On the singular impulsive functional integral equations involving nonlocal conditions. (English) Zbl 1270.45011
The authors consider a singular impulsive functional integral equation of the form
\[
u(t)= T(t) H(u)+{1\over \Gamma(\alpha)} \int^t_0 (t-s)^{\alpha-1} T(t-s) F(s,u(s))\,ds
\]
for \(t\in[0, T]\), but \(t\) not equal to \(t_i\), and \(\Delta u(t_i)= I_i(u(t_i))\), \(i= 1,\dots,n\).
Here \(T(t)\) is a compact semigroup on the Banach space \(X\), \(0<\alpha <1\), \(H\) is a nonlocal operator mapping \(\text{PC}([0,T]; X)\) into \(X\). (PC = piecewise continuous) and \(I_i: X\to X\), \(\Delta u(t_i)= u(t_i^+)- u(t_i^-)\), \(i= 1,\dots,n\).
Under certain growth conditions on \(H\), \(I_i\) and \(F\) (including a Lipschitz condition on \(H\) but not on \(I_i\)), the authors prove the existence of a solution. The proof is based on Schauders fixed point theorem.
Here \(T(t)\) is a compact semigroup on the Banach space \(X\), \(0<\alpha <1\), \(H\) is a nonlocal operator mapping \(\text{PC}([0,T]; X)\) into \(X\). (PC = piecewise continuous) and \(I_i: X\to X\), \(\Delta u(t_i)= u(t_i^+)- u(t_i^-)\), \(i= 1,\dots,n\).
Under certain growth conditions on \(H\), \(I_i\) and \(F\) (including a Lipschitz condition on \(H\) but not on \(I_i\)), the authors prove the existence of a solution. The proof is based on Schauders fixed point theorem.
Reviewer: Stig-Olof Londen (Aalto)
MSC:
| 45N05 | Abstract integral equations, integral equations in abstract spaces |
| 45G05 | Singular nonlinear integral equations |
| 45D05 | Volterra integral equations |
Keywords:
Volterra equations; nonlocal initial conditions; singular impulsive functional integral equation; Banach space; Schauders fixed point theoremReferences:
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