Fixed point results for Su-type contractive mappings with an application

Subject Areas : Fixed point theory

A.  Ali ¹ (Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan)
H.  Işık ² (Department of Mathematics, Faculty of Science and Arts, Mus Alparslan University, Mus 49250, Turkey)
F.  Uddin ³ (Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan)
M.  Arshad ⁴ (Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan)

Received: 2019-11-04 Accepted : 2020-03-18 Published : 2020-03-01

Keywords: Altering distance function‎, ‎Su-type contraction‎, ‎extended partial $b$-metric space‎, ‎integral equation,

Abstract :

‎In this paper‎, ‎we introduce the concept of Su-type contractive mapping and establish fixed point theorems for such mappings in the setting of ordered‎‎extended partial $b$-metric space‎. ‎We also develop an‎‎application for Fredholm type integral equations to‎‎validate our main result and a non-trivial example is given to‎‎elucidate our work.

References:

[1] I. Altun, A. Erduran, Fixed point theorems for monotone mappings on partial metric spaces, Fixed Point Theory Appl. 2011, 2011:508730.
[2] A. Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory Appl. 2012, 2012:204.
[3] M. Arshad, M. Abbas, A. Hussain, N. Hussain, Generalized dynamic process for generalized (f,L)almost
F-contraction with applications, J. Nonlinear Sci. Appl. 9 (2016), 1702-1715.
[4] H. Aydi, E. Karapınar, C. Vetro, On Ekeland’s variational principle in partial metric spaces, Appl. Math.
Inf. Sci. 9 (1) (2015), 257-262.
[5] I. Beg, A. R. Butt, Common fixed point and coincidence point of generalized contractions in ordered metric spaces, Fixed Point Theory Appl. 2012, 2012:229.
[6] L. B. Ciric, M. Abbas, R. Saadati, N. Hussain, Common fixed points of almost generalized contractive mappings in ordered metric spaces, Appl. Math. Comput. 217 (12) (2011), 5784-5789.
[7] M. Cosentino, P. Vetro, Fixed point results for F-contractive mappings of Hardy-Rogers-type, Filomat. 28 (4) (2014), 715-722.
[8] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav. 1 (1993), 5-11.
[9] S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena. 46 (1998), 263-276.
[10] D. Gopal, M. Abbas, D. K. Patel, C. Vetro, Fixed points of α-type F-contractive mappings with an application to nonlinear fractional differential equation, Acta Math. Sci. 36 (3) (2016), 957-970.
[11] R. Gubran, M. Imdad, Results on coincidence and common fixed points for (ψ,φ)g-generalized weakly contractive mappings in ordered metric spaces, Mathematics. 2016, 4:68.
[12] N. Hussain, H. Isık, M. Abbas, Common fixed point results of generalized almost rational contraction mappings with an application, J. Nonlinear Sci. Appl. 9 (2016), 2273-2288.
[13] N. Hussain, P. Salimi, Suzuki–Wardowski type fixed point theorems for α-GF-contractions, Taiwan. J. Math. 18 (6) (2014), 1879-1895.
[14] N. Hussain, M. H. Shah, KKM mappings in cone b-metric spaces, Comput. Math. Appl. 62 (2011), 1677-1684.
[15] H. Isık, Solvability to coupled systems of functional equations via fixed point theory, TWMS J. App. Eng. Math. 8 (1a) (2018), 230-237.
[16] H. Isık, M. Imdad, D. Turkoglu, N. Hussain, Generalized Meir-Keeler type ψ-contractive mappings and applications to common solution of integral equations, International Journal of Analysis and Applications. 13 (2) (2017), 185-197.
[17] H. Isık, C. Ionescu, New type of multivalued contractions with related results and applications, U.P.B. Sci. Bull. Ser. A. 80 (2) (2018), 13-22.
[18] H. Isık, D. Turkoglu, Common fixed points for (ψ,α,β)-weakly contractive mappings in generalized metric spaces, Fixed Point Theory Appl. 2013, 2013:131.
[19] H. Isık, D. Turkoglu, Some fixed point theorems in ordered partial metric spaces, Journal of Inequalities and Special Functions. 4 (2) (2013), 13-18.
[20] T. Kamran, M. Samreen, Q. UL Ain, A generalization of b-metric space and some fixed point theorems, Mathematics. 2017, 5:19.
[21] M. S. Khan, M. Swaleh, S. Sessa, Fixed point theorems by altering distances between the points, Bulletin of the Australian Mathematical Society. 30 (1) (1984), 1-9.
[22] D. Klim, D. Wardowski, Fixed points of dynamic processes of set-valued F-contractions and application to functional equations, Fixed Point Theory Appl. 2015, 2015:22.
[23] A. Latif, M. Abbas, A. Hussain, Coincidence best proximity point of Fg-weak contractive mappings in partially ordered metric spaces, J. Nonlinear Sci. Appl. 9 (5) (2016), 2448-2457.
[24] S. G. Matthews, Partial metric topology, N. Y. Acad. Sci. 728 (1994), 183-197.
[25] Z. Mustafa, J. R. Roshan, V. Parvaneh, Z. Kadelburg, Some common fixed point results in ordered partial b-metric spaces, J. Inequal. Appl. 2013, 2013:562.
[26] V. Parvaneh, N. Hussain, Z. Kadelburg, Generalized Wardowski type fixed point theorems via α-admissible FG-contractions in b-metric spaces, Acta Math. Sci. 36 (5) (2016), 1445-1456.
[27] V. Parvaneh, Z. Kadelburg, Extended partial b-metric spaces and some fixed point results, Filomat. 32 (8) (2018), 2837-2850.
[28] M. Sgroi, C. Vetro, Multi-valued F-contractions and the solution of certain functional and integral equations, Filomat. 27 (2013), 1259-1268.
[29] S. Shukla, Partial b-metric spaces and fixed point theorems, Mediterr. J. Math. 11 (2014), 703-711.
[30] S. Shukla, S. Radenovic, Z. Kadelburg, Some fixed point theorems for ordered F-generalized contractions in 0-f-orbitally complete partial metric spaces, Theory Appl. Math. Comput. Sci. 4 (1) (2014), 87-98.
[31] Y. Su, Contraction mapping principle with generalized altering distance function in ordered metric spaces and applications to ordinary differential equations, Fixed Point Theory Appl. 2014, 2014:227.
[32] N. Van Dung, V. T. Le Hang, A fixed point theorem for generalized F-contractions on complete metric spaces, Vietnam J. Math. 4 (43) (2015), 743-753.
[33] F. Vetro, F-contractions of Hardy-Rogers type and application to multistage decision processes, Nonlinear Anal. Model. Control. 21 (4) (2016), 531-546.
[34] D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 2012, 2012:94.
[35] D. Wardowski, N. Van Dung, Fixed points of F-weak contractions on complete metric space, Demonstr. Math. XLVII (1) (2014), 146-155.