Kurchatov-type methods for non-differentiable Hammerstein-type integral equations

We consider a generic type of nonlinear Hammerstein-type integral equations with the particularity of having non-differentiable kernel of Nemystkii type. So, in order to solve it we consider a uniparametric family of iterative processes derivative free, with the main advantage that for a special value of the involved parameter the iterative method obtained coincides with Newton’s method, that is due to the fact of evaluating the divided difference operator when the two values are the same. We perform a qualitative convergence study by choosing an auxiliary point, that allow us to obtain the existence and separation of solutions of the given equation, that is, local and semilocal convergence balls can be obtained.


Introduction
In this paper, we focus on the study of the existence of a solution for nonlinear Hammerstein-type integral equations, as well as the uniqueness of this solution and its approximation. So, we consider nonlinear Hammerstein-type integral equations of the form [5,28,31] where h ∈ C[a, b] , the kernel K(s,t) is a known function in [a,b] × [a,b], N is the Nemytskii operator N ∶ C[a, b] → C [a, b] such that N( )(x) = N( (x)) , where N ∶ ℝ → ℝ , and z is the unknown function to be determined. In our case, we will require that the Nemystkii operator N be a simply continuous operator.
A commonly used procedure to prove the existence and uniqueness of a solution of the (1) consists in transforming said equation into an equivalent fixed point problem (see [30,31]). Thus, considering the operator T ∶ Ω ⊆ C[a, b] → C[a, b] with by using a fixed point theorem, it is proved that the method of Successive Approximations converges to a fixed point of (2) and therefore a solution of (1). Other iterative methods are also used for this purpose. Thus, for example, the Picard method has been used (see [10]), testing its convergence to a solution of the equation G(z) = 0 , equivalent to (1), where we consider the operator G ∶ Ω ⊆ C[a, b] → C[a, b] with Other methods such as Newton's method [12] and Newton-type methods [7,25,29] have been used to prove the existence and uniqueness of a solution for the (1). These studies are based on the qualitative results, of existence and uniqueness of solution, provided by the study of the convergence of iterative processes considered.
In our case, since the kernel of Nemystkii N is a continuous operator, there may be a possibility that it is not differentiable. In [16], the non-differentiable case has been studied. Also in this work, we will consider an iterative process that does not use derivatives in its algorithm, that is, a derivative-free iterative process. So, one of the aims of this paper is the qualitative study that we can obtain from the uniparametric family of iterative processes (4) ⎧ ⎪ ⎨ ⎪ ⎩ z 0 , z −1 given in Ω, ∈ [0, 1], x n = (1 − )z n + z n−1 , y n = (1 + )z n − z n−1 , z n+1 = z n − � x n , y n ;G � −1 G(z n ), n ≥ 0.
Note that this uniparametric family of iterative processes can be considered as a combination of the Kurchatov method [4,22](λ = 1) and, for differentiable case, Newton's method [8] (λ = 0). But, the main advantage for considering this scheme is that we have a family of derivative-free iterative methods that can be used in the non differential case. We use a first-order divided difference [1,6,14]. It is well known that, if we denote by L(X, Y) the space of bounded linear operators from X to Y, an operator x, y;D ∈ L(X, Y) is called a first-order divided difference for the operator D ∶ Ω ⊆ X → Y on the points x and y (x≠y) if To realize the qualitative study for the uniparametric family of iterative processes (4), we do an analysis of the convergence for (4), so that we can obtain the existence of a solution of (1) in a certain domain. Moreover, we obtain a result on the uniqueness of solution that allows separating solutions of (1). We analyze the convergence of the method by a technique based on recurrence relations that use an auxiliary function [9,11]. So, we use the theoretical results obtained from the convergence of the method (4) to draw conclusions about the existence and separation of solutions of (1).
Other of the aims of this paper is to approximate a solution of (1). To obtain this objective in an appropriate way, we have considered the family of iterative processes given in (4), which is formed by iterative processes with quadratic convergence and low operational cost, therefore efficient iterative processes that also have good accessibility [19] Our main result in the paper is to perform a complete convergence study of the scheme that we state and prove in Section 2.1 along with some necessary lemmas. In Section 2.2 we introduce the assumptions to make the semilocal convergence analysis and give some preliminary results before to set the main theorem. Then, in Section 2.3 by choosing an adequate auxiliary point we get the local convergence results.
In Section 3, we apply the theoretical results obtained in previous sections for obtaining domains of existence and uniqueness of solutions for the nonlinear Hammerstein-type integral equation, giving in the last subsection a numerical experiment.
Finally, in Section 4 we drawn some conclusions.

