Motivated by the work ^[7], in which the author studied the convexity of nonlinear mappings defined between bounded linear operator spaces, our research extends this inquiry. In this work, we continue the study of the convexity of nonlinear mappings defined between bounded linear operator spaces and we establish a characterization in terms of the second order directional derivative. We apply the main result to prove the convexity and the nonconvexity of well-known nonlinear mappings. The case of nondifferentiable mappings is also treated in the last section.

We witness an increased emphasis on the integration of different areas of manufacturing firms with the intention of avoiding suboptimal solutions. In particular, due to new technologies which make it easy to change the price of a product in real time, the integration of pricing and production planning may be garnering the most interest. We are proposing in this paper a way to model the dynamics of the price. Thus, the price and the inventory level are considered as state variables whereas the supply (production) rate is the control variable. The demand rate is dynamic and state-dependent. Using a model predictive control approach, the optimal supply rate, and thus the optimal price and inventory level, are obtained. Different examples are provided under different scenarios for the supply rate and for the demand rate.

In the setting of $2$-uniformly convex Banach spaces, we prove the existence of solutions for a variant of the implicit state-dependent convex sweeping processes. Our approach is based on a differential equation associated with the generalized projection operator.

Using an optimal control framework, the optimization of production and maintenance processes is investigated in this paper. We focused on analyzing system behavior, managing costs, and controlling quality. Differential equations are utilized to model the relationship between inventory level, production, and maintenance strategies. A cost function is built combining different cost elements, and the optimal production and maintenance rates were obtained. A sensitivity analysis is performed to assess the impact of different parameters on the total cost and on the optimal solution. The key findings demonstrated that higher initial inventory levels significantly decreased long-term costs and improved production efficiency. The optimal preventive maintenance strategy emphasized the importance of early investments in quality and maintenance, leading to sustained operational stability.

We continue the study of the properties of the $V$-Moreau envelope and generalized $(f,\lambda )$-projection that we started in ^[5]. In this paper, we study the differentiability and the regularity of the $V$-Moreau envelope and the Hölder continuity of the generalized $(f,\lambda )$-projection. Our results extend many existing results from the convex case to the nonconvex case and from Hilbert spaces to Banach spaces. Even on Hilbert spaces and for convex functions and sets, we derived new results.

This paper establishes a successive approximation method designed to compute a point belonging simultaneously to the fixed-point set of a given mapping (relatively nonexpansive) and to the solution set of a nonconvex equilibrium problem within a Banach space. Our present findings provide a unifying generalization of numerous previously obtained results, beginning with the transition from Hilbertian structures to the more general Banach framework, and further encompassing the passage from convex analytical settings to their considerably more delicate nonconvex counterparts. We mainly extend the results proved in ^[27] from convex to nonconvex settings.