Existence and Numerical Simulation of Solutions for Fractional Equations Involving Two Fractional Orders with Nonlocal Boundary Conditions

We study a boundary value problem for fractional equations involving two fractional orders. By means of a fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. In addition, we describe the dynamic behaviors of the fractional Langevin equation by using theG2algorithm.

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On fractional Langevin equation involving two fractional orders in different intervals

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2019.6.3 ◽

2019 ◽

Vol 24 (6) ◽

Cited By ~ 3

Author(s):

Hamid Baghani ◽

J. Nieto

Keyword(s):

Langevin Equation ◽

Existence And Uniqueness ◽

Initial Conditions ◽

Sufficient Conditions ◽

Uniqueness Of Solutions ◽

Numerical Examples ◽

Fractional Equations ◽

Fractional Langevin Equation ◽

Nonlinear Langevin Equation ◽

Fractional Orders

In this paper, we study a nonlinear Langevin equation involving two fractional orders α ∈ (0; 1] and β ∈ (1; 2] with initial conditions. By means of an interesting fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions for the fractional equations. Some illustrative numerical examples are also discussed.

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Existence and Uniqueness of Solutions to the Wage Equation of Dixit-Stiglitz-Krugman Model with No Restriction on Transport Costs

Discrete Dynamics in Nature and Society ◽

10.1155/2017/9341502 ◽

2017 ◽

Vol 2017 ◽

pp. 1-7 ◽

Cited By ~ 2

Author(s):

Minoru Tabata ◽

Nobuoki Eshima

Keyword(s):

Fixed Point ◽

Existence And Uniqueness ◽

Nonlinear Operator ◽

Sufficient Conditions ◽

Wage Equation ◽

Discrete Equation ◽

Transport Costs ◽

Economic Activities ◽

Uniqueness Of Solutions ◽

Double Nonlinear

In spatial economics, the distribution of wages is described by a solution to the wage equation of Dixit-Stiglitz-Krugman model. The wage equation is a discrete equation that has a double nonlinear singular structure in the sense that the equation contains a discrete nonlinear operator whose kernel itself is expressed by another discrete nonlinear operator with a singularity. In this article, no restrictions are imposed on the maximum of transport costs of the model and on the number of regions where economic activities are conducted. Applying Brouwer fixed point theorem to this discrete double nonlinear singular operator, we prove sufficient conditions for the wage equation to have a solution and a unique one.

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Existence and Uniqueness of Solutions to Third-Order Boundary Value Problems

Trends in Computational and Applied Mathematics ◽

10.5540/tcam.2021.022.02.00221 ◽

2021 ◽

Vol 22 (2) ◽

pp. 221-240

Author(s):

S. S. Almuthaybiri ◽

J. M. Jonnalagadda ◽

C. C. Tisdell

Keyword(s):

Fixed Point ◽

Boundary Value Problems ◽

Existence And Uniqueness ◽

Boundary Value ◽

Sufficient Conditions ◽

Uniqueness Of Solutions ◽

Third Order ◽

Fixed Point Methods ◽

Bounded Sets

The purpose of this research is to connect fixed point methods with certain third-order boundary value problems in new and interesting ways. Our strategy involves an analysis of the problem under consideration within closed and bounded sets. We develop sufficient conditions under which the associated mappings will be contractive and invariant in these sets, which generates new advances concerning the existence, uniqueness and approximation of solutions.

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Some Existence Results for Systems of Impulsive Stochastic Differential Equations

Annales Mathematicae Silesianae ◽

10.2478/amsil-2020-0028 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Sliman Mekki ◽

Tayeb Blouhi ◽

Juan J. Nieto ◽

Abdelghani Ouahab

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Stochastic Differential Equations ◽

Banach Spaces ◽

Existence And Uniqueness ◽

Fixed Point Theorems ◽

Sufficient Conditions ◽

Existence Results ◽

Impulsive Systems ◽

Uniqueness Of Solutions

Abstract In this paper we study a class of impulsive systems of stochastic differential equations with infinite Brownian motions. Sufficient conditions for the existence and uniqueness of solutions are established by mean of some fixed point theorems in vector Banach spaces. An example is provided to illustrate the theory.