Convergence of Kurchatov-type methods
In this section, to make our study of convergence as general as possible, we consider a nonlinear equation where G ∶ Ω ⊆ X → Y is a continuous operator defined on a nonempty convex subset Ω of a Banach space X to a Banach space Y.
In previous works [2,3,8,13], the convergence analysis for these kind of iterative methods, (4), have been analyzed from two different points of view. On the one hand, the semilocal convergence study, where we assume conditions on the initial guess z 0 and on the operator G, in order to obtain the existence ball, that is, this process assures the existence of solution z * of the equation G(z) = 0 remaining all the iterates and the solution in the cited ball. On the other hand, the local convergence study, where we must assume the existence of a solution z * of G(z) = 0 and then, with additional assumptions on the involved operators, we obtain the convergence domain, that is the ball centered at the z * where we can take a possible starting guess for the iterative process. Moreover, we also have in the literature the technique based on auxiliary points [9,10]. We use it in this paper. So, we assume some conditions on the operator G and on an auxiliary point z in Ω for getting the existence of a solution z * of G(z) = 0 and to prove the convergence of (4) to z * . So, this technique is much general driving us to obtain results of semilocal and local convergence for iterative process (4) by choosing adequately two particular auxiliary point z.

Convergence from an auxiliary point
Throughout our study, we will assume that there is a first-order divided difference in the Banach space X. Now, we perform a convergence result for the iterative process defined in (4), for a fixed value λ ∈ (0,1], by assuming that G is a continuous operator in Ω and such that the following conditions are satisfied. ‖ ≤ (‖x − u‖, ‖y − v‖) , where ∶ ℝ + × ℝ + → ℝ + is a continuous nondecreasing function in its two arguments.
We notice that from (C3), we deduce the following condition: Therefore, (C3 ′ ) is not an extra condition.
Secondly, we will consider that z k+ 1 ≠z k for all k ⩾ 0 , because, in other case, z k+ 1 = z k for some k ⩾ 0 , and then the sequence {z n } converges to z * with z * = z n = z k+ 1 = z k for all n ⩾ k + 2. Moreover, if z k+ 1 ≠z k we obtain that x k+ 1 ≠y k+ 1 . Therefore, the operators [x k+ 1 ,y k+ 1 ;G] are always well defined.
Finally, with respect to the first-order divided differences, [1], we include the boundedness process by means of ω-functions, that is used in the non-differentiable case, (see [17]), and it is a generalization of the case in which [x,y;G] is Lipschitz-continuous or Hölder-continuous condition [20]. In the above cases, the Fréchet derivative of G exists in Ω and satisfies [x, x;G] = G � (x) , see [1]. Moreover, it is already well known, see [18], that if ψ(0,0) = 0 then G is differentiable, so in non-differentiable situations we have that ψ(0,0) > 0.
We will begin our convergence study by considering n = 0. Firstly, notice that and So, it follows that x 0 , y 0 ∈ B(z, + r + ), and as x 0 ≠y 0 then [x n ,y n ;G] is well defined.
Secondly, by using (C3 ′ ), we have that Therefore, by applying Banach Lemma, one gets the existence of [x 0 ,y 0 ;G] − 1 and Next, we consider two technical lemmas that we use later.
Lemma 1 If z n ,z n+ 1 ∈Ω, then Proof As z n+ 1 ≠z n and x n ≠y n , then [z n+ 1 ,z n ;G] and [x n ,y n ;G] are well defined.

