Existence Of Solutions Research Articles

Granular fuzzy calculus on time scales and its applications to fuzzy dynamic equations

This paper introduces the foundational theory of fuzzy calculus on time scales, utilizing granular arithmetic operations between fuzzy intervals. These operations are developed based on the concept of the horizontal membership function (HMF), which is applied in multidimensional fuzzy arithmetic (MFA). Furthermore, the paper explores the existence of a unique solution and the continuous dependence of the solution to fuzzy dynamic equations on initial data, employing the Banach fixed-point theorem under a new metric for fuzzy functions in time scales involving the generalized exponential function. Finally, to highlight the practical significance of these results and their potential applications, the paper presents mathematical models relevant to nuclear physics and biology.

Multiplicity of solutions for a Kirchhoff equation with non‐linearity concave at the origin

We are concerned with the Kirchhoff problem where is a bounded domain with a regular boundary , and , with for and for . Given appropriate conditions on the function , we successfully applied variational methods to establish the existence of multiple solutions, including both a ground state and a nodal solution, for the specified problem, provided that is sufficiently small and for certain values of .

A study on existence and stability analysis of implicit k,Ψ$$ \left(k,\Psi \right) $$‐Hilfer fractional differential equations and application to RC$$ \mathcal{RC} $$ circuit model

In the present study, we establish the existence and uniqueness of solutions for nonlinear initial value problems and nonlocal boundary value problems associated with implicit fractional differential equations involving ‐Hilfer derivative operator. Furthermore, we explore the stability properties of these solutions in the sense of the Ulam–Hyers, Ulam–Hyers–Rassias, and their generalized stability concepts by proving and applying generalized Gronwall inequality. Lastly, we present the fractional electric circuit model as an example to show the practical applicability of our main results.

Existence and forms of entire solutions to system of non-linear partial differential equations

Existence and forms of entire solutions to system of non-linear partial differential equations

Global Mild Solutions For Hilfer Fractional Neutral Evolution Equation

Abstract In this paper, we present the existence of global mild solutions for the Hilfer fractional neutral evolution equations (HFNEEs), regardless of whether the semigroups are compact or noncompact. We achieve our main results by utilizing the generalization Ascoli-Arzelà theorem, Krasnoselskii’s fixed point theorem, Laplace transform, and measures of noncompactness. To demonstrate the feasibility of our method, we provide an example.

Navier-Stokes: Singularities and Bifurcations

This article presents substantial advances in the analysis of the Navier-Stokes equations for both compressible and incompressible fluids, focusing on the formation of singularities, hypercomplex bifurcations, and regularity in Sobolev and Besov spaces. Through new theorems, we extend the theory of singularities in fluid dynamics and introduce quaternionic bifurcations, representing an innovative extension of classical bifurcation theory. Moreover, we delve into the investigation of the regularity of compressible fluids, exploring the conditions under which solutions remain smooth or develop singularities. These contributions are fundamental to the understanding of global regularity issues, directly linked to the renowned Millennium Prize problem, which seeks definitive answers on the existence and smoothness of solutions to the Navier-Stokes equations. Additionally, we discuss how these theoretical advancements offer new approaches to unresolved problems related to the formation of singularities in turbulent flows and the multiscale behavior of solutions, which are crucial for a comprehensive understanding of fluid dynamics. This work not only broadens the scope of traditional mathematical analysis of the Navier-Stokes equations but also establishes a robust theoretical framework for the investigation of bifurcations and regularity in advanced functional spaces, fostering a deeper understanding of global regularity phenomena and the complex dynamics governing fluid systems.

A Comprehensive View of the Solvability and Stability of a Feedback Control Problem with a State-Dependent Delay Implicit Pantograph Equation

In this paper, we examine the existence of a unique solution of a feedback control problem with an implicit state-dependent pantograph equation. Additionally, the study implements the problem’s Hyers-Ulam stability and the continuous dependence of the unique solution on the initial data and the parameters. Furthermore, we investigate this problem in the absence of feedback control. We also provide some examples to illustrate our results.

On a fractional Cauchy problem with singular initial data

Abstract This article is dedicated to establishing the existence and uniqueness of solutions for the following problem: D α x ( t ) = F ( t , x ( t ) ) x ( 0 ) = x 0 , \left\{\begin{array}{l}{D}^{\alpha }x\left(t)=F\left(t,x\left(t))\hspace{1.0em}\\ x\left(0)={x}_{0},\hspace{1.0em}\end{array}\right. where x 0 {x}_{0} is the singular generalized function and F satisfies L ∞ {L}^{\infty } logarithmic type, D α {D}^{\alpha } is the Caputo derivative of order m − 1 < α < m m-1\lt \alpha \lt m with m ∈ N * m\in {{\mathbb{N}}}^{* } , which we will confirm to be present in Colombeau algebra. The Gronwall lemma is used in Colombeau’s algebra to establish the main results. To illustrate our theoretical analysis, we ended our work with an example.

