Existence and uniqueness of solutions for a class of fractional nonlinear boundary value problems under mild assumptions
Copyright policy )Existence and uniqueness of solutions for a class of fractional nonlinear boundary value problems under mild assumptions
AbstractWe deal with the following Riemann–Liouville fractional nonlinear boundary value problem: $$ \textstyle\begin{cases} \mathcal{D}^{\alpha }v(x)+f(x,v(x))=0, & 2< \alpha \leq 3, x\in (0,1), \\ v(0)=v^{\prime }(0)=v(1)=0. \end{cases} $$ { D α v ( x ) + f ( x , v ( x ) ) = 0 , 2 < α ≤ 3 , x ∈ ( 0 , 1 ) , v ( 0 ) = v ′ ( 0 ) = v ( 1 ) = 0 . Under mild assumptions, we prove the existence of a unique continuous solution v to this problem satisfying $$ \bigl\vert v(x) \bigr\vert \leq cx^{\alpha -1}(1-x)\quad\text{for all }x \in [ 0,1]\text{ and some }c>0. $$ | v ( x ) | ≤ c x α − 1 ( 1 − x ) for all x ∈ [ 0 , 1 ] and some c > 0 . Our results improve those obtained by Zou and He (Appl. Math. Lett. 74:68–73, 2017).
- King Abdulaziz University Saudi Arabia
- King Saud University Saudi Arabia
- Tunis El Manar University Tunisia
Banach's contraction principle, Fractional ordinary differential equations, Positive solutions to nonlinear boundary value problems for ordinary differential equations, Green’s function, Green's function, Fractional differential equation, existence and uniqueness of solution, Banach’s contraction principle, Integro-ordinary differential equations, Fractional derivatives and integrals, fractional differential equation, QA1-939, Existence and uniqueness of solution, Mathematics
Banach's contraction principle, Fractional ordinary differential equations, Positive solutions to nonlinear boundary value problems for ordinary differential equations, Green’s function, Green's function, Fractional differential equation, existence and uniqueness of solution, Banach’s contraction principle, Integro-ordinary differential equations, Fractional derivatives and integrals, fractional differential equation, QA1-939, Existence and uniqueness of solution, Mathematics
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