Convergence of a Two-Step Iterative Method for Nondifferentiable Operators in Banach Spaces

The semilocal and local convergence analyses of a two-step iterativemethod for nonlinear nondifferentiable operators are described in Banach spaces. The recurrence relations are derived under weaker conditions on the operator. For semilocal convergence, the domain of the parameters is obtained to ensure guaranteed convergence under suitable initial approximations. The applicability of local convergence is extended as the differentiability condition on the involved operator is avoided. The region of accessibility and a way to enlarge the convergence domain are provided.Theorems are given for the existence-uniqueness balls enclosing the unique solution. Finally, some numerical examples including nonlinear Hammerstein type integral equations are worked out to validate the theoretical results.


Introduction
Let  * be the unique solution of where  : D ⊆ X → Y is a continuous nonlinear operator defined on a nonempty convex subset D of a Banach X with values in a Banach space Y.This problem frequently occurs in numerical analysis.Many scientific and real-life problems can be formulated mathematically in terms of integral equations, boundary value problems, equilibrium theory, optimization, and differential equations whose solutions require solving (1).The solutions of discrete dynamical systems also require solving them in order to represent the equilibrium states of these systems.With the existence of high-speed computational devices which solve them faster and with more accuracy, the problem for solving nonlinear equations has further gained and added advantages.Generally, iterative methods along with their convergence analysis are used to find the solutions of these equations.Many researchers [1][2][3][4][5][6] have extensively studied these problems and proposed many direct and iterative methods for their solutions along with their semilocal [1][2][3]7], local [8,9], and global convergence analysis [10].In semilocal convergence analysis, we impose conditions on starting points while local convergence analysis requires the condition on the solution.
Global study of convergence generally depends on the type of operators involved.The well-known quadratically convergent Newton's method [11,12] used for (1) is given by where  0 ∈ D and Γ  =   (  ) −1 ∈ (Y, X).Here, (Y, X) denotes the set of bounded linear operators from Y into X.
In [13], a family representing third-order iterative methods for (1) is given for  ≥ 0 by where  ,0 is the starting iterate and   () =   () −1   ()  () −1 ().This family contains the Chebyshev method ( = 0), the Halley method ( = 1/2), 2 Complexity and the Super-Halley method ( = 1), respectively.These methods and many others use the differentiability of .Not much work is done by using nondifferentiability of  which can be expressed in the form  () =  () +  () = 0, where ,  : D ⊆ X → Y are continuous and nonlinear operators. is differentiable while  is continuous only and not differentiable.In [14], a quadratic order iterative method for (4) is given by where  −1 and  0 are two starting points and [, ; ] ∈ (X, Y) satisfying [, ; ](−) = ()−() for ,  ∈ D and  ̸ = .Consider the following two-step difference differential method [15] for solving (4) given by where  0 ,  0 ∈ D are two starting points and Its local convergence analysis with super quadratic order is described.The differentiability condition on   ( * ) which restricts the applicability of ( 6) is also used.Moreover, no numerical examples were worked out.Recently, some special cases of (6) are also studied.One special case of ( 6) is given in [1,16] for  = 0. Another one is given in [17] for  = 0.The importance of (6) lies in the fact that it uses the memory of (  ) in each iterate.If we consider the evaluation of (  ) as one function evaluation, then the total number of function evaluations of (6) for solving (4) coincides with that given by (5).However, the rate of convergence of ( 6) is much faster.This is shown by the following example.
Example 1.Consider where A comparison of the absolute error approximation obtained by ( 6) and ( 5) with tolerance ‖ * −   ‖ < 10 −20 is given in Table 1.
In this study, the semilocal and local convergence analyses of a two-step iterative method for nonlinear nondifferentiable operators are described in Banach spaces.The recurrence relations are derived under weaker conditions on the operator.For semilocal convergence, the domain of parameters is obtained to ensure guaranteed convergence under suitable initial approximations.The applicability of local convergence is extended as the differentiability condition on the involved operator is avoided.The region of accessibility and a way to enlarge the convergence domain are provided.Theorems are given for the existence-uniqueness balls enclosing the unique solution.Finally, some numerical examples including nonlinear Hammerstein type integral equations are worked out to validate the theoretical results.
The paper is organized as follows.The Introduction forms Section 1.In Section 2, semilocal convergence of ( 6) is established.In Section 2.1, some special cases and domain of parameters are given to ensure the initial points for guaranteed convergence of (6).In Section 3, local convergence analysis of ( 6) is established.Some special cases and region of accessibility are also discussed.In Section 4, some numerical examples including nonlinear Hammerstein type integral equations are given to validate the theoretical results.Finally, conclusions are given in Section 5.

