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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.

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 [

Let us consider the equation,

where

Along with Equation (1), we will consider the equation

where 0 < ε is a small parameter,

Everywhere we assume that

where

Various questions of the theory of the integral equations were investigated in many works. In particular, in [

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:

a)

b) If the condition

where

c)

at

at

where

Here

with norm

We will denote

where M is a positive constant depending on

In further the lemmas 1, 2 and 3 are used,

Lemma 1.

Let conditions a) holds and

If

where

Lemma 2.

Let conditions a), b) hold and

The following estimate is fair

Lemma 3.

Let conditions a), c) hold and

If that, the following estimation is fair

Theorem 1.

Let the conditions a), b), c) be satisfied, and Equation (1) has a solution

Then solution

holds. Where

Further let us consider that function

where

Let us consider the equation

From (2) by subtracting formula (11) and introducing the notation

We have

Equation (13) can be written in the form

Using the kernel resolvents

where

It is not hard to be convinced that

Taking into account condition c) and identity (18), from (17) we have

Based on the Equation (10), from (16) we have

Based on Lemma 2,

By estimating

Based on the estimate (20), (6), (21) and taking into account (12), from (15) we have

Further, based on the generalized Gronwall-Bellman inequality [

where

It is known that

Here taking into account (23), we have

where number

where

Assuming

where numbers

Thus, Theorem 2 was proved.

Theorem 2. Let conditions a), b), c) be satisfied, and Equation (1) has a solution

Then the solution

Wherein, Estimate (27) is fair.

Example. Let us consider Equations (1) at

i.e., let us look at the following equation

In this case, conditions a), b), c) of Theorems 1 and 2 are satisfied. Since at conditions

Here

At

the following estimate is fair

After choosing the regularization parameter for solving nonlinear Volterra-Stieltjes integral equations of the third kind, we made the following conclusions:

1) Sufficient uniqueness conditions and regularization of solutions of nonlinear Volterra-Stieltjes integral equations of the third kind were found;

2) The choice of the regularization parameter for solving a class of Volterra-Stieltjes nonlinear equations of the third kind was considered;

3 Uniqueness theorems for solutions proved for the nonlinear Volterra-Stieltjes integral equations of the third kind.

The authors are thankful to Professor A. Asanov for discussions and advice in solving equations.

The authors declare no conflicts of interest regarding the publication of this paper.

Bedelova, N., Asanov, A., Orozmamatova, Z. and Abdullaeva, Z. (2021) Regularization and Choice of the Parameter for the Third Kind Nonlinear Volterra-Stieltjes Integral Equation Solutions. International Journal of Modern Nonlinear Theory and Application, 10, 81-90. https://doi.org/10.4236/ijmnta.2021.102006