# A Nonconstant Gradient Constrained Problem for Nonlinear Monotone Operators

## Abstract

**:**

## 1. Introduction

## 2. Results

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 3. Preliminary Results

**Theorem**

**4**

**.**Assume that the composite mapping $(F,G):S\to \mathbb{R}\times Y$ is convex-like with respect to product cone ${\mathbb{R}}_{+}\times C$ in $\mathbb{R}\times Y$, $\mathbb{K}$ is nonempty and the ordering cone C has a nonempty interior $int\left(C\right)$. If the primal problem (11) is solvable and the generalized Slater condition is satisfied, namely there is a vector $\overline{v}\in S$ with $G\left(\overline{v}\right)\in -int\left(C\right)$, then the dual problem (12) is also solvable and the extremal values of the two problems are equal. Moreover, if u is the optimal solution to problem (11) and $\overline{\nu}\in {C}^{\ast}$ is a solution to problem (12), it follows that

**Theorem**

**5**

**.**Under the same assumptions as in Theorem 4, if the ordering cone C is closed, then a point $(u,\overline{\nu})\in S\times {C}^{\ast}$ is a saddle point of the Lagrange functional $\mathcal{L}(v,\nu )$, namely

**Theorem**

**6.**

**Theorem**

**7**

**.**Under the same assumptions as in Theorem 6, ${v}_{0}\in \mathbb{K}$ solves problem (16) if and only if there exist ${\lambda}^{\ast}\in {C}^{\ast}$ and ${\mu}^{\ast}\in {Z}^{\ast}$ such that $({x}_{0},{\lambda}^{\ast},{\mu}^{\ast})$ is a saddle point of the Lagrange functional, namely

**Theorem**

**8.**

**Proof.**

## 4. The Equivalence of the Two Variational Problems

## 5. Lagrange Multipliers

## 6. Discussions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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Giuffrè, S.
A Nonconstant Gradient Constrained Problem for Nonlinear Monotone Operators. *Axioms* **2023**, *12*, 605.
https://doi.org/10.3390/axioms12060605

**AMA Style**

Giuffrè S.
A Nonconstant Gradient Constrained Problem for Nonlinear Monotone Operators. *Axioms*. 2023; 12(6):605.
https://doi.org/10.3390/axioms12060605

**Chicago/Turabian Style**

Giuffrè, Sofia.
2023. "A Nonconstant Gradient Constrained Problem for Nonlinear Monotone Operators" *Axioms* 12, no. 6: 605.
https://doi.org/10.3390/axioms12060605