## On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order.(English)Zbl 1207.45014

The authors consider the four-point nonlocal boundary value problem in a Banach space $$X$$, i.e.
$^cD^qx(t)=f(t,x(t),(\phi x)(t),(\psi x)(t)),\;\;0<t<1, \;\;1<q<2,$
$x'(0)+ax(\eta_1)=0, \;\;bx'(1)+x(\eta_2)=0,\;\;0<\eta_1\leq \eta_2<1,$
where $$^cD$$ is the Caputo’s fractional derivative, $$f:[0,1]\times X\times X\times X\times X\to X$$ is continuous, $$\phi$$, $$\psi$$ are Volterra integral operators and $$a,c\in (0,1)$$. By using fixed point arguments they prove an existence and uniqueness result of solutions for the problem above.

### MSC:

 45N05 Abstract integral equations, integral equations in abstract spaces 45J05 Integro-ordinary differential equations 45G10 Other nonlinear integral equations 26A33 Fractional derivatives and integrals
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### References:

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