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 Existence results of a Schr$\ddot{o}$dinger equation involving a nonlinear operator Yan Sun Keywords:Nonlinear Schr$\ddot{o}$dinger equation; Nondecreasing operator; Positive solution; Existence; Monotonicity Abstract: In this article, by making use of a completely different approach, i. e. monotone operator theory and analysis techniques, we present the existence of positive solutions for a nonlinear Schr$\ddot{o}$dinger equations in a unique way under reasonable conditions. Moreover, illustrative examples are also presented.