Global diffeomorphism theorem applied to the solvability of discrete and continuous boundary value problems

Abstract

Using the global diffeomorphism theorem we consider the existence of solutions to the following Dirichlet problemẍ (t) = f (t, x (t)) + v(t), x (0) = x (1) = 0, where f : [0, 1] × R → R is a jointly continuous function subject to some further growth conditions. Together with a given problem we consider the family of its discretizations and as a result we prove the existence of a non-spurious solution.

Solvability of Abstract Semilinear Equations by a Global Diffeomorphism Theorem

“…Coining the above observations together, we can now show that equality (5) implies Au n − Au 0 → 0 as n → ∞ which means, by (2), that (u n ) n∈N converges strongly to u 0 in X. This shows that ϕ y satisfies (PS) condition.…”

“…There was also an attempt to examine second order Dirichlet problem for O.D.E. in [2], but for some specific problem and without any abstract scheme allowing for considering boundary value problems in some unified manner. Results for continuous problem in [2] are related to the existence result obtained in [19], although the methods are different, both yield the existence with similar assumptions.…”

On solvability of elliptic boundary value problems via global invertibility

“…Moreover, for every n ∈ N there exists e n ∈ L 2 (Ω) such that Le n = λ n e n and (e i ) i∈N is an orthonormal basis of L 2 (Ω). As a consequence, for every…”

“…We have already considered global invertibility of mappings with applications to solvability of nonlinear boundary value problems in [2], [3] using the global inversion result due to [5]. The methodology used there was much more complicated and pertained to the application of tools from critical point theory, namely it required the Palais-Smale condition to be checked in addition to local invertibility and coercivity.…”

Sobolev spaces on Gelfand pairs

“…Even at the author's alma mater, Łódź University of Technology, there is a research group studying Sobolev's legacy (comp. [9,10,11]).…”