Turkish Journal of Analysis and Number Theory

ISSN (Print): 2333-1100

ISSN (Online): 2333-1232

Frequency: bimonthly

Editor-in-Chief: Mehmet Acikgoz, Feng Qi, Cenap ozel

Website: http://www.sciepub.com/journal/TJANT

Google-based Impact Factor: 2.54   Citations

Article

A Nonlinear Extension of Fibonacci Sequence

1Department of Mathematics, Goa University, Taleigaon Plateau, Goa, India


Turkish Journal of Analysis and Number Theory. 2016, 4(4), 109-112
doi: 10.12691/tjant-4-4-4
Copyright © 2016 Science and Education Publishing

Cite this paper:
M. Tamba, Y.S. Valaulikar. A Nonlinear Extension of Fibonacci Sequence. Turkish Journal of Analysis and Number Theory. 2016; 4(4):109-112. doi: 10.12691/tjant-4-4-4.

Correspondence to: M.  Tamba, Department of Mathematics, Goa University, Taleigaon Plateau, Goa, India. Email: tamba@unigoa.ac.in

Abstract

A new extension of Fibonacci sequence which yields a nonlinear second order recurrence relation is defined. Some identities and congruence properties for the new sequence are obtained.

Keywords

References

[1]  Z.Akyuz, S. Halici, On Some Combinatorial Identities involving the terms of generalized Fibonacci and Lucas sequences, Hacettepe Journal of Mathematics and Statistics, Volume 42 (4) (2013), 431-435.
 
[2]  A.T. Benjmin, J.J. Quinn, Proofs that really count: The Art of Combinatorial Proof, Mathematical Association of America, Washington, D.C., 2003.
 
[3]  A.T. Benjmin, J.J. Quinn, The Fibonacci Numbers Exposed More Discretely, Mathematics Magazine 76:3 (2003), 182-192.
 
[4]  D. Burton, Elementary Number Theory, 6th edition, Tata McGraw-Hill, 2006.
 
[5]  M. Edson, O. Yayenie, A new generalization of Fibonacci sequence and extended Binet's formula, Integers, Volume 9, Issue 6, Pages 639-654, ISSN (Print) 1867-0652.
 
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[6]  J. Kappraff, G.W. Adamson, Generalized Binet Formulas, Lucas polynomials and Cyclic constants, Forma,19,(2004) 355-366.
 
[7]  M. Renault, The Fibonacci sequence under various moduli, Masters Thesis, 1996.
 
[8]  S. Vajda, Fibonacci and Lucas numbers and the Golden section: Theory and Applications, Dover Publications, 2008.
 
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Article

Generalization of Horadam’s Sequence

1Department of Mathematics, G.V.M’s College of Commerce & Economics, Ponda, Goa 403401, India

2Department of Mathematics, Goa University Taleigao Plateau, 403206, Goa, India


Turkish Journal of Analysis and Number Theory. 2016, 4(4), 113-117
doi: 10.12691/tjant-4-4-5
Copyright © 2016 Science and Education Publishing

Cite this paper:
C.N. Phadte, Y.S. Valaulikar. Generalization of Horadam’s Sequence. Turkish Journal of Analysis and Number Theory. 2016; 4(4):113-117. doi: 10.12691/tjant-4-4-5.

Correspondence to: C.N.  Phadte, Department of Mathematics, G.V.M’s College of Commerce & Economics, Ponda, Goa 403401, India. Email: dbyte09@gmail.com

Abstract

In this paper a new class of Fibonacci like sequence is introduced. Here we consider non-homogeneous recurrence relation to obtain generalization of Horadam’s Sequence. Some identities concerning this new sequence are obtained and proved. Some examples are given in support of the results.

Keywords

References

[1]  A. F. Horadam, “A Generalized Sequence of Numbers”, The American Mathematical Monthly, 68 No. 5,(1961), pp.455-459.
 
[2]  A. F. Horadam, “Basic Properties of a certain Generalized Sequence of Numbers”, The Fibonacci Quarterly, 3, No.3(1965), pp.161-176.
 
[3]  A. F. Horadam, “Generating functions for power of a certain Generalized Sequence of numbers”, Duke Math J. 32, No.3(1965), pp. 437-446.
 
[4]  A. F. Horadam, “Special Properties of the Sequence Wn(a,b;p,q)” , The Fibonacci Quarterly, 5, No. 5 (1967), pp. 424-434.
 
