### Abstract

him, the fixed-point results in various settings play an important role in the optimization theory and applications. This technique has proven to be indispensable for the proofs of multiple results related to the existence of solutions to numerous problems in the areas of optimization and approximation theory, differential equations, variational inequalities, complementary problems, equilibrium theory, game theory, mathematical economics, etc. It is also worthwhile to mention that the majority of problems of finding solutions (zero-points) of functions (operators) can be easily reduced to that of discovering of fixed points of properly modified mappings. Not only theoretical but also practical (algorithmic) developments are based on the fixed-point theory. For

instance, the well-known simplicial (triangulation) algorithms help one to find the

desired fixed points in a constructive way. That approach allows one to investigate the solvability of complicated problems arising in theory and applications. In this paper, making use of the triangulation technique, we extend some antipodal and fixed point theorems to the case of non-convex, more exactly, star-shaped sets. Also, similar extensions are made for set-valued mappings defined over star-shaped sets.

Original language | English |
---|---|

Pages (from-to) | 399-412 |

Journal | Journal of Optimization Theory and Applications |

Volume | 177 |

Issue number | 2 |

DOIs | |

Publication status | Published - May 2018 |

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### Keywords

- Antipodal theorems
- Star-shaped subsets
- Trianggulations techniques
- Set-valued mappings

### Cite this

*Journal of Optimization Theory and Applications*,

*177*(2), 399-412. https://doi.org/10.1007/s10957-018-1283-8

}

*Journal of Optimization Theory and Applications*, vol. 177, no. 2, pp. 399-412. https://doi.org/10.1007/s10957-018-1283-8

**Extended antipodal theorems.** / Kalashnikov, V.V.; Talman, Dolf; Alanis-Lopez, Lilia; Kalashnykova, Nataliya I.

Research output: Contribution to journal › Article › Scientific › peer-review

TY - JOUR

T1 - Extended antipodal theorems

AU - Kalashnikov, V.V.

AU - Talman, Dolf

AU - Alanis-Lopez, Lilia

AU - Kalashnykova, Nataliya I.

PY - 2018/5

Y1 - 2018/5

N2 - Since 1909 when Brouwer proved the first fixed-point theorem named afterhim, the fixed-point results in various settings play an important role in the optimization theory and applications. This technique has proven to be indispensable for the proofs of multiple results related to the existence of solutions to numerous problems in the areas of optimization and approximation theory, differential equations, variational inequalities, complementary problems, equilibrium theory, game theory, mathematical economics, etc. It is also worthwhile to mention that the majority of problems of finding solutions (zero-points) of functions (operators) can be easily reduced to that of discovering of fixed points of properly modified mappings. Not only theoretical but also practical (algorithmic) developments are based on the fixed-point theory. Forinstance, the well-known simplicial (triangulation) algorithms help one to find thedesired fixed points in a constructive way. That approach allows one to investigate the solvability of complicated problems arising in theory and applications. In this paper, making use of the triangulation technique, we extend some antipodal and fixed point theorems to the case of non-convex, more exactly, star-shaped sets. Also, similar extensions are made for set-valued mappings defined over star-shaped sets.

AB - Since 1909 when Brouwer proved the first fixed-point theorem named afterhim, the fixed-point results in various settings play an important role in the optimization theory and applications. This technique has proven to be indispensable for the proofs of multiple results related to the existence of solutions to numerous problems in the areas of optimization and approximation theory, differential equations, variational inequalities, complementary problems, equilibrium theory, game theory, mathematical economics, etc. It is also worthwhile to mention that the majority of problems of finding solutions (zero-points) of functions (operators) can be easily reduced to that of discovering of fixed points of properly modified mappings. Not only theoretical but also practical (algorithmic) developments are based on the fixed-point theory. Forinstance, the well-known simplicial (triangulation) algorithms help one to find thedesired fixed points in a constructive way. That approach allows one to investigate the solvability of complicated problems arising in theory and applications. In this paper, making use of the triangulation technique, we extend some antipodal and fixed point theorems to the case of non-convex, more exactly, star-shaped sets. Also, similar extensions are made for set-valued mappings defined over star-shaped sets.

KW - Antipodal theorems

KW - Star-shaped subsets

KW - Trianggulations techniques

KW - Set-valued mappings

U2 - 10.1007/s10957-018-1283-8

DO - 10.1007/s10957-018-1283-8

M3 - Article

VL - 177

SP - 399

EP - 412

JO - Journal of Optimization Theory and Applications

JF - Journal of Optimization Theory and Applications

SN - 0022-3239

IS - 2

ER -