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The aim of the paper is to introduce the spaces $\ell_{\infty}^{\lambda}(\widehat{F})$ and $\ell_{p}^{\lambda}(\widehat{F})$ derived by the composition of the two infinite matrices $\Lambda=(\lambda_{nk})$ and $\widehat{F}=\left( f_{nk} \right),$ which are the $BK$-spaces of non-absolute type and also derive some inclusion relations. Further, we determine the $\alpha$-, $\beta$-, $\gamma$-duals of those spaces and also construct the basis for $\ell_{p}^{\lambda}(\widehat{F}).$ Additionally, we characterize some matrix classes on the spaces $\ell_{\infty}^{\lambda}(\widehat{F})$ and $\ell_{p}^{\lambda}(\widehat{F}).$ We also investigate some geometric properties concerning Banach-Saks type $p.$ Here we characterize the subclasses $\mathcal{K}(X:Y)$ of compact operators, where $X\in\{\ell_{\infty}^{\lambda}(\widehat{F}),\ell_{p}^{\lambda}(\widehat{F})\}$ and $Y\in\{c_{0},c, \ell_{\infty}, \ell_{1}, bv\}$ by applying the Hausdorff measure of non-compactness, and $1\leq p<\infty.$, Comment: 19 pages. arXiv admin note: text overlap with arXiv:1604.07396

In the present paper, by using the band matrix F defined by the Fibonacci sequence, we introduce the sequence sequence spaces c_0(F) and c(F). Also, we give some inclusion relations and construct the bases of the spaces c_0(F) and c(F). Finally, we compute the alpha-, beta-, gamma-duals of these spaces and characterize the classes (c_0(F),X) and (c(F),X) for certain choice of the sequence space X.