an:05964256
Zbl 1235.54024
Berinde, Vasile
Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces
EN
Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 18, 7347-7355 (2011).
00287895
2011
j
54H25 54E50 54F05 34B15
metric space; mixed monotone operator; contractive condition; coupled fixed point; periodic boundary value problem
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 \(h(t,u)= f(t,u)+ g(t,u)\).
Klaus R. Schneider (Berlin)