Geometric Fixed Point Theorems on Transversal Spaces
Mathematica Moravica, Tome 8 (2004) no. 2.
Voir la notice de l'article dans eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts
This paper presents new fixed point theorems on lower and upper transversal spaces. The following main result is proved that if $T$ is a self-map on an orbitally DS-complete transversal lower space $(X, \rho)$, if there exists an upper semicontinuous function $G : X \to \mathbb{R}$ such that $\rho [x, Tx] \geq G(Tx) − G(x) \geq 0$ for every $x \in X$ and if $G(T^na) \rightarrow +\infty$ as $n \rightarrow \infty$ for some $a \in X$, then $T$ has a fixed point in $X$. For the lower transversal spaces are essential the mappings $T : X \to X$ which are unbounded variation, i.e., if there exists a function $A : X \times X \to \mathbb{R}^0_{+}$ such that $\sum_{n=0}^{\infty} A(T^nx, T^{n+1}x)= +\infty$ for arbitrary $x \in X$. On the other hand, for upper transversal spaces are essential the mappings $T : X \to X$ which are bounded variation.
Mots-clés :
Fixed point theorems, lower and upper transversal spaces, orbitally DS-complete and CS-complete spaces, functions of unbounded variation, function of bounded variation, geometric fixed point theorems
@article{MM3_2004_8_2_a3, author = {Milan Taskovi\'c}, title = {Geometric {Fixed} {Point} {Theorems} on {Transversal} {Spaces}}, journal = {Mathematica Moravica}, pages = {29 - 52}, publisher = {mathdoc}, volume = {8}, number = {2}, year = {2004}, url = {https://geodesic-test.mathdoc.fr/item/MM3_2004_8_2_a3/} }
Milan Tasković. Geometric Fixed Point Theorems on Transversal Spaces. Mathematica Moravica, Tome 8 (2004) no. 2. https://geodesic-test.mathdoc.fr/item/MM3_2004_8_2_a3/