Convergence Analysis of Algorithms for Variational Inequalities Involving Strictly Pseudo-Contractive Operators

Nandal, Ashish; Chugh, Renu; Kumari, Sudesh

doi:10.46753/pjaa.2019.v06i02.006

Cited by 6 publications

(2 citation statements)

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“…Using Equations (20) and 23 x n − a n = 0 and boundedness of {a n } implies there is a subsequence {a n k } of {a n } such that a n k p. Since lim n→+∞ a n − Ua n = 0, so by demiclosedness of I − U at zero, we have p ∈ Fix(U). Proceeding similarly as for case ρ n = 0 and using Equations (20)- (24) and Lemma 8, we conclude that p ∈…”

Section: Proofmentioning

confidence: 96%

See 1 more Smart Citation

A Cyclic Iterative Algorithm for Multiple-Sets Split Common Fixed Point Problem of Demicontractive Mappings without Prior Knowledge of Operator Norm

Gupta¹,

Postolache

Nandal³

et al. 2021

Mathematics

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The aim of this paper is to formulate and analyze a cyclic iterative algorithm in real Hilbert spaces which converges strongly to a common solution of fixed point problem and multiple-sets split common fixed point problem involving demicontractive operators without prior knowledge of operator norm. Significance and range of applicability of our algorithm has been shown by solving the problem of multiple-sets split common null point, multiple-sets split feasibility, multiple-sets split variational inequality, multiple-sets split equilibrium and multiple-sets split monotone variational inclusion.

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Section: Proofmentioning

confidence: 96%

“…Recently, various split type problems have been introduced by many authors (see [4,[7][8][9][10][11][12][13]). For important results in this direction or in similar subjects, see [14][15][16][17][18][19][20][21][22][23][24].…”

Section: Introductionmentioning

confidence: 99%

A Cyclic Iterative Algorithm for Multiple-Sets Split Common Fixed Point Problem of Demicontractive Mappings without Prior Knowledge of Operator Norm

Gupta¹,

Postolache

Nandal³

et al. 2021

Mathematics

Self Cite

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An Application of Viscosity Approximation Type Iterative Method in the Generation of Mandelbrot and Julia Fractals

Kumari

Gdawiec

Nandal³

et al. 2022

Aequat. Math.

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In this paper, we present an application of the viscosity approximation type iterative method introduced by Nandal et al. (Iteration Process for Fixed Point Problems and Zeros of Maximal Monotone Operators, Symmetry, 2019) to visualize and analyse the Julia and Mandelbrot sets for a complex polynomial of the type $$T(z) = z^{n} + p z + r$$ T ( z ) = z n + p z + r , where $$p, r\in {\mathbb {C}}$$ p , r ∈ C , and $$n \ge 2$$ n ≥ 2 . This iterative method has many applications in solving various fixed point problems. We derive an escape criterion to visualize Julia and Mandelbrot sets via the proposed viscosity approximation type method. Moreover, we present several graphical examples of the fractals generated with the proposed iteration method.

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On the viscosity approximation type iterative method and its non-linear behaviour in the generation of Mandelbrot and Julia sets

Kumari,

Gdawiec,

Nandal

et al. 2023

Numer Algor

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In this paper, we visualise and analyse the dynamics of fractals (Julia and Mandelbrot sets) for complex polynomials of the form $$T(z) = z^{n} + mz + r$$ T ( z ) = z n + m z + r , where $$n \ge 2$$ n ≥ 2 and $$m, r \in \mathbb {C}$$ m , r ∈ C , by adopting the viscosity approximation type iteration process which is most widely used iterative method for finding fixed points of non-linear operators. We establish a convergence condition in the form of escape criterion which allows to adapt the escape-time algorithm to the considered iteration scheme. We also present some graphical examples of the Mandelbrot and Julia fractals showing the dependency of Julia and Mandelbrot sets on complex polynomials, contraction mappings, and iteration parameters. Moreover, we propose two numerical measures that allow the study of the dependency of the set shape change on the values of the iteration parameters. Using these two measures, we show that the dependency for the considered iteration method is non-linear.

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Convergence Analysis of Algorithms for Variational Inequalities Involving Strictly Pseudo-Contractive Operators

Cited by 6 publications

References 0 publications

A Cyclic Iterative Algorithm for Multiple-Sets Split Common Fixed Point Problem of Demicontractive Mappings without Prior Knowledge of Operator Norm

A Cyclic Iterative Algorithm for Multiple-Sets Split Common Fixed Point Problem of Demicontractive Mappings without Prior Knowledge of Operator Norm

An Application of Viscosity Approximation Type Iterative Method in the Generation of Mandelbrot and Julia Fractals

On the viscosity approximation type iterative method and its non-linear behaviour in the generation of Mandelbrot and Julia sets

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