Impulsively hybrid fractional quantum Langevin equation with boundary conditions involving Caputo q_k-fractional derivatives

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• 作者：Weerawat Sudsutad^a ; ^{wrw.sst@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Bashir Ahmad ; ^b ; ^{bashirahmad_qau@yahoo.com" class="auth_mail" title="E-mail the corresponding author} ; ^{bahmad@kau.edu.sa" class="auth_mail" title="E-mail the corresponding author} ; Sotiris K. Ntouyas^c ; ^b ; ^{sntouyas@uoi.gr" class="auth_mail" title="E-mail the corresponding author} ; Jessada Tariboon^a ; ^{jessada.t@sci.kmutnb.ac.th" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Quantum calculus ; Impulsive fractional qk-difference equations ; Existence ; Uniqueness ; Fixed point
• 刊名：Chaos, Solitons & Fractals
• 出版年：2016
• 出版时间：October 2016
• 年：2016
• 卷：91
• 期：Complete
• 页码：47-62
• 全文大小：598 K

文摘

We obtain some existence and uniqueness results for an impulsively hybrid fractional quantum Langevin (q_k-difference) equation involving a new q_k  -shifting operator class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0960077916301552&_mathId=si1.gif&_user=111111111&_pii=S0960077916301552&_rdoc=1&_issn=09600779&md5=3f83fc05a0752f8c36d2f19cfdfa16f0" title="Click to view the MathML source">_aΦ_qk(m)=q_km+(1−q_k)aclass="mathContainer hidden">class="mathCode">${}_a{\Phi }_{q_k}\left(m\right)=q_{k}m+(1-q_k)a$ and supplemented with non-separated boundary conditions containing Caputo q_k-fractional derivatives. Our first result, relying on Banach’s fixed point theorem, is concerned with the existence of a unique solution of the problem. The existence results are established by means of Leray–Schauder nonlinear alternative and a fixed point theorem due to O’Regan. We construct some examples for the applicability of the obtainedresults. The paper concludes with interesting observations.