Lemma 2
Let G be a continuous operator in Ω such that the conditions (C1)-(C5) are satisfied, z n ,z n− 1 ∈ B(z 0 ,r) and ∥z n − z n− 1 ∥≤∥z 1 − z 0 ∥, for n ⩾ 1 , then x n , y n ∈ B(z, + r + ), and there exists [x n ,y n ;G] − 1 with Proof Firstly, notice that and It follows that x n , y n ∈ B(z, + r + ), and as x n ≠y n then [x n ,y n ;G] is well defined.
Secondly, by using (C3 ′ ) and (10), we have that Therefore, by applying Banach Lemma, one gets the existence of [x n ,y n ;G] − 1 and the result is obtained.
On the one hand, we have On the other hand, To continue, we prove (iv 3 ), this is that iterate z 3 remains in the ball B(z 0 ,r). For this, noting that the fact of being g(r) < 1 allow us to sum a geometric progression of reason g(r), we have In order to complete the proof we apply an inductive procedure. So, we suppose that the items are satisfied, for k ≥ j ≥ 3 and similarly to the case j = 3, we prove that these items hold for j = k + 1.
Proof From the recurrence relations given in Lemma 3 we only have to prove the convergence of the sequence {z n }, given by (4). As X is a Banach space, we will see that {z n } is a Cauchy sequence. For this, we consider By taking limits when n → ∞ , then ‖z n+k − z n ‖ → 0. Hence, {z n } is a Cauchy sequence which converges to z * ∈ B(z 0 , r).
Moreover, from the following by the continuity of the operator G and as g(r) k− 1 tends to zero when k tends to infinity, we have that G(z * ) = 0.
Notice that B(z 0 , r) is a domain of existence of solution for G(z) = 0.
To prove the uniqueness we give the following result, We assume that there exists r 1 ≥ r such that Hence, by Banach lemma the operator Q − 1 exists and as then z * = y * .
Note that it may happen that the hypotheses necessary to ensure convergence are not verified for all values of λ ∈ (0,1].

Semilocal convergence
To establish another result for semilocal convergence, we consider z = z −1 under the following conditions: with x≠y and u≠v, where ∶ ℝ + × ℝ + → ℝ + is a continuous non decreasing function in its both arguments. (S3) The scalar equation where we have η > 0, for and has at least one positive real root and we denote by r the smallest positive root. (S4) B(z 0 , r + ) ⊆ Ω and 0 < g(r) < 1.
Proof To prove the convergence of the sequence, it is sufficient to prove that the sequence {z n } is a Cauchy sequence. For this, we consider As n → ∞ , then ‖z n+k − z n ‖ → 0, hence, {z n } is a Cauchy sequence which converges to z * ∈ B(z 0 , r). Now, As n → ∞ , we obtain G(z * ) = 0 by using the continuity of G. To prove the uniqueness, let y * be another solution of G(z) = 0 in B(z 0 , r 1 + ) ∩ Ω. Define Q = [z * ,y * ;G], then Thus, by the Banach Lemma, Q − 1 exists and hence, z * = y * .
As previously, we notice that from (LC3), we deduce the following condition: with x≠y, where 0 ∶ ℝ + × ℝ + → ℝ + is a continuous non decreasing function in both arguments.
Firstly, we present a result on the inverse of the divided difference of the operator G.

Lemma 10
Assume that the conditions (LC1)-(LC4) hold, then, for n ≥ 1, it follows: Next, by using Lemma 10, we obtain a local convergence result for the sequence {z n } given by (4).
Proof From the previous Lemma, the sequence {z n } is well defined remains in B(z * ,μ) and converges to z * . To prove the uniqueness of solution, suppose that y * ∈ B(z * ,r 1 ) ∩Ω be such that G(y * ) = 0. Then by using (LC2 ′ ), we get Therefore, [z * ,y * ;G] − 1 exists and hence z * = y * .

