相关论文: Ekeland's variational principle in weak and strong…

Motivated by the recent work on conditional risk measures, this paper studies the Ekeland's variational principle for a proper, lower semicontinuous and lower bounded $\bar{L}^{0}-$valued function, where $\bar{L}^{0}$ is the set of…

In the setting of real vector spaces, we establish a general set-valued Ekeland variational principle (briefly, denoted by EVP), where the objective function is a set-valued map taking values in a real vector space quasi-ordered by a convex…

We establish a pre-order principle. From the principle, we obtain a very general set-valued Ekeland variational principle, where the objective function is a set-valued map taking values in a quasi ordered linear space and the perturbation…

Roughly speaking, Ekeland's Variational Principle (EkVP) (J. Math. Anal. Appl. 47 (1974), 324--353) asserts the existence of strict minima of some perturbed versions of lower semicontinuous functions defined on a complete metric space.…

This paper is concerned with an elliptic system of Kirchhoff type, driven by the variable-order fractional $p(x)$-operator. With the help of the direct variational method and Ekeland variational principle, we show the existence of a weak…

In this paper, we propose a weak regularity principle which is similar to both weak K\"onig's lemma and Ramsey's theorem. We begin by studying the computational strength of this principle in the context of reverse mathematics. We then…

For proper lower semi-continuous functionals bounded below which do not increase upon polarization, an improved version of Ekeland's variational principle can be formulated in Banach spaces, which provides almost symmetric points.

In this short communication, we present a generalization of the Ekeland variational principle. The main result is established through standard tools of functional analysis and calculus of variations. The novelty here is a result involving…

We prove conservativity results for weak K\H{o}nig's lemma that extend the celebrated result of Harrington (for $\Pi^1_1$-statements) and are somewhat orthogonal to the extension by Simpson, Tanaka and Yamazaki (for statements of the form…

The present paper is concerned with Ekeland Variational Principle (EkVP) and its equivalents (Caristi-Kirk fixed point theorem, Takahashi minimization principle, Oettli-Th\'era equilibrium version of EkVP) in quasi-uniform spaces. These…

In this note, we remark, with sufficient mathematical rigor, that many weak generalizations of the usual minimum available in the literature are not true generalizations. Motivated by the Ekeland Variational Principle, we provide, first…

It is well-known that any finite $\Pi^{0}_{1}$-class of $2^{\mathbb N}$ has a computable member. Then, how can we understand this in the context of reverse mathematics? In this note, we consider several very weak fragments of K\H{o}nig's…

This paper uses the framework of reverse mathematics to investigate the strength of two recurrence theorems of topological dynamics. It establishes that one of these theorems, the existence of an almost periodic point, lies strictly between…

In mathematical logic there are two seemingly distinct kinds of principles called "reflection principles." Semantic reflection principles assert that if a formula holds in the whole universe, then it holds in a set-sized model. Syntactic…

The Jordan decomposition theorem states that every function $f \colon [0,1] \to \mathbb{R}$ of bounded variation can be written as the difference of two non-decreasing functions. Combining this fact with a result of Lebesgue, every function…

We consider several formalizations in the language of second-order arithmetic of "The formula $\phi$ is a theorem of $\omega$-logic", including some which have been studied in the literature and a new variant defined via a least fixed…

We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood, we say that a…

The strong continuity principle reads "every pointwise continuous function from a complete separable metric space to a metric space is uniformly continuous near each compact image." We show that this principle is equivalent to the fan…

We prove that the statement "there is a $k$ such that for every $f$ there is a $k$-bounded diagonally non-recursive function relative to $f$" does not imply weak K\"onig's lemma over $\mathrm{RCA}_0 + \mathrm{B}\Sigma^0_2$. This answers a…

In this paper, we present some implicit function theorems for set-valued mappings between Fr\'echet spaces. The proof relies on Lebesgue's Dominated Convergence Theorem and on Ekeland's variational principle. An application to the existence…