Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales

In this paper we prove the existence of weak solutions of the dynamic Cauchy problem \begin{equation*} \begin{split} x^{(\Delta m)}(t)&=f(t,x(t)),\quad t\in T, \\ x(0)&=0, \\ x^\Delta (0)&=\eta _1 ,\dots,x^{(\Delta (m-1))}(0)=\eta _{m-1},\quad \eta _1 ,\dots,\eta _{m-1} \in E, \end{split...
Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Aneta Sikorska-Nowak [verfasserIn] Samir Saker [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2014

Schlagwörter:	cauchy dynamic problem banach space measure of weak noncompactness weak solutions time scales fixed point

Übergeordnetes Werk:	In: Electronic Journal of Qualitative Theory of Differential Equations - University of Szeged, 2003, (2014), 21, Seite 13
Übergeordnetes Werk:	year:2014 ; number:21 ; pages:13

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.14232/ejqtde.2014.1.21

Katalog-ID:	DOAJ058225730

Internformat


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allfields	10.14232/ejqtde.2014.1.21 doi (DE-627)DOAJ058225730 (DE-599)DOAJ08b7e152dfb74b3987688ec4c3ba6e10 DE-627 ger DE-627 rakwb eng QA1-939 Aneta Sikorska-Nowak verfasserin aut Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In this paper we prove the existence of weak solutions of the dynamic Cauchy problem \begin{equation} \begin{split} x^{(\Delta m)}(t)&=f(t,x(t)),\quad t\in T, \\ x(0)&=0, \\ x^\Delta (0)&=\eta _1 ,\dots,x^{(\Delta (m-1))}(0)=\eta _{m-1},\quad \eta _1 ,\dots,\eta _{m-1} \in E, \end{split} \end{equation} where $x^{(\Delta m)}$ denotes a weak $m$-th order $\Delta$-derivative, $T$ denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence $(a_n )$ in $T$ and $a_n \to \infty )$, $E$ is a Banach space and $f$ is weakly -- weakly sequentially continuous and satisfies some conditions expressed in terms of measures of weak noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. As dynamic equations are a unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions. cauchy dynamic problem banach space measure of weak noncompactness weak solutions time scales fixed point Mathematics Samir Saker verfasserin aut In Electronic Journal of Qualitative Theory of Differential Equations University of Szeged, 2003 (2014), 21, Seite 13 (DE-627)320518264 (DE-600)2014234-1 14173875 nnns year:2014 number:21 pages:13 https://doi.org/10.14232/ejqtde.2014.1.21 kostenfrei https://doaj.org/article/08b7e152dfb74b3987688ec4c3ba6e10 kostenfrei http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=2693 kostenfrei https://doaj.org/toc/1417-3875 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2031 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2061 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_2190 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2014 21 13
spelling	10.14232/ejqtde.2014.1.21 doi (DE-627)DOAJ058225730 (DE-599)DOAJ08b7e152dfb74b3987688ec4c3ba6e10 DE-627 ger DE-627 rakwb eng QA1-939 Aneta Sikorska-Nowak verfasserin aut Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In this paper we prove the existence of weak solutions of the dynamic Cauchy problem \begin{equation} \begin{split} x^{(\Delta m)}(t)&=f(t,x(t)),\quad t\in T, \\ x(0)&=0, \\ x^\Delta (0)&=\eta _1 ,\dots,x^{(\Delta (m-1))}(0)=\eta _{m-1},\quad \eta _1 ,\dots,\eta _{m-1} \in E, \end{split} \end{equation} where $x^{(\Delta m)}$ denotes a weak $m$-th order $\Delta$-derivative, $T$ denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence $(a_n )$ in $T$ and $a_n \to \infty )$, $E$ is a Banach space and $f$ is weakly -- weakly sequentially continuous and satisfies some conditions expressed in terms of measures of weak noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. As dynamic equations are a unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions. cauchy dynamic problem banach space measure of weak noncompactness weak solutions time scales fixed point Mathematics Samir Saker verfasserin aut In Electronic Journal of Qualitative Theory of Differential Equations University of Szeged, 2003 (2014), 21, Seite 13 (DE-627)320518264 (DE-600)2014234-1 14173875 nnns year:2014 number:21 pages:13 https://doi.org/10.14232/ejqtde.2014.1.21 kostenfrei https://doaj.org/article/08b7e152dfb74b3987688ec4c3ba6e10 kostenfrei http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=2693 kostenfrei https://doaj.org/toc/1417-3875 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2031 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2061 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_2190 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2014 21 13