Study of a non-linear Volterra integro-differential equation with a non-linear unknown source term

In this paper, we propose a new class of integro-differential nonlinear Volterra equations with a non-linear unknown source term. Our study focuses on both theoretical analysis and numerical approximation of solutions to these equations. In the theoretical analysis partition, we define the specific assumptions on the integral kernel's characteristics, on the source term function and also on the derivative of the solutions of equations, under which the Krasnoselskii's fixed point theorem can be applied and in order to ensures the existence and uniqueness of solutions to the proposed integro-differential equations. In the numerical approximation partition, we use a well-known numerical technique for solving integral equations called Nyström method. This method allows us to get an approximate solutions of the proposed equations. Furthermore, we provide some illustrative examples and according to the numerical method described in this paper we approach them, where the comparison results between the exact and approximate solutions of these examples confirm the effectiveness of the numerical Nyström method for approaching the solutions of this class of integro-differential equations.

Analysis of multi-term arbitrary order implicit differential equations with variable type delay

Due to their capacity to simulate intricate dynamic systems containing memory effects and non-local interactions, fractional differential equations have attracted a great deal of attention lately. This study examines multi-term fractional differential equations with variable type delay with the goal of illuminating their complex dynamics and analytical characteristics. The introduction to fractional calculus and the justification for its use in many scientific and technical domains sets the stage for the remainder of the essay. It describes the importance of including variable type delay in differential equations and then applying it to model more sophisticated and realistic behaviours of real-world phenomena. The research study then presents the mathematical formulation of variable type delay and multi-term fractional differential equations. The system’s novelty stems from its unique combination of variable delay, generalized multi terms fractional differential operators (n and m), and integral implicit parameters, and studying the stability of the the newly formulated system as compared to the work in the existing literature. While the variable type delay is introduced as a function of time to describe instances where the delay is not constant, the fractional order derivatives are generated using the Caputo approach. The existence, uniqueness, and stability of solutions are the main topics of the theoretical analysis of the suggested differential equations. In order to establish important mathematical features, the inquiry makes use of spectral techniques, and fixed-point theorems. The study finishes by summarizing the major discoveries and outlining potential future research avenues in this developing field. It highlights the potential contribution of multi-term fractional differential equations with variable type delay to improving the control and design of complex systems. Overall, this study adds to the growing body of knowledge in the field of fractional calculus and provides insightful information about the investigation of multi-term fractional differential equations with variable type delay, making it pertinent for academics and practitioners from a variety of fields.

Formation of Alfvénic supernonlinear waves in a plasma containing double spectral distributed electrons

The characteristics of nonlinear and supernonlinear Alfvén waves propagating in a multicomponent plasma composed of a double spectral electron distribution and positive and negative ions were investigated. The Sagdeev technique was employed, and an energy equation was derived. Our findings show that the proposed system reveals the existence of a double-layer solution, periodic, supersoliton, and superperiodic waves. The phase portrait and potential analysis related to these waves were investigated to study the main features of existing waves. It was also found that decreasing the electron temperature helps the superperiodic structure to be excited in our plasma model. Our results help interpret the nonlinear and supernonlinear features of the recorded Alfvén waves propagating in the ionosphere D-region.

A stochastic hepatitis B model with media coverage and Lévy noise

Globally, it is evident that vaccination and social media significantly influence the spread of diseases. Research has demonstrated that the spread of diseases can be affected by unforeseen events. Therefore, a stochastic hepatitis B virus (HBV) dynamical model is constructed that incorporates media coverage, while also Lévy noise is included. Itô's formula and the Lyapunov function approach are employed to analyze the existence of a positive global solution. Furthermore, the necessary conditions for disease eradication as well as long-term survival are also established to facilitate the general understanding. The statistics indicate that the media's impact is not immediate and does not lead to disease eradication. To effectively manage the spread of diseases, it is essential to employ a combination of significant noise disruption and a low rate of decline in vaccine effectiveness. Numerical simulations are conducted to demonstrate the analytical findings.

Global solutions and their limit behavior for parabolic inclusions with an unbounded right-hand part

We study global resolvability for parabolic inclusions with an upper semicontinuous multi-valued right-hand part of more than linear growth. Theorems about the existence of global mild solutions in different phase spaces are proved. Limit sets for the obtained global solutions in the corresponding phase spaces are investigated.

Weak and strong solutions for a fluid‐poroelastic‐structure interaction via a semigroup approach

A filtration system comprising a Biot poroelastic solid coupled to an incompressible Stokes free‐flow is considered in 3D. Across the flat 2D interface, the Beavers‐Joseph‐Saffman coupling conditions are taken. In the inertial, linear, and non‐degenerate case, the hyperbolic‐parabolic coupled problem is posed through a dynamics operator on a chosen energy space, adapted from Stokes‐Lamé coupled dynamics. A semigroup approach is utilized to circumvent issues associated to mismatched trace regularities at the interface. The generation of a strongly continuous semigroup for the dynamics operator is obtained via a non‐standard maximality argument. The latter employs a mixed‐variational formulation in order to invoke the Babuška‐Brezzi theorem. The Lumer‐Philips theorem then yields semigroup generation, and thereby, strong and generalized solutions are obtained. For the linear dynamics, density obtains the existence of weak solutions; we extend to the case where the Biot compressibility of constituents degenerates. Thus, for the inertial linear Biot‐Stokes filtration system, we provide a clear elucidation of strong solutions and a construction of weak solutions, as well as their regularity through associated estimates.

