Existence for Certain Systems of Nonlinear Fractional Differential Equations

By establishing a comparison result and using the monotone iterative technique, combining with the method of upper and lower solutions, the existence of solutions for systems of nonlinear fractional differential equations is considered. An example is given to demonstrate the applicability of our results.

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Monotone iterative method for a class of nonlinear fractional differential equations

Fractional Calculus and Applied Analysis ◽

10.2478/s13540-012-0018-z ◽

2012 ◽

Vol 15 (2) ◽

Cited By ~ 32

Author(s):

Guotao Wang ◽

Dumitru Baleanu ◽

Lihong Zhang

Keyword(s):

Differential Equations ◽

Iterative Method ◽

Fractional Differential Equations ◽

Monotone Iterative Technique ◽

Monotone Iterative Method ◽

Iterative Technique ◽

Nonlinear Fractional Differential Equations ◽

Monotone Iterative ◽

Liouville Fractional Derivative ◽

Fractional Differential

AbstractBy applying the monotone iterative technique and the method of lower and upper solutions, this paper investigates the existence of extremal solutions for a class of nonlinear fractional differential equations, which involve the Riemann-Liouville fractional derivative D q x(t). A new comparison theorem is also build. At last, an example is given to illustrate our main results.

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Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations

Applied Mathematics Letters ◽

10.1016/j.aml.2011.09.078 ◽

2012 ◽

Vol 25 (6) ◽

pp. 1019-1024 ◽

Cited By ~ 80

Author(s):

Guotao Wang ◽

Ravi P. Agarwal ◽

Alberto Cabada

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Existence Results ◽

Monotone Iterative Technique ◽

Iterative Technique ◽

Nonlinear Fractional Differential Equations ◽

Monotone Iterative ◽

Fractional Differential

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Monotone iterative technique via initial time different coupled lower and upper solutions for fractional differential equations

Filomat ◽

10.2298/fil1704031y ◽

2017 ◽

Vol 31 (4) ◽

pp. 1031-1039 ◽

Cited By ~ 5

Author(s):

Ali Yakar ◽

Hadi Kutlay

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Upper And Lower Solutions ◽

Monotone Iterative Technique ◽

Extremal Solutions ◽

Lower And Upper Solutions ◽

Initial Time ◽

Iterative Technique ◽

Monotone Iterative ◽

Fractional Differential

In this paper, we investigate the extremal solutions for a class of nonlinear fractional differential equations with order q 2 (0; 1) by means of monotone iterative technique via initial time different coupled upper and lower solutions.

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General uniqueness and monotone iterative technique for fractional differential equations

Applied Mathematics Letters ◽

10.1016/j.aml.2007.09.006 ◽

2008 ◽

Vol 21 (8) ◽

pp. 828-834 ◽

Cited By ~ 161

Author(s):

V. Lakshmikantham ◽

A.S. Vatsala

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Monotone Iterative Technique ◽

Iterative Technique ◽

Monotone Iterative ◽

Fractional Differential

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Analysis of a system of autonomous fractional differential equations

International Journal of Biomathematics ◽

10.1142/s1793524517500942 ◽

2017 ◽

Vol 10 (07) ◽

pp. 1750094 ◽

Cited By ~ 1

Author(s):

Xiaojun Zhou ◽

Chuanju Xu

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Upper And Lower Solutions ◽

Coupled System ◽

Monotone Iterative Technique ◽

Metapopulation Model ◽

Uniqueness Of Solutions ◽

Predator Prey ◽

Monotone Iterative ◽

Fractional Differential

In this work, we study a system of autonomous fractional differential equations. The differential operator is taken in the Caputo sense. Using the monotone iterative technique combined with the method of upper and lower solutions, we investigate the existence and uniqueness of solutions for coupled system which are nonlinear fractional differential equations, moreover, we obtain the dependence of the solution on the initial values. In addition, we give an important example that is a two-patch subdiffusive predator–prey metapopulation model, investigate the solvability and give the numerical results with this model. The numerical simulation indicates that the results of the subdiffusive model approximate to the two-patch predator–prey metapopulation model with the order [Formula: see text] approach to 1.

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Monotone Iterative Technique for Conformable Fractional Differential Equations with Deviating Arguments

Discrete Dynamics in Nature and Society ◽

10.1155/2020/5827127 ◽

2020 ◽

Vol 2020 ◽

pp. 1-9

Author(s):

Huiling Chen ◽

Shuman Meng ◽

Yujun Cui

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Sufficient Conditions ◽

Monotone Iterative Technique ◽

Extremal Solutions ◽

Iterative Technique ◽

Deviating Arguments ◽

Comparison Principles ◽

Monotone Iterative ◽

Fractional Differential

This paper is concerned with the existence of extremal solutions for periodic boundary value problems for conformable fractional differential equations with deviating arguments. We first build two comparison principles for the corresponding linear equation with deviating arguments. With the help of new comparison principles, some sufficient conditions for the existence of extremal solutions are established by combining the method of lower and upper solutions and the monotone iterative technique. As an application, an example is presented to enrich the main results of this article.

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Monotone iterative technique for impulsive Riemann-Liouville fractional differential equations

Filomat ◽

10.2298/fil1809381c ◽

2018 ◽

Vol 32 (9) ◽

pp. 3381-3395 ◽

Cited By ~ 3

Author(s):

Renu Chaudhary ◽

Dwijendra Pandey

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Fractional Differential Equations ◽

Existence And Uniqueness ◽

Measure Of Noncompactness ◽

Semigroup Theory ◽

Monotone Iterative Technique ◽

Iterative Technique ◽

Monotone Iterative ◽

Fractional Differential

In this article, Monotone iterative technique coupled with the method of lower and upper solutions is employed to discuss the existence and uniqueness of mild solution to an impulsive Riemann-Liouville fractional differential equation. The results are obtained using the concept of measure of noncompactness, semigroup theory and generalized Gronwall inequality for fractional differential equations. At last, an example is given to illustrate the applications of the main results.

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