Existence and uniqueness of solutions for a class of initial value problems of fractional differential systems on half lines

In this paper, we research CFR fractional differential equations with the derivative of order 3<α<4. We prove existence and uniqueness theorems for CFR-type initial value problem. By Green’s function and its corresponding maximum value, we obtain the Lyapunov-type inequality of corresponding equations. As for application, we study the eigenvalue problem in the sense of CFR.

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Existence and uniqueness of solutions of initial value problems for nonlinear langevin equation involving two fractional orders

Communications in Nonlinear Science and Numerical Simulation ◽

10.1016/j.cnsns.2013.09.035 ◽

2014 ◽

Vol 19 (6) ◽

pp. 1661-1668 ◽

Cited By ~ 32

Author(s):

Tao Yu ◽

Ke Deng ◽

Maokang Luo

Keyword(s):

Langevin Equation ◽

Existence And Uniqueness ◽

Initial Value Problems ◽

Initial Value ◽

Uniqueness Of Solutions ◽

Nonlinear Langevin Equation ◽

Fractional Orders

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Existence-uniqueness of positive solutions to nonlinear impulsive fractional differential systems and optimal control

Boundary Value Problems ◽

10.1186/s13661-020-01461-x ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Shu Song ◽

Lingling Zhang ◽

Bibo Zhou ◽

Nan Zhang

Keyword(s):

Differential Equation ◽

Optimal Control ◽

Fixed Point ◽

Fixed Point Theorem ◽

Positive Solutions ◽

Existence And Uniqueness ◽

Mixed Monotone Operator ◽

Differential Systems ◽

Fractional Differential Systems ◽

Fractional Differential

Abstract In this thesis, we investigate a kind of impulsive fractional order differential systems involving control terms. By using a class of φ-concave-convex mixed monotone operator fixed point theorem, we obtain a theorem on the existence and uniqueness of positive solutions for the impulsive fractional differential equation, and the optimal control problem of positive solutions is also studied. As applications, an example is offered to illustrate our main results.

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Existence and Uniqueness of Solutions for Initial Value Problem of Nonlinear Fractional Differential Equations

Abstract and Applied Analysis ◽

10.1155/2012/615230 ◽

2012 ◽

Vol 2012 ◽

pp. 1-14

Author(s):

Qiuping Li ◽

Shurong Sun ◽

Ping Zhao ◽

Zhenlai Han

Keyword(s):

Differential Equation ◽

Initial Value Problem ◽

Existence And Uniqueness ◽

Upper And Lower Solutions ◽

Initial Value ◽

Uniqueness Of Solutions ◽

Nonlinear Fractional Differential Equation ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Fractional Differential

We discuss the initial value problem for the nonlinear fractional differential equationL(D)u=f(t,u), t∈(0,1], u(0)=0, whereL(D)=Dsn-an-1Dsn-1-⋯-a1Ds1,0<s1<s2<⋯<sn<1, andaj<0,j=1,2,…,n-1,Dsjis the standard Riemann-Liouville fractional derivative andf:[0,1]×ℝ→ℝis a given continuous function. We extend the basic theory of differential equation, the method of upper and lower solutions, and monotone iterative technique to the initial value problem. Some existence and uniqueness results are established.

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Impulsive Multiorders Riemann-Liouville Fractional Differential Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2015/603893 ◽

2015 ◽

Vol 2015 ◽

pp. 1-9 ◽

Cited By ~ 6

Author(s):

Weera Yukunthorn ◽

Sotiris K. Ntouyas ◽

Jessada Tariboon

Keyword(s):

Differential Equations ◽

Fractional Differential Equations ◽

Existence And Uniqueness ◽

Initial Value Problems ◽

Weighted Space ◽

Initial Value ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Banach’S Fixed Point Theorem ◽

Fractional Differential

Impulsive multiorders fractional differential equations are studied. Existence and uniqueness results are obtained for first- and second-order impulsive initial value problems by using Banach’s fixed point theorem in an appropriate weighted space. Examples illustrating the main results are presented.

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