Fixed Point and Common Fixed Point Results of \(D_F\)-Contractions via Measure of Non-compactness with Applications

Asmat Ullah, Israr Ali Khan, Nayyar Mehmood


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 \(D_F\)-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.


\(D_F\)-contraction; Fixed point; Common fixed points; Measure of non-compactness

Full Text:



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. 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.



  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905