Research Article | Open Access
Influence of the Center Condition on the Two-Step Secant Method
The aim of this paper is to present a new improved semilocal and local convergence analysis for two-step secant method to approximate a locally unique solution of a nonlinear equation in Banach spaces. This study is important because starting points play an important role in the convergence of an iterative method. We have used a combination of Lipschitz and center-Lipschitz conditions on the Fréchet derivative instead of only Lipschitz condition. A comparison is established on different types of center conditions and the influence of our approach is shown through the numerical examples. In comparison to some earlier study, it gives an improved domain of convergence along with the precise error bounds. Finally, some numerical examples including nonlinear elliptic differential equations and integral equations validate the efficacy of our approach.
Consider the problem to approximate a locally unique solution ofwhere is a nonlinear operator. are Banach spaces and is an open nonempty convex subset of . This is one of the very important problems in applied mathematics and engineering science. Many real life problems in diverse areas such as equilibrium theory and elasticity often reduce to solving these equations depending on one or more parameters. Mathematical modeling of many problems uses integral equations, boundary value problems, differential equations, and so forth, whose solutions are obtained by solving scalar equations or a system of equations. Many nonlinear differential equations can be solved by transforming them to matrix equations which give a system of nonlinear equations in . Many researchers [1–4] have extensively studied these problems and many methods, both direct and iterative, are developed for their solutions. Good convergence properties, efficiency, and numerical stability are the requirements of all these methods. It is a common problem to choose the good starting points for the iterative methods which ensure the convergence of the iterative method. The semilocal convergence [2, 5, 6] uses information given at the initial point whereas local convergence [7, 8] uses information around the solution. The quadratically convergent Newton’s iteration [9, 10] is used to solve (1). It is defined for bywhere is the starting point and (the set of bounded linear operators from into ). Sufficient conditions for the semilocal convergence with existence ball and error estimates of (2) are given in . The secant iteration [1, 12, 13] is the simplification of (2) used to solve (1) and is given for bywhere are two starting points and is the divided difference of order one for on the points and satisfies the equality . In case of operators, this equality does not hold uniquely unless is one-dimensional. In , it is defined by a matrixfor and . So, many real life problems that require the solution of matrix equations can also be solved by the abovementioned methods.
Recently, an iteration known as the King-Werner iteration originally proposed by King  is discussed in [15, 16] along with its local and semilocal convergence using majorizing sequences under the Lipschitz continuous Fréchet derivative of . It is given for by where, are the starting iterates. Its order is equal to . A two-step secant iteration with order of convergence same as (5) with its semilocal and local convergence under combination of Lipschitz and center-Lipschitz continuous divided differences of order one using majorizing sequences for solving (1) is described in Banach space setting in . It is defined for by where are starting iterates.
In this paper, iteration (6) is considered for solving (1) along with its semilocal and local convergence analysis under weaker Lipschitz continuity condition on divided differences of order one on the involved operator in Banach space setting. The influence on the domain by our approach is shown by some numerical examples. It provides the improved error estimations along with the better information on the location of solutions. Semilocal convergence of (6) is studied, which improves the applicability of the method corresponding to some earlier study [17, 18]. It is shown by our work that earlier studies for (6) do not hold while the new convergence criteria hold. For local convergence analysis, weaker center-Lipschitz continuity condition is used in place of a combination of Lipschitz and center-Lipschitz continuity conditions. Larger convergence ball is obtained through this study in comparison to the older one.
The paper is arranged as follows. Introduction forms Section 1. In Section 2, the semilocal convergence analysis of (6) under weaker convergence conditions on divided differences of operator is established. In Section 3, local convergence analysis of (6) is established using only center-Lipschitz continuity condition on divided differences. In Section 4, numerical examples are given to validate the theoretical results obtained by us. Finally, conclusions and references are included in Section 5.
2. Semilocal Convergence
In this section, firstly, we provide a lemma that will be used to provide the semilocal convergence theorem of (6).
