Existence Of Solutions Research Articles

A study on the existence of unique stable approximate solutions: fractal-fractional-based model of an energy supply–demand system

We investigate a fractional energy supply–demand system (ES–DS) model using power-law-type kernels and advanced operators called fractal-fractional operators with a couple of fractal and fractional orders. It is established that for the fractal-fractional model of the ES–DS, a solution exists and it is unique. One of the principal innovations is to conduct the stability of the fractal-fractional-based ES–DS model. Moreover, after some theorems on the existence theory, we upgrade the Adams–Bashforth (AB) method in the context of the fractal-fractional-based operators and simulate the graphs for varied data on the fractal and fractional orders when we explored the stability requirements in four variants to approach the required numerical solutions. Next, by removing the fractal order, we transform our fractal-fractional-based ES–DS model to a fractional Caputo-type model, and then, a numerical methodology known as the Taylor operational matrix method is applied to simulate new graphs and compare them to previous fractal-fractional-based model.

Qualitative analysis for k-fractional integral boundary value problems of quasilinear equation with p-Laplacian operator

In this paper, we are devoted to considering the existence of solutions for k-fractional integral boundary value problems of Hadamard fractional order quasilinear equation with p-Laplacian operator. Based on the coincidence degree theory, some new results are presented. Moreover, an example is shown to verify our main conclusions.

Modulational instability and discrete quantum droplets in a deep quasi-one-dimensional optical lattice

We study the properties of modulational instability and discrete breathers arising in a quasi-one-dimensional discrete Gross-Pitaevskii equation with Lee-Huang-Yang corrections. Conditions for modulation instability and instability regions of nonlinear plane waves are determined in parameter space. We analytically investigate the existence of different quantum droplet solutions, including intersite, onsite, front-like, flat-top, and dark localized modes, using the Page method and variational approach. Their stability is checked using linear stability analyses and numerical simulations. The analytical predictions corroborated with the numerical simulations. Published by the American Physical Society 2025

Monotonicity of Limit Wave Speed in the pgKdV Equation with Nonlinear Terms of Arbitrarily High Degrees

In this paper, the existence and number of periodic wave and solitary wave solutions of a pgKdV equation are established by applying the geometric singular perturbation theory and studying the limit cycle bifurcations of a perturbed hyperelliptic Hamiltonian system of arbitrarily high-order terms. For a general integer [Formula: see text], we provide a simple and rigorous proof for the monotonicity of the limit wave speed of traveling wave solutions. For the equivalent form of this equation, our results provide the complete answer to the open question posted by [Yan et al., 2014] and the conjecture proposed by [Ouyang et al., 2014].

Boundedness in a one-dimensional quasilinear attraction–repulsion chemotaxis system with flux limitation

This paper is concerned with a quasilinear attraction–repulsion chemotaxis system with flux limitation in the one-dimensional setting. The main results assert global existence and boundedness of classical solutions to the system under smallness conditions on initial data and certain assumptions on parameters.

Sufficient Conditions for Optimal Stability in Hilfer–Hadamard Fractional Differential Equations

The primary objective of this study is to explore sufficient conditions for the existence, uniqueness, and optimal stability of positive solutions to a finite system of Hilfer–Hadamard fractional differential equations with two-point boundary conditions. Our analysis centers around transforming fractional differential equations into fractional integral equations under minimal requirements. This investigation employs several well-known special control functions, including the Mittag–Leffler function, the Wright function, and the hypergeometric function. The results are obtained by constructing upper and lower control functions for nonlinear expressions without any monotonous conditions, utilizing fixed point theorems, such as Banach and Schauder, and applying techniques from nonlinear functional analysis. To demonstrate the practical implications of the theoretical findings, a pertinent example is provided, which validates the results obtained.

An existence results of a product type fractional functional integral equations using Petryshyn's fixed point theorem

In this paper, we investigate the existence of solutions for a new class of nonlinear product-type fractional functional integral equations (FFIEs) involving the Riemann–Liouville fractional integral operator. To establish the existence of at least one solution, we employ Petryshyn's fixed-point theorem (PFPT) combined with the concept of the measure of noncompactness (MNC) in the Banach space C [ 0 , a ] of continuous functions. Unlike other approaches based on Darbo's or Schauder's fixed-point theorems in Banach algebras, our method does not require the operator to map a closed convex subset onto itself, nor does it rely on the commonly assumed “sublinear condition” for the functional involved in the equation. Therefore, our results generalize and unify several existing results in the literature under fewer conditions. Additionally, to support our theoretical findings, we provide an example of such nonlinear FFIEs, thereby illustrating the applicability of the proposed results.