Periodically modulated solitary waves of the CH–KP-I equation

We consider the CH–KP-I equation. For this equation, we prove the existence of steady solutions, which are solitary in one horizontal direction and periodic in the other. We show that such waves bifurcate from the line solitary wave solutions, i.e. solitary wave solutions to the Camassa–Holm equation, in a dimension-breaking bifurcation. This is achieved through reformulating the problem as a dynamical system for a perturbation of the line solitary wave solutions, where the periodic direction takes the role of time, then applying the Lyapunov–Iooss theorem.

Anisotropic error estimator for the Stokes–Biot system

This paper presents an a posteriori error analysis for the problem defining the interaction between a free fluid and poroelastic structure approximated by finite element methods on anisotropic meshes in Rd, d=2 or 3. Korn’s inequality for piecewise linear vector fields on anisotropic meshes is established and is applied to nonconforming finite element method. Then the existence and uniqueness of the approximation solution are deduced for conforming and nonconforming cases. With the obtained finite element solutions, local error indicators and a global estimator are generated, demonstrating reliability and efficiency. The efficiency is proved by means of bubble functions and the corresponding anisotropic inverse inequalities. In order to prove the reliability, it is vital that an anisotropic mesh corresponds to the anisotropic function under consideration. To measure this correspondence, a so-called matching function is defined, and its discussion shows it to be useful tool. With its help, the upper error bound is shown by means of the corresponding anisotropic interpolation estimates and a special Helmholtz decomposition in both media.

Mild Solutions of the (ρ1,ρ2,k1,k2,φ)-Proportional Hilfer–Cauchy Problem

Inspired by prior research on fractional calculus, we introduce new fractional integral and derivative operators: the (ρ1,ρ2,k1,k2,φ)-proportional integral and the (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional derivative. Numerous previous studied fractional integrals and derivatives can be considered as particular instances of the novel operators introduced above. Some properties of the (ρ1,ρ2,k1,k2,φ)-proportional integral are discussed, including mapping properties, the generalized Laplace transform of the (ρ1,ρ2,k1,k2,φ)-proportional integral and (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional derivative. The results obtained suggest that the most comprehensive formulation of this fractional calculus has been achieved. Under the guidance of the findings from earlier sections, we investigate the existence of mild solutions for the (ρ1,ρ2,k1,k2,φ)-proportional Hilfer fractional Cauchy problem. An illustrative example is provided to demonstrate the main results.

Existence and uniqueness for a mixed fractional differential system with slit-strips conditions

This paper studies the existence and uniqueness of solutions for a new kind of mixed fractional differential systems with slit-strips conditions, containing Caputo-type fractional derivatives. The first result on the existence and uniqueness is based on Banach’s fixed point theorem, and the second result on the existence and uniqueness of the solution is proved by using a Schaefer-type fixed point theorem. The applicability of our primary results is finally illustrated by some examples.

On Small-Amplitude Asymmetric Water Waves

AbstractWe generalize the method used by Mæhlen and Seth (Asymmetric traveling wave solutions of the capillary–gravity Whitham equation, 2023, arxiv:2312.15343) used to prove the existence of small-amplitude asymmetric solutions to the capillary–gravity Whitham equation, so that it can be applied directly to a class of similar equations. The purpose is to prove or disprove the existence of asymmetric waves for the water wave problem or other model equations for water waves. Our main result in this paper is a theorem that gives both necessary and sufficient conditions for the existence of small-amplitude periodic asymmetric solutions for this class of equations. The result is then applied to an infinite depth capillary–gravity Whitham equation and an infinite depth capillary–gravity Babenko equation to show a nonexistence result of small-amplitude waves for these equations. This example also highlights the similarities between these equations suggesting the potential existence of small-amplitude asymmetric waves for the finite depth capillary–gravity Babenko equation.

Sign-changing solutions for a class of fractional Choquard equation with the Sobolev critical exponent in [formula omitted]

In this paper, we study the following fractional Choquard equation with critical exponent(−Δ)σu+u=(Iα⁎|u|p)|u|p−2u+|u|2σ⁎−2uinR3, where σ∈(34,1), α∈(2σ,3) and α+2σ3−2σ<p<2α,σ⁎:=3+α3−2σ, 2α,σ⁎ is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, 2σ⁎:=63−2σ is the fractional Sobolev critical exponent and the operator (−Δ)σ stands for the fractional Laplacian of order σ. Based on the above assumptions, we establish the existence of positive ground state solutions, and if p∈(α+2σ3−2σ,2α3−2σ), we also have the corresponding regular property. Subsequently, by introducing some other additional hypotheses on σ, α, p and with the help of quantitative deformation lemma, we employ constrained minimization arguments on the sign-changing Nehari manifold to obtain the existence of ground state sign-changing solutions.