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Solvability of a coupled system of fractional differential equations with nonlocal and integral boundary conditions. (English) Zbl 1381.34010

In this article, the authors study the coupled system \[ ^{C}D^{\alpha}x(t)=f\big(t,y(t),D^{\gamma}y(t)\big) \] and \[ ^{C}D^{\beta}x(t)=g\big(t,x(t),D^{\delta}x(t)\big), \] where \(t\in[0,T]\), and the fractional derivative orders satisfy \(\gamma\), \(\delta\in(0,1)\) and \(\alpha\), \(\beta\in(1,2]\). Note that in the above equations the fractional derivatives utilized are a combination of Caputo-type and Riemann-Liouville-type. In addition, the preceding system is associated with the following nonlocal-type boundary condition: \[ x(0)=h(y),\;\int_0^Tx(s)\;ds=d \] and \[ y(0)=\phi(x),\;\int_0^Ty(s)\;ds=c. \] In the above boundary conditions we note that \(h\) and \(\phi\) are functionals.
Having stated this problem the authors then study the existence and uniqueness of solution to this problem by means of two classical fixed-point methodologies. The first is the Banach contraction theorem, whereas the second is the Leray-Schauder alternative. After stating and proving these abstract existence and uniqueness theorems, the authors provide some examples, which serve to illustrate the application of these results.
All in all, the paper is nicely written and very easy to follow. For researchers interested in either the fractional calculus or boundary value problems (particularly, with nonlocal boundary conditions), this article may be of interest.

MSC:

34A08 Fractional ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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