Existence and uniqueness of solutions for a class of fractional nonlinear boundary value problems under mild assumptions

Autor: Imed Bachar, Habib Mâagli, Hassan Eltayeb
Jazyk: angličtina
Rok vydání: 2021
Předmět:
Zdroj: Advances in Difference Equations, Vol 2021, Iss 1, Pp 1-11 (2021)
Druh dokumentu: article
ISSN: 1687-1847
DOI: 10.1186/s13662-020-03176-w
Popis: Abstract We deal with the following Riemann–Liouville fractional nonlinear boundary value problem: { D α v ( x ) + f ( x , v ( x ) ) = 0 , 2 < α ≤ 3 , x ∈ ( 0 , 1 ) , v ( 0 ) = v ′ ( 0 ) = v ( 1 ) = 0 . $$ \textstyle\begin{cases} \mathcal{D}^{\alpha }v(x)+f(x,v(x))=0, & 2< \alpha \leq 3, x\in (0,1), \\ v(0)=v^{\prime }(0)=v(1)=0. \end{cases} $$ Under mild assumptions, we prove the existence of a unique continuous solution v to this problem satisfying | v ( x ) | ≤ c x α − 1 ( 1 − x ) for all x ∈ [ 0 , 1 ] and some c > 0 . $$ \bigl\vert v(x) \bigr\vert \leq cx^{\alpha -1}(1-x)\quad\text{for all }x \in [ 0,1]\text{ and some }c>0. $$ Our results improve those obtained by Zou and He (Appl. Math. Lett. 74:68–73, 2017).
Databáze: Directory of Open Access Journals
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