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Zbl 1207.45014
Ahmad, Bashir; Sivasundaram, S.
On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. (English)
[J] Appl. Math. Comput. 217, No. 2, 480-487 (2010). ISSN 0096-3003

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.
[Ioan I. Vrabie (Iaşi)]

MSC 2000:: *45N05 Integral equations in abstract spaces
45J05 Integro-ordinary differential equations
45G10 Nonsingular nonlinear integral equations
26A33 Fractional derivatives and integrals (real functions)

Keywords: nonlinear fractional differential equations; nonlocal boundary conditions; fixed point arguments; Banach space; Volterra integral operators

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