Best Proximity Results with Applications to Nonlinear Dynamical Systems
Best proximity point theorem furnishes sufficient conditions for the existence and computation of an approximate solution <inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula> that is...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2019-09-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/7/10/900 |
| _version_ | 1818199158912188416 |
|---|---|
| author | Hamed H Al-Sulami Nawab Hussain Jamshaid Ahmad |
| author_facet | Hamed H Al-Sulami Nawab Hussain Jamshaid Ahmad |
| author_sort | Hamed H Al-Sulami |
| collection | DOAJ |
| description | Best proximity point theorem furnishes sufficient conditions for the existence and computation of an approximate solution <inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula> that is optimal in the sense that the error <inline-formula> <math display="inline"> <semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>ω</mi> <mo>,</mo> <mi mathvariant="script">J</mi> <mi>ω</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> assumes the global minimum value <inline-formula> <math display="inline"> <semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϑ</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. The aim of this paper is to define the notion of Suzuki <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-<inline-formula> <math display="inline"> <semantics> <mo>Θ</mo> </semantics> </math> </inline-formula>-proximal multivalued contraction and prove the existence of best proximity points <inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula> satisfying <inline-formula> <math display="inline"> <semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>ω</mi> <mo>,</mo> <mi mathvariant="script">J</mi> <mi>ω</mi> <mo>)</mo> <mo>=</mo> <mi>σ</mi> <mo>(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϑ</mi> <mo>)</mo> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">J</mi> </semantics> </math> </inline-formula> is assumed to be continuous or the space <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">M</mi> </semantics> </math> </inline-formula> is regular. We derive some best proximity results on a metric space with graphs and ordered metric spaces as consequences. We also provide a non trivial example to support our main results. As applications of our main results, we discuss some variational inequality problems and dynamical programming problems. |
| first_indexed | 2024-12-12T02:17:19Z |
| format | Article |
| id | doaj.art-bd9fdf137b9c49ef84f7afb10e3e4f72 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-12-12T02:17:19Z |
| publishDate | 2019-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-bd9fdf137b9c49ef84f7afb10e3e4f722022-12-22T00:41:46ZengMDPI AGMathematics2227-73902019-09-0171090010.3390/math7100900math7100900Best Proximity Results with Applications to Nonlinear Dynamical SystemsHamed H Al-Sulami0Nawab Hussain1Jamshaid Ahmad2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, University of Jeddah, P.O.Box 80327, Jeddah 21589, Saudi ArabiaBest proximity point theorem furnishes sufficient conditions for the existence and computation of an approximate solution <inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula> that is optimal in the sense that the error <inline-formula> <math display="inline"> <semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>ω</mi> <mo>,</mo> <mi mathvariant="script">J</mi> <mi>ω</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> assumes the global minimum value <inline-formula> <math display="inline"> <semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϑ</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. The aim of this paper is to define the notion of Suzuki <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-<inline-formula> <math display="inline"> <semantics> <mo>Θ</mo> </semantics> </math> </inline-formula>-proximal multivalued contraction and prove the existence of best proximity points <inline-formula> <math display="inline"> <semantics> <mi>ω</mi> </semantics> </math> </inline-formula> satisfying <inline-formula> <math display="inline"> <semantics> <mrow> <mi>σ</mi> <mo>(</mo> <mi>ω</mi> <mo>,</mo> <mi mathvariant="script">J</mi> <mi>ω</mi> <mo>)</mo> <mo>=</mo> <mi>σ</mi> <mo>(</mo> <mi>θ</mi> <mo>,</mo> <mi>ϑ</mi> <mo>)</mo> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">J</mi> </semantics> </math> </inline-formula> is assumed to be continuous or the space <inline-formula> <math display="inline"> <semantics> <mi mathvariant="script">M</mi> </semantics> </math> </inline-formula> is regular. We derive some best proximity results on a metric space with graphs and ordered metric spaces as consequences. We also provide a non trivial example to support our main results. As applications of our main results, we discuss some variational inequality problems and dynamical programming problems.https://www.mdpi.com/2227-7390/7/10/900nonlinear dynamical systemsbest proximity point<i>α</i>-proximal contractionmulti-valued mappingsgraphs |
| spellingShingle | Hamed H Al-Sulami Nawab Hussain Jamshaid Ahmad Best Proximity Results with Applications to Nonlinear Dynamical Systems Mathematics nonlinear dynamical systems best proximity point <i>α</i>-proximal contraction multi-valued mappings graphs |
| title | Best Proximity Results with Applications to Nonlinear Dynamical Systems |
| title_full | Best Proximity Results with Applications to Nonlinear Dynamical Systems |
| title_fullStr | Best Proximity Results with Applications to Nonlinear Dynamical Systems |
| title_full_unstemmed | Best Proximity Results with Applications to Nonlinear Dynamical Systems |
| title_short | Best Proximity Results with Applications to Nonlinear Dynamical Systems |
| title_sort | best proximity results with applications to nonlinear dynamical systems |
| topic | nonlinear dynamical systems best proximity point <i>α</i>-proximal contraction multi-valued mappings graphs |
| url | https://www.mdpi.com/2227-7390/7/10/900 |
| work_keys_str_mv | AT hamedhalsulami bestproximityresultswithapplicationstononlineardynamicalsystems AT nawabhussain bestproximityresultswithapplicationstononlineardynamicalsystems AT jamshaidahmad bestproximityresultswithapplicationstononlineardynamicalsystems |