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Existence and uniqueness of solutions for nonlinear impulsive differential equations with three-point boundary conditions

e-Journal of Analysis and Applied Mathematics ◽

10.2478/ejaam-2018-0003 ◽

2018 ◽

Vol 1 (1) ◽

pp. 21-36 ◽

Cited By ~ 1

Author(s):

Mısır J. Mardanov ◽

Yagub A. Sharifov ◽

Kamala E. Ismayilova

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Differential Equations ◽

Boundary Conditions ◽

Point Theorem ◽

Existence And Uniqueness ◽

Sufficient Conditions ◽

Impulsive Differential Equations ◽

Point Boundary ◽

Uniqueness Of Solutions

AbstractThis paper is devoted to a system of nonlinear impulsive differential equations with three-point boundary conditions. The Green function is constructed and considered original problem is reduced to the equivalent impulsive integral equations. Sufficient conditions are found for the existence and uniqueness of solutions for the boundary value problems for the first order nonlinear system of the impulsive ordinary differential equations with three-point boundary conditions. The Banach fixed point theorem is used to prove the existence and uniqueness of a solution of the problem and Schaefer’s fixed point theorem is used to prove the existence of a solution of the problem under consideration. We illustrate the application of the main results by two examples.

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Fractional Langevin Equation Involving Two Fractional Orders: Existence and Uniqueness Revisited

Mathematics ◽

10.3390/math8050743 ◽

2020 ◽

Vol 8 (5) ◽

pp. 743 ◽

Cited By ~ 2

Author(s):

Hossein Fazli ◽

HongGuang Sun ◽

Juan J. Nieto

Keyword(s):

Fixed Point ◽

Langevin Equation ◽

Existence And Uniqueness ◽

Initial Conditions ◽

Representation Formula ◽

Fixed Point Problem ◽

Point Problem ◽

Contraction Mapping Theorem ◽

Fractional Langevin Equation ◽

Fractional Orders

We consider the nonlinear fractional Langevin equation involving two fractional orders with initial conditions. Using some basic properties of Prabhakar integral operator, we find an equivalent Volterra integral equation with two parameter Mittag–Leffler function in the kernel to the mentioned equation. We used the contraction mapping theorem and Weissinger’s fixed point theorem to obtain existence and uniqueness of global solution in the spaces of Lebesgue integrable functions. The new representation formula of the general solution helps us to find the fixed point problem associated with the fractional Langevin equation which its contractivity constant is independent of the friction coefficient. Two examples are discussed to illustrate the feasibility of the main theorems.

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On a two-dimensional fractional thermoelastic system with nonlocal constraints describing a fractional Kirchhoff plate

Advances in Difference Equations ◽

10.1186/s13662-020-03188-6 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Said Mesloub ◽

Faten Aldosari

Keyword(s):

Existence And Uniqueness ◽

Energy Inequality ◽

Nonlocal Boundary ◽

Nonlocal Boundary Conditions ◽

Kirchhoff Plate ◽

Two Dimensional ◽

Uniqueness Of Solutions ◽

Fractional Caputo Derivative ◽

Kirchhoff Plate Model ◽

Nonlocal Constraints

AbstractWe show herein the existence and uniqueness of solutions for coupled fractional order partial differential equations modeling a thermoelastic fractional Kirchhoff plate model associated with initial, Dirichlet, and nonlocal boundary conditions involving fractional Caputo derivative. Some efficient results of existence and uniqueness are obtained by employing the energy inequality method.

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Nonlocal Boundary Value Problem for Nonlinear Impulsiveqk-Integrodifference Equation

Abstract and Applied Analysis ◽

10.1155/2014/478185 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Lihong Zhang ◽

Dumitru Baleanu ◽

Guotao Wang

Keyword(s):

Fixed Point ◽

Boundary Value Problem ◽

Existence And Uniqueness ◽

Fixed Point Theorems ◽

Boundary Value ◽

Nonlocal Boundary ◽

Nonlocal Boundary Value Problem ◽

Uniqueness Of Solutions ◽

Integrodifference Equation ◽

Quantum Calculus

A nonlinear impulsive integrodifference equation within the frame ofqk-quantum calculus is investigated by applying using fixed point theorems. The conditions for existence and uniqueness of solutions are obtained.

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Existence of Solutions for Nonlinear Impulsive Fractional Differential Equations of Orderα∈(2,3]with Nonlocal Boundary Conditions

Abstract and Applied Analysis ◽

10.1155/2012/717235 ◽

2012 ◽

Vol 2012 ◽

pp. 1-26 ◽

Cited By ~ 2

Author(s):

Lihong Zhang ◽

Guotao Wang ◽

Guangxing Song

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Fixed Point Theorems ◽

Existence Of Solutions ◽

Nonlocal Boundary ◽

Sufficient Conditions ◽

Nonlocal Boundary Conditions ◽

Uniqueness Of Solutions ◽

Impulsive Fractional Differential Equations ◽

Fractional Differential

We investigate the existence and uniqueness of solutions to the nonlocal boundary value problem for nonlinear impulsive fractional differential equations of orderα∈(2,3]. By using some well-known fixed point theorems, sufficient conditions for the existence of solutions are established. Some examples are presented to illustrate the main results.

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