Global existence of solutions of integral equations with delay: progressive contractions
In the theory of progressive contractions an equation such as \[ x(t) = L(t)+\int^t_0 A(t-s)[ f(s,x(s)) + g(s,x(s-r(s))]ds, \] with initial function $\omega$ with $\omega (0) =L(0)$ defined by $ t\leq 0 \implies x(t) =\omega (t)$ is studied on an interval $[0,E]$ with $r(t) \geq \alpha >0$. The...
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| Format: | Article |
| Language: | English |
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University of Szeged
2017-06-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
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| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5827 |
| _version_ | 1797830574505721856 |
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| author | Theodore Burton Ioannis Purnaras |
| author_facet | Theodore Burton Ioannis Purnaras |
| author_sort | Theodore Burton |
| collection | DOAJ |
| description | In the theory of progressive contractions an equation such as
\[
x(t) = L(t)+\int^t_0 A(t-s)[ f(s,x(s)) + g(s,x(s-r(s))]ds,
\]
with initial function $\omega$ with $\omega (0) =L(0)$ defined by $ t\leq 0 \implies x(t) =\omega (t)$ is studied on an interval $[0,E]$ with $r(t) \geq \alpha >0$. The interval $[0,E]$ is divided into parts by $0=T_0<T_1<\dots<T_n=E$ with $T_i-T_{i-1} <\alpha$. It is assumed that $f$ satisfies a Lipschitz condition, but there is no growth condition on $g$. When we try for a contraction on $[0,T_1]$ the terms with $g$ add to zero and we get a unique solution $\xi_1$ on $[0,T_1]$. Then we get a complete metric space on $[0,T_2]$ with all functions equal to $\xi_1$ on $[0,T_1]$ enabling us to get a contraction. In $n$ steps we have obtained a solution on $[0,E]$. When $r(t) >0$ on $[0,\infty)$ we obtain a unique solution on that interval as follows. As we let $E= 1,2,\dots$ we obtain a sequence of solutions on $[0,n]$ which we extend to $[0,\infty)$ by a horizontal line, thereby obtaining functions converging uniformly on compact sets to a solution on $[0,\infty)$. Lemma 2.1 extends progressive contractions to delay equations |
| first_indexed | 2024-04-09T13:38:15Z |
| format | Article |
| id | doaj.art-5056a9398eab4a938eb0c0fc45b278d3 |
| institution | Directory Open Access Journal |
| issn | 1417-3875 |
| language | English |
| last_indexed | 2024-04-09T13:38:15Z |
| publishDate | 2017-06-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj.art-5056a9398eab4a938eb0c0fc45b278d32023-05-09T07:53:07ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752017-06-012017491610.14232/ejqtde.2017.1.495827Global existence of solutions of integral equations with delay: progressive contractionsTheodore Burton0Ioannis Purnaras1Northwest Research Institute, Port Angeles, WA, U.S.A.University of Ioannina, Ioannina, GreeceIn the theory of progressive contractions an equation such as \[ x(t) = L(t)+\int^t_0 A(t-s)[ f(s,x(s)) + g(s,x(s-r(s))]ds, \] with initial function $\omega$ with $\omega (0) =L(0)$ defined by $ t\leq 0 \implies x(t) =\omega (t)$ is studied on an interval $[0,E]$ with $r(t) \geq \alpha >0$. The interval $[0,E]$ is divided into parts by $0=T_0<T_1<\dots<T_n=E$ with $T_i-T_{i-1} <\alpha$. It is assumed that $f$ satisfies a Lipschitz condition, but there is no growth condition on $g$. When we try for a contraction on $[0,T_1]$ the terms with $g$ add to zero and we get a unique solution $\xi_1$ on $[0,T_1]$. Then we get a complete metric space on $[0,T_2]$ with all functions equal to $\xi_1$ on $[0,T_1]$ enabling us to get a contraction. In $n$ steps we have obtained a solution on $[0,E]$. When $r(t) >0$ on $[0,\infty)$ we obtain a unique solution on that interval as follows. As we let $E= 1,2,\dots$ we obtain a sequence of solutions on $[0,n]$ which we extend to $[0,\infty)$ by a horizontal line, thereby obtaining functions converging uniformly on compact sets to a solution on $[0,\infty)$. Lemma 2.1 extends progressive contractions to delay equationshttp://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5827progressive contractionsintegral and differential equations with delayglobal existencefixed points |
| spellingShingle | Theodore Burton Ioannis Purnaras Global existence of solutions of integral equations with delay: progressive contractions Electronic Journal of Qualitative Theory of Differential Equations progressive contractions integral and differential equations with delay global existence fixed points |
| title | Global existence of solutions of integral equations with delay: progressive contractions |
| title_full | Global existence of solutions of integral equations with delay: progressive contractions |
| title_fullStr | Global existence of solutions of integral equations with delay: progressive contractions |
| title_full_unstemmed | Global existence of solutions of integral equations with delay: progressive contractions |
| title_short | Global existence of solutions of integral equations with delay: progressive contractions |
| title_sort | global existence of solutions of integral equations with delay progressive contractions |
| topic | progressive contractions integral and differential equations with delay global existence fixed points |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=5827 |
| work_keys_str_mv | AT theodoreburton globalexistenceofsolutionsofintegralequationswithdelayprogressivecontractions AT ioannispurnaras globalexistenceofsolutionsofintegralequationswithdelayprogressivecontractions |