Boundary Exponential Stabilization for the Linear KP-II Equation without Critical Size Restrictions

In this paper, we delve into the intricacies of boundary stabilization for the linearized KP-II equation within the constraints of a bounded domain, a phenomenon known as “critical length.” Our primary aim is to design a feedback law that ensures the existence and exponential stabilization of solutions in the energy space, without length restrictions on the domain Ω=(0,L)×(0,L), L>0 0 $$\\end{document}]]>. Furthermore, we examine the interaction between the drift term ux under these constraints.

Existence and multiplicity of solutions for a class of quasilinear anisotropic elliptic systems via fibering method

Existence and multiplicity of solutions for a class of quasilinear anisotropic elliptic systems via fibering method

A novel approach to Caputo–Hadamard pantograph problems in Lp-spaces

In this paper, we investigate the existence and uniqueness of solutions for Caputo–Hadamard pantograph fractional differential equations with boundary conditions in Lp-spaces, by applying fixed-point theorems in Lp-spaces to prove existence and uniqueness. Stability is analyzed through the Ulam–Hyers stability and Ulam–Hyers–Rassias stability, offering key insights into the reliability of the system. Practical examples are also provided for illustration.

A computer-assisted proof for solutions of Elkouh’s equation related to Navier–Stokes equations

The present paper presents a computer-assisted proof of the existence of solutions for Elkouh’s equation, which is representative of fluid flow between parallel circular disks. The proposed approach is based on an infinite-dimensional fixed-point theorem with interval arithmetic. Various existing proofs with mathematically rigorous error bounds show the validity of the presented numerical verification method.

Emergent Dynamics of an Orientation Flocking Model With Distance‐Dependent Delay

ABSTRACTIn this paper, we investigate the dynamical behaviors of an orientation flocking model that incorporates state‐dependent delay. The research unfolds in three principal phases: First, we establish the global existence and uniqueness of solutions for our system. Subsequently, we develop a comprehensive theoretical framework that provides sufficient conditions for the emergence of orientation flocking phenomena. In particular, this framework remains valid regardless of the number of particles . In the final phase of our analysis, we perform a systematic derivation of the mean‐field limit for the discrete system, accompanied by a proof of global well‐posedness for the resulting mean‐field equations. Furthermore, we demonstrate that our theoretical results and analytical framework naturally extend to and encompass the case of orientation flocking models with constant time delays.

Semiclassical solutions for critical Schrödinger–Poisson systems involving multiple competing potentials

AbstractIn this paper, a class of Schrödinger–Poisson system involving multiple competing potentials and critical Sobolev exponent is considered. Such a problem cannot be studied using the same argument for the nonlinear term with only a positive potential, because the weight potentials set contains nonpositive, sign changing, and nonnegative elements. By introducing the ground energy function and subtle analysis, we first prove the existence of ground state solution in the semiclassical limit via the Nehari manifold and concentration–compactness principle. Then we show that converges to the ground state solution of the associated limiting problem and concentrates at a concrete set characterized by the potentials. At the same time, some properties for the ground state solution are also studied. Moreover, a sufficient condition for the nonexistence of the ground state solution is obtained.

Well‐Posedness of the Fluid–Structure Interactions Problem for Magnetofluid in the Flat Domain

ABSTRACTA 2D nonlinear fluid–structure interaction (FSI) problems in which the elastic body fully immersed in the magnetofluid is considered. The coupling takes place on the interface via the continuity of the normal components of the Cauchy stress tensor and the displacement (or velocity). First, a divergence‐free weak solution that is an auxiliary, equivalent formulation to the solution of FSI problem is introduced. Then, based on Galerkin approximations and energy estimates, the global‐in‐time existence of the weak solution is obtained. Further, the regularity on time and space derivatives of weak solution is established by using the estimates on fractional tangential derivative. Finally, the uniqueness is proved under additional assumptions on the data.

Existence and multiplicity of solutions for subquadratic fractional Hamiltonian systems

Existence and multiplicity of solutions for subquadratic fractional Hamiltonian systems

