The aftermath of the intermediate value theorem
- PDF / 552,856 Bytes
- 8 Pages / 468 x 680 pts Page_size
- 95 Downloads / 200 Views
The solvability of nonlinear equations has awakened great interest among mathematicians for a number of centuries, perhaps as early as the Babylonian culture (3000–300 B.C.E.). However, we intend to bring to our attention that some of the problems studied nowadays appear to be amazingly related to the time of Bolzano’s era (1781–1848). Indeed, this Czech mathematician or perhaps philosopher has rigorously proven what is known today as the intermediate value theorem, a result that is intimately related to various classical theorems that will be discussed throughout this work. 1. Introduction The main motivation of this paper is to establish a close connection between a classical theorem from real analysis (discovered over two centuries ago) and recent works in monotone operator theory for reflexive Banach spaces. Throughout this presentation, we give a brief description of how the original problem has evolved in time, passing through various generalizations obtained in the last thirty years. However, the main purpose of this paper is to generalize Theorem 1 of Minty [17], where the convexity condition on the domain of the operator is no longer required. We also obtain a new result on monotone operators perturbed by compact mappings. The study of existence of solutions for nonlinear functional equations involving monotone operators has been extensively discussed for forty years or so. Concerning the study of existence of zeros under the boundary condition (2.3), we find, among many contributions, the work of Va˘ınberg and Kaˇcurovski˘ı [27], Minty [16, 17], Browder [4, 5, 6], and Shinbrot [25]. For related work, we also mention Br´ezis et al. [3], Kaˇcurovski˘ı [11], Leray and Lions [15], and Rockafellar [24]. However, the connection between Bolzano’s boundary condition and this most recent condition (2.3) (known by the early 1950s) has not been explicitly observed. Therefore, our main interest is to identify some of the work done in the contour of this condition (2.3) that was, perhaps, first observed by this mathematician of the 19th century. Copyright © 2004 Hindawi Publishing Corporation Fixed Point Theory and Applications 2004:3 (2004) 243–250 2000 Mathematics Subject Classification: 47H10 URL: http://dx.doi.org/10.1155/S1687182004310053
244
The aftermath of the intermediate value theorem
We begin with a result for the Euclidean finite-dimensional space Rn , where the symbol ·, · represents the corresponding Euclidean inner product. We continue using the same symbol, although the space of definition will change, passing through Hilbert spaces to end with reflexive Banach spaces. Since, indeed, most of the results will be given for this latter class of spaces, we may assume that both the reflexive Banach space X and its dual X ∗ are locally uniformly convex after renorming [26]. This very fact implies that the duality mapping J is single valued and strictly monotone. In addition, we will present the main results for demicontinuous operators (i.e., continuous mappings from the strong topology into the weak t
Data Loading...
