Perturbation theory for fractional evolution equations in a Banach space

Ahmadova, Arzu; Huseynov, Ismail T.; Mahmudov, Nazım I.

doi:10.1007/s00233-022-10322-1

Cited by 2 publications

(1 citation statement)

References 21 publications

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“…An important method in studying the fractional evolution equation is the resolvent operator. The research publications [5,32] of the fractional resolvent operator theory of abstract integral equations can be reviewed by the readers. Remarkably, Zhou et al [37,38] considered the abstract Cauchy problem by using the property of the C 0 -semigroup and probability density function when the order α ∈ (0, 1).…”

Section: Introductionmentioning

confidence: 99%

Mild solution for $(\rho ,k,\Psi )$-proportional Hilfer fractional Cauchy problem

Wang

2024

Adv Cont Discr Mod

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Hilfer fractional derivative is an important and interesting operator in fractional calculus, and it can be applicable in pure theories and other fields. It yields to other notable definitions, Ψ-Hilfer, $(k,\Psi )$ ( k , Ψ ) -Hilfer derivatives, etc. Motivated by the concepts of the proportional fractional derivative and $(k,\Psi )$ ( k , Ψ ) -Hilfer fractional derivative, we first introduce new definitions of integral and derivative, termed the $(\rho ,k,\Psi )$ ( ρ , k , Ψ ) -proportional integral and $(\rho ,k,\Psi )$ ( ρ , k , Ψ ) -proportional Hilfer fractional derivative. This type of fractional derivative is advantageous as it aligns with earlier studies on fractional differential equations. Additionally, we present a more generalized version of the $(\rho ,\alpha ,\beta ,k,r)$ ( ρ , α , β , k , r ) -resolvent family, followed by an exploration of its properties. By analyzing the generalized resolvent family, we examine the existence of mild solutions to the $(\rho ,k,\Psi )$ ( ρ , k , Ψ ) -proportional Hilfer fractional Cauchy problem, supported by an illustrative example to show the main result.

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Section: Introductionmentioning

confidence: 99%

Mild solution for $(\rho ,k,\Psi )$-proportional Hilfer fractional Cauchy problem

Wang

2024

Adv Cont Discr Mod

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Extremal solutions for fractional evolution equations of order $ 1 < \gamma < 2 $

Li,

Zhao

2023

MATH

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<abstract><p>This manuscript considers a class of fractional evolution equations with order $ 1 < \gamma < 2 $ in ordered Banach space. Based on the theory of cosine operators, this paper extends the application of monotonic iterative methods in this type of equation. This method can be applied to some physical problems and phenomena, providing new tools and ideas for academic research and practical applications. Under the assumption that the linear part is an $ m $-accretive operator, the positivity of the operator families of fractional power solutions is obtained by using Mainardi's Wright-type function. By virtue of the positivity of the family of fractional power solution operators, we establish the monotone iterative technique of the solution of the equation and obtain the existence of extremal mild solutions under the assumption that the upper and lower solutions exist. Moreover, we investigate the positive mild solutions without assuming the existence of upper and lower solutions. In the end, we give an example to illustrate the applied value of our study.</p></abstract>

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Perturbation theory for fractional evolution equations in a Banach space

Cited by 2 publications

References 21 publications

Mild solution for $(\rho ,k,\Psi )$-proportional Hilfer fractional Cauchy problem

Mild solution for $(\rho ,k,\Psi )$-proportional Hilfer fractional Cauchy problem

Extremal solutions for fractional evolution equations of order $ 1 < \gamma < 2 $

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Perturbation theory for fractional evolution equations in a Banach space

Cited by 2 publications

References 21 publications

Mild solution for $(\rho ,k,\Psi )$-proportional Hilfer fractional Cauchy problem

Mild solution for $(\rho ,k,\Psi )$-proportional Hilfer fractional Cauchy problem

Extremal solutions for fractional evolution equations of order $ 1 &lt; \gamma &lt; 2 $

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Extremal solutions for fractional evolution equations of order $ 1 < \gamma < 2 $