On the oscillation of q-fractional difference equations
Abstract In this paper, sufficient conditions are established for the oscillation of solutions of q-fractional difference equations of the form { ∇ 0 α q x ( t ) + f 1 ( t , x ) = r ( t ) + f 2 ( t , x ) , t > 0 , lim t → 0 + q I 0 j − α x ( t ) = b j ( j = 1 , 2 , … , m ) , $$ \left \{ \textstyl...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2017-08-01
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| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13662-017-1316-x |
| _version_ | 1828809597621633024 |
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| author | Bahaaeldin Abdalla |
| author_facet | Bahaaeldin Abdalla |
| author_sort | Bahaaeldin Abdalla |
| collection | DOAJ |
| description | Abstract In this paper, sufficient conditions are established for the oscillation of solutions of q-fractional difference equations of the form { ∇ 0 α q x ( t ) + f 1 ( t , x ) = r ( t ) + f 2 ( t , x ) , t > 0 , lim t → 0 + q I 0 j − α x ( t ) = b j ( j = 1 , 2 , … , m ) , $$ \left \{ \textstyle\begin{array}{l} {}_{q}\nabla_{0}^{\alpha}x(t)+f_{1}(t,x)=r(t)+f_{2}(t,x), \quad t>0 ,\\ \lim_{t \to0^{+}}{{}_{q}I_{0}^{j-\alpha}x(t)}=b_{j} \quad(j=1,2,\ldots,m), \end{array}\displaystyle \right . $$ where m = ⌈ α ⌉ $m=\lceil\alpha\rceil$ , ∇ 0 α q ${}_{q}\nabla_{0}^{\alpha}$ is the Riemann-Liouville q-differential operator and I 0 m − α q ${}_{q}I_{0}^{m-\alpha}$ is the q-fractional integral. The results are also obtained when the Riemann-Liouville q-differential operator is replaced by Caputo q-fractional difference. Examples are provided to demonstrate the effectiveness of the main result. |
| first_indexed | 2024-12-12T08:59:04Z |
| format | Article |
| id | doaj.art-3ad99673753a45a4903ee65891fd19c4 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-12T08:59:04Z |
| publishDate | 2017-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-3ad99673753a45a4903ee65891fd19c42022-12-22T00:29:53ZengSpringerOpenAdvances in Difference Equations1687-18472017-08-012017111110.1186/s13662-017-1316-xOn the oscillation of q-fractional difference equationsBahaaeldin Abdalla0Department of Mathematics and Physical Sciences, Prince Sultan UniversityAbstract In this paper, sufficient conditions are established for the oscillation of solutions of q-fractional difference equations of the form { ∇ 0 α q x ( t ) + f 1 ( t , x ) = r ( t ) + f 2 ( t , x ) , t > 0 , lim t → 0 + q I 0 j − α x ( t ) = b j ( j = 1 , 2 , … , m ) , $$ \left \{ \textstyle\begin{array}{l} {}_{q}\nabla_{0}^{\alpha}x(t)+f_{1}(t,x)=r(t)+f_{2}(t,x), \quad t>0 ,\\ \lim_{t \to0^{+}}{{}_{q}I_{0}^{j-\alpha}x(t)}=b_{j} \quad(j=1,2,\ldots,m), \end{array}\displaystyle \right . $$ where m = ⌈ α ⌉ $m=\lceil\alpha\rceil$ , ∇ 0 α q ${}_{q}\nabla_{0}^{\alpha}$ is the Riemann-Liouville q-differential operator and I 0 m − α q ${}_{q}I_{0}^{m-\alpha}$ is the q-fractional integral. The results are also obtained when the Riemann-Liouville q-differential operator is replaced by Caputo q-fractional difference. Examples are provided to demonstrate the effectiveness of the main result.http://link.springer.com/article/10.1186/s13662-017-1316-xq-fractional difference equationsoscillation theory |
| spellingShingle | Bahaaeldin Abdalla On the oscillation of q-fractional difference equations Advances in Difference Equations q-fractional difference equations oscillation theory |
| title | On the oscillation of q-fractional difference equations |
| title_full | On the oscillation of q-fractional difference equations |
| title_fullStr | On the oscillation of q-fractional difference equations |
| title_full_unstemmed | On the oscillation of q-fractional difference equations |
| title_short | On the oscillation of q-fractional difference equations |
| title_sort | on the oscillation of q fractional difference equations |
| topic | q-fractional difference equations oscillation theory |
| url | http://link.springer.com/article/10.1186/s13662-017-1316-x |
| work_keys_str_mv | AT bahaaeldinabdalla ontheoscillationofqfractionaldifferenceequations |