Compatible mappings and its variants satisfying generalized (ψ, φ)−weak contraction

Authors

  • Kavita Lather Deenbandhu Chhotu Ram University of Science and Technology, Murthal, Sonepat
  • Sanjay Kumar Deenbandhu Chhotu Ram University of Science and Technology, Murthal, Sonepat
  • Rajesh Kumar Institute of Higher Learning, BPS Mahila Vishwavidyalaya, Khanpur Kalan, Sonipat
  • Nikita Kadian Deenbandhu Chhotu Ram University of Science and Technology, Murthal, Sonepat

DOI:

https://doi.org/10.23755/rm.v48i0.1224

Keywords:

generalized (ψ, φ)-weak contraction, compatible mappings, minimal commutative mappings, functional equations

Abstract

Banach contraction principle behaves as a mathematical tool to solve various practical problems arising during mathematical formulation of many theoretical problems. In present work, the existence of a unique common fixed point for pairs of minimal commutative mappings is discussed, which satisfy a generalized (ψ, φ)−weak contraction involving cubic terms of distance functions. Examples are given in support of the obtained results and as an application the existence of solution of system of certain functional equations arising in dynamic programming is discussed.

Author Biographies

  • Kavita Lather, Deenbandhu Chhotu Ram University of Science and Technology, Murthal, Sonepat

    Research Scholar

    Department of Mathematics

  • Sanjay Kumar, Deenbandhu Chhotu Ram University of Science and Technology, Murthal, Sonepat

    Professor

    Department of Mathematics

  • Rajesh Kumar, Institute of Higher Learning, BPS Mahila Vishwavidyalaya, Khanpur Kalan, Sonipat

    Assistant Professor

    Department of Mathematics

  • Nikita Kadian, Deenbandhu Chhotu Ram University of Science and Technology, Murthal, Sonepat

    Research scholar

    Department of Mathematics

References

Y. I. Alber, S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces. In: New Results in Operator Theory and its Applications, Adv. Appl. Math.(Y. Gahbery, Yu. Librich, eds.), Birkhauser Verlag Basel, 1997, pp.7-22., 98(1997), 7-22.

S. Banach, Surles operations dans les ensembles abstraites et leaursapplications, Fundam. Math., 3(1922), 133-181.

R. Baskaran, P. V. Subrahmanayam, A note on the solution of a class of functional equations, Appl. Anal., 22(1986), 235-241.

R. Bellman, Methods of Nonlinear Analysis, Vol. II, vol. 61 of Mathematics in Science and Engineering, Acadmic Press, New York, USA, (1973).

R. Bellman, B.S. Lee, Functional equations arising in dynamic programming, Aequationes Math., 17(1978), 1-18.

H. Bouhadjera, Common fixed point theorems for compatible mappings of type (C), Sarajevo J. Math., 1(4)(2005), 261-270.

D. W. Boyd, J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math.Soc., 20(2)(1969), 458-464.

Lj. B. Ciri ´ c,´ Generalized contractions and fixed point theorems, Publ. Inst. Math., 26(1971),19-26.

P.N. Dutta, B.S. Choudhury, A generalization of contraction principle in metric spaces, Fixed Point Theory Appl., 2008(2008), 1-8.

D. Jain, S. Kumar, S. M. Kang, Weak contraction condition for mappings involving cubic terms of the metric function, Int. J. Pure Appl. Math., 116(2017), 1115-1126.

D. Jain, S. Kumar, S. Kang, Generalized weak contraction condition for compatible mappings of types involving cubic terms under the fixed point consideration, Far East. J. Math. Sci. 103(2018), 1363-1377.

D. Jain, S. Kumar, S. Kang and C. Jung, Weak contraction condition for compatible mappings involving cubic terms of the metric function, Far East. J. Math. Sci., 103(4)(2018), 799-818.

K. Jha, V. Popa, K. B. Manandhar, Common fixed point theorem for compatible of type (K) in metric space, Int. J. Math. Sci. Engg. Appl. 8(2014), 383-391.

C. Y. Jung, D. Jain, S. Kumar and S. M. Kang, Generalized weak contraction condition for compatible mappings of types involving cubic terms of the metric function, Int. J. Pure. Appl. Math., 119(1)(2019), 9-30.

G. Jungck, Commuting mappings and fixed points, Amer. Math. Monthly, 83(1976), 261-263.

G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Math. Sci., 9(1986), 771-779.

G. Jungck, P. P. Murthy, Y. J. Cho, Compatible mappings of type (A) and common fixed points, Math. Jpn., 38(1993), 381-390.

S. M. Kang, Y. J. Cho, G. Jungck, Common fixed points of compatible mappings, Int. J. Math. Math. Sci., 13(1)(1990), 61-66.

Kavita, Sanjay Kumar, Fixed points for Intimate mappings, J. Math. Comput. Sci., 12(2022), Paper No. 48.

M. S. Khan, M. Swalek, S. Sessa, Fixed point theorems by altering distances between two points, Bull. Austra. Math. Soc., 30(1984), 1-9.

P. P. Murthy, K. N. V. V. V. Prasad, Weak contraction condition involving cubic terms of d(x, y) under the fixed point consideration, J. Math., Art. Id 967045,5 pages. doi: 10.1155/2013/967045.

R. P. Pant, A common fixed point theorem under a new condition, Indian J. Pure Appl. Math. 30(2)(1999), 147-152.

H. K. Pathak, M. S. Khan, Compatible mappings of type (B) and common fixed point theorems of Gregus type, Czechoslov. Math. J., 45(1995), 685-698.

H. K. Pathak, Y. J. Cho, S. M. Kang and B. S. Lee, Fixed point theorems for compatible mappings of type (P) and applications to dynamic programming, Matematiche, 50(1995), 15-33.

H. K. Pathak, Y. J. Cho, S. M. Kang and B.Madharia, Compatible mappings of type (C) and common fixed point theorems of Gregus type, Demonstr. Math., 31(1998), 499-518.

B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Analysis, 47(4)(2001), 2683-2693.

Y. Rohen, M.R. Singh, Common fixed point of compatible mappings of type(R) in complete metric spaces, Int. J. Math. Sci. Engg. Appl. 2(4)(2008), 295-303.

S. Sessa, On a weak commutativity conditions of mappings in fixed point consideration, Publ. Inst. Math. Beograd, 32(46)(1982), 146-153.

B. Singh, S. Jain, A fixed point theorem in Menger space through weak compatibility, J. Math. Anal. Appl., 301(2005), 439-448.

M. R. Singh, Y. M Singh, Compatible mappings of type (E) and common fixed point theorem of Meir-Keeler type, Int. J. Math. Sci. Engg. Appl., 1(2007), 299-315.

M. R. Singh, Y. M Singh, On various types of compatible maps and common fixed point theorems for non-continuous maps, Hacettepe J. Math. Statics, 40(4)(2011), 503-513.

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Published

2023-12-30

Issue

Vol. 48 (2023)

Section

Volume 48-2023

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