Unique solutions for a new coupled system of fractional differential equations

Abstract

In this article, we discuss a new coupled system of fractional differential equations with integral boundary conditions { D α u ( t ) + f ( t , v ( t ) ) = a , 0 < t < 1 , D β v ( t ) + g ( t , u ( t ) ) = b , 0 < t < 1 , u ( 0 ) = 0 , u ( 1 ) = ∫ 0 1 ϕ ( t ) u ( t ) d t , v ( 0 ) = 0 , v ( 1 ) = ∫ 0 1 ψ ( t ) v ( t ) d t , $$\textstyle\begin{cases} D^{\alpha}u(t)+f(t,v(t))=a,\quad 0< t< 1,\\ D^{\beta}v(t)+g(t,u(t))=b,\quad 0< t< 1,\\ u(0)=0,\qquad u(1)=\int_{0}^{1} \phi(t)u(t)\,dt,\\ v(0)=0,\qquad v(1)=\int_{0}^{1} \psi(t)v(t)\,dt, \end{cases} $$ where 1 < α , β ≤ 2 , f , g ∈ C ( [ 0 , 1 ] × ( − ∞ , + ∞ ) , ( − ∞ , + ∞ ) ) , ϕ , ψ ∈ L 1 [ 0 , 1 ] $1< \alpha,\beta\le2, f,g \in C([0,1]\times(-\infty,+\infty ),(-\infty,+\infty)), \phi,\psi\in L^{1}[0,1]$ , a , b $a,b$ are constants and D denotes the usual Riemann-Liouville fractional derivative. Based upon a fixed point theorem of increasing φ- ( h , e ) $(h,e)$ -concave operators, we establish the existence and uniqueness of solutions for the new coupled system dependent on two constants. And then the obtained result is well demonstrated with the aid of an interesting example.