allfields_unstemmed	10.14232/ejqtde.2014.1.21 doi (DE-627)DOAJ058225730 (DE-599)DOAJ08b7e152dfb74b3987688ec4c3ba6e10 DE-627 ger DE-627 rakwb eng QA1-939 Aneta Sikorska-Nowak verfasserin aut Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In this paper we prove the existence of weak solutions of the dynamic Cauchy problem \begin{equation} \begin{split} x^{(\Delta m)}(t)&=f(t,x(t)),\quad t\in T, \\ x(0)&=0, \\ x^\Delta (0)&=\eta _1 ,\dots,x^{(\Delta (m-1))}(0)=\eta _{m-1},\quad \eta _1 ,\dots,\eta _{m-1} \in E, \end{split} \end{equation} where $x^{(\Delta m)}$ denotes a weak $m$-th order $\Delta$-derivative, $T$ denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence $(a_n )$ in $T$ and $a_n \to \infty )$, $E$ is a Banach space and $f$ is weakly -- weakly sequentially continuous and satisfies some conditions expressed in terms of measures of weak noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. As dynamic equations are a unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions. cauchy dynamic problem banach space measure of weak noncompactness weak solutions time scales fixed point Mathematics Samir Saker verfasserin aut In Electronic Journal of Qualitative Theory of Differential Equations University of Szeged, 2003 (2014), 21, Seite 13 (DE-627)320518264 (DE-600)2014234-1 14173875 nnns year:2014 number:21 pages:13 https://doi.org/10.14232/ejqtde.2014.1.21 kostenfrei https://doaj.org/article/08b7e152dfb74b3987688ec4c3ba6e10 kostenfrei http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=2693 kostenfrei https://doaj.org/toc/1417-3875 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2031 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2061 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_2190 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2014 21 13
allfieldsGer	10.14232/ejqtde.2014.1.21 doi (DE-627)DOAJ058225730 (DE-599)DOAJ08b7e152dfb74b3987688ec4c3ba6e10 DE-627 ger DE-627 rakwb eng QA1-939 Aneta Sikorska-Nowak verfasserin aut Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In this paper we prove the existence of weak solutions of the dynamic Cauchy problem \begin{equation} \begin{split} x^{(\Delta m)}(t)&=f(t,x(t)),\quad t\in T, \\ x(0)&=0, \\ x^\Delta (0)&=\eta _1 ,\dots,x^{(\Delta (m-1))}(0)=\eta _{m-1},\quad \eta _1 ,\dots,\eta _{m-1} \in E, \end{split} \end{equation} where $x^{(\Delta m)}$ denotes a weak $m$-th order $\Delta$-derivative, $T$ denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence $(a_n )$ in $T$ and $a_n \to \infty )$, $E$ is a Banach space and $f$ is weakly -- weakly sequentially continuous and satisfies some conditions expressed in terms of measures of weak noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. As dynamic equations are a unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions. cauchy dynamic problem banach space measure of weak noncompactness weak solutions time scales fixed point Mathematics Samir Saker verfasserin aut In Electronic Journal of Qualitative Theory of Differential Equations University of Szeged, 2003 (2014), 21, Seite 13 (DE-627)320518264 (DE-600)2014234-1 14173875 nnns year:2014 number:21 pages:13 https://doi.org/10.14232/ejqtde.2014.1.21 kostenfrei https://doaj.org/article/08b7e152dfb74b3987688ec4c3ba6e10 kostenfrei http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=2693 kostenfrei https://doaj.org/toc/1417-3875 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2031 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2061 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_2190 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2014 21 13
allfieldsSound	10.14232/ejqtde.2014.1.21 doi (DE-627)DOAJ058225730 (DE-599)DOAJ08b7e152dfb74b3987688ec4c3ba6e10 DE-627 ger DE-627 rakwb eng QA1-939 Aneta Sikorska-Nowak verfasserin aut Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier In this paper we prove the existence of weak solutions of the dynamic Cauchy problem \begin{equation} \begin{split} x^{(\Delta m)}(t)&=f(t,x(t)),\quad t\in T, \\ x(0)&=0, \\ x^\Delta (0)&=\eta _1 ,\dots,x^{(\Delta (m-1))}(0)=\eta _{m-1},\quad \eta _1 ,\dots,\eta _{m-1} \in E, \end{split} \end{equation} where $x^{(\Delta m)}$ denotes a weak $m$-th order $\Delta$-derivative, $T$ denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence $(a_n )$ in $T$ and $a_n \to \infty )$, $E$ is a Banach space and $f$ is weakly -- weakly sequentially continuous and satisfies some conditions expressed in terms of measures of weak noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. As dynamic equations are a unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions. cauchy dynamic problem banach space measure of weak noncompactness weak solutions time scales fixed point Mathematics Samir Saker verfasserin aut In Electronic Journal of Qualitative Theory of Differential Equations University of Szeged, 2003 (2014), 21, Seite 13 (DE-627)320518264 (DE-600)2014234-1 14173875 nnns year:2014 number:21 pages:13 https://doi.org/10.14232/ejqtde.2014.1.21 kostenfrei https://doaj.org/article/08b7e152dfb74b3987688ec4c3ba6e10 kostenfrei http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1&paramtipus_ertek=publication&param_ertek=2693 kostenfrei https://doaj.org/toc/1417-3875 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2031 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2061 GBV_ILN_2088 GBV_ILN_2111 GBV_ILN_2190 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 2014 21 13
language	English