Non-differentiable Hammerstein-type integral equations
One of the most important mathematical tool to describe applied problems is related with nonlinear integral equations. We can mention among others science applied problems, fracture mechanics problems, aerodynamics, the theory of porous filtering, antenna problems in electromagnetic theory and others. These complex and practical situations can be formulated as integral equations of the first, second and third kind. It is well known that, the obtainment of a solution of these equations are used to be very difficult and sometimes impossible, so numerical procedures to approximate the solutions are the angular stone in Numerical Methods. In the literature, we find different Fredholm-type integral equations [27,31], Volterra-Fredholm integral equations [15,26], nonlinear Fredholm integro-differential equations [23], systems of Fredholm-Volterra integral equations [24], etc. Next, we deal with the following case of nonlinear Hammerstein-type integral equation [7,25,27]: In this section, our first objective is to carry out a qualitative study of (22), obtaining domains of existence and uniqueness of solutions. The second objective is to use an iterative process of (4) for a fixed value λ ∈ (0,1], and by direct application to approximate a solution of (22). For this, we observe that the (22) can be defined as Obviously, a solution of G(z) = 0 is a solution of (22).
It is clear that, in order to apply an iterative process of (4) for a fixed value λ ∈ (0,1], we will first need to define a first-order divided difference for the G operator. Taking into account that N is a non-derivable real function, we can define the first-order divided difference in [a, b] ⊂ ℝ given by Then, from this definition, we can define [x, where we consider Therefore, for the continuous real functions x and y(x≠y), obviously x, y;G ∈ L(Ω, C([a, b])) and Then, x, y;G is a first-order divided difference for the operator G ∶ Ω ⊆ C([a, b]) → C ([a, b]). In our study, the max-norm has been considered in C([a, b]).

Domain of existence of solution
As a consequence of Theorem 4 we saw that for each z 0 ∈ B(z, ) and z − 1 ∈ B(z 0 ,α) such that the conditions (C1)-(C5) are satisfied, from the iterative processes given in (4), we obtain a domain of existence of solution, B(z 0 , r) , for the equation G(z) = 0. Therefore, we study what conditions must be verified on the operator G , given in (23) from the integral (22), so that the corresponding conditions (C1)-(C5) are satisfied and then, we apply the Theorem 4.
Firstly, we suppose that z 0 ,z ∈ Ω, withz 0 ≠z, such that[z, z 0 ;G] −1 exists. Notice that as z 0 ≠z then, [z, z 0 ;N] is well defined and if I is the identity on C ([a, a]) , we have x, y;G (x − y) = G(x) − G(y).
Thus, if ‖[z, z 0 ;N]‖ < 1∕(� �M), by the Banach Lemma for inverse operators [21], we obtain that there exists [z, z 0 ;G] −1 with so, we denote = Secondly, in order to analyze the domain of existence of iterative processes that do not use derivatives in their algorithms, the conditions are usually required on the operator divided difference. For this, we have that for each pair of distinct points x,y ∈Ω, there exists a first-order divided difference of G at these points given in (24). Then, if we suppose that the following condition hold: where ∶ ℝ + × ℝ + → ℝ + is a continuous nondecreasing function in its two arguments, then we deduce that and therefore Notice that, from (26), we deduce the following condition: For all x,y ∈Ω, with x≠y holds: where 0 ∶ ℝ + × ℝ + → ℝ + is a continuous nondecreasing function in its two arguments.
Next, we establish the following convergence result for the sequence {z n }, given in (4). This sequence converges to a solution of equation G(z) = 0 with G given in (23). (23). Fixed λ ∈ (0,1], we suppose that the following conditions are satisfied:

Theorem 12 Let G be the continuous operator in Ω ⊆ C([a, b]) given in
where and has at least one positive real root and we denote by R the smallest positive real root.

Proof
The idea of the proof is to apply Theorem 5 to the operator G given in (23). To finish, note that, for α = βδ, the real functions ψ(−,−) and ψ 0 (−,−) indicated previously and r = R, from conditions (IV) and (V), it follows that the conditions (C4) and (C5) are also satisfied. Therefore we can apply the Theorem 5 and the result is proved.
Note that, the ball B(z 0 , R) is the domain of existence of solution for the equation G(z) = 0.

Domain of uniqueness of solution
To obtain the domain of uniqueness of solution for the (23), it is enough to apply the Theorem 5 taking into account the Theorem 12 that we have just proved.
Theorem 13 Let G be the continuous operator in Ω given in (23), z 0 ∈ B(z, ) and z − 1 ∈ B(z 0 ,βδ) such that the conditions (I)-(V) are satisfied. We assume that there exists R 1 ≥ R such that then, the limit point z * is the only solution of equation Therefore, B(z 0 , R 1 + ) ∩ Ω is the domain of uniqueness of solution for the equation G(z) = 0.

