Semnan University International Journal of Nonlinear Analysis and Applications 2008-6822 2026 02 10 Approximation results of non-homogeneous Cauchy problem of semigroup of linear operators 10474 10.22075/ijnaa.2025.33612.5023 EN Akinola Yussuff Akinyele Department of Mathematics, University of Ilorin, Ilorin, Nigeria 0000-0002-0328-0787 Ayodele Olakiitan Owolanke Department of Mathematical Sciences, Olusegun Agagu University of Science and Technology, Okitipupa, Nigeria Aminat Olabisi Ajiboye Department of Mathematics, Federal University, Oye-Ekiti, Nigeria Journal Article 2024 03 25 In this paper, results of $\omega$-order reversing partial contraction mapping generated by approximation results of non-homogeneous Cauchy problem were presented. We investigated the concepts of $C'$, strong and, respectively, $C^0$-solution. We established that $Z:D(Z)\subseteq X\to X$ is the infinitesimal generator of a $C_0$-semigroup of contraction, $\xi\in X$ and $f\in L^2(a,b;X)$. Furthermore, we deduced that the unique $C^0$-solution is strong and that the class is absolutely continuous on $[a,b]$.\\ $\omega$-$ORCP_n$ Absolutely Continuous $C'$-solution Mild solution https://ijnaa.semnan.ac.ir/article_10474_a44cd04fe23d5e3d3b5d6fc9e817695e.pdf