Iterative Procedures for Solving Equations in Abstract Fractional Calculus

The concept of asymptotically nonexpansive mappings is an important generalization of the class of nonexpansive mappings. Implicit midpoint procedures are extremely fundamental for solving equations involving nonlinear operators. This paper studies the convergence analysis of the class of asymptotically nonexpansive mappings by the implicit midpoint iterative procedures. The necessary conditions for the convergence of the class of asymptotically nonexpansive mappings are established, by using a well-known iterative algorithm which plays important roles in the computation of fixed points of nonlinear mappings. A numerical example is presented to illustrate the convergence result. Under relaxed conditions on the parameters, some algorithms and strong convergence results were derived to obtain some results in the literature as corollaries.

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A necessary and sufficient condition for the convergence of iterative procedures for solving equations of nonlinear monotone resistive networks

Proceedings of 1997 IEEE International Symposium on Circuits and Systems. Circuits and Systems in the Information Age ISCAS '97 ◽

10.1109/iscas.1997.621849 ◽

2002 ◽

Author(s):

Y. Jiang ◽

O. Wing

Keyword(s):

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Resistive Networks ◽

Solving Equations ◽

Iterative Procedures ◽

Necessary And Sufficient

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High Convergence Order Iterative Procedures for Solving Equations Originating from Real Life Problems

Mathematics ◽

10.3390/math7090855 ◽

2019 ◽

Vol 7 (9) ◽

pp. 855

Author(s):

Ramandeep Behl ◽

Ioannis K. Argyros ◽

Ali Saleh Alshomrani

Keyword(s):

Real Life ◽

Nonlinear Problems ◽

Local Study ◽

Precise Information ◽

Numerical Examples ◽

First Order ◽

Life Problems ◽

Solving Equations ◽

Iterative Procedures ◽

Good Number

The foremost aim of this paper is to suggest a local study for high order iterative procedures for solving nonlinear problems involving Banach space valued operators. We only deploy suppositions on the first-order derivative of the operator. Our conditions involve the Lipschitz or Hölder case as compared to the earlier ones. Moreover, when we specialize to these cases, they provide us: larger radius of convergence, higher bounds on the distances, more precise information on the solution and smaller Lipschitz or Hölder constants. Hence, we extend the suitability of them. Our new technique can also be used to broaden the usage of existing iterative procedures too. Finally, we check our results on a good number of numerical examples, which demonstrate that they are capable of solving such problems where earlier studies cannot apply.

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