Lemma 1. Let , , , , , and be nonnegative parameters and be the unique root of the polynomial defined byand sequences and defined for by , , , and for all by Supposingthen sequences are well defined, increasing, and bounded above by and converge to their least upper bound which satisfies . Moreover, the following estimates hold for all :
Proof. From (7), and . Using intermediate value theorem, has at least one root in ; also, it is increasing in this interval. So, it has a unique root in this interval which is denoted by . Suppose ; then, all terms of sequences and will be equal to and Lemma 1 holds in this case. Taking , then (11) is true iffor each .
This implies that and . Now, instead of showing (12), it will be sufficient to show thatFrom (13), we are motivated to construct a recurrent polynomialReplacing by in (14), this givesNow, from (7), (14), and (15) and the help of some algebraic manipulations, we haveUsing (16), we get ; also, is an increasing function in . Let us define a function on by Now, we need to show only Using (10), this assertion can be proved easily and, thus, Lemma 1 is established.
Next, we provide a semilocal convergence theorem followed by Lemma 1 for (6).
Theorem 2. Let be a nonlinear operator; , and are given parameters. Denote for . Under the hypothesis of Lemma 1, the following assumptions hold in :Staring with suitable , sequences and defined in (6) are well defined, remain in , and converge to a solution in of (1). Moreover, the following estimates hold for each :Further, if there exists such that and , then is the only solution of (1) in
Proof. Using mathematical induction on , we shall show that (19) hold true. For , this follows directly from (18) which shows that . Using Banach lemma and (18), we get . Next, This shows that Using Banach lemma  on invertible operators, we getNow, This implies and thus (19) is true for . Now, from (6), This shows that . Thus, replacing by and proceeding in a similar manner, this gives the notion that is a complete sequence in Banach space such that it converges to some Now, to show that is a solution of (1),So, Suppose is another solution of (1) such that . Let be an operator and It follows that and this establishes Theorem 2.
Lemma 3 (see ). Let , and be nonnegative parameters and be the unique root of the polynomial defined by and sequences and defined for , by and for all by Supposing then sequences are well defined, increasing, and bounded above by and converge to their least upper bound which satisfies Moreover, the following estimates hold for all :
Theorem 4 (see ). Let be a nonlinear operator; , and are given parameters. Denote for . Under the hypothesis of Lemma 3, the following assumptions hold in :Starting with suitable , sequences and defined in (6) are well defined, remain in , and converge to a unique solution in of (1). Moreover, the following estimates hold for each : Further, if there exists such that and , then is the only solution of (1) in .
Previous assertions  are made for iteration (6) as follows: One can easily see that our conditions are more general than (35), and with conditions (35), the following majorizing sequences are obtained:
3. Local Convergence
Theorem 5. Let be given parameters and be a nonlinear divided difference operator such that for all and , where . Then, sequences of (6) starting from are well defined, remain in for each , and converge to . Moreover, the following error estimates hold:Additionally, if there exists such that and , then is the only solution of (1) in .
Proof. For and using (37), we getSo, by Banach lemma, exists andUsing (37) and (40) and hypothesis of Theorem 5, we haveAgain, using (37)–(41) and hypothesis of Theorem 5, we getThis shows that . Clearly, using induction on ,This shows (38). Now, let be another solution of (1) in such that . Using (37), this gives This shows that is the unique solution of (1) in .
4. Numerical Examples
Example 1. Let , and define function on by We take and free in order to find a relation between and for which all the criteria for ensuring the convergence are satisfied. In Figure 1, we have taken horizontal axis for and vertical axis for . With the help of (6), we obtain , , , , , , , , , , and . The efficacy of our approach can be seen in Figure 1.
For comparing the error estimation where all approaches including the older approach are satisfied, we take the domain and fix . In this case, we getand , , , , , , , , , and . Comparison of error estimations with different approaches is given in Table 1.
From Figure 1, it can be seen that there exists some combination of and where the condition used earlier fails. When all conditions hold, then it gives the precise error bounds. Thus, the claim made by us in the abstract and the Introduction is justified here.
Example 2. Let , the space of all continuous functions defined in equipped with the max-norm. Let such that and define on bywhere is a given function and the kernel is Green’s function: Now, one can represent a linear operator by Choose and ; we obtain It can be easily seen that , , , , , , , , , and . Comparison of the error estimation with the older one is given in Table 2.