A class of distributed order semilinear equationsin Banach spaces

Fedorov, Vladimir E.; Phuong, Ta Duy; Kien, B. T.; Boyko, Kseniya V.; Izhberdeeva, E.M.

doi:10.47475/2500-0101-2020-15308

Cited by 6 publications

(8 citation statements)

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“…hence, x f is a solution to problem (5), (7). The proof of the uniqueness of a solution is the same as in Theorem 1.…”

Section: Integro-differential Equation Of Gerasimov Typementioning

confidence: 86%

“…Suppose that for all λ ∈ Ω R 0 there exists K(λ) −1 ∈ L(X ), f ∈ C([0, T ]; X ). Then there exists an unique solution to problem ( 5), (7) with…”

Section: Integro-differential Equation Of Gerasimov Typementioning

confidence: 99%

“…Note that, by similar methods, various fractional differential equations in Banach spaces, including degenerate ones, were researched in the works [7][8][9][10], see the references therein also. In this sense, it is necessary to mention the monograph by J. Prüss [11] on evolution integral equations in Banach spaces.…”

Section: Introductionmentioning

confidence: 99%

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Integro-differential equations with bounded operators in Banach spaces

Федоров¹,

Godova²,

Kien³

2022

Bul. Kar. Univ. - Math.

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The paper investigates integro-differential equations in Banach spaces with operators, which are a composition of convolution and differentiation operators. Depending on the order of action of these two operators, we talk about integro-differential operators of the Riemann—Liouville type, when the convolution operator acts first, and integro-differential operators of the Gerasimov type otherwise. Special cases of the operators under consideration are the fractional derivatives of Riemann—Liouville and Gerasimov, respectively. The classes of integro-differential operators under study also include those in which the convolution has an integral kernel without singularities. The conditions of the unique solvability of the Cauchy type problem for a linear integro-differential equation of the Riemann—Liouville type and the Cauchy problem for a linear integrodifferential equation of the Gerasimov type with a bounded operator at the unknown function are obtained. These results are used in the study of similar equations with a degenerate operator at an integro-differential operator under the condition of relative boundedness of the pair of operators from the equation. Abstract results are applied to the study of initial boundary value problems for partial differential equations with an integro-differential operator, the convolution in which is given by the Mittag-Leffler function multiplied by a power function.

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“…hence, x f is a solution to problem (5), (7). The proof of the uniqueness of a solution is the same as in Theorem 1.…”

Section: Integro-differential Equation Of Gerasimov Typementioning

confidence: 86%

“…Suppose that for all λ ∈ Ω R 0 there exists K(λ) −1 ∈ L(X ), f ∈ C([0, T ]; X ). Then there exists an unique solution to problem ( 5), (7) with…”

Section: Integro-differential Equation Of Gerasimov Typementioning

confidence: 99%

See 1 more Smart Citation

Integro-differential equations with bounded operators in Banach spaces

Федоров¹,

Godova²,

Kien³

2022

Bul. Kar. Univ. - Math.

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“…Abstract results for non-degenerate and degenerate equations in Banach spaces are applied to the investigation of a class of initial boundary value problems for partial differential equations with a time-fractional derivative and with polynomials in a self-adjoint elliptical differential operator with respect to spatial variables. This article is a continuation of the previous work of the authors, who investigated equations in Banach spaces with other fractional derivatives [14][15][16][17] with applications to initial boundary value problems for partial differential equations and systems of equations.…”

Section: Introductionmentioning

confidence: 86%

Initial Value Problems of Linear Equations with the Dzhrbashyan–Nersesyan Derivative in Banach Spaces

2021

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Among the many different definitions of the fractional derivative, the Riemann–Liouville and Gerasimov–Caputo derivatives are most commonly used. In this paper, we consider the equations with the Dzhrbashyan–Nersesyan fractional derivative, which generalizes the Riemann–Liouville and the Gerasimov–Caputo derivatives; it is transformed into such derivatives for two sets of parameters that are, in a certain sense, symmetric. The issues of the unique solvability of initial value problems for some classes of linear inhomogeneous equations of general form with the fractional Dzhrbashyan–Nersesyan derivative in Banach spaces are investigated. An inhomogeneous equation containing a bounded operator at the fractional derivative is considered, and the solution is presented using the Mittag–Leffler functions. The result obtained made it possible to study the initial value problems for a linear inhomogeneous equation with a degenerate operator at the fractional Dzhrbashyan–Nersesyan derivative in the case of relative p-boundedness of the operator pair from the equation. Abstract results were used to study a class of initial boundary value problems for equations with the time-fractional Dzhrbashyan–Nersesyan derivative and with polynomials in a self-adjoint elliptic differential operator with respect to spatial variables.

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“…Note that unique solvability issues for the Cauchy problem to multi-term linear equation of form (1) with Gerasimov-Caputo derivatives and bounded operators at them were studied in [8], various classes of nonlinear equations with Gerasimov-Caputo derivatives [9][10][11], or with a unique Riemann-Liouville derivative in a linear part of an equation [12,13] have been studied before.…”

Section: Introductionmentioning

confidence: 99%

A Class of Quasilinear Equations with Riemann–Liouville Derivatives and Bounded Operators

Fedorov

Turov

Kien

2022

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The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives and bounded operators at them. Nonlinearity in the equation is assumed to be Lipschitz continuous and dependent on lower order fractional derivatives, which orders have the same fractional part as the order of the highest fractional derivative. The obtained abstract result is applied to study a class of initial-boundary value problems to time-fractional order equations with polynomials of an elliptic self-adjoint differential operator with respect to spatial variables as linear operators at the time-fractional derivatives. The nonlinear operator in the considered partial differential equations is assumed to be smooth with respect to phase variables.

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A class of distributed order semilinear equationsin Banach spaces

Cited by 6 publications

References 0 publications

Integro-differential equations with bounded operators in Banach spaces

Integro-differential equations with bounded operators in Banach spaces

Initial Value Problems of Linear Equations with the Dzhrbashyan–Nersesyan Derivative in Banach Spaces

A Class of Quasilinear Equations with Riemann–Liouville Derivatives and Bounded Operators

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