source	In Electronic Journal of Qualitative Theory of Differential Equations (2014), 21, Seite 13 year:2014 number:21 pages:13
sourceStr	In Electronic Journal of Qualitative Theory of Differential Equations (2014), 21, Seite 13 year:2014 number:21 pages:13
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	cauchy dynamic problem banach space measure of weak noncompactness weak solutions time scales fixed point Mathematics
isfreeaccess_bool	true
container_title	Electronic Journal of Qualitative Theory of Differential Equations
authorswithroles_txt_mv	Aneta Sikorska-Nowak @@aut@@ Samir Saker @@aut@@
publishDateDaySort_date	2014-01-01T00:00:00Z
hierarchy_top_id	320518264
id	DOAJ058225730
language_de	englisch
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callnumber-first	Q - Science
author	Aneta Sikorska-Nowak
spellingShingle	Aneta Sikorska-Nowak misc QA1-939 misc cauchy dynamic problem misc banach space misc measure of weak noncompactness misc weak solutions misc time scales misc fixed point misc Mathematics Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales
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topic_title	QA1-939 Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales cauchy dynamic problem banach space measure of weak noncompactness weak solutions time scales fixed point
topic	misc QA1-939 misc cauchy dynamic problem misc banach space misc measure of weak noncompactness misc weak solutions misc time scales misc fixed point misc Mathematics
topic_unstemmed	misc QA1-939 misc cauchy dynamic problem misc banach space misc measure of weak noncompactness misc weak solutions misc time scales misc fixed point misc Mathematics
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title	Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales
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title_full	Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales
author_sort	Aneta Sikorska-Nowak
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title_sort	weak solutions for the dynamic equations $x^{\delta(m)}(t) = f (t; x(t))$ on time scales
callnumber	QA1-939
title_auth	Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales
abstract	In this paper we prove the existence of weak solutions of the dynamic Cauchy problem \begin{equation} \begin{split} x^{(\Delta m)}(t)&=f(t,x(t)),\quad t\in T, \\ x(0)&=0, \\ x^\Delta (0)&=\eta _1 ,\dots,x^{(\Delta (m-1))}(0)=\eta _{m-1},\quad \eta _1 ,\dots,\eta _{m-1} \in E, \end{split} \end{equation} where $x^{(\Delta m)}$ denotes a weak $m$-th order $\Delta$-derivative, $T$ denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence $(a_n )$ in $T$ and $a_n \to \infty )$, $E$ is a Banach space and $f$ is weakly -- weakly sequentially continuous and satisfies some conditions expressed in terms of measures of weak noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. As dynamic equations are a unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions.
abstractGer	In this paper we prove the existence of weak solutions of the dynamic Cauchy problem \begin{equation} \begin{split} x^{(\Delta m)}(t)&=f(t,x(t)),\quad t\in T, \\ x(0)&=0, \\ x^\Delta (0)&=\eta _1 ,\dots,x^{(\Delta (m-1))}(0)=\eta _{m-1},\quad \eta _1 ,\dots,\eta _{m-1} \in E, \end{split} \end{equation} where $x^{(\Delta m)}$ denotes a weak $m$-th order $\Delta$-derivative, $T$ denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence $(a_n )$ in $T$ and $a_n \to \infty )$, $E$ is a Banach space and $f$ is weakly -- weakly sequentially continuous and satisfies some conditions expressed in terms of measures of weak noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. As dynamic equations are a unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions.
abstract_unstemmed	In this paper we prove the existence of weak solutions of the dynamic Cauchy problem \begin{equation} \begin{split} x^{(\Delta m)}(t)&=f(t,x(t)),\quad t\in T, \\ x(0)&=0, \\ x^\Delta (0)&=\eta _1 ,\dots,x^{(\Delta (m-1))}(0)=\eta _{m-1},\quad \eta _1 ,\dots,\eta _{m-1} \in E, \end{split} \end{equation} where $x^{(\Delta m)}$ denotes a weak $m$-th order $\Delta$-derivative, $T$ denotes an unbounded time scale (nonempty closed subset of R such that there exists a sequence $(a_n )$ in $T$ and $a_n \to \infty )$, $E$ is a Banach space and $f$ is weakly -- weakly sequentially continuous and satisfies some conditions expressed in terms of measures of weak noncompactness. The Sadovskii fixed point theorem and Ambrosetti's lemma are used to prove the main result. As dynamic equations are a unification of differential and difference equations our result is also valid for differential and difference equations. The results presented in this paper are new not only for Banach valued functions but also for real valued functions.
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title_short	Weak solutions for the dynamic equations $x^{\Delta(m)}(t) = f (t; x(t))$ on time scales
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Vorhandene Bände

Nicht das Richtige dabei?