Let $\mathrm{X}$ be a real Banach space and $I$ a function on $\mathrm{X}$ such that $I=\Phi +\psi$ with $\mathrm{\Phi }\in {\mathrm{C}}^1\left(\mathrm{X},\mathrm{R}\right)$ and $\psi :X\to (-\infty ,+\infty ]$ convex, proper and lower semicontinuous. A point $u\in \mathrm{X}$ is said to be critical if $\psi (u)\ne +\infty$ and $\langle {\Phi }^{\prime }(u),v-u\rangle +\psi (v)-\psi (u)\geqq 0$ $\forall v\in \mathrm{X}$. The paper contains a number of existence theorems for critical points of functions of the above mentioned type. Critical levels of saddle type are characterized by minimax principles. The results are applied to variational inequalities and variational equations with single- and multivalued operators, which arise from studying certain elliptic boundary value problems.

Fundamentals of Set Theory

Set theory is a branch of mathematical logic that studies collections of objects, known as sets. It serves as a foundational system for mathematics. The modern study of set theory was initiated by Georg Cantor in the 1870s, and it was further developed by many mathematicians throughout the twentieth century.

A set is an unordered collection of distinct objects, which may be numbers, symbols, points, or even other sets. The objects that make up a set are called its elements or members. Membership is denoted by the symbol ∈. For example, if A = {1, 2, 3}, then 2 ∈ A and 4 ∉ A.

Basic Operations

The union of two sets A and B, written A ∪ B, is the set of all elements that belong to A or to B (or to both). The intersection A ∩ B consists of elements that belong to both A and B. The difference A − B (or A \ B) contains elements that are in A but not in B. The complement of A relative to a universal set U is the set of all elements of U that are not in A.

The empty set, denoted ∅ or {}, contains no elements. The power set of A, written 𝒫(A), is the set of all subsets of A, including the empty set and A itself.

Cardinality and Infinite Sets

The cardinality of a finite set is simply the number of elements it contains. Cantor showed that infinite sets can have different sizes. The set of natural numbers ℕ is countably infinite, while the set of real numbers ℝ is uncountable. This distinction is expressed by the inequality |ℕ| < |ℝ|.

Cantor’s continuum hypothesis conjectures that there is no set whose cardinality is strictly between that of the integers and that of the real numbers. The continuum hypothesis was later shown to be independent of the standard Zermelo–Fraenkel axioms of set theory (with the axiom of choice).

Axiomatic Foundations

Naive set theory leads to paradoxes such as Russell’s paradox. To avoid these difficulties, mathematicians developed axiomatic systems. The most widely used system is Zermelo–Fraenkel set theory with the axiom of choice (ZFC). Other systems include von Neumann–Bernays–Gödel set theory and various alternatives that restrict the notion of a set more severely.

Set theory provides the language in which nearly all of contemporary mathematics is expressed. Concepts such as functions, relations, numbers, and geometric spaces are routinely defined in terms of sets. Consequently, a solid understanding of elementary set theory is essential for advanced study in almost every branch of pure mathematics.

Applications Beyond Pure Mathematics

Outside pure mathematics, set-theoretic ideas appear in computer science (databases, type theory, formal verification), linguistics, philosophy, and the foundations of probability. The notion of a measurable set is central to modern analysis and probability theory.