Topological Methods in the Study of Boundary Value Problems (Universitext)

Topological Methods in the Study of Boundary Value Problems (Universitext)

Pablo Amster

Language: English

Pages: 226

ISBN: 1461488923

Format: PDF / Kindle (mobi) / ePub


This textbook is devoted to the study of some simple but representative nonlinear boundary value problems by topological methods. The approach is elementary, with only a few model ordinary differential equations and applications, chosen in such a way that the student may avoid most of the technical difficulties and focus on the application of topological methods. Only basic knowledge of general analysis is needed, making the book understandable to non-specialists. The main topics in the study of boundary value problems are present in this text, so readers with some experience in functional analysis or differential equations may also find some elements that complement and enrich their tools for solving nonlinear problems. In comparison with other texts in the field, this one has the advantage of a concise and informal style, thus allowing graduate and undergraduate students to enjoy some of the beauties of this interesting branch of mathematics. Exercises and examples are included throughout the book, providing motivation for the reader.

Numerical Mathematics (Undergraduate Texts in Mathematics / Readings in Mathematics)

A Concept of Limits (Dover Books on Mathematics)

Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century

Fundamentals of Differential Geometry (Graduate Texts in Mathematics, Volume 191)

Mathematik für Informatiker

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

obtaining better or sharper theorems. We focus all the time on the methods and on the specific problems; in particular, this is one of the reasons for which all examples refer only to ordinary differential equations, which makes it possible to avoid some technicalities. Also, we have chosen to work always in the spaces of continuous or continuously differentiable functions, although better results can be obtained using, for example, Sobolev spaces. In many cases, the same problem is studied using

(0, 1] such that w(t0 ) = maxt∈[0,1] w(t); then w (t0 ) ≥ 0. If w(t0 ) > 0, then w (t0 ) ≤ −λ w(t0 ) < 0, a contradiction. The case λ < 0 follows in a similar way. The preceding lemma will allow us to adapt the idea of upper and lower solutions to best suit our purposes. Assume that of class f is C1 with respect to u and that α , β ∈ X verify α ≤ β and α (t) ≤ f (t, α (t)), β (t) ≥ f (t, β (t)). (3.9) Taking λ > 0 large enough, for each t the function f (t, u) + λ u is nondecreasing for u ∈

Lower Solutions Strike Again 95 choose R satisfying R ≥ β ∞ : in this case, Q((u (t0 )) = Q(β (t0 )) = β (t0 ). Analogous considerations for the remaining case also impose the condition R ≥ α ∞ ; summarizing, if we take R ≥ max{ α ∞ , β ∞ }, then α ≤ u ≤ β . As a consequence, we deduce that u (t) = f (t, u(t), Q(u (t))). Now we want to find a condition that will allow us to prove that u ∞ ≤ R. Although there exist several variants, the most famous is the one known in the literature as the

distinguish as geometrically different those solutions that do not differ by a multiple of 2π . The previous comments motivate us to write the forcing term p as p0 + c, where p0 = p − p has zero average and c = p is a constant. Henceforth, we consider the equivalent problem u (t) + au (t) + sin u(t) = p0 (t) + c, u(0) = u(1), u (0) = u (1). (4.9) 4.3 Upper and Lower Solutions Strike Again 97 We shall prove the following theorem. Theorem 4.7. Let p0 ∈ C([0, 1]) be such that p0 = 0. Then

deg(ψ , B1(0), y). But y ∈ / ϕ (B1 (0)), so deg(ψ , B1 (0), y) = 0, a contradiction. There is a very popular application of the Borsuk–Ulam theorem for the specific case n = 2, again of a meteorological character: at any given moment there exists on Earth a pair of antipodal points with the same temperature and pressure. In addition, the Borsuk–Ulam theorem provides a different proof of Brouwer’s theorem. Suppose that f : B1 (0) → B1 (0) is continuous with no fixed points, and let F : B1 (0) → Rn

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