Abstract
In this paper, we prove the existence of solutions of initial value problems for nth order nonlinear impulsive integro-differential equations of mixed type on an infinite interval with an infinite number of impulsive times in Banach spaces. Our results are obtained by introducing a suitable measure of noncompactness.
Highlights
The branch of modern applied analysis known as ”impulsive” differential equations furnishes a natural framework to mathematically describe some ”jumping processes”
We will use the technique associated with measures of noncompactness to consider the boundary value problem (BVP) for nth-order
Throughout this section we will work in the Banach space DP Cn−1[J, E] and our considerations are placed in the Banach space DP Cn−1[J, E] considered previously
Summary
The branch of modern applied analysis known as ”impulsive” differential equations furnishes a natural framework to mathematically describe some ”jumping processes”. The theory of nth order nonlinear impulsive integro-differential equations of mixed type has received attention and has been achieved significant development in recent years (see [8,9,10]). Guo [9] and [10] have established the existence of solutions for the above nth order problems on an infinite interval with an infinite number of impulsive times in Banach spaces by means of the Schauder fixed point theorem and the fixed point index theory of completely continuous operators, respectively. 1 nonlinear impulsive integro-differential equation of mixed type as follows: u(n)(t) = f (t, u(t), u (t), · · · , u(n−1)(t), (T u)(t), (Su)(t)), ∀t ∈ J. [10] has shown that B∗P C[J, E] is a Banach space with norm u B = sup{e−t u(t) : t ∈ J}. Rzepka [13] for a nonlinear functional-integral equation
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