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Fixed point for set-valued mappings satisfying an implicit relation in partially ordered metric spaces. (English) Zbl 1176.54028
Let $(X,d;\le)$ be a complete partially ordered metric space and $F,G:B(x_0,r)\to C(X)$ be two maps with bounded values, fulfilling {\parindent=8mm\item{(i)} $T(D(Fx,Gy),d(x,y),d(x,Fx),d(y,Gy),d(x,Gy),d(y,Gx))\le 0$ for all $x,y\in B(x_0,r)$, where $T:\bbfR_+^6\to\bbfR_+$ satisfies some mild conditions, \item{(ii)} for each $x\in X$, there exists $y\in Lx$ with $x\le y$ such that $d(x,y)\le d(x,Lx)+\varepsilon$, where $L\in \{F,G\}$, \item{(iii)} if $(x_n)\subseteq B(x_0,r)$ fulfills $x_n\le x_{n+1}$ for all $n$ and $x_n\to x$ as $n\to \infty$, then $x_n\le x$ for all $n$. \par}In addition, assume that there exists a continuous strictly increasing function $\Phi:\bbfR_+\to \bbfR_+$ with $\Phi(t)< t$ for all $t> 0$, such that {\parindent=8mm\item{(iv)} $d(x_0,x_1)< r-\Phi(r)$ for some $x_1\in Fx_0$ with $x_0\le x_1$, \item{(v)} $\sum_n \Phi^n(r-\Phi(r))\le \Phi(r)$.\par} Then there exists in $B(x_0,r)$ a common fixed point for $F$ and $G$. Reviewer’s remark: The seminal 2004 fixed point result of {\it A. C. M. Ran} and {\it M. C. B. Reurings} [Proc. Am. Math. Soc. 132, No. 5, 1435--1443 (2004; Zbl 1060.47056)] was obtained in 1986 by the reviewer [J. Math. Anal. Appl. 117, 100--127 (1986; Zbl 0613.47037)].

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 54F05 Linearly, generalized, and partial ordered topological spaces 54C60 Set-valued maps (general topology)
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##### References:
 [1] I. Altun, Fixed point and homotopy results for multivalued maps satisfying an implicit relation, J. Fixed Point Theory and Appl. (in press) · Zbl 1205.54036 [2] Altun, I.; Hancer, H. A.; Türkoğlu, D.: A fixed point theorem for multi-maps satisfying an implicit relation on metrically convex metric spaces. Math. commun. 11, 17-23 (2006) · Zbl 1105.54303 [3] Altun, I.; Türkoğlu, D.: Fixed point and homotopy results for mappings satisfying an implicit relation. Discuss. math. Differ. incl. Control optim. 27, 349-363 (2007) [4] Altun, I.; Türkoğlu, D.; Rhoades, B. E.: Fixed points of weakly compatible maps satisfying a general contractive condition of integral type. Fixed point theory appl. (2007) [5] Sedghi, S.; Altun, I.; Shobe, N.: A fixed point theorem for multi-maps satisfying an implicit relation on metric spaces. Appl. anal. Discrete math. 2, 189-196 (2008) · Zbl 1199.54245 [6] V. Lakshmikantham, L. Ciric, Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces, Nonlinear Anal., in press (doi:10.1016/j.na.2008.09.020) [7] Nieto, J. J.; Rodríguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223-239 (2005) · Zbl 1095.47013 [8] Nieto, J. J.; Rodríguez-López, R.: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta. math. Sin. (English ser.) 23, 2205-2212 (2007) · Zbl 1140.47045 [9] Nieto, J. J.; Pouso, R. L.; Rodríguez-López, R.: Fixed point theorems in ordered abstract spaces. Proc. amer. Math. soc. 135, 2505-2517 (2007) · Zbl 1126.47045 [10] Nieto, J. J.: Applications of contractive-like mapping principles to fuzzy equations. Rev. mat. Complut. 19, 361-383 (2006) · Zbl 1113.26030 [11] Nieto, J. J.; Rodríguez-López, R.: Existence of extremal solutions for quadratic fuzzy equations. Fixed point theory appl. 2005, No. 3, 321-342 (2005) · Zbl 1102.54004 [12] Ran, A. C. M.; Reurings, M. C. B.: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. amer. Math. soc. 132, 1435-1443 (2003) · Zbl 1060.47056 [13] O’regan, D.; Petrusel, A.: Fixed point theorems for generalized contractions in ordered metric spaces. J. math. Anal. appl. 341, 1241-1252 (2008) [14] Cabada, A.; Nieto, J. J.: Fixed points and approximate solutions for nonlinear operator equations. J. comput. Appl. math. 113, 17-25 (2000) · Zbl 0954.47038 [15] Drici, Z.; Mcrae, F. A.; Devi, J. V.: Fixed point theorems in partially ordered metric space for operators with PPF dependence. Nonlinear anal. 67, 641-647 (2007) · Zbl 1127.47049 [16] Petrusel, A.; Rus, I. A.: Fixed point theorems in ordered L-spaces. Proc. amer. Math. soc. 134, 411-418 (2005) · Zbl 1086.47026 [17] Kirk, W. A.; Goebel, K.: Topics in metric fixed point theory. (1990) · Zbl 0708.47031 [18] Tarski, A.: A lattice theoretical fixed point theorem and its application. Pacific J. Math. 5, 285-309 (1955) · Zbl 0064.26004 [19] Echenique, F.: A short and constructive proof of Tarski’s fixed-point theorem. Internat. J. Game theory 33, No. 2, 215-218 (2005) · Zbl 1071.91002 [20] Granas, A.; Dugundji, J.: Fixed point theory. (2003) · Zbl 1025.47002