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In the present paper, we first investigate the construction of compact sets of DCn[J,E]$\mathit{DC}^{n}[J,E]$ and C0n[J,E]$C^{n}_{0}[J,E]$, 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.

advances in difference equations

Construction of measures of noncompactness of DC n [ J , E ] $\mathit{DC}^{n}[J,E]$ and C 0 n [ J , E ] $C^{n}_{0}[J,E]$ with application to the solvability of nth-order integro-differential equations in Banach spaces