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 nonlinear Volterra-Stieltjes integral equations of the third kind.

Let us consider the equation,

where are given functions, , are non-decreasing continuous functions on, is an unknown function on, is an increasing continuous function.

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

where 0 < ε is a small parameter,.

Everywhere we assume that is representable as

,

where

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 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:

a), , at

b) If the condition for any function, the following equation is fair:

where is a known positive number.

c)

at,

at for any following equation is fair:

where is a known positive number.

Here is the space of all continuous functions, determined on

with norm

We will denote linear space of all functions, determined on and satisfying condition

where M is a positive constant depending on, but not on thet and s.

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

Lemma 1.

Let conditions a) holds and

If, then

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 of the Equation (2) converges in the norm to for and the estimate

holds. Where

Further let us consider that function and number, in agreement with

where and are constant values.

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, and generalized Dirichlet formula [15], Equation (14) is reduced to the following equivalent equation

where was determined in the lemma 2,

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, estimate (6) is fair.

By estimating. Taking into account conditions a) and c) from (19) we obtain

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

Further, based on the generalized Gronwall-Bellman inequality [6], from (22) we obtain the following estimate

where

It is known that

Here taking into account (23), we have

where number determined by the formula (24).

where , numbers K, M and were determined in Theorem 1.

Assuming from (26) we obtain

where numbers are determined in Equations (24) and (26).

Thus, Theorem 2 was proved.

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

Then the solution in Equation (11) converges at the norm

to.

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, , the following estimate is fair

Here.

At for

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