About Existence and Estimates of Solutions of the Implicit Differential Equation With Autoadjustable Deviation Argument

Serova, Irina D.; Repin, A. A.

doi:10.20310/1810-0198-2018-23-123-566-574

Cited by 3 publications

(2 citation statements)

References 6 publications

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“…It is possible that statements about the existence of solutions and their estimates and dependence on parameters can be obtained with the use of contemporary results on covering maps (see [17][18][19][20]), which have recently been successfully applied to ordinary differential equations unsolved with respect to the derivative (see, e.g., articles [21][22][23][24][25] and other works by the same authors). But until now, such studies of "implicit" FDEs were fragmentary; we note only the works [26][27][28]. Here, it is proposed to apply the above Theorem 1 on equations in vector metric spaces to the study of Equation (11).…”

Section: The Functionmentioning

confidence: 99%

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Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations

Zhukovskiy,

Panasenko

2023

Mathematics

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The equation G(x,x)=y˜, where G:X×X→Y, and X,Y are vector metric spaces (meaning that the values of a distance between the points in these spaces belong to some cones E+,M+ of a Banach space E and a linear space M, respectively), is considered. This operator equation is compared with a “model” equation, namely, g(t,t)=0, where a continuous map g:E+×E+→M+ is orderly covering in the first argument and antitone in the second one. The idea to study equations comparing them with “simpler” ones goes back to the Kantorovich fixed-point theorem for an operator acting in a Banach space. In this paper, the conditions under which the solvability of the “model” equation guarantees the existence of solutions to the operator equation are obtained. The statement proved extends the recent results about fixed points and coincidence points to more general equations in more general vector metric spaces. The results obtained for the operator equation are then applied to the study of the solvability, as well as to finding solution estimates, of the Cauchy problem for a functional differential equation.

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Section: The Functionmentioning

confidence: 99%

“…We compare this equation with "model" integral Equation (19) (see Example 5), the right-hand side of which is defined by relation (28).…”

mentioning

confidence: 99%

Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations

Zhukovskiy,

Panasenko

2023

Mathematics

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On Estimates of Solutions of Boundary-Value Problems for Implicit Differential Equations with Deviating Argument

Zhukovskaya,

Serova

2024

J Math Sci

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Study of the boundary value problem for a differential inclusion

Serova

2023

Russian Universities Reports. Mathematics

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The boundary value problem with respect to an absolutely continuous function x:[a,b]→R^n for the differential inclusion F(t,x,x ̇,x ̇)∋0,t∈[a,b], with the condition αx(a)+βx(b)=γ ̃ and additional restriction on the derivative of the desired function (Lx)(t)≐x ̇(t)-λx(t)∈B(t), t∈[a,b] is under discussion. It is assumed that the boundary value problem with the same conditions for the linear differential equation Lx=y is uniquely solvable for any summable function y. Using Green’s function of this «auxiliary» linear boundary value problem, the original problem is reduced to an equivalent integral inclusion with respect to the summable function x ̇. To the inclusion obtained, the results on operator inclusion with an orderly covering multivalued mapping are applied. In the first section of the work, the information about multivalued mappings of partially ordered spaces used in this study is given. In the main section of the work, conditions for the existence and estimates of solutions to the boundary value problem under investigation are obtained in the form of a statement similar to Chaplygin’s theorem on differential inequality. These results are illustrated by an example of studying a periodic boundary value problem for a differential equation which is not resolved with respect to the derivative.

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About Existence and Estimates of Solutions of the Implicit Differential Equation With Autoadjustable Deviation Argument

Abstract: Conditions of a solubility and assessment of solutions of an implicit differential equation with autoadjustable (that is depending on required function) argument deviation are received. Results about the covering displays of partially ordered spaces are used.

Cited by 3 publications

References 6 publications

Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations

Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations

On Estimates of Solutions of Boundary-Value Problems for Implicit Differential Equations with Deviating Argument

Study of the boundary value problem for a differential inclusion

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