Abstract
In the present paper, we first investigate the construction of compact sets of $\mathit{DC}^{n}[J,E]$ and $C^{n}_{0}[J,E]$ , and then we introduce new measures of noncompactness on these spaces. In addition, as an application, we discuss the existence of solutions of initial value problems for nth-order nonlinear integro-differential equations of mixed type on an infinite interval in Banach spaces. We will also state an interesting example which shows that our results can apply for solving infinite systems of integro-differential equations.
Highlights
The integro-differential equation (IDE) can be considered in different branches of sciences and engineering
Measures of noncompactness are very useful tools in functional analysis, for instance in metric fixed point theory and in the theory of operator equations in Banach spaces. They are used in the studies of functional equations, ordinary and partial differential equations, integral and integro-differential equations, fractional partial differential equations, and optimal control theory
In this paper, we shall investigate the existence of solutions of an initial value problems (IVP) for nth-order nonlinear integro-differential equations of mixed type on an infinite interval in E by a new Allahyari et al Advances in Difference Equations (2015) 2015:376 measure of noncompactness
Summary
The integro-differential equation (IDE) can be considered in different branches of sciences and engineering. Let us denote by ME the family of nonempty bounded subsets of E and by NE its subfamily consisting of all relatively compact sets. [ ] Let be a nonempty, bounded, closed, and convex subset of a Banach space E and let F : → be a continuous mapping such that there exists a constant k ∈ [ , ) with the property μ(FX) ≤ kμ(X)
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