Abstract
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 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.
1. Introduction
Let be the unique solution ofwhere is a continuous nonlinear operator defined on a nonempty convex subset of a Banach with values in a Banach space . 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 [1–6] have extensively studied these problems and proposed many direct and iterative methods for their solutions along with their semilocal [1–3, 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 and . Here, denotes the set of bounded linear operators from into . In [13], a family representing third-order iterative methods for (1) is given for by where is the starting iterate and . This family contains the Chebyshev method , the Halley method , and the Super-Halley method , 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 formwhere 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 bywhere and are two starting points and satisfying for and .
Consider the following two-step difference differential method [15] for solving (4) given bywhere 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 . Another one is given in [17] for . 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 .
Take and .
A comparison of the absolute error approximation obtained by (6) and (5) with tolerance 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.
2. Semilocal Convergence
This section describes the semilocal convergence of (6) for solving (1). Let and denote the open and closed balls with center at and radius , respectively. Let and be suitably chosen initial approximations and , , , and be some positive real numbers. Define a class , where and are to be defined. The triplet if(C1) for ;(C2) such that and ;(C3);(C4), where is a continuous and nondecreasing function for ;(C5),where is a continuous and nondecreasing function in its both arguments for , ;(C6)the equation where , , , and , has at least one positive root. The smallest positive root is denoted by ;(C7);(C8), where .
Lemma 2. For operators , , and defined in (4) and (5), it follows that, for all ,
Proof. The proof is simple and hence omitted here.
Lemma 3. Under conditions – and parameters introduced in Section 2, we have that if , then the following recurrence relations hold for :(i)There exists such that .(ii).(iii).(iv).(v).
Proof. This lemma can be proved by mathematical induction. Clearly, we have , and using Lemma 2, we get Now, Thus, . Let us assume that the induction is true for some . Then, This gives . Now, Using Banach Lemma on invertible operators [4], we get 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.
Theorem 4. Let verify conditions – such that . Then, starting with , , the sequences and generated by (6) are well defined, remain in , and converge to the solution of (1). Moreover, is unique in .
Proof. From Lemmas 2 and 3, it is clear that and are well defined and belong to . First, we shall show that is a cauchy sequence. For fixed and , we get This gives as . Now, to show that is a solution of (1), we have and as . This gives . Next, to show the uniqueness of , let us assume that is another solution of (1) in . For , we get Using Banach Lemma on invertible operators, we get that exists and . Taking norm on both sides on , we get that . This shows the uniqueness and thus the theorem is proved.
2.1. 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 . Consider , given by where is a nonlinear vector function of size , is a matrix of size , , and . It can be observed thatwith and . Earlier studies [1, 15, 16] for nondifferentiable operators do not satisfy this condition. For , this condition holds for differentiable operators. Now, we can take as a special case of condition (C5) of class .
Theorem 5. Let and be the smallest positive real number satisfyingwhere , , and Suppose that , , and . Starting with and , the sequences and generated by (6) are well defined and converge to of . Moreover, is unique in .
Proof. From , we get . Take . The proof is given by mathematical induction. Let , and for . Now, Using Banach Lemma, it is given that Again, This givesThis shows that . Similarly, we can have that . Now, we obtain that is a Cauchy sequence. For fixed and , we get Thus, as . Now, to show that is a solution of (4), we get and as . This gives . 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 equationFollowing [2], it is easy to see that (29) has two positive real roots, if Using (30), it is necessary to take for the existence of positive real roots. Moreover, the smallest positive real root is given by
Next, we take , , and 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 (differentiable case) and the second case is when (nondifferentiable case).
It can be seen in Figure 1 that, with the increasing value of , we get a smaller region of the domain of parameters. By treating 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.
(a)
(b)
3. Local Convergence
In this section, we shall establish the local convergence analysis of (6). Recently, this is given in [15] under the condition on , which is much more restrictive as it includes the differentiability of divided difference terms involved in (6). This restriction is removed and the following set of conditions is defined as class . We say that the triplet if we have the following.(L1), where is a continuous and nondecreasing function defined on with values in for .(L2), where is a continuous and nondecreasing function in its both arguments defined on with values in for .(L3)Let be such that . Take and so that the operator is invertible and .(L4)The equation has a positive real root. Let the minimum positive root be denoted by .(L5) and .
Lemma 6. Let . If , and , then the operator is invertible and .
Proof. We prove this lemma using the definition of class . We consider Using Banach lemma on invertible operators, it follows that exists and
Theorem 7. If and Lemma 6 holds, then starting with , the sequences and generated by (6) are well defined and converge to the solution of (1). Moreover, the solution is unique in . Furthermore, the following holds: where and .
Proof. Clearly, and exists from Lemma 6. Now, From , we get and . Now, Thus, and . Proceeding in a similar way, it follows that for each . This gives and consequently as . It remains to show the uniqueness of . It can be proved in a similar manner as proved in Theorem 4. This proves the theorem.
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 and 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 and replace the condition (L2) byIt 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 whenTo verify condition (40), we draw the region by taking and 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 (L5) is equivalent to the condition for such , which isObviously, 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.
Let us discuss the local convergence of (6) which does not use the differentiability condition.
Theorem 8. Let be a nonlinear operator. Suppose that (D1)there exists such that ; choose so that so that the operator is invertible and ;(D2);(D3). Then, starting with , the sequences and given by (6) are well defined and converge to the unique solution of (4), where .
Proof. The proof follows the same lines as the theorems given above. So, we omit the proof here.
4. Numerical Experiments
In this section, some numerical examples are given to demonstrate the applicability and efficacy of our work.
Example 9. Consider the nonlinear system for .
Consider the operator , where . We take , where and such that , , , and . Here, we take the norm for vectors and the corresponding norm for the matrix . Now, by the definition of divided difference for , for . We get We get and . We take the starting points and . Now, using Theorem 4, we can easily obtain that , , , , , and . It can be easily seen that . So, all conditions of Theorem 4 are true and hence (6) can be applied to Example 9. After 4 iterations, we get that the iteration converges to the solution . Absolute error is given in Table 2 with tolerance .
Example 10 (see [18]). Consider
where are given functions and is the solution of (45), to be determined. This problem can be transformed to solve , where , given byConsider as a green function in , and we use a discretization process to transform (46) into a finite dimensional problem. For this, we implement the Gauss-Legendre formula, given by where is the number of nodes. are weights and nodes, respectively, to be determined. We denote the approximation of and by and , respectively, for . Then, (46) is equivalent to solving a system of nonlinear equations, given bywhere With this transformation, (48) is equivalent to , , , and . Now, we consider the nonlinear integral equation using the transformation defined above, by taking , this equation can be transformed to a nonlinear system of equations:where , , , and . Choosing , and with the help of (21) and Theorem 5, we get , , , , , , and . From (22), we get . All the conditions of Theorem 5 are true as and . So, we can ensure the semilocal convergence of (6). After 3 iterations, we get the approximate solution, which is given in Table 3 with tolerance . An approximate error to the solution is given in Table 4.
Example 11. Turning to Example 1, it can be easily seen that . Now, to apply Theorem 7, we take , and then and . We consider and . Also, there exists that verifies (32) and . Thus, all the conditions of Theorem 7 hold true and thus the ball of convergence along with its domain of uniqueness of ball is given by .
5. 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 existence-uniqueness 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.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported in part by the project of Generalitat Valenciana Prometeo/2016/089 and MTM2014-52016-C2-2-P of the Spanish Ministry of Science and Innovation.