R. P. Pant
Dalhousie Villa Compound, Ayarpata, Nainital, Uttarakhand, India.

DOI https://doi.org/10.33889/IJMEMS.2024.9.4.049

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Abstract

We extend the Kannan contraction principle and obtain a result that holds for both contractive and non-expansive mappings. Such mappings admit multiple fixed-points and the fixed-point sets and domains of these mappings display interesting algebraic, geometric and dynamical features. Since contraction mappings admit only one fixed-point, almost all the existing results on contractive mappings can be generalized in the light of our theorem. As an application of our main theorem, we obtain the integral solutions of a nonlinear Diophantine equation; the solutions are Pythagorean triples, which represent right angled triangles, and each integer of the triple belongs to a Fibonacci type sequence. These results can be generalised to obtain integral solutions of Diophantine equations of the type (n+k)2 – n2 = p2, k > 1, and to check whether the related sequences are Fibonacci sequences.

Keywords- Contraction mappings, Eventual fixed points, Fibonacci sequence, Fixed points, Pythagorean triple.

Citation

Pant, R. P (2024). An Extended Kannan Contraction Mapping and Applications. International Journal of Mathematical, Engineering and Management Sciences, 9(4), 931-942. https://doi.org/10.33889/IJMEMS.2024.9.4.049.