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Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators. (English) Zbl 1220.45009

The paper deals with the following partial functional integro-differential equation:

x ' (t)=Ax(t)+Ft,x(σ 1 (t)),,x(σ n (t)), 0 t h(t,s,x(σ n+1 (s)))ds,t[0,b],tt k ,k=1,m ¯,

where A is the infinitesimal generator of a compact, analytic semigroup, t k [0,b], and F,h,σ k are some given functions. The equation is considered here together with the conditions:

x(0)+g(x)=x 0 andx(t k + )-x(t k - )=I k (x(t k )),k=1,m ¯·

It is shown that, under suitable conditions on the functions F,h,g,σ k , and for any x 0 X α , the above Cauchy problem has at least one mild solution on [0,b]. The proof of this result employs the Leray-Schauder alternative. The author also presents an illustrative example at the end of the paper.

45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations
26A33Fractional derivatives and integrals (real functions)
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