# Oscillation of impulsive conformable fractional differential equations

Open Mathematics (2016)

• Volume: 14, Issue: 1, page 497-508
• ISSN: 2391-5455

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In this paper, we investigate oscillation results for the solutions of impulsive conformable fractional differential equations of the form tkDαpttkDαxt+rtxt+qtxt=0,t≥t0,t≠tk,xtk+=akx(tk−),tkDαxtk+=bktk−1Dαx(tk−),k=1,2,…. $\left\{\begin{array}{c}{t}_{k}{D}^{\alpha }\left(p\left(t\right)\left[{t}_{k}{D}^{\alpha }x\left(t\right)+r\left(t\right)x\left(t\right)\right]\right)+q\left(t\right)x\left(t\right)=0,\phantom{\rule{1.0em}{0ex}}t\ge {t}_{0},\phantom{\rule{0.277778em}{0ex}}t\ne {t}_{k},\hfill \\ x\left({t}_{k}^{+}\right)={a}_{k}x\left({t}_{k}^{-}\right),\phantom{\rule{1.0em}{0ex}}{t}_{k}{D}^{\alpha }x\left({t}_{k}^{+}\right)={b}_{k\phantom{\rule{0.277778em}{0ex}}{t}_{k-1}}{D}^{\alpha }x\left({t}_{k}^{-}\right),\phantom{\rule{1.0em}{0ex}}\phantom{\rule{0.277778em}{0ex}}k=1,2,....\hfill \end{array}\right$/extract_itex] Some new oscillation results are obtained by using the equivalence transformation and the associated Riccati techniques. ## How to cite top Jessada Tariboon, and Sotiris K. Ntouyas. "Oscillation of impulsive conformable fractional differential equations." Open Mathematics 14.1 (2016): 497-508. <http://eudml.org/doc/285658>. @article{JessadaTariboon2016, abstract = {In this paper, we investigate oscillation results for the solutions of impulsive conformable fractional differential equations of the form tkDαpttkDαxt+rtxt+qtxt=0,t≥t0,t≠tk,xtk+=akx(tk−),tkDαxtk+=bktk−1Dαx(tk−),k=1,2,…. \[\left\lbrace \begin\{array\}\{l\} \{t\_k\}\{D^\alpha \}\left( \{p\left( t \right)\left[ \{\{t\_k\}\{D^\alpha \}x\left( t \right) + r\left( t \right)x\left( t \right)\} \right]\} \right) + q\left( t \right)x\left( t \right) = 0,\quad t \ge \{t\_0\},\;t \ne \{t\_k\},\\ x\left( \{t\_k^ + \} \right) = \{a\_k\}x(t\_k^ - ),\quad \{t\_k\}\{D^\alpha \}x\left( \{t\_k^ + \} \right) = \{b\_\{k\;\{t\_\{k - 1\}\}\}\}\{D^\alpha \}x(t\_k^ - ),\quad \;k = 1,2, \ldots . \end\{array\} \right.$ Some new oscillation results are obtained by using the equivalence transformation and the associated Riccati techniques.},
author = {Jessada Tariboon, Sotiris K. Ntouyas},
journal = {Open Mathematics},
keywords = {Fractional differential equations; Impulsive differential equations; Conformable fractional derivative; Oscillation; fractional differential equations; impulsive differential equations; conformable fractional derivative; oscillation},
language = {eng},
number = {1},
pages = {497-508},
title = {Oscillation of impulsive conformable fractional differential equations},
url = {http://eudml.org/doc/285658},
volume = {14},
year = {2016},
}

TY - JOUR
AU - Sotiris K. Ntouyas
TI - Oscillation of impulsive conformable fractional differential equations
JO - Open Mathematics
PY - 2016
VL - 14
IS - 1
SP - 497
EP - 508
AB - In this paper, we investigate oscillation results for the solutions of impulsive conformable fractional differential equations of the form tkDαpttkDαxt+rtxt+qtxt=0,t≥t0,t≠tk,xtk+=akx(tk−),tkDαxtk+=bktk−1Dαx(tk−),k=1,2,…. $\left\lbrace \begin{array}{l} {t_k}{D^\alpha }\left( {p\left( t \right)\left[ {{t_k}{D^\alpha }x\left( t \right) + r\left( t \right)x\left( t \right)} \right]} \right) + q\left( t \right)x\left( t \right) = 0,\quad t \ge {t_0},\;t \ne {t_k},\\ x\left( {t_k^ + } \right) = {a_k}x(t_k^ - ),\quad {t_k}{D^\alpha }x\left( {t_k^ + } \right) = {b_{k\;{t_{k - 1}}}}{D^\alpha }x(t_k^ - ),\quad \;k = 1,2, \ldots . \end{array} \right.$ Some new oscillation results are obtained by using the equivalence transformation and the associated Riccati techniques.
LA - eng
KW - Fractional differential equations; Impulsive differential equations; Conformable fractional derivative; Oscillation; fractional differential equations; impulsive differential equations; conformable fractional derivative; oscillation
UR - http://eudml.org/doc/285658
ER -

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