scholarly journals Existence for Time-Fractional Semilinear Diffusion Equation on the Sphere

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
N. D. Phuong ◽  
Ho Duy Binh ◽  
Ho Thi Kim Van ◽  
Le Dinh Long
Keyword(s):  
Fixed Point ◽  
Fractional Diffusion ◽  
Analytical Tool ◽  
Banach Fixed Point Theorem ◽  
Solution Existence ◽  
Large Role ◽  
Initial Value ◽  
New Techniques ◽  
Physical Phenomena ◽  
The Solution Existence

Fractional diffusion on the sphere plays a large role in the study of physical phenomena customs and meteorology and geophysics. In this paper, we examine two types of the sphere problem: the initial value problem and the end value problem. We are interested in focus on the solution existence in a local or global form. In order to overcome difficult evaluations when evaluating, we need some new techniques. The main analytical tool is the use of the Banach fixed point theorem.

scholarly journals Regularity of a Stochastic Fractional Delayed Reaction-Diffusion Equation Driven by Lévy Noise

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Tianlong Shen ◽  
Jianhua Huang ◽  
Jin Li
Keyword(s):  
Fixed Point ◽  
Diffusion Equation ◽  
Mild Solution ◽  
Existence And Uniqueness ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equation ◽  
Banach Fixed Point Theorem ◽  
Fractional Operator ◽  
Initial Value ◽  
Delayed Reaction

The current paper is devoted to the regularity of the mild solution for a stochastic fractional delayed reaction-diffusion equation driven by Lévy space-time white noise. By the Banach fixed point theorem, the existence and uniqueness of the mild solution are proved in the proper working function space which is affected by the delays. Furthermore, the time regularity and space regularity of the mild solution are established respectively. The main results show that both time regularity and space regularity of the mild solution depend on the regularity of initial value and the order of fractional operator. In particular, the time regularity is affected by the regularity of initial value with delays.

scholarly journals On a Kirchhoff diffusion equation with integral condition

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Danh Hua Quoc Nam ◽  
Dumitru Baleanu ◽  
Nguyen Hoang Luc ◽  
Nguyen Huu Can
Keyword(s):  
Fixed Point ◽  
Mild Solution ◽  
Parabolic Problem ◽  
Local Existence ◽  
Integral Condition ◽  
Banach Fixed Point Theorem ◽  
Homogeneous Case ◽  
New Techniques ◽  
Biological Phenomena ◽  
Nonlocal Integral Condition

Abstract This paper is devoted to Kirchhoff-type parabolic problem with nonlocal integral condition. Our problem has many applications in modeling physical and biological phenomena. The first part of our paper concerns the local existence of the mild solution in Hilbert scales. Our results can be studied into two cases: homogeneous case and inhomogeneous case. In order to overcome difficulties, we applied Banach fixed point theorem and some new techniques on Sobolev spaces. The second part of the paper is to derive the ill-posedness of the mild solution in the sense of Hadamard.

scholarly journals On The Initial Value Problem For Fractional Volterra Integrodifferential Equations With A Caputo - Fabrizio Derivative

2021 ◽  
Author(s):  
Nguyen Huy Tuan ◽  
Nguyen Anh Tuan ◽  
Donal O’regan ◽  
Vo Viet Tri
Keyword(s):  
Fixed Point ◽  
Existence And Uniqueness ◽  
Source Term ◽  
Main Tool ◽  
Integrodifferential Equations ◽  
Banach Fixed Point Theorem ◽  
Initial Value ◽  
Locally Lipschitz ◽  
Source Case ◽  
Well Posed

In this paper, time fractional integrodifferential equations with the Caputo - Fabrizio type derivative will be considered. In the case of a globally Lipschitz source term we obtain a global well - posed result on $\R^N$ for our problem. For the locally Lipschitz source case, we present existence and uniqueness and a finite time blow result for the solution. Our main tool is the Banach fixed point theorem and we extend  a  recent  paper of N.H. Tuan and Y. Zhou.

