Differential equations are used to represent different scientific problems are handled efficiently by integral transformations, where integral transforms represent an easy and effective tool for solving many problems in the mentioned fields. This work utilizes the integral transform of the Complex SEE integral transformation to provide an efficient solution method for the Difference and differential-Difference equations by benefiting from the properties of this complex transform to solve some problems related to Difference and differential-Difference equations. The 3D, contour and 2D surfaces, as well as the related density plot surfaces of some acquired data, are used to draw the physical aspect of the obtained findings. The proposed approach offers an efficient and rapid solution for addressing the inherent complexity of differential-Difference problems with initial conditions.

The theory of interval-valued Difference equations under $gH$-Difference is an interesting topic, since it can be applied to study numerical solutions to interval-valued or fuzzy-valued differential equations. In this paper, we estimate the number of solutions to a class of first-order interval-valued Difference equations under $gH$-Difference, which reveals the complexity of the stability analysis in this area, as well as the difficulty for prediction and control problems. Then, based on the relative positions of initial values and equilibrium points, we provide sufficient conditions for the existence of convergent solutions. We also provide examples to illustrate the validity of our results.

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Classification, boundedness, and existence of solutions of a second order nonlinear Difference equation are investigated. First, it is proved that all solutions are eventually monotone. Then, the necessary and sufficient conditions for the boundedness of all solutions are established. Finally, the existence of different types of monotonic solutions are presented. The obtained results have extended and improved some existing ones.