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Weak Base Solutions03:21

Weak Base Solutions

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Some compounds produce hydroxide ions when dissolved by chemically reacting with water molecules. In all cases, these compounds react only partially and so are classified as weak bases. These types of compounds are also abundant in nature and important commodities in various technologies. For example, global production of the weak base ammonia is typically well over 100 metric tons annually, being widely used as an agricultural fertilizer, a raw material for chemical synthesis of other...
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Weak Acid Solutions04:02

Weak Acid Solutions

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Few compounds act as strong acids. A far greater number of compounds behave as weak acids and only partially react with water, leaving a large majority of dissolved molecules in their original form and generating a relatively small amount of hydronium ions. Weak acids are commonly encountered in nature, being the substances partly responsible for the tangy taste of citrus fruits, the stinging sensation of insect bites, and the unpleasant smells associated with body odor. A familiar example of a...
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Solution Equilibrium and Saturation01:59

Solution Equilibrium and Saturation

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Imagine adding a small amount of sugar to a glass of water, stirring until all the sugar has dissolved, and then adding a bit more. You can repeat this process until the sugar concentration of the solution reaches its natural limit, a limit determined primarily by the relative strengths of the solute-solute, solute-solvent, and solvent-solvent attractive forces. You can be certain that you have reached this limit because, no matter how long you stir the solution, undissolved sugar remains. The...
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Alternative Sets of Equilibrium Equations01:31

Alternative Sets of Equilibrium Equations

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When analyzing the behavior of structures, engineers often rely on the concept of equilibrium. This refers to the state where all forces and moments acting on a system balance each other, resulting in no net movement or rotation. In many cases, equilibrium can be described by a set of standard equations. However, in some situations, alternative sets of equilibrium equations must be used to describe the system's behavior accurately.
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Determining the pH of Salt Solutions04:08

Determining the pH of Salt Solutions

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The pH of a salt solution is determined by its component anions and cations. Salts that contain pH-neutral anions and the hydronium ion-producing cations form a solution with a pH less than 7. For example, in ammonium nitrate (NH4NO3) solution, NO3− ions do not react with water whereas NH4+ ions produce the hydronium ions resulting in the acidic solution.  In contrast, salts that contain pH-neutral cations and the hydroxide ion-producing anions form a solution with a pH greater than 7. For...
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Titration Calculations: Weak Acid - Strong Base03:55

Titration Calculations: Weak Acid - Strong Base

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Calculating pH for Titration Solutions: Weak Acid/Strong Base
For the titration of 25.00 mL of 0.100 M CH3CO2H with 0.100 M NaOH, the reaction can be represented as:
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Approximate weakly efficient solutions of set-valued vector equilibrium problems.

Jian Chen1, Yihong Xu1, Ke Zhang1

  • 1Department of Mathematics, Nanchang University, Nanchang, China.

Journal of Inequalities and Applications
|August 24, 2018
PubMed
Summary
This summary is machine-generated.

This study introduces approximate weakly efficient solutions for constrained set-valued vector equilibrium problems. It establishes optimality conditions, linking weak and approximate solutions in topological vector spaces.

Keywords:
Approximate weakly efficient solutionNear cone-subconvexlikenessOptimality conditionSet-valued vector equilibrium problem

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Area of Science:

  • Optimization Theory
  • Vector Equilibrium Problems
  • Set-Valued Analysis

Background:

  • Vector equilibrium problems (VEPs) are fundamental in game theory and economics.
  • Existing solution concepts for VEPs often lack robustness to small perturbations.
  • Locally convex Hausdorff topological vector spaces provide a general framework for studying VEPs.

Purpose of the Study:

  • To introduce and analyze approximate weakly efficient solutions for constrained set-valued VEPs.
  • To explore the relationship between weakly efficient and approximate weakly efficient solutions.
  • To derive novel optimality conditions for these problems.

Main Methods:

  • Definition of approximate weakly efficient solutions.
  • Application of separation theorems for convex sets.
  • Development of Kuhn-Tucker-type and Lagrange-type conditions.

Main Results:

  • A new class of approximate weakly efficient solutions is proposed.
  • A relationship connecting weak and approximate weak efficiency is established.
  • Optimality conditions are derived under the near cone-subconvexlikeness assumption.

Conclusions:

  • The introduced approximate solutions offer a more flexible approach to VEPs.
  • The derived optimality conditions provide valuable tools for analysis and computation.
  • This work extends the theory of equilibrium problems in topological vector spaces.