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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

### Article

**A Nonlinear Extension of Fibonacci Sequence**

^{1}Department 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

### 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. | ||

[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. | ||

### Article

**Generalization of Horadam’s Sequence**

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

^{2}Department 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

### 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 W_{n}(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. | ||

[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. | ||

### Article

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

^{1}Department 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

### Keywords

**Partial metric space, metric-like space, b-metric space, b-metric-like space, fixed point, integral equation**

### 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). | ||

[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). | ||