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We establish e-soft fixed point results for soft set-valued maps under some integral contractive conditions on a complete metric space. Starting from Branciari integral contraction, the presented results are soft set extensions of many existing results on point-to-point and point-to-set mappings. In particular, the established idea herein improves the recently introduced concepts of soft set-valued maps. Moreover, examples and applications to nonlinear neutral differential equations are provided to support the usability of the derived results.