Regularization and Choice of the Parameter for the Third Kind Nonlinear Volterra-Stieltjes Integral Equation Solutions

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.


Introduction
Differential and integral equations theory considering fractional order are relevant in mathematics nowadays, which have numerous applications in various fields, physics, mechanics, control theory, engineering, electrochemistry, bioengineering, viscoelasticity, porous media [1] [2] [3]. Solution of the nonlinear integral equation of Volterra-Stieltjes type, and the method is based on an equivalence relation between the fractional differential equation, and Volterra-Stieltjes integral equation of the second kind was also reported in our previous works [4] [5].
Here we are describing regularization and the choice of the parameter for nonli- where 0 < ε is a small parameter, ( ) ( ) Everywhere we assume that ( ) , , K t s u is representable as Various questions of the theory of the integral equations were investigated in many works. In particular, in [6] linear integral equations of the second kind and their systems on finite and infinite intervals were studied. A survey of results on Volterra integral equations of the second kind was described [7]. The existence of a multiparameter family of solutions proved for linear Volterra integral equations of the first and third kind with smooth kernels [8]. But the fundamental results for the Fredholm integral equations of the first kind were obtained [9], where regularizing operators according to M.M. Lavrentyev were constructed for solving the linear Fredholm integral equations of the first kind. In [10] and [11], Volterra equations of the first kind and inverse problems were investigated. The uniqueness theorems were proved and regularizing operators were constructed according to M.M. Lavrentyev for systems of linear and nonlinear Volterra integral equations of the first kind with nonsmooth matrix kernels [12] [13]. The systems of nonlinear Volterra integral equations of the third kind, uniqueness theorems were proved and regularizing operators were constructed according to M.M. Lavrentyev [14]. In [15], the uniqueness theorems were proved for systems of linear Fredholm integral equations of the third kind, and regularizing operators were constructed according to M.M. Lavrentyev. In [16], based on a new approach, the questions of existence and uniqueness of solutions for systems of linear Fredholm integral equations of the third kind with a singularity at one point on a finite interval were investigated. Based on the approach proposed in [17], the class of Fredholm integral equations of the third kind on a finite interval was studied. Based on the approaches proposed in [18] [19], an improved new approach was developed for studying systems of linear and nonlinear Fredholm integral equations of the third kind with multipoint singularities on a finite interval. In [20], according to the concept of the derivative of a function concerning an increasing function introduced in [19], linear and nonlinear Volterra-Stieltjes integral equations of the first and second kind were investigated. For the solution of one class of linear Volterra, and Volterra-Stieltjes Int. J. Modern Nonlinear Theory and Application integral equations of the third kind, a regularizing operator was constructed according to MM. Lavrentyev and proved the uniqueness theorem [21] [22]. The regularization parameter is chosen for solving the linear Volterra-Stieltjes integral equation of the third kind [4].
Here, to solve the nonlinear Volterra-Stieltjes integral equation of the third kind (1), a regularizing operator which was constructed according to M.M. Lavrentyev, a uniqueness theorem proved, and a regularization parameter was chosen.
Suppose the following conditions are met: , , , t s s G τ ∈ , the following equation is fair: l is a known positive number.
The following estimate is fair