Complexity
Using Lemma 2 and (6), we get Following in a similar manner, it can be derived that This implies (i)-(iii).(iv) and (v) can easily be derived using  and (i)-(iii) recursively.Thus, the lemma is proved.

Some Special Cases and Domain of Parameters.
In this subsection, some special cases of Theorem 4 and the iterative method (6) are presented.We find the domain of parameters to get the set of initial approximations for the guaranteed convergence of (6) for  = 0. Consider  : R  → R  , given by where  is a nonlinear vector function of size  × 1,  is a matrix of size  × ,  = ( where  1 = ( This gives This shows that  +1 ∈ B( 0 , ).Similarly, we can have that  +1 ∈ B( 0 , ).Now, we obtain that {  } is a Cauchy sequence.For fixed  and  ≥ 1, we get Thus,   →  * as  → ∞.Now, to show that  * is a solution of (4), we get      (  )     ≤        −  −1     , where  = max ( 1 ,  2 ) (28) and ‖  −  −1 ‖ → 0 as  → ∞.This gives ( * ) = 0. Uniqueness of  * can be shown in a similar manner given in Theorem 4. This proves the theorem.Now, we present the domain of the parameters associated with Theorem 5.The domain of the parameters represents the set of all those points in  plane that allow the guaranteed convergence of (6) from the initial conditions used in Theorem 5. Transform (22) into the quadratic equation Following [2], it is easy to see that (29) has two positive real roots, if Using (30), it is necessary to take 2 1  +  2  < 1 for the existence of positive real roots.Moreover, the smallest positive real root is given by  6) for  = 0.5 and  1 = 0, 0.2, 0.4, 0.6, 0.8 represented by orange, blue, yellow, red, and green colors, respectively.
Next, we take  = ,  2  = , and  =  2  and draw the domain of the parameters which gives the relation between some initial estimations.For this, we follow the criteria of Theorem 5 distinguished by two different cases.The first case is when  1 = 0 (differentiable case) and the second case is when  1 ̸ = 0 (nondifferentiable case).It can be seen in Figure 1 that, with the increasing value of  1 , we get a smaller region of the domain of parameters.By treating  2 as a constant, it can be concluded here that the decrease of the value of  increases the domain of the parameter.This can be verified from Figure 2.
3.1.On the Accessibility and Some Special Cases.In this section, we present some special cases of Theorem 7 and ( 6).We establish the region of accessibility for  * .A solution  * is said to be accessible from those points  0 and  0 if the sequences {  } and {  } given by ( 6) converge to  * .The set of a combination of all such points for which the sequences {  } and {  } converge to  * is called the region of accessibility of  * .We use here Theorem 7 to find the region of accessibility of (6).We consider here  = 0 and replace the condition (L 2 ) by It is indicated in Section 2.1 that this type of condition arises for nondifferentiable operators.Now, we present the local convergence analysis of (6) using condition (38).If condition (38) is used, then * should be the real positive number which is possible only when To verify condition (40), we draw the region by taking  1 =  and  2 =  and then taking different values of  to see the difference between the convergence regions.This can be seen by Figure 3. Now, using (40), it is observed that the condition (L 5 ) is equivalent to the condition for such  * , which is Obviously, the condition satisfying (40) satisfies (41).This can be seen in Figure 4. We come to the conclusion that, with a smaller distance between  * and , a larger domain is achieved.
Proof.The proof follows the same lines as the theorems given above.So, we omit the proof here.

Numerical Experiments
In this section, some numerical examples are given to demonstrate the applicability and efficacy of our work.

Conclusions
In this paper, we established a new convergence analysis of the two-step iterative method for solving nonlinear nondifferentiable equations.Using some recurrences, we analyze the semilocal as well as local convergence analysis for this method.The novelty of this work lies in the fact that it avoids the differentiability condition in the convergence analysis for nondifferentiable operators, a contrast with earlier studies to take into account.This is very important for the practical purpose.In semilocal convergence analysis, theorems are given for existence-uniqueness balls.Moreover, the domain of the parameters is given to show the guaranteed convergence of the method and suitability of the starting points.
In local convergence analysis, we avoid the differentiability condition on the involved operator as a contrast to the earlier study.This way, the applicability of local convergence theorem is extended.Theorems are provided for the existenceuniqueness ball.Furthermore, its region of accessibility is given and an idea for enlarging the convergence domain is provided.Finally, some numerical examples including nonlinear Hammerstein type integral equations are given to validate the theoretical results obtained by us.

Table 1 :
Comparison of absolute error.