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Remarks on Caristi’s fixed point theorem and Kirk’s problem. (English) Zbl 1201.54036
Let $(X,d)$ be a complete metric space. An operator $T: X\to T$ is said to be a Caristi type mapping if the following condition is satisfied: $\eta(d(x, Tx))\le\varphi(x)- \varphi(Tx)$ for all $x\in X$, where $\eta: [0,+\infty)\to (-\infty,+\infty)$ and $\varphi: X\to (-\infty,+\infty)$. In the present paper, the author, motivated by {\it M. A.\thinspace Khamsi} [Nonlinear Anal., Theory Methods Appl. 71, No.  1--2, A, 227--231 (2009; Zbl 1175.54056)], proves the following theorems. Theorem 1. Suppose that $\eta :[0,+\infty)\to [0,+\infty)$ with $\eta(0)= 0$, $\varphi: X\to(-\infty,+\infty)$ is lower-semicontinuous on $X$, and there exist $x_0\in X$ and two real numbers $a< 0$, $-\infty<\beta<+\infty$, such that $\varphi(x)\ge ad(x,x_0)+\beta$. Suppose that one of the following conditions is satisfied: (i) $a\ge 0$, $\eta$ is nonnegative and nondecreasing on $W= \{d(x,y): x,y\in X\}$, and there exists $c> 0$ and $\varepsilon> 0$ such that $\eta(t)\ge ct$ for all $t\in\{t\ge 0:\eta(t)\le\varepsilon\}\cap W$; (ii) $a< 0$, $\eta(t)+ at$ is nonnegative and nondecreasing on $W$, and there exist $c> 0$ and $\varepsilon> 0$ such that $\eta(t)+ at\ge ct$ for all $t\in\{t\ge 0:\eta(t)+ at\le\varepsilon\}\cap W$. Then each Caristi type mapping $T: X\to X$ has a fixed point in $X$. Theorem 2. Suppose that $\eta:[0,+\infty)\to [0,+\infty)$ with $\eta(0)= 0$, $\varphi: X\to(-\infty,+\infty)$ is lower-semicontinuous on $X$ and bounded below on each bounded subset of $X$, and there exist $x_0\in X$ and a real number $-\infty< a<+\infty$ such that $$\liminf_{d(x,x_0)\to+\infty}\ {\varphi(x)\over d(x,x_0)}> a.$$ Suppose that one of the following conditions is satisfied: (i) $a\ge 0$, $\eta$ is nondecreasing on $[0,+\infty)$ and $$\liminf_{t\to 0^+}\ {\eta(t)\over t}> 0,$$ (ii) $a< 0$, $\eta(t)+ at$ is nonnegative and nondecreasing on $[0,+\infty)$ and $$\liminf_{t\to 0^+}\ {\eta(t)\over t}\ge -a.$$ Then each Caristi type mapping $T: X\to X$ has a fixed point in $X$.

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 06A06 Partial order
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##### References:
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