiments that a decrease in both fractal dimensions and leads to an increase in the number of people living with diabetes mellitus.

Novel projection neurodynamic approaches for constrained convex optimization

<abstract><p>In this work, we present existence results for some problems which arise in image processing namely image restoration. Our essential tools are Picard's fixed point theorem for a strict contraction and Mountain-pass Theorem for critical point.</p></abstract>

This paper establishes a version of Nevanlinna theory based on Jackson difference operator $D_{q}f(z)=\frac{f(qz)-f(z)}{qz-z}$ for meromorphic functions of zero order in the complex plane $\mathbb{C}$. We give the logarithmic difference lemma, the second fundamental theorem, the defect relation, Picard theorem and five-value theorem in sense of Jackson $q$-difference operator. By using this theory, we investigate the growth of entire solutions of linear Jackson $q$-difference equations $D^{k}_{q}f(z)+A(z)f(z)=0$ with meromorphic coefficient $A,$ where $D^k_q$ is Jackson $k$-th order difference operator, and estimate the logarithmic order of some $q$-special functions.

In this paper, we consider fractional differential equations with the new fractional derivative involving a nonsingular kernel, namely, the Caputo-Fabrizio fractional derivative. Using a successive approximation method, we prove an extension of the Picard-Lindelöf existence and uniqueness theorem for fractional differential equations with this derivative, which gives a set of conditions, under which a fractional initial value problem has a unique solution.

Smoothing inertial neurodynamic approach for sparse signal reconstruction via [formula omitted]-norm minimization

Generalizations of Picard's theorem with moving hypersurfaces

This paper aims at treating a study of Banach fixed point theorem for mapping results that introduced in the setting of normed space. The classical Banach fixed point theorem is a generalization of this work. A fixed point theory is a beautiful mixture of Mathematical analysis to explain some conditions in which maps give excellent solutions. Here later many mathematicians used this fixed point theory to establish their results, see for instance, Picard-Lindel of Theorem, The Picard theorem, Implicit function theorem etc. Also, we developed ideas that many of known fixed point theorems can easily be derived from the Banach theorem. It extends some recent works on the extension of Banach contraction principle to metric space with norm spaces.

In this paper, we will prove a big Picard’s theorem for holomorphic curves from a punctured disc into $${\mathbb {P}}^n({\mathbb {C}})$$ with $$q \ (q>(N-n+1)(n+1))$$ hypersurfaces which are located in N-subgeneral position.