Coupled fixed point theorems with respect to binary relations in metric spaces
Authors
Mohammad Sadegh Asgari
 Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch, Tehran, Iran.
Baharak Mousavi
 Department of Mathematics, Faculty of Science, Islamic Azad University, Central Tehran Branch, Tehran, Iran.
Abstract
In this paper we present a new extension of coupled fixed point theorems in metric spaces endowed with
a reflexive binary relation that is not necessarily neither transitive nor antisymmetric. The key feature in
this coupled fixed point theorems is that the contractivity condition on the nonlinear map is only assumed
to hold on elements that are comparable in the binary relation. Next on the basis of the coupled fixed
point theorems, we prove the existence and uniqueness of positive definite solutions of a nonlinear matrix
equation.
Keywords
 Coupled fixed point
 reflexive relation
 matrix equations
 positive define solution.
MSC
References

[1]
M. Abbas, W. Sintunavarat, P. Kumam, Coupled fixed point of generalized contractive mappings on partially ordered Gmetric spaces, Fixed Point Theory Appl., 2012 (2012), 14 pages.

[2]
W. N. Anderson, T. D. Morley, G. E. Trapp, Ladder networks, fixed points and the geometric mean, Circuits Systems Signal Process, 3 (1983), 259268.

[3]
T. Ando , Limit of cascade iteration of matrices, Numer. Funct. Anal. Optim., 21 (1980), 579589

[4]
M. Berzig, B. Samet, Solving systems of nonlinear matrix equations involving Lipshitzian mappings, Fixed Point Theory Appl., 2011 (2011), 10 pages.

[5]
M. Berzig, Solving a class of matrix equations via the BhaskarLakshmikantham coupled fixed point theorem, Appl. Math. Lett., 25 (2012), 16381643.

[6]
T. G. Bhaskar, V. Lakshmikantham, Fixed point theorems in partially ordered metric spaces and applications, Nonlinear Analysis., 65 (2006), 13791393.

[7]
B. L. Buzbee, G. H. Golub, C. W. Nielson , On direct methods for solving Poisson's equations, SIAM J. Numer. Anal., 7 (1970), 627656.

[8]
S. Chandok, W. Sintunavarat, P. Kumam , Some coupled common fixed points for a pair of mappings in partially ordered Gmetric spaces, Mathematical Sciences, 7 (2013), 7 pages.

[9]
X. Duan, A. Liao, B. Tang, On the nonlinear matrix equation \(X  \Sigma^m_{ i=1} A^*_i X^{\delta_i}A_i = Q\), Linear Algebra Appl., 429 (2008), 110121.

[10]
J. C. Engwerda, On the existence of a positive solution of the matrix equation \(X +A^TX^{1}A = I\), Linear Algebra Appl., 194 (1993), 91108.

[11]
W. L. Green, E. Kamen, Stabilization of linear systems over a commutative normed algebra with applications to spatially distributed parameter dependent systems, SIAM J. Control Optim., 23 (1985), 118.

[12]
E. Karapinar, W. Sintunavarat, P. Kumam, Coupled fixed point theorems in cone metric spaces with a cdistance and applications, Fixed Point Theory Appl., 2012 (2012), 19 pages.

[13]
J. H. Long, X. Y. Hu, L. Zhang, 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 (2008), 371386.

[14]
W. Pusz, S. L. Woronowitz, Functional calculus for sequilinear forms and the purification map, Rep. Math. Phys., 8 (1975), 159170.

[15]
A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 132 (2003), 14351443.

[16]
W. Sintunavarat, Y. J. Cho, P. Kumam, Coupled fixed point theorems for contraction mapping induced by cone ballmetric in partially ordered spaces, Fixed Point Theory Appl., 12 (2012), 18 pages.

[17]
W. Sintunavarat, P. Kumam, Coupled fixed point results for nonlinear integral equations, J. Egyptian Math. Soc., 21 (2013), 266272.

[18]
W. Sintunavarat, P. Kumam, Y. J. Cho, Coupled fixed point theorems for nonlinear contractions without mixed monotone property, Fixed Point Theory Appl., 2012 (2012), 16 pages.

[19]
W. Sintunavarat, A. Petruşel, P. Kumam, Coupled common fixed point theorems for \(w^*\)compatible mappings without mixed monotone property, Rend. Circ. Mat. Palermo, 61 (2012), 361383.

[20]
W. Sintunavarat, S. Radenović, Z. Golubović, P. Kumam, Coupled fixed point theorems for Finvariant set and applications, Appl. Math. Inf. Sci., 7 (2013), 247255.