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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.12231 |
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| _version_ | 1866912485988106240 |
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| author | Khachatryan, A. Kh. Khachatryan, Kh. A. Petrosyan, H. S. |
| author_facet | Khachatryan, A. Kh. Khachatryan, Kh. A. Petrosyan, H. S. |
| contents | This paper investigates two classes of quasilinear and essentially nonlinear integral equations with a sum-difference kernel on the half-line. Such equations arise in various areas of physics, including the theory of radiative transfer in spectral lines, the dynamic theory of p-adic strings, and the kinetic theory of gases within the framework of the modified Bhatnagar-Gross-Krook (BGK) model. Under specific conditions on the kernel and the nonlinear terms, we establish constructive existence theorems for non-negative, nontrivial, and continuous solutions. For the quasilinear case, we construct a one-parameter family of non-negative, nontrivial, linearly growing, and continuous solutions. For the class of essentially nonlinear equations, we prove the uniform convergence-at a geometric rate-of a specially constructed sequence of successive approximations to a non-negative, nontrivial, continuous, and bounded solution. We also study the asymptotic behavior of the constructed solutions at infinity. Additionally, for the second class of equations, a uniqueness theorem is established within a certain subclass of non-negative, bounded, and nontrivial functions. The paper concludes with concrete examples of kernels and nonlinearities that satisfy all conditions of the proven theorems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_12231 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Solvability Issues of Two Classes of Quasilinear and Nonlinear Integral Equations with a Sum-Difference Kernel on the Half-Line Khachatryan, A. Kh. Khachatryan, Kh. A. Petrosyan, H. S. Functional Analysis This paper investigates two classes of quasilinear and essentially nonlinear integral equations with a sum-difference kernel on the half-line. Such equations arise in various areas of physics, including the theory of radiative transfer in spectral lines, the dynamic theory of p-adic strings, and the kinetic theory of gases within the framework of the modified Bhatnagar-Gross-Krook (BGK) model. Under specific conditions on the kernel and the nonlinear terms, we establish constructive existence theorems for non-negative, nontrivial, and continuous solutions. For the quasilinear case, we construct a one-parameter family of non-negative, nontrivial, linearly growing, and continuous solutions. For the class of essentially nonlinear equations, we prove the uniform convergence-at a geometric rate-of a specially constructed sequence of successive approximations to a non-negative, nontrivial, continuous, and bounded solution. We also study the asymptotic behavior of the constructed solutions at infinity. Additionally, for the second class of equations, a uniqueness theorem is established within a certain subclass of non-negative, bounded, and nontrivial functions. The paper concludes with concrete examples of kernels and nonlinearities that satisfy all conditions of the proven theorems. |
| title | Solvability Issues of Two Classes of Quasilinear and Nonlinear Integral Equations with a Sum-Difference Kernel on the Half-Line |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2507.12231 |