A class of integral equations of convolution type

In this paper, we study a class of integral equations with power nonlinearity on the entire real line. This class of equations arises in the-adic theory of open-closed strings. Using the method of successive approximations and justifying their convergence, we prove the existence of a nontrivial continuous odd bounded solution on the entire line. The asymptotic behavior of the solution is studied as the argument increases unboundedly. We obtain integral estimates and some properties of approximations of the solution to the considered equation. Under some additional restrictions, we also establish the uniqueness of the constructed solution in a certain class of continuous functions. We provide examples of integral kernels of the equation satisfying all assumptions of the formulated theorems. As the nuclear function is a Gaussian distribution, from the proven results we obtain Vladimirov-Volovich theorem as a special case.

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“…since the convolution of summable and bounded functions tends to zero at infinity, see [7]. Therefore, lim we conclude that 1 − ∈ 1 (R + ).…”

Section: Existence Of a Nontrivial Continuous Bounded Solution On Thementioning

confidence: 71%

On solvability of class of nonlinear integral equations in $p$-adic string theory

Andriyan¹,

Kroyan²,

Khachatryan³

2018

Ufimskii Matematicheskii Zhurnal

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“…(see [18]). Now using (4.12), Theorem 2, condition C), and estimate (4.1), we have Examples of kernels {K ij (x)} n×n i,j=1 .…”

Section: Examples Of Functions {µmentioning

confidence: 99%

On non-trivial solvability of one system of non-linear integral equations on the real axis

Khachatryan,

Petrosyan

2023

Izv. Math.

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A system of singular integral equations with monotonic and convex non-linearity on the entire real line is considered. System of this form have applications in many areas of natural science. In particular, such systems arise in the theory of $p$-adic open-closed strings, in the mathematical theory of spatial-temporal epidemic spread within the framework of the well known Diekmann-Kaper model, in the kinetic theory of gases, in the radiative transfer theory. An existence theorem for a non-trivial and bounded solution is proved. The asymptotic behaviour of the constructed solution at $\pm\infty$ is also studied. Specific examples of non-linearities and kernel functions having an applied character are given.

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“…On some properties of the operation of convolution. In what follows we make essential use of the following property of the operation of convolution (see [27]). …”

mentioning

confidence: 99%

Positive solubility of some classes of non-linear integral equations of Hammerstein type on the semi-axis and on the whole line

Khachatryan¹

2015

Izv. Math.

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We study certain classes of non-linear Hammerstein integral equations on the semi-axis and the whole line. These classes of equations arise in the theory of radiative transfer in nuclear reactors, in the kinetic theory of gases, and for travelling waves in non-linear Richer competition systems. By combining special iteration methods with the methods of construction of invariant cone segments for the appropriate nonlinear operator, we are able to prove constructive existence theorems for positive solutions in various function spaces. We give illustrative examples of equations satisfying all the hypotheses of our theorems.

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A class of integral equations of convolution type

Cited by 23 publications

References 6 publications

On solvability of class of nonlinear integral equations in $p$-adic string theory

On solvability of class of nonlinear integral equations in $p$-adic string theory

On non-trivial solvability of one system of non-linear integral equations on the real axis

Positive solubility of some classes of non-linear integral equations of Hammerstein type on the semi-axis and on the whole line

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