Numerical experiment
Next, we present a numerical experiment where we illustrate all the above results. We consider the following nonlinear and non-differentiable integral equation of Fredholm-type of the form given in (22): this is, for a fixed value ∈ ℝ , we consider K(s, t) = st and N(z)(t) = z(t) 2  On the one hand, note that, if we consider z 0 ≠z with z 0 ,z ∈ Ω and I is the identity on C ( On the other hand, in this case, N( ) = 2 − | | 5 and from (25), it is easy to check that therefore, for condition (III) we have ( , ) = 2 5 + + , a non decreasing real function in its two variables.
Firstly, fixed μ = 1, and taking different values for z − 1 and z 0 , by applying the theoretical results obtained in Theorems 6 and 7 we get the results given in Tables 1 and 2 . In this case, we observe that taking smaller values for the output points, better domains of existence and uniqueness are obtained. Besides, as one can check, when λ decreases the values slightly improve. The ball of existence (see R) is smaller, that is, we locate the solution better. While the ball of uniqueness (see R 1 ) grows, so we better separate the solutions.
Secondly, for μ = 1, μ = 1/3 and μ = 1/5 with z = 0, z 0 = 1∕2 and z − 1 = 1/3μ, by applying the theoretical results obtained in Theorems 6 and 7 we get the results given in Tables 2, 3 and 4. This study allows us to affirm that, reducing the value of μ better domains of existence and uniqueness are obtained. Furthermore, in each case, we can observe that when λ decreases the radii of existence and uniqueness balls slightly improve. In general, since this situation is not differentiable, we observe that by approaching Newton's method (λ = 0) with the iterative processes given in (4), we obtain better qualitative results.
Next, our interest is focused on approximating a solution of the (22), for this we will apply the iterative processes (4) to the operator G , given in (23), and thus obtain a solution of G(z) = 0 . In first place, we will need to calculate [x n , y n ;G] −1 . For this, taking into account (24), for u, v, x, y ∈ Ω = C([0, 1]) , with x≠y, we consider where w(s) = s 2 . So, we obtain and then as long as ∫    (22) by means the iterative process (4), with λ ∈ (0,1], we take z − 1 ,z 0 ∈Ω and as x n (t) + y n (t) = 2z n (t), fixed ∈ ℝ we apply the following algorithm for n ⩾ 0: We implement this algorithm working with Matlab R2019a setting variable precision arithmetic with 60 digits, by choosing = 1/10, z 0 = 1/2 and z − 1 = 1/3. So, by imposing as a stopping criteria ∥z n+ 1 (s) − z n (s)∥≤ 10 − 30 we obtain the exact solution z * . With the results of Table 5, it can be checked that for smaller values of λ the results show a slight improvement in accuracy although in all cases computational order of convergence p = 2 is reached with the same number of iterations, iter. Also, we can compare the distance between the last 2 iterates, ||z n+ 1 (s) − z n (s)||, the value of the function at the approximation solution, x n (t) − y n (t) [G(z n )](t))dt.
C n = ∫ Now, by following the theoretical study performed in section 2.3 we obtain the convergence ball centered at the exact solution z * (s) = s − 1/2. We take the values z 0 (s) = 1/2 and z − 1 (s) = 1/3 so, α = ||z − 1 (s) − z 0 (s)|| = 1/6 and then by imposing condition (LC4) we solve the equation: which smallest positive root gives us the value of μ = 1.7548, then we verify that 2α ≤ μ, that is we obtain the local convergence ball B(z * ,1.7) in case λ = 0.8. In Table 6 we can check the radius for other values of parameter λ. We notice that the domain of convergence is better for smaller values ofλ.

Conclusions
In this work, we consider a uniparametric family of iterative processes that are derivative free, so we can apply them to solve non-differentiable problems, as is the case of nonlinear Hammerstein-type integral equations with the Nemystkii operator continuous but maybe non-differentiable. We perform a qualitative convergence study with the particularity of using the technique based on auxiliary points, that allow us to obtain local and semilocal convergence balls. Finally, we apply the theoretical results for proving the existence of solution of an apply problem, obtaining the domain of existence and uniqueness. Moreover, the accessibility region is also provided.

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Competing interests
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