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# Existence criterion for the solutions of fractional order p-Laplacian boundary value problems

 dc.contributor.author Jafari, Hossein dc.contributor.author Baleanu, Dumitru dc.contributor.author Khan, Hasib dc.contributor.author Khan, Rahmat A dc.contributor.author Khan, Aziz dc.date.accessioned 2017-02-10T17:32:06Z dc.date.available 2017-02-10T17:32:06Z dc.date.issued 2015-09-17 dc.identifier.citation Boundary Value Problems. 2015 Sep 17;2015(1):164 dc.identifier.uri http://dx.doi.org/10.1186/s13661-015-0425-2 dc.identifier.uri http://hdl.handle.net/10500/21983 dc.description.abstract Abstract The existence criterion has been extensively studied for different classes in fractional differential equations (FDEs) through different mathematical methods. The class of fractional order boundary value problems (FOBVPs) with p-Laplacian operator is one of the most popular class of the FDEs which have been recently considered by many scientists as regards the existence and uniqueness. In this scientific work our focus is on the existence and uniqueness of the FOBVP with p-Laplacian operator of the form: D γ ( ϕ p ( D θ z ( t ) ) ) + a ( t ) f ( z ( t ) ) = 0 $D^{\gamma}(\phi_{p}(D^{\theta}z(t)))+a(t)f(z(t)) =0$ , 3 < θ $3<{\theta}$ , γ ≤ 4 $\gamma\leq{4}$ , t ∈ [ 0 , 1 ] $t\in[0,1]$ , z ( 0 ) = z ‴ ( 0 ) $z(0)=z'''(0)$ , η D α z ( t ) | t = 1 = z ′ ( 0 ) $\eta D^{\alpha}z(t)|_{t=1}= z'(0)$ , ξ z ″ ( 1 ) − z ″ ( 0 ) = 0 $\xi z''(1)-z''(0)=0$ , 0 < α < 1 $0<\alpha<1$ , ϕ p ( D θ z ( t ) ) | t = 0 = 0 = ( ϕ p ( D θ z ( t ) ) ) ′ | t = 0 $\phi_{p}(D^{\theta}z(t))|_{t=0}=0 =(\phi_{p}(D^{\theta}z(t)))'|_{t=0}$ , ( ϕ p ( D θ z ( t ) ) ) ″ | t = 1 = 1 2 ( ϕ p ( D θ z ( t ) ) ) ″ | t = 0 $(\phi_{p}(D^{\theta} z(t)))''|_{t=1} = \frac{1}{2}(\phi_{p}(D^{\theta} z(t)))''|_{t=0}$ , ( ϕ p ( D θ z ( t ) ) ) ‴ | t = 0 = 0 $(\phi_{p}(D^{\theta}z(t)))'''|_{t=0}=0$ , where 0 < ξ , η < 1 $0<\xi, \eta<{1}$ and D θ $D^{\theta}$ , D γ $D^{\gamma}$ , D α $D^{\alpha}$ are Caputo’s fractional derivatives of orders θ, γ, α, respectively. For this purpose, we apply Schauder’s fixed point theorem and the results are checked by illustrative examples. dc.title Existence criterion for the solutions of fractional order p-Laplacian boundary value problems dc.type Journal Article dc.date.updated 2017-02-10T17:32:06Z dc.language.rfc3066 en dc.rights.holder Jafari et al.
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