Nonlinear Fredholm Functional-Integral Equation of First Kind with Degenerate Kernel and Integral Maxima


  • T. K. Yuldashev Tashkent State University of Economics (TashSUE), Karimov street, 49, Tashkent, 100066 Uzbekistan, Tashkent, Uzbekistan
  • Z. K. Eshkuvatov ᵇFaculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu (UMT), Kuala Terengganu, Terengganu, Malaysia; ᶜFaculty of Applied Mathematics and Intellectual Technologies, National University of Uzbekistan (NUUz), Tashkent, Uzbekistan
  • N. M. A. Nik Long Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia (UPM), Serdang, Selangor, Malaysia



Fredholm functional-integral equation, nonlinear equation, degenerate kernel, integral maxima, boundary conditions, regularization, small parameter


In this note, the problems of solving by the aid of regularization method for a nonlinear Fredholm functional-integral equation of the first kind with degenerate kernel and integral maxima are considered. In using the regularization method, we denote the nonlinear function as a new unknown function. This method we combine with the method of the degenerate kernel. Fredholm functional-integral equation of the first kind is ill-posed (non-correct) problem. We have used boundary conditions to ensure the uniqueness of the solution. We transform the implicit functional equation with integral maxima to another type of nonlinear functional-integral equation of the second kind by some integral transforms. Using the method of successive approximations and the method of compressing mapping, the theorem on single valued solvability of the problem is proved. It is obtained the necessary and sufficient conditions of existence and uniqueness of the solution of the problem with integral maxima. Two simple examples are analyzed with an exact and approximate solution.


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