Fixed Point and Common Fixed Point Results of DF-Contractions via Measure of Non-compactness with Applications

Authors

  • Asmat Ullah Department of Mathematics and Statistics, International Islamic University, Islamabad
  • Israr Ali Khan Department of Electrical Engineering, Namal College Mianwali, Talagang Road, Mianwali 42250
  • Nayyar Mehmood Department of Mathematics and Statistics, International Islamic University, Islamabad

DOI:

https://doi.org/10.26713/cma.v9i1.926

Keywords:

DF-contraction, Fixed point, Common fixed points, Measure of non-compactness

Abstract

In this paper, we study a new contraction mapping inspired by the concept of F-contraction, which was recently introduced by Wardowski [20]. We find common fixed points for a sequence of mappings by introducing DF-contractive operators in Banach space using the concept of measure of
non-compactness. As an application, we prove some results on the existence of solutions for a system of an infinite fractional order differential equations in the space c, where space c consists of real sequences having the finite limits.

Downloads

References

R.P. Agarwal, D. O'Regan and N. Shahzad, Fixed point theorems for generalized contractive maps of Mei-Keeler type, Mathematische Nachrichten 276 (2004), 3 – 12.

R. Agarwal, M. Maheen and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press (2004).

A. Aghajani, R. Allahyari and M. Mursaleen, A generalization of Darbo's theorem with application to the solvability of systems of integral equations, Journal of Computational and Applied Mathematics 260 (2014), 68 – 77, doi:10.1016/j.cam.2013.09.039.

J. Banas and M. Mursaleen, Sequence spaces and measure of noncompactness with applications to differential and integral equations, Springer India, doi:10.1007/978-81-322-1886-9.

S. Banach, Sur les opérations dens les énsembles abstraits et leur applications aux équations integrales, Fund. Math. 3 (1922), 133 – 181.

V. Berinde, Iterative Approximation of Fixed Points, Springer-Verlag, Berlin ” Heidelberg (2007).

V. Berinde, On the approximation of fixed points of weak contractive mappings, Carpathian Journal of Mathematics 19 (1) (2003), 7 – 22.

D.W.Boyd and J.S.W. Wong, On nonlinear contractions, Proceedings of the American Mathematical Society 20 (1969), 458 – 464.

Lj.B. Ciric, A generalization of Banach's contraction principle, Proceedings of the American Mathematical Society 45 (1974), 267 – 273.

G. Darbo, Punti uniti in transformazioni a condominio non compatto, Rendiconti del Seminario Mathematico della Universita di Padova 24 (1955), 84 – 92.

L.S. Goldenstein and A.S. Markus, On a measure of non-compactness of bounded sets and linear operators, in: Studies in Algebra and Mathematical Analysis, Kishinev, pp. 45 – 54 (1964).

L.S. Goldenstein, I.T. Gohberg and A. Markus, Investigation of some properties of bounded linear operators with their q-norms, Ucen. Zap. Kishinevsk. Univ. 29 (1957), 29 – 36.

V.I. Istratescu, On a measure of non-compactness, Bull. Math. Soc. Sci. Math. R.S. Roumanie (1972), 195 – 197.

R.Khalil, M.A. Horani, A. Yousef and M. Sababheh, A new definition of fractional derivative, Journal of Computational and Applied Mathematics 264 (2014), 65 – 70.

K. Kuratowski, Sur les espaces completes, Fund. Math. 5 (1930), 301 – 309.

J. Matkowski, Fixed point theorems for mappings with a contractive iterate at a point, Proceedings of the American Mathematical Society 62 (1977), 344 – 348.

S. Reich, Kannan's fixed point theorem, Bollettino della Unione Matematica Italiana 4 (4) (1971), 1 – 11.

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

T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proceedings of the American Mathematical Society 136 (5) (2008), 1861 – 1869.

D. Wardowski, Fixed Points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory and Applications 94 (2012), 1 – 6, doi:10.1186/1687-1812-2012-94.

Downloads

Published

30-04-2018
CITATION

How to Cite

Ullah, A., Khan, I. A., & Mehmood, N. (2018). Fixed Point and Common Fixed Point Results of DF-Contractions via Measure of Non-compactness with Applications. Communications in Mathematics and Applications, 9(1), 53–62. https://doi.org/10.26713/cma.v9i1.926

Issue

Vol. 9 No. 1 (2018)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.