Global Dynamics of a Lotka–Volterra Competition Diffusion System with Nonlocal Effects

2020 ◽  
Vol 30 (05) ◽  
pp. 2050066 ◽  
Author(s):  
Bang-Sheng Han ◽  
Yinghui Yang ◽  
Wei-Jian Bo ◽  
Huiling Tang
Keyword(s):  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Diffusion System ◽  
Upper And Lower Bounds ◽  
Global Dynamics ◽  
Nonlocal Effects ◽  
Monotone Iteration ◽  
Existence And Nonexistence ◽  
Uniqueness Of The Solution ◽  
Competition Diffusion

This paper is concerned with the global dynamics of a Lotka–Volterra competition diffusion system having nonlocal intraspecies terms. Based on the reconstructed comparison principle and monotone iteration, the existence and uniqueness of the solution for the corresponding Cauchy problem are established. In addition, the spreading speed of the system with compactly supported initial data is considered, which admits uniform upper and lower bounds. Finally, some sufficient conditions for guaranteeing the existence and nonexistence of Turing bifurcation are given, which depend on the intensity of nonlocality. Comparing with the classical Lotka–Volterra competition diffusion system, our results indicate that a nonconstant periodic solution may exist if the nonlocality is strong enough, which are also illustrated numerically.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

scholarly journals Conditions to Guarantee the Existence of the Solution to Stochastic Differential Equations of Neutral Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1613
Author(s):  
Mun-Jin Bae ◽  
Chan-Ho Park ◽  
Young-Ho Kim
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Exponential Estimates ◽  
Analysis Theory ◽  
Linear Growth Condition ◽  
Uniqueness Of The Solution ◽  
The Uniqueness Theorem

The main purpose of this study was to demonstrate the existence and the uniqueness theorem of the solution of the neutral stochastic differential equations under sufficient conditions. As an alternative to the stochastic analysis theory of the neutral stochastic differential equations, we impose a weakened Ho¨lder condition and a weakened linear growth condition. Stochastic results are obtained for the theory of the existence and uniqueness of the solution. We first show that the conditions guarantee the existence and uniqueness; then, we show some exponential estimates for the solutions.

scholarly journals Existence Theory and Novel Iterative Method for Dynamical System of Infectious Diseases

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Gauhar Ali ◽  
Ghazala Nazir ◽  
Kamal Shah ◽  
Yongjin Li
Keyword(s):  
Dynamical System ◽  
Existence And Uniqueness ◽  
Integral Transform ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Qualitative Theory ◽  
Existence Theory ◽  
Uniqueness Of The Solution ◽  
Respective Theory

This manuscript is devoted to investigate qualitative theory of existence and uniqueness of the solution to a dynamical system of an infectious disease known as measles. For the respective theory, we utilize fixed point theory to construct sufficient conditions for existence and uniqueness of the solution. Some results corresponding to Hyers–Ulam stability are also investigated. Furthermore, some semianalytical results are computed for the considered system by using integral transform due to the Laplace and decomposition technique of Adomian. The obtained results are presented by graphs also.

scholarly journals On existence and uniqueness of the solution of the equation of motion for constrained mechanical systems

1995 ◽  
Vol 1 (1) ◽  
pp. 37-39
Author(s):  
Firdaus E. Udwadia ◽  
Robert E. Kalaba ◽  
M. Zuhair Nashed
Keyword(s):  
Existence And Uniqueness ◽  
Mechanical Systems ◽  
Sufficient Conditions ◽  
Equation Of Motion ◽  
Constrained Mechanical Systems ◽  
Uniqueness Of The Solution

In this paper we provide sufficient conditions for the existence and uniqueness of the solution of the newly obtained equation of motion for constrained mechanical systems.

scholarly journals Study of a nonlinear multi-terms boundary value problem of fractional pantograph differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Muhammad Bahar Ali Khan ◽  
Thabet Abdeljawad ◽  
Kamal Shah ◽  
Gohar Ali ◽  
Hasib Khan ◽  
...  
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Research Work ◽  
Sufficient Conditions ◽  
Point Of View ◽  
Uniqueness Of The Solution ◽  
Delay Differential ◽  
Ulam Type Stability

AbstractIn this research work, a class of multi-term fractional pantograph differential equations (FODEs) subject to antiperiodic boundary conditions (APBCs) is considered. The ensuing problem involves proportional type delay terms and constitutes a subclass of delay differential equations known as pantograph. On using fixed point theorems due to Banach and Schaefer, some sufficient conditions are developed for the existence and uniqueness of the solution to the problem under investigation. Furthermore, due to the significance of stability analysis from a numerical and optimization point of view Ulam type stability and its various forms are studied. Here we mention different forms of stability: Hyers–Ulam (HU), generalized Hyers–Ulam (GHU), Hyers–Ulam Rassias (HUR) and generalized Hyers–Ulam–Rassias (GHUR). After the demonstration of our results, some pertinent examples are given.

scholarly journals The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Huanting Li ◽  
Yunfei Peng ◽  
Kuilin Wu
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Initial State ◽  
Switching Line ◽  
Uniqueness Of The Solution ◽  
The Stability ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>In this paper, we deal with the qualitative theory for a class of nonlinear differential equations with switching at variable times (SSVT), such as the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters and the stability. Firstly, we obtain the existence and uniqueness of a global solution by defining a reasonable solution (see Definition 2.1). Secondly, the continuous dependence and differentiability of the solution with respect to the initial state and the switching line are investigated. Finally, the global exponential stability of the system is discussed. Moreover, we give the necessary and sufficient conditions of SSVT just switching <inline-formula><tex-math id="M1">\begin{document}$ k\in \mathbb{N} $\end{document}</tex-math></inline-formula> times on bounded time intervals.</p>

scholarly journals AN EXTENSION OF THE PRODUCT INTEGRATION METHOD TO L1 WITH APPLICATIONS IN ASTROPHYSICS

2016 ◽  
Vol 21 (6) ◽  
pp. 774-793 ◽  
Author(s):  
Laurence Grammont ◽  
Mario Ahues ◽  
Hanane Kaboul
Keyword(s):  
Integral Equation ◽  
Existence And Uniqueness ◽  
Integration Method ◽  
Sufficient Conditions ◽  
Iterative Refinement ◽  
Weakly Singular Kernel ◽  
Product Integration ◽  
Weakly Singular ◽  
Uniqueness Of The Solution ◽  
Product Integration Method

A Fredholm integral equation of the second kind in L1([a, b], C) with a weakly singular kernel is considered. Sufficient conditions are given for the existence and uniqueness of the solution. We adapt the product integration method proposed in C0 ([a, b], C) to apply it in L1 ([a, b], C), and discretize the equation. To improve the accuracy of the approximate solution, we use different iterative refinement schemes which we compare one to each other. Numerical evidence is given with an application in Astrophysics.

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