Abstract:MSC: 26A18 47H10 54C05Keywords: Continuous functions Convergence theorem Fixed point Nondecreasing functions Rate of convergence a b s t r a c tIn this paper, we propose a new iteration, called the SP-iteration, for approximating a fixed point of continuous functions on an arbitrary interval. Then, a necessary and sufficient condition for the convergence of the SP-iteration of continuous functions on an arbitrary interval is given. We also compare the convergence speed of Mann, Ishikawa, Noor and SP-iterations… Show more
“…Let K be a nonempty closed convex subset of an arbitrary Banach space X and be a mapping satisfying (1.9). Let {s n } be defined through the Ishikawa iteration (1.4) and 0 Several authors [5,[8][9][10][11][12][13][14][15][16][17] have studied the equivalence between various iterative schemes. S. M. Solutz [15,16] proved that for quasi-contractive operators the itérative processes Picard, Mann, Ishikawa and Noor are équi-valent.…”
Section: Introductionmentioning
confidence: 99%
“…S. L. Singh [19] extended the work of Rhoades. Very recently, Phuengrattana and Suantai [5] proved that SP iterative scheme is equivalent to and faster than Mann, Ishikawa and Noor iterative schemes for increasing functions. Now, we introduce the following CR iterative process: Let X be a Banach space, a self map of X : T X X and 0…”
In this paper, we suggest a new type of three step iterative scheme called the CR iterative scheme and study the strong convergence of this iterative scheme for a certain class of quasi-contractive operators in Banach spaces. We show that for the aforementioned class of operators, the CR iterative scheme is equivalent to and faster than Picard, Mann, Ishikawa, Agarwal et al., Noor and SP iterative schemes. Moreover, we also present various numerical examples using computer programming in C++ for the CR iterative scheme to compare it with the other above mentioned iterative schemes. Our results show that as far as the rate of convergence is concerned 1) for increasing functions the CR iterative scheme is best, while for decreasing functions the SP iterative scheme is best; 2) CR iterative scheme is best for a certain class of quasi-contractive operators.
“…Let K be a nonempty closed convex subset of an arbitrary Banach space X and be a mapping satisfying (1.9). Let {s n } be defined through the Ishikawa iteration (1.4) and 0 Several authors [5,[8][9][10][11][12][13][14][15][16][17] have studied the equivalence between various iterative schemes. S. M. Solutz [15,16] proved that for quasi-contractive operators the itérative processes Picard, Mann, Ishikawa and Noor are équi-valent.…”
Section: Introductionmentioning
confidence: 99%
“…S. L. Singh [19] extended the work of Rhoades. Very recently, Phuengrattana and Suantai [5] proved that SP iterative scheme is equivalent to and faster than Mann, Ishikawa and Noor iterative schemes for increasing functions. Now, we introduce the following CR iterative process: Let X be a Banach space, a self map of X : T X X and 0…”
In this paper, we suggest a new type of three step iterative scheme called the CR iterative scheme and study the strong convergence of this iterative scheme for a certain class of quasi-contractive operators in Banach spaces. We show that for the aforementioned class of operators, the CR iterative scheme is equivalent to and faster than Picard, Mann, Ishikawa, Agarwal et al., Noor and SP iterative schemes. Moreover, we also present various numerical examples using computer programming in C++ for the CR iterative scheme to compare it with the other above mentioned iterative schemes. Our results show that as far as the rate of convergence is concerned 1) for increasing functions the CR iterative scheme is best, while for decreasing functions the SP iterative scheme is best; 2) CR iterative scheme is best for a certain class of quasi-contractive operators.
“…(i) The Picard iteration [17] converges to p ∈ F (T ), (ii) The Mann iteration [13] converges to p ∈ F (T ), (iii) The SP iteration [16] converges to p ∈ F (T ), (iv) The multistep iteration (2.5) converges to p ∈ F (T ), (v) The multistep Picard-Mann iteration (2.8) converges to p ∈ F (T ).…”
Section: Resultsmentioning
confidence: 99%
“…If we take r = 2 and r = 3 in (2.7), respectively, we obtain the two-step iteration procedure given in [23] and SP iteration method in [16].…”
Section: Remark 22 ([4]mentioning
confidence: 99%
“…In 2007, Şoltuz [21] proved that the Mann [13], Ishikawa [9], Noor [14] and multistep (2.5) iterations are equivalent for quasi-contractive mappings in a normed space. In 2011, Chugh and Kumar [6] showed that the Picard [17], Mann [13], Ishikawa [9], new two step [23], Noor [14] and SP [16] iterations are equivalent for quasi-contractive mappings in a Banach space. In 2013, Karakaya et al [10] proved the data dependence results for the multistep (2.5) and CR [7] iteration processes for the class of contractive-like operators satisfying (2.4).…”
In this paper, we introduce a new iteration process and prove the convergence of this iteration process to a fixed point of contractive-like operators. We also present a data dependence result for such mappings. Our results unify and extend various results in the existing literature.
Dynamical systems are one of the interesting concepts where iteration algorithms and chaos can be considered together. In iteration algorithms, one of the basic concepts of fixed point theory, it is well known that the behavior of the iteration mechanism is chaotic if the original transformation is taken as chaotic. One of the natural ways to transform a chaotic system into a dynamical system is through control mechanisms. In this paper, we first consider an iteration class defined on Banach spaces, which is prominent in the literature in terms of both speed and convergence rate. Then, we consider the transformation constituting the iteration class as chaotic and obtain the stability and unstability behaviors of the iterations according to the operator norm by using Gâteaux and Fréchet derivatives representing the direction‐dependent derivative. In this way, we aimed to obtain the chaos control intervals obtained with functions in real space with the help of operators in Banach spaces. Using the operator norm obtained with the Fréchet derivative, we derive interesting dynamical system intervals from a chaotic system with parameter variables of iteration algorithms. It is noteworthy that the parameter variables of the studied iteration classes are the same in some transformation classes. In addition, analytical proofs are followed by computer simulations of parameter‐dependent control intervals considering the logistic operator with a chaotic structure. At the same time, the periodic behavior of the iteration algorithms used in our study is illustrated by the Lyapunov exponent with parameter‐dependent control intervals according to operator norm. Finally, as a real‐life problem, it has been shown that the chaos of the logistic population growth model with dynamic features can be controlled by chaos control mechanisms established by fixed point iteration methods.
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