This work considers LFIVP defined under gH-differentiability with the aim of establishing a method and subsequently, a fuzzy-valued function, which is a solution to the LFIVP. The conditions for a fuzzy function to be H- differentiable and gH-differentiability are defined. It is however established that gH-differentiability concept to fuzzy-valued functions is more generalized than H-differentiable. Characterization and Stacking theorems are also used to translate LFIVP to a system of specialized ODEs. Method of variation of parameters is also extended to LFIVP. Examples are constructed to test the applicability of the methods and graphical presentation of the behavior of solutions of the constructed examples is done with the aid of Matlab 8.5 version. We recommended that further studies should be made on second and higher order LFIVP.
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