Berinde, Vasile Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces. (English) Zbl 1235.54024 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 18, 7347-7355 (2011). Let \(X\) be a complete metric space with metric \(d\), which is partially ordered. A mapping \(F: X\times X\to X\) is called mixed monotone if \(F(x,y)\) is monotone nondecreasing in \(x\) and monotone nonincreasing in \(y\). A pair \((\overline x,\overline y)\in X\times X\) is called a coupled fixed point of \(F\) if \(F(\overline x,\overline y)=\overline x\), \(F(\overline y,\overline x)=\overline y\). The main result of the paper is the following theorem. Theorem. Let \(X\) be a partialy ordered complete metric space, let \(F: X\times X\to X\) be mixed monotone and such that (i) There is a constant \(k\in [0,1)\) such that for each \(x\geq u\), \(y\leq v\) \[ d(F(x,y), F(u,v))+ d(F(y, x), F(v,u))\leq k[d(x, u)+ d(y,v)]. \] (ii) There exist \(x_0,y_0\in X\) with \[ x_0\leq F(x_0, y_0)\quad\text{and}\quad y_0\leq F(y_0, x_0) \] or \[ x_0\geq F(x_0, y_0)\quad\text{and}\quad y_0\leq F(y_0, x_0). \] Then \(F\) has a coupled fixed point \((\overline x,\overline y)\). The author also gives conditions under which there exists a unique coupled fixed point. Finally, he applies this theorems to the periodic boundary value problem \[ u'= h(t,u),\quad t\in (0,T),\quad u(0)= u(T) \] with \(h(t,u)= f(t,u)+ g(t,u)\). Reviewer: Klaus R. Schneider (Berlin) Cited in 12 ReviewsCited in 115 Documents MSC: 54H25 Fixed-point and coincidence theorems (topological aspects) 54E50 Complete metric spaces 54F05 Linearly ordered topological spaces, generalized ordered spaces, and partially ordered spaces 34B15 Nonlinear boundary value problems for ordinary differential equations Keywords:metric space; mixed monotone operator; contractive condition; coupled fixed point; periodic boundary value problem PDFBibTeX XMLCite \textit{V. Berinde}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 18, 7347--7355 (2011; Zbl 1235.54024) Full Text: DOI arXiv References: [1] Rus, I. A.; Petruşel, A.; Petruşel, G., Fixed Point Theory (2008), Cluj University Press: Cluj University Press Cluj-Napoca · Zbl 1171.54034 [2] Agarwal, R. P.; El-Gebeily, M. A.; O’Regan, D., Generalized contractions in partially ordered metric spaces, Appl. Anal., 87, 1-8 (2008) · Zbl 1140.47042 [3] Drici, Z.; McRae, F.; Devi, J. 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