[5]  C. N. Phadte, S.P. Pethe, “Generalization of the Fibonacci Sequence”, Applications of Fibonacci Numbers,5, Kluwer Academic Pub. 1993, 465-472.
 
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[6]  C. N. Phadte, S. P. Pethe, “On Second Order Non-Homogeneous Recurrence Relation”, Annales Mathematicae et Informaticae vol.41 (2013) pp.205-210.
 
[7]  C. N. Phadte, “Trigonometric Pseudo Fibonacci Sequence”, Notes on Number Theory and Discrete Mathematics,21 No.3, (2015) pp.70-76.
 
[8]  J. E. Walton, A. F. Horadam, “Some Aspect of Fibonacci Numbers”, The Fibonacci Quarterly, 12.
 
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Article

Some Fixed Point Results on Multiplicative (b)-metric-like Spaces

1Department of Mathematics, University of Peshawar, Peshawar, Pakistan


Turkish Journal of Analysis and Number Theory. 2016, 4(5), 118-131
doi: 10.12691/tjant-4-5-1
Copyright © 2016 Science and Education Publishing

Cite this paper:
Bakht Zada, Usman Riaz. Some Fixed Point Results on Multiplicative (b)-metric-like Spaces. Turkish Journal of Analysis and Number Theory. 2016; 4(5):118-131. doi: 10.12691/tjant-4-5-1.

Correspondence to: Bakht  Zada, Department of Mathematics, University of Peshawar, Peshawar, Pakistan. Email: bakhtzada56@gmail.com, bakhtzada56@yahoo.com

Abstract

We give the concept of multiplicative partial metric space, multiplicative metric-like space, multiplicative b-metric space and multiplicative b-metric-like space. Then we build the existence and uniqueness of fixed points in a multiplicative b-metric-like space as well as in a partially ordered multiplicative b-metric-like space. We derive some fixed point results in multiplicative partial metric spaces, multiplicative metric-like spaces and multiplicative b-metric spaces as an application, some examples and an application to existence of solution of integral equations.

Keywords

References

[1]  M. A. Alghamdi, N. Hussain, and P. Salimi, Fixed point and coupled fixed point theorems on b-metric-like spaces. Journal of Inequalities and Applications, article 402, (2013).
 
[2]  Agarwal, El-Gebeily, ORegan, Generalized contractions in partially ordered metric spaces. Appl. Anal., (2008).
 
[3]  A. E. Bashirov, E. M. Kurpinar, A. Ozyapici, Multiplicative calculus and its applications, J. Math. Anal. Appl., 337 (2008), 36-48.
 
[4]  S. Czerwik, Contraction mappings in b-metric spaces. Acta Mathematica et Informatica Universitatis Ostraviensis, vol. 1, (1993).
 
[5]  Daffer, Kaneko, On expansive mappings. Math. Jpn. (1992).
 
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[6]  Ozavsar, Cevikel, Fixed points of multiplicative contraction mappings on multiplicative metric spaces. arXiv:1205.5131v1 [math.GM] (2012).
 
[7]  Edelstein, On fixed and periodic points under contractive mappings. J. Lond. Math. Soc. 37, (1962).
 
[8]  S. Gaulyaz, E. Karapinar and V. Rakocevic and P. Salimi, Existence of a solution of integral equations via fixed point theorem. Journal of Inequalities and Appl, (2013).
 
[9]  Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points. Fixed Point Theory and Applications, (2012).
 
[10]  Kirk,WA, Srinavasan, PS, Veeramani, Fixed points for mapping satisfying cyclical contractive conditions. Fixed Point Theory 4, (2003).
 
[11]  Kumar, Garg, Expansion mapping theorems in metric spaces. Int. J. Contemp. Math. Sci. (2009).
 
[12]  S. G. Matthews, Partial metric topology. Annals of the New York Academy of Sciences. General Topology and Applications, vol. 728, (1994).
 
[13]  H. K. Pathak, M. S. Khan and R. Tiwari, A common fixed point theorem and its application to nonlinear integral equations. Computers and Mathematics with Applications, (2007).
 
[14]  H. K. Pathak, S. N. Mishra and A. K. Kalinde, Common xed point theorems with applications to nonlinear integral equations. Demonstratio Math., XXXII (1999).
 
[15]  S. Shukla, Partial b-metric spaces and fixed point theorems. Mediterranean Journal of Mathematics, (2013).
 
[16]  Suzuki, A new type of fixed point theorem in metric spaces. Nonlinear Anal. 71(11), (2009).
 
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