An existence result for a new class of impulsive functional differential equations with delay. (English) Zbl 1353.34092
The authors study the existence of bounded solutions for the following impulsive delay differential system
\[
x'(t)-A(t)x(t)=f(t,x(t),x(\sigma(t))\text{ for a.e. }t\in \bigcup_{i=1}^{\infty} (s_i,t_{i+1}],
\]
\[ x(t)=(Kx)(t)\text{ for }t\in[-r,0]\cup\bigcup_{i=1}^{\infty} (t_i,s_{i}], \] where \(A\) is the generator of a uniformly bounded and compact evolution system, \(K\) is a bounded and completely continuous operator.
\[ x(t)=(Kx)(t)\text{ for }t\in[-r,0]\cup\bigcup_{i=1}^{\infty} (t_i,s_{i}], \] where \(A\) is the generator of a uniformly bounded and compact evolution system, \(K\) is a bounded and completely continuous operator.
Reviewer: Leonid Berezanski (Beer-Sheva)
MSC:
| 34K30 | Functional-differential equations in abstract spaces |
| 34K45 | Functional-differential equations with impulses |
| 47N20 | Applications of operator theory to differential and integral equations |
Keywords:
functional differential equation; evolution operator; non-instantaneous impulse; delay; strong solution; Schaefer’s fixed point theoremReferences:
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