Abstract
In (Fixed Point Theory Appl 94:6, 2012), the author introduced a new kind of contractions, called F-contractions, that extended the Banach contractions in a newfangled way. In this work, we introduce the notion of \(F_\mathfrak {R}\)-contraction where \(\mathfrak {R}\) is a binary relation on its domain that has not to be neither transitive nor a partial order. Consequently, we establish some fixed point results for such contractions in complete metric spaces that improve the Wardowski’s original idea and we also give illustrative examples. Furthermore, we show some results to guarantee existence and uniqueness of fixed point of N-order. As an application, we apply our main result to study a class of nonlinear matrix equation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
\(\mathcal {G}\) is order preserving if \(A,B\in H(n)\) with \(A\preceq B\) implies that \(\mathcal {G}(A)\preceq \mathcal {G}(B)\).
References
Agarwal, R.P., O’Regan, D., Shahzad, N.: Fixed point theory for generalized contractive maps of Meir-Keeler type. Math. Nachr. 276, 3–22 (2004)
Berinde, V.: On the approximation of fixed points of weak contractive mappings. Carpath. J. Math. 19(1), 7–22 (2003)
Berinde, V.: Iterative Approximation of Fixed Points. Springer, Berlin (2007)
Boyd, D.W., Wong, J.S.W.: On nonlinear contractions. Proc. Am. Math. Soc. 20, 458–464 (1969)
Ćirić, L.B.: A generalization of Banachs contraction principle. Proc. Am. Math. Soc. 45, 267–273 (1974)
Du, W.-S.: Some new results and generalizations in metric fixed point theory. Nonlinear Anal. 73(5), 1439–1446 (2010)
Hardy, G.E., Rogers, T.D.: A generalization of a fixed point theorem of Reich. Can. Math. Bull. 16, 201–206 (1973)
Matkowski, J.: Fixed point theorems for mappings with a contractive iterate at a point. Proc. Am. Math. Soc. 62, 344–348 (1977)
Reich, S.: Kannan’s fixed point theorem. Bollettino della Unione Matematica Italiana 4(4), 1–11 (1971)
Suzuki, T.: A generalized Banach contraction principle that characterizes metric completeness. Proc. Am. Math. Soc. 136(5), 1861–1869 (2008)
Rhoades, B.E.: Some theorems on weakly contractive maps. Nonlinear Anal. 47(4), 2683–2693 (2001)
Aguirre Salazar, L., Reich, S.: A remark on weakly contractive mappings. J. Nonlinear Convex. Anal. 16, 767–773 (2015)
Wardowski, D.: Fixed points of a new type of contractive mappings in complete metric spaces. Fixed Point Theory Appl. 2012, 94 (2012)
Durmaz, G., Minak, G., Altun, I.: Fixed points of ordered \(F\)-contractions. Hacet. J. Math. Stat. 45(1), 15–21 (2016)
Lipschutz, S.: Schaums Outlines of Theory and Problems of Set Theory and Related Topics. McGraw-Hill, New York (1964)
Alam, A., Imdad, M.: Relation-theoretic contraction principle. Fixed Point Theory Appl. 17(4), 693–702 (2015)
Kolman, B., Busby, R.C., Ross, S.: Discrete Mathematical Structures, 3rd edn. PHI Pvt. Ltd., New Delhi (2000)
Roldán López de Hierro: A.F.: A unified version of Ran and Reurings theorem and Nieto and Rodríguez-Lópezs theorem and low-dimensional generalizations. Appl. Math. Inf. Sci. 10(2), 383–393 (2016)
Samet, B., Vetro, C.: Coupled fixed point, \(f\)-invariant set and fixed point of N-order. Ann. Funct. Anal. 1(2), 46–56 (2010)
Ran, A.C.M., Reurings, M.C.B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 132, 1435–1443 (2003)
Berzig, M., Samet, B.: Solving systems of nonlinear matrix equations involving Lipschitzian mappings. Fixed Point Theory Appl. 2011, 89 (2011)
Berzig, M.: Solving a class of matrix equations via the Bhaskar-Lakshmikantham coupled fixed point theorem. Appl. Math. Lett. 25, 1638–1643 (2012)
Long, J.H., Hu, X.Y., Zhang, L.: On the Hermitian positive definite solution of the nonlinear matrix equation \(X + A^*x^{-1}A + B^*X^{-1}B = I\). Bull. Braz. Math. Soc. 39(3), 317–386 (2015)
Acknowledgments
The authors gratefully acknowledge the financial support provided by Thammasat University Research Fund under the TU Research Scholar, Contract No. 2/11/2559.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sawangsup, K., Sintunavarat, W. & de Hierro, A.F.R.L. Fixed point theorems for \(F_\mathfrak {R}\)-contractions with applications to solution of nonlinear matrix equations. J. Fixed Point Theory Appl. 19, 1711–1725 (2017). https://doi.org/10.1007/s11784-016-0306-z
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-016-0306-z