The article is considering the third kind of nonlinear Volterra-Stieltjes integral equations with the solution by Lavrentyev regularizing operator. A uniqueness theorem was proved, and a regularization parameter was chosen. This can be used in further development of the theory of the integral equations in non-standard problems, classes in the numerical solution of third kind Volterra-Stieltjes integral equations, and when solving specific problems that lead to equations of the third kind.
References
[1]
Asanov, A., Hazar, E., Eroz, M., Matanova, K. and Abdyldaeva, E. (2016) Approximate Solution of Volterra-Stieltjes Linear Integral Equations of the Second Kind with the Generalized Trapezoid Rule. Advances in Mathematical Physics, 2016, Article ID: 1798050. https://doi.org/10.1155/2016/1798050
[2]
Ezzati, R., Salahshour, S., Yager, R.R. and Khodabin, M. (2014) Fuzzy Linear and Nonlinear Integral Equations: Numerical Methods. Abstract and Applied Analysis, 2014, Article ID: 147351. https://doi.org/10.1155/2014/147351
[3]
Gordji, M.E., Baghani, H. and Baghani, O. (2011) On Existence and Uniqueness of Solutions of a Nonlinear Integral Equation. Journal of Applied Mathematics, 2011, Article ID: 743923. https://doi.org/10.1155/2011/743923
[4]
Bedelova, N. (2020) The Choice of the Regularization Parameter for Solving Linear Volterra-Stieltjes Integral Equations of the Third Kind. In: Popkova, E. and Sergi, B., Eds., Scientific and Technical Revolution: Yesterday, Today and Tomorrow, ISC 2019, Lecture Notes in Networks and Systems, Vol. 129, Springer, Cham, 321-328. https://doi.org/10.1007/978-3-030-47945-9_36
[5]
Asanov, A., Almeida, R. and Malinowska, A.B. (2019) Fractional Differential Equations and Volterra-Stieltjes Integral Equations of the Second Kind. Computational and Applied Mathematics, 38, Article No. 160. https://doi.org/10.1007/s40314-019-0941-2
[6]
Mikhlin, S.G. (1959) Lectures on Linear Integral Equations. Fizmatgiz, Moscow, 234 p.
[7]
Tsalyuk, Z.B. (1977) Volterra Integral Equations. Itogi Nauki i Tekhniki. Ser. Mat. Analysis. Moscow, 15, 131-198.
[8]
Lavrentyev, M.M. (1959) Integral Equations of the First Kind. Dokl. USSR Academy of Sciences, 127, 31-33.
[9]
Bughgеim, A.L. (1999) Volterra Equations and Inverse Problems. VSP, Utrecht, 204 p. https://doi.org/10.1515/9783110943245
[10]
Denisov, A.M. (1999) Elements to the Theory of Inverse Problems. VSP, Utrecht, 272 p. https://doi.org/10.1515/9783110943252
[11]
Asanov, A. (1998) Regularization, Uniqueness and Existence of Solutions of Volterra Equations of the First Kind. VSP, Utrecht, 276 p. https://doi.org/10.1515/9783110943238
[12]
Imanaliev, M.I. and Asanov, A. (1989) On Solutions of Systems of Volterra Nonlinear Integral Equations of the first KIND. Dokl. USSR Academy of Sciences, 309, 1052-1055.
[13]
Imanaliev, M.I., and Asanov, A. (2007) Regularization and Uniqueness of Solutions of Systems of Nonlinear Volterra Integral Equations of the Third Kind. Doklady Mathematics, 76, 490-493. https://doi.org/10.1134/S1064562407040035
[14]
Imanaliev, M.I. and Asanov, A. (2010) Solutions to Systems of Linear Fredholm Integral Equations of the Third Kind. Doklady Mathematics, 81, 115-118. https://doi.org/10.1134/S1064562410010321
[15]
Imanaliev, M.I., Asanov, A. and Asanov, R.A. (2011) A Class of Systems of Linear Fredholm Integral Equations of the Third Kind. Doklady Mathematics, 83, 227-231.
[16]
Asanov, A., Matanova, K.B. and Asanov, R.A. (2017) A Class of Linear and Nonlinear Fredholm Integral Equations of the Third Kind. Kuwait Journal of Science, 44, 17-28.
[17]
Imanaliev, M.I., Asanov, A. and Asanov, R.A. (2017) Solutions to Systems of Linear Fredholm Integral Equations of the Third Kind with Multipoint Singularities. Doklady Mathematics, 95, 235-239.
[18]
Imanaliev, M.I., Asanov, A. and Asanov, R.A. (2018) On a Class of Systems of Linear and Nonlinear Fredholm Integral Equations of the Third Kind with Multipoint Singularities. Differential Equations, 54, 381-391. https://doi.org/10.1134/S0012266118030096
[19]
Asanov, A. (2001) Derivative Function with Respect to Increasing Function. Journal of Natural Sciences KTUM, 1, 18-64.
[20]
Asanov, A. (2002) Volterra-Stieltjes Integral Equations of the Second and First Kind. Journal of Natural Sciences, KTUM, 2, 79-95.
[21]
Asanov, A. and Bedelova, N.S. (2014) One Class of Linear Integral Volterra-Stieltjes Equations of the Third Kind. Bulletin of KazNPU Named after Abay, Almaty, 4, 8-13.
[22]
Liu, Z., Lee, S. and Kang, S.M. (2012) Solvability of Nonlinear Integral Equations of Volterra Type. Abstract and Applied Analysis, 2012, Article ID: 932019. https://doi.org/10.1155/2012/932019
