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Unique solutions for a new coupled system of fractional differential equations. (English) Zbl 1445.34030

Summary: In this article, we discuss a new coupled system of fractional differential equations with integral boundary conditions \[ \begin{cases} D^{\alpha}u(t)+f(t,v(t))=a, \quad 0< t< 1,\\ D^{\beta}v(t)+g(t,u(t))=b,\quad 0< t< 1,\\ u(0)=0,\qquad u(1)=\int_{0}^{1} \phi(t)u(t)\,dt,\\ v(0)=0,\qquad v(1)=\int_{0}^{1} \psi(t)v(t)\,dt, \end{cases} \] where \(1< \alpha,\beta\leq2, f,g \in C([0,1]\times(-\infty,+\infty ),(-\infty,+\infty)), \phi,\psi\in L^{1}[0,1]\), \(a,b\) are constants and \(D\) denotes the usual Riemann-Liouville fractional derivative. Based upon a fixed point theorem of increasing \(\varphi\)-\((h,e)\)-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.

MSC:

34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations
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