On a Class of Functional Equations over the Real and over the Complex Fields
Keywords:functional calculus., self-adjoint operators, analyt, function defined implicitlicityy, constructive solutions
In the present review-paper, we start by recalling some of our earlier results on the construction of a nontrivial function defined implicitly by the equation (1.1), without using the implicit function theorem. This is the first aim of the paper. Here the function is given, satisfying some conditions. All these considerations work in the real case, for functions and a class of operators. The second aim is to consider the complex case, proving the analyticity of the function defined implicitly, under the hypothesis that is analytic and verifies natural conditions, related to the real case. Some consequences are deduced. Finally, one illustrates the preceding results by an application to a concrete functional and respectively operatorial equation. Related examples are given, some of them pointing out elementary functions for which equation (1.1) leads to nontrivial solutions which can be expressed by means of elementary functions.
Cristescu, R., Ordered Vector Spaces and Linear Operators, Academiei, Bucharest, and Abacus Press, Tunbridge Wells, Kent, 1976. www.ear.ro
Olariu, V.V., Olteanu, C.O., Orthogonality and Mathematical Physics, LAMBERT Academic Publishing, Beau Bassin, 2018. www.lap-publishing.com
Olteanu, A. and Olteanu, O., Solving some special functional equations by a general “geometrical” method, and an approach to the complex case, Revue Roumaine de Mathématiques Pures et Appliquées