Some $\psi-$fixed point theorems of Wardowski kind in $\mathcal{G}$-metric spaces with application to integral equations

Document Type : Research Paper


Department of Mathematics, K.R.M.D.A.V. College, Nakodar-144040, Punjab, India


In this manuscript, we introduce new notions of generalized ($\mathfrak{f^{*}}, \psi)$-contraction and utilize this concept to prove some fixed point results for lower semi-continuous $\psi$-mapping satisfying certain conditions in the frame of G-metric spaces. Our results improve the results of [6] and [8] by omitting the continuity condition of $F\in \Im$ with the aid of the $\psi$-fixed point. We give an illustrative example to help accessibility of the got results and to show the genuineness of our results. Also, many existing results in the frame of metric spaces are established. Moreover, as an application, we employ the achieved result to earn the existence and uniqueness criteria of the solution of a type of non-linear integral equation.


[1] B. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales, Fundam. Math. 3 (1922), 133–145.
[2] L. Ciric, Fixed points for generalized multi-valued mappings, Mat. Vesnik 24 (1972), 265–272.
[3] N.V. Dung and V.L. Hang, A fixed point theorem for generalized F-contractions on complete metric spaces, Vietnam J. Math. 43 (2015), 743–753.
[4] J. Gornicki, Fixed point theorems for F-expanding mappings, Fixed Point Theory Appl. 9 (2017), 1–10.
[5] M. Jleli, B. Samet, and C. Vetro, Fixed point theory in partial metric spaces via φ-fixed point concept in metric spaces, J. Inequal. Appl. 426 (2014), no. 1, 1–9.
[6] M. Kumar and S. Arora, Fixed point theorems for modified generalized F-contraction in G-metric spaces, Bol. Soc. Paran. Mat. 40 (2022), 1–8.
[7] Z. Mustafa and B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal. 7 (2006), no. 2, 289–297.
[8] H.N. Saleh, M. Imdad and W.M. Alfaqih, Some metrical φ-fixed point results of Wardowski type with applications to integral equations, Bol. Soc. Paran. Mat. 40 (2022), 1–11.
[9] H. Piri and P. Kumam, Wardowski type fixed point theorems in complete metric spaces, Fixed Point Theory Appl. 45 (2016), 1–12.
[10] H. Piri and P. Kumam, Some fixed point theorems concerning F-contraction in complete metric spaces, Fixed Point Theory Appl. 210 (2014), 1–13.
[11] D. Wardowski and N.V. Dung, Fixed points of F-weak contractions on complete metric spaces, Demonst. Math. 47 (2014), 146–155.
[12] D. Wardowski, Fixed point theory of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl. 94 (2012), 1–6.
Volume 14, Issue 6
June 2023
Pages 335-343
  • Receive Date: 23 February 2021
  • Revise Date: 19 January 2023
  • Accept Date: 22 January 2023
  • First Publish Date: 29 January 2023