Dynamics of convex mean curvature flow

Abstract There is an extensive and growing body of work analyzing convex ancient solutions to mean curvature flow (MCF), or equivalently of rescaled mean curvature flow (RMCF). The goal of this paper is to complement the existing literature, which analyzes ancient solutions one at a time, by considering the space 𝑋 of all convex hypersurfaces M ⊂ R n + 1 M\subset\mathbb{R}^{n+1} , regard RMCF as a semiflow on this space, and study the dynamics of this semiflow. To this end, we first extend the well-known existence and uniqueness of solutions to MCF with smooth compact convex initial data to include the case of arbitrary noncompact and nonsmooth initial convex hypersurfaces. We identify a suitable weak topology with good compactness properties on the space 𝑋 of convex hypersurfaces and show that RMCF defines a continuous local semiflow on 𝑋 whose fixed points are the shrinking cylinder solitons S k × R n − k S^{k}\times\mathbb{R}^{n-k} , and for which the Huisken energy is a Lyapunov function. Ancient solutions to MCF are then complete orbits of the RMCF semiflow on 𝑋. We consider the set of all hypersurfaces that lie on an ancient solution that, in backward time, is asymptotic to one of the shrinking cylinder solitons and prove various topological properties of this set. We show that this space is a path connected, compact subset of 𝑋, and considering only point symmetric hypersurfaces, that it is topologically trivial in the sense of Čech cohomology. We also prove that the space of all convex ancient solutions with point symmetry in R n + 1 \mathbb{R}^{n+1} is homeomorphic to an ( n − 1 ) (n-1) -dimensional simplex, in the case when n = 2 n=2 or n = 3 n=3 , and conjecture that it holds true for any n ≥ 2 n\geq 2 .

Existence of solutions to Dirichlet boundary value problems of the stationary relativistic Boltzmann equation

Abstract In this paper, we study the Dirichlet boundary value problem of steady-state relativistic Boltzmann equation in half-line with hard potential model, given the data for the incoming particles at the boundary and a relativistic global Maxwellian with nonzero macroscopic velocities at the far field. We first explicitly address the sound speed for the relativistic Maxwellian in the far field, according to the eigenvalues of an finite dimensional operator based on macroscropic projection. Then we demonstrate that the solvability of the problem varies with the Mach number M ∞ . If M ∞ < − 1 , a unique solution exists connecting the Dirichlet data and the far field Maxwellian when the boundary data is sufficiently close to the Maxwellian. If M ∞ > − 1 , such a solution exists only if the inflow boundary data is small and satisfies certain solvability conditions. The proof is based on the macro–micro decomposition of solutions combined with an artificial damping term. A singular in velocity (at p 1 = 0 and | p | ≫ 1 ) and spatially exponential decay weight is chosen to carry out the energy estimates. The result extends the previous work (Ukai et al 2003 Commun. Math. Phys. 236 373–93) to the relativistic problem.

Dynamic Analysis and Optimal Control of a Stochastic SVEIRS Model With Saturation Incidence

ABSTRACTThis paper proposes a stochastic version based on a deterministic SVEIRS (Susceptible‐Vaccinated‐Exposed‐Infectious‐Recovered‐Susceptible) model with saturation incidences and analyzes their dynamic properties and control strategies. For the deterministic model, we simply study the asymptotic properties of disease‐free and endemic equilibria and discuss the parametric branching. Focusing on the stochastic model, we prove the existence and uniqueness of the global positive solution. Further, the extinction and persistence are obtained under certain threshold conditions. By constructing suitable Lyapunov functions, we explore the existence of an ergodic distribution. Moreover, the control variables are introduced into the stochastic model to prevent the spread of the epidemic. Finally, numerical simulations illustrate the theoretical results, such as the dynamic curves of extinction, persistence, and ergodic distribution under given parameters. The results show the treatment and vaccine controls significantly inhibit the disease.

A Single and Multi‐Valued Problems Involving Mixed (k1,η)$$ \left({k}_1,\eta \right) $$‐Hilfer and (k2,ϕ)$$ \left({k}_2,\phi \right) $$‐Hilfer Fractional Derivatives for the Fractional Navier Problem

ABSTRACTThis paper investigates fractional differential equations and inclusions involving two distinct fractional derivatives, specifically the ‐Hilfer and ‐Hilfer fractional derivatives, within the framework of the Navier boundary value problem. For the single‐valued case, we establish the existence and uniqueness of solutions using classical iterative methods, including the Leray–Schauder, Banach, and Krasnoselskii fixed‐point theorems. In the multi‐valued case, we apply the Leray–Schauder nonlinear alternative when the right‐hand side (RHS) of the inclusion has convex values, while the Covitz–Nadler fixed‐point theorem is employed for non‐convex RHS mappings. To illustrate the practical applicability of our results, we present numerical examples that validate our theoretical findings.

Unique Solution Analysis for Generalized Caputo-Type Fractional BVP via Banach Contraction

In this manuscript, we investigate the existence of a unique solution to a boundary value problem (BVP) involving generalized fractional derivatives of the Caputo type. Our approach is grounded in the Banach contraction mapping theorem, which provides a rigorous framework for proving the existence of a fixed point and, consequently, a solution to the BVP. We extend this methodology to explore analogous problems, offering further insights and interpretations of the results derived from the main theorem. This work not only contributes to the theoretical understanding of fractional differential equations but also demonstrates how these techniques can be applied to a broader class of problems in mathematical physics and engineering. Through detailed analysis and extrapolation, we aim to establish a deeper connection between fractional calculus and fixed-point theory, providing a foundation for future research in this area.