Modification of the application of a contraction mapping method on a class of fractional differential equation

El-Raheem, Zaki F. A.

doi:10.1016/s0096-3003(02)00136-4

Cited by 19 publications

(14 citation statements)

References 3 publications

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“…Let us point out that from the construction of constant A, it follows that it depends solely on the values of orders α β and parameter given in (7). This ends the proof of Lemma 2.9.…”

Section: There Exists a Constant A (Dependent On α And β) Such That Tsupporting

confidence: 53%

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Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

Klimek¹,

Błasik²

2012

Open Mathematics

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Two-term semi-linear and two-term nonlinear fractional differential equations (FDEs) with sequential Caputo derivatives are considered. A unique continuous solution is derived using the equivalent norms/metrics method and the Banach theorem on a fixed point. Both, the unique general solution connected to the stationary function of the highest order derivative and the unique particular solution generated by the initial value problem, are explicitly constructed and proven to exist in an arbitrary interval, provided the nonlinear terms fulfil the corresponding Lipschitz condition. The existence-uniqueness results are given for an arbitrary order of the FDE and an arbitrary partition of orders between the components of sequential derivatives. MSC:26A33, 34A08, 45Jxx

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“…Let us point out that from the construction of constant A, it follows that it depends solely on the values of orders α β and parameter given in (7). This ends the proof of Lemma 2.9.…”

Section: There Exists a Constant A (Dependent On α And β) Such That Tsupporting

confidence: 53%

“…The equivalent norm and metric are chosen so that the mapping becomes a contraction on a new complete metric space. This approach was extended to fractional differential equations by El-Raheem in [7], where he considered a one-term FDE of order α ∈ (0 1). Then, Lakshmikantham et al in [16] [12,13].…”

Section: Remark 28mentioning

confidence: 99%

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

Klimek¹,

Błasik²

2012

Open Mathematics

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“…He used exponential functions to introduce an equivalent metric and to show the existence of global solutions of certain ordinary and partial differential equations in respective function spaces. A similar technique was proposed in [17,18] for some simple nonlinear fractional differential equations. Then Lakshmikantham et al [1,19] space which are equivalent to standard metric (4) generated by norm (3).…”

Section: Preliminariesmentioning

confidence: 99%

Sequential fractional differential equations with Hadamard derivative

Klimek

2011

Communications in Nonlinear Science and Numerical Simulation

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“…The modified norm is equivalent to the original norm and a contraction in the new metric is also a contraction in the original metric. This method has been used by several authors (see [9] [45] [46] [47]). …”

Section: T E Atmentioning

confidence: 99%

“…There are several articles devoted to the general analysis of stability of equilibria of non-integer order differential equations (see for instance the articles by [23] [45] [46] [48]- [58] and applications of delay fractional differentail equations to epidemic analysis(see [59] [60]) The asymptotic behaviour of solutions of fractional order systems can be quite different from the corresponding integer order systems indicating a change in the asymptotic behavior of a fractionalized dynamic system from its integer order counterpart. For instance an unstable integer order system can become a stable one when the system is fractionalized; also new asymptotic behaviour not found in the integer order system can emerge in fractionalized …”

Section: T E Atmentioning

confidence: 99%

Fractionalization of a Class of Semi-Linear Differential Equations

Leung¹,

Gopalsamy

2017

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The dynamics of a fractionalized semi-linear scalar differential equation is considered with a Caputo fractional derivative. By using a symbolic operational method, a fractional order initial value problem is converted into an equivalent Volterra integral equation of second kind. A brief discussion is included to show that the fractional order derivatives and integrals incorporate a fading memory (also known as long memory) and that the order of the fractional derivative can be considered to be an index of memory. A variation of constants formula is established for the fractionalized version and it is shown by using the Fourier integral theorem that this formula reduces to that of the integer order differential equation as the fractional order approaches an integer. The global existence of a unique solution and the global Mittag-Leffler stability of an equilibrium are established by exploiting the complete monotonicity of one and two parameter Mittag-Leffler functions. The method and the analysis employed in this article can be used for the study of more general systems of fractional order differential equations.

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Modification of the application of a contraction mapping method on a class of fractional differential equation

Cited by 19 publications

References 3 publications

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

Existence and uniqueness of solution for a class of nonlinear sequential differential equations of fractional order

Sequential fractional differential equations with Hadamard derivative

Fractionalization of a Class of Semi-Linear Differential Equations

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