Fixed-point theorems of ${F}^{\star}- ( \psi,\phi )$ integral-type contractive conditions on $1_{E}$-complete multiplicative partial cone metric spaces over Banach algebras and applications

Abstract In this paper, we introduce some user-friendly versions of integral-type fixed-point results and give some modifications of the classical Banach contraction principle by constructing a special type of contractive restrictions of integral forms for weak contraction mappings defined on $1_{E}$ 1 E -complete multiplicative partial cone metric spaces over Banach algebras and formulate some existence and uniqueness results regarding the fixed-point theorems using some integrative conditions. Moreover, we validate the significance our results and exploit them to find the unique solution of a fractional nonlinear differential equation of Caputo type, which complements some previously well-known generalizations found in the literature.