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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.

c D q x(t)=f(t,x(t),(ϕx)(t),(ψx)(t)),0<t<1,1<q<2,
x ' (0)+ax(η 1 )=0,bx ' (1)+x(η 2 )=0,0<η 1 η 2 <1,

where c D is the Caputo’s fractional derivative, f:[0,1]×X×X×X×XX is continuous, ϕ, ψ are Volterra integral operators and a,c(0,1). By using fixed point arguments they prove an existence and uniqueness result of solutions for the problem above.

MSC:
45N05Abstract integral equations, integral equations in abstract spaces
45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations
26A33Fractional derivatives and integrals (real functions)
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