Optimal approximate fixed point results in locally convex spaces

Let $C$ be a convex subset of a locally convex space. We provide optimal approximate fixed point results for sequentially continuous maps $f\colon C\to\bar{C}$. First we prove that if $f(C)$ is totally bounded, then it has an approximate fixed point net. Next, it is shown that if $C$ is bounded but not totally bounded, then there is a uniformly continuous map $f\colon C\to C$ without approximate fixed point nets. We also exhibit an example of a sequentially continuous map defined on a compact convex set with no approximate fixed point sequence. In contrast, it is observed that every affine (not-necessarily continuous) self-mapping a bounded convex subset of a topological vector space has an approximate fixed point sequence. Moreover, it is constructed a affine sequentially continuous map from a compact convex set into itself without fixed points.Comment: 12 page

Application of bifurcation theory for existence of travelling waves in examples of semilinear and quasilinear wave equations

Wir untersuchen wandernde Wellen für semilineare bzw. quasilineare Wellengleichungen. Wir beweisen die Existenz von speziellen wandernden Wellen, welche periodisch in der Ausbreitungsrichtung und lokalisiert in eine der orthogonalen Richtungen sind. Die Gleichungen werden durch die Maxwell-Gleichungen im optischen Fall mit Kerr-Nichtlinearität im Polarisationsfeld motiviert. Die wandernden Wellen verzweigen von den linear geführten Moden

The Trouble with Farmouts: The Problem of the Innocent, Nonperforming Farmee

The oil market is a volatile universe. The price of the commodity has a profound impact both on the national and global economies and on the lives of everyday consumers. Consider the high prices of 2022 compared with the record lows seen in 2020, the price of oil affects almost everything. The United States is one of the top oil producing nations in the world. The size and importance of the industry has led to a somewhat unique area of the legal practice known as oil and gas law. Among its many tenants is an instrument known as a farmout contract. Farmout contracts have steadily grow in use by the industry since their inception, supplementing and even replacing the oil and gas lease, which traditionally has been the primary legal mechanism under which oil drilling takes place. Oil and gas leases generally obligate lessees to drill continually or else face breach liability. In response, many courts eased this requirement of strict performance by way of novel and established legal doctrines when the lessee’s nonperformance was caused by circumstances outside of its control. However, even though, farmees – the parties that are obligated to drill under farmout contracts – are generally subject to the same drilling requirements, they do not enjoy the same judicial protections as oil and gas lessees. No current legal doctrine can readily excuse their nonperformance, no matter how blameless the farmee may be. This is the problem of the innocent, nonperformance farmee. The following comment examines the problem’s relevant background, analyzes the shortcomings in the current law, and suggests two solutions for future parties and courts to consider