scholarly journals On the Solution Existence of Variational-Like Inequalities Problems for Weakly Relaxedη−αMonotone Mapping

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Marwan Amin Kutbi ◽  
Wutiphol Sintunavarat
Keyword(s):  
Fixed Point ◽  
Banach Spaces ◽  
Existence Of Solutions ◽  
Monotone Mapping ◽  
Existence Of Solution ◽  
Monotone Mappings ◽  
Solution Existence ◽  
Reflexive Banach Spaces ◽  
New Concepts ◽  
The Solution Existence

We introduce two new concepts of weakly relaxedη-αmonotone mappings and weakly relaxedη-αsemimonotone mappings. Using the KKM technique, the existence of solutions for variational-like problems with weakly relaxedη-αmonotone mapping in reflexive Banach spaces is established. Also, we obtain the existence of solution for variational-like problems with weakly relaxedη-αsemimonotone mappings in arbitrary Banach spaces by using the Kakutani-Fan-Glicksberg fixed-point theorem.

scholarly journals On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Jing Cui ◽  
Litan Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Banach Fixed Point Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Composition Theorem

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.

scholarly journals Laplacian Matrix-Based Power Flow Formulation for LVDC Grids with Radial and Meshed Configurations

Energies ◽  
2021 ◽  
Vol 14 (7) ◽  
pp. 1866
Author(s):  
Zahid Javid ◽  
Ulas Karaagac ◽  
Ilhan Kocar ◽  
Ka Wing Chan
Keyword(s):  
Fixed Point ◽  
Power Flow ◽  
Low Voltage ◽  
Mathematical Formulation ◽  
Distribution Networks ◽  
Load Flow ◽  
Banach Fixed Point Theorem ◽  
Loading Conditions ◽  
Flow Equations ◽  
Uniqueness Of The Solution

There is an increasing interest in low voltage direct current (LVDC) distribution grids due to advancements in power electronics enabling efficient and economical electrical networks in the DC paradigm. Power flow equations in LVDC grids are non-linear and non-convex due to the presence of constant power nodes. Depending on the implementation, power flow equations may lead to more than one solution and unrealistic solutions; therefore, the uniqueness of the solution should not be taken for granted. This paper proposes a new power flow solver based on a graph theory for LVDC grids having radial or meshed configurations. The solver provides a unique solution. Two test feeders composed of 33 nodes and 69 nodes are considered to validate the effectiveness of the proposed method. The proposed method is compared with a fixed-point methodology called direct load flow (DLF) having a mathematical formulation equivalent to a backward forward sweep (BFS) class of solvers in the case of radial distribution networks but that can handle meshed networks more easily thanks to the use of connectivity matrices. In addition, the convergence and uniqueness of the solution is demonstrated using a Banach fixed-point theorem. The performance of the proposed method is tested for different loading conditions. The results show that the proposed method is robust and has fast convergence characteristics even with high loading conditions. All simulations are carried out in MATLAB 2020b software.

scholarly journals The analysis of some special results of a Lasota–Wazewska model with mixed variable delays

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ramazan Yazgan ◽  
Osman Tunç
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Existence And Uniqueness ◽  
Banach Fixed Point Theorem ◽  
Time Varying ◽  
Differential Inequalities ◽  
Almost Periodic ◽  
Variable Delays ◽  
Pseudo Almost Periodic ◽  
Mixed Variable

AbstractThis study is about getting some conditions that guarantee the existence and uniqueness of the weighted pseudo almost periodic (WPAP) solutions of a Lasota–Wazewska model with time-varying delays. Some adequate conditions have been obtained for the existence and uniqueness of the WPAP solutions of the Lasota–Wazewska model, which we dealt with using some differential inequalities, the WPAP theory, and the Banach fixed point theorem. Besides, an application is given to demonstrate the accuracy of the conditions of our main results.

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