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# On the existence of solution for fractional differential equations of order 3 < δ 1 ≤ 4 $3<\delta_{1}\leq4$

 dc.contributor.author Baleanu, Dumitru dc.contributor.author Agarwal, Ravi P dc.contributor.author Khan, Hasib dc.contributor.author Khan, Rahmat A dc.contributor.author Jafari, Hossein dc.date.accessioned 2017-02-10T17:32:13Z dc.date.available 2017-02-10T17:32:13Z dc.date.issued 2015-11-26 dc.identifier.citation Advances in Difference Equations. 2015 Nov 26;2015(1):362 dc.identifier.uri http://dx.doi.org/10.1186/s13662-015-0686-1 dc.identifier.uri http://hdl.handle.net/10500/21985 dc.description.abstract Abstract In this paper, we deal with a fractional differential equation of order δ 1 ∈ ( 3 , 4 ] $\delta_{1}\in(3,4]$ with initial and boundary conditions, D δ 1 ψ ( x ) = − H ( x , ψ ( x ) ) $\mathcal{D}^{\delta_{1}}\psi(x)=-\mathcal{H}(x,\psi(x))$ , D α 1 ψ ( 1 ) = 0 = I 3 − δ 1 ψ ( 0 ) = I 4 − δ 1 ψ ( 0 ) $\mathcal{D}^{\alpha_{1}} \psi(1)=0=\mathcal{I}^{3-\delta_{1}}\psi(0)= \mathcal{I}^{4-\delta_{1}}\psi(0)$ , ψ ( 1 ) = Γ ( δ 1 − α 1 ) Γ ( ν 1 ) I δ 1 − α 1 H ( x , ψ ( x ) ) ( 1 ) $\psi(1) = \frac{\Gamma(\delta_{1}-\alpha_{1})}{\Gamma(\nu_{1})}\mathcal{I}^{\delta _{1}-\alpha_{1}} \mathcal{H}(x,\psi(x))(1)$ , where x ∈ [ 0 , 1 ] $x\in[0,1]$ , α 1 ∈ ( 1 , 2 ] $\alpha_{1} \in(1,2]$ , addressing the existence of a positive solution (EPS), where the fractional derivatives D δ 1 $\mathcal{D}^{\delta_{1}}$ , D α 1 $\mathcal{D}^{\alpha_{1}}$ are in the Riemann-Liouville sense of the order δ 1 $\delta_{1}$ , α 1 $\alpha_{1}$ , respectively. The function H ∈ C ( [ 0 , 1 ] × R , R ) $\mathcal{H}\in C([0,1]\times{R} , {R})$ and I δ 1 − α 1 H ( x , ψ ( x ) ) ( 1 ) = 1 Γ ( δ 1 − α 1 ) ∫ 0 1 ( 1 − z ) δ 1 − α 1 − 1 H ( z , ψ ( z ) ) d z $\mathcal{I}^{\delta_{1}-\alpha_{1}}\mathcal{H}(x,\psi(x))(1)=\frac {1}{\Gamma(\delta_{1}-\alpha_{1})} \int_{0}^{1}(1-z)^{\delta_{1}-\alpha_{1}-1}\mathcal{H}(z,\psi(z))\,dz$ . To this aim, we establish an equivalent integral form of the problem with the help of a Green’s function. We also investigate the properties of the Green’s function in the paper which we utilize in our main result for the EPS of the problem. Results for the existence of solutions are obtained with the help of some classical results. dc.title On the existence of solution for fractional differential equations of order 3 < δ 1 ≤ 4 $3<\delta_{1}\leq4$ dc.type Journal Article dc.date.updated 2017-02-10T17:32:13Z dc.language.rfc3066 en dc.rights.holder Baleanu et al.
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