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A System of Inhomogeneous Integral Equations of Convolution Type with Power Nonlinearity

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Abstract

We consider a system of the inhomogeneous integral equations of convolution type with power nonlinearity which arises in describing fluid infiltration from a cylindrical reservoir into an isotropic homogeneous porous medium, propagation of shock waves in pipes filled with gas and cooling bodies under radiation in accord with the Stefan–Boltzmann law, etc. We seek for nonnegative solutions continuous on the positive semiaxis and obtain two-sided a priori estimates for a solution to the system of use for constructing a complete metric space and proving the unique solvability of this system in this space by the method of weight metrics (an analog of Bielicki’s method). We show that a solution can be found by the successive approximations of the Picard type. We also estimate the convergence rate and establish that a solution is unique in the class of positive continuous functions. Furthermore, the existence of nontrivial solutions is also studied for the corresponding homogeneous systems of integral equations of convolution type with power nonlinearity.

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References

  1. Askhabov S.N., Karapetyants N.K., and Yakubov A.Ya., “Integral equations of convolution type with power nonlinearity and systems of such equations,” Dokl. Akad. Nauk SSSR, vol. 311, no. 5, 1035–1039 (1990).

    MathSciNet  Google Scholar 

  2. Askhabov S.N. and Betilgiriev M.A., “Nonlinear integral equations of convolution type with almost increasing kernels in cones,” Differ. Equ., vol. 27, no. 2, 234–242 (1991).

    MathSciNet  MATH  Google Scholar 

  3. Askhabov S.N., Nonlinear Equations of Convolution Type, Fizmatlit, Moscow (2009) [Russian].

    Google Scholar 

  4. Okrasinski W., “On the existence and uniqueness of nonnegative solutions of a certain nonlinear convolution equation,” Ann. Polon. Math., vol. 36, no. 1, 61–72 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  5. Okrasinski W., “Nonlinear Volterra equations and physical applications,” Extracta Math., vol. 4, no. 2, 51–80 (1989).

    MathSciNet  Google Scholar 

  6. Keller J.J., “Propagation of simple nonlinear waves in gas filled tubes with friction,” J. Appl. Math. Phys., vol. 32, no. 2, 170–181 (1981).

    MATH  Google Scholar 

  7. Tikhonov A.N., “On the cooling of bodies during radiation, following the Stefan–Boltzmann law,” Izv. Akad. Nauk SSSR. Ser. Geogr. Geophiz., vol. 3, 461–479 (1937).

    Google Scholar 

  8. Ermentrout G.B. and Cowan J.D., “Secondary bifurcation in neuronal nets,” SIAM J. Appl. Math., vol. 39, no. 2, 323–340 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  9. Askhabov S.N., “Integro-differential equation of the convolution type with a power nonlinearity and inhomogeneity in the linear part,” Differ. Equ., vol. 56, no. 6, 775–784 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  10. Askhabov S.N., “Nonlinear convolution integro-differential equation with variable coefficient,” Fract. Calc. Appl. Anal., vol. 24, no. 3, 848–864 (2021).

    Article  MathSciNet  MATH  Google Scholar 

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Funding

The study was carried out in the framework of the State Task (Project 075–03–2021–071).

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Authors and Affiliations

  1. Chechen State Pedagogical University, Grozny, Russia

    S. N. Askhabov

  2. Kadyrov Chechen State University, Grozny, Russia

    S. N. Askhabov

Authors
  1. S. N. Askhabov
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Correspondence to S. N. Askhabov.

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Translated from Vladikavkazskii Matematicheskii Zhurnal, 2022, Vol. 24, No. 1, pp. 5–14. https://doi.org/10.46698/w9450-6663-7209-q

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Askhabov, S.N. A System of Inhomogeneous Integral Equations of Convolution Type with Power Nonlinearity. Sib Math J 64, 691–698 (2023). https://doi.org/10.1134/S0037446623030163

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  • DOI: https://doi.org/10.1134/S0037446623030163

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