In this paper we establish a variant and generalized weak linking theorem, which contains more delicate result and insures the existence of bounded Palais–Smale sequences of a strongly indefinite functional. The abstract result will be used to study the semilinear Schrödinger equation $-\Delta u+V\left(x\right)u={K\left(x\right)\left|u\right|}^{2^{*}-2}u+g(x,u),u\in W^{1,2}\left({𝐑}^N\right)$, where N ≥ 4; V,K,g are periodic in xj for 1 ≤ j ≤ N and 0 is in a gap of the spectrum of -Δ + V; K>0. If $0<g(x,u)u\le {c\left|u\right|}^{2^{*}}$ for an appropriate constant c, we show that this equation has a nontrivial solution.

We use tighter majorizing sequences than in earlier studies to provide a semilocal convergence analysis for the secant method. Our sufficient convergence conditions are also weaker. Numerical examples are provided where earlier conditions do not hold but for which the new conditions are satisfied.

Some sufficient conditions are provided that guarantee that the difference of a compact mapping and a proper mapping defined between any two Banach spaces over $𝕂$ has at least one zero. When conditions are strengthened, this difference has at most a finite number of zeros throughout the entire space. The proof of the result is constructive and is based upon a continuation method.

The paper is a supplement to a survey by J. Franců: Monotone operators, A survey directed to differential equations, Aplikace Matematiky, 35(1990), 257–301. An abstract existence theorem for the equation $Au=b$ with a coercive weakly continuous operator is proved. The application to boundary value problems for differential equations is illustrated on two examples. Although this generalization of monotone operator theory is not as general as the M-condition, it is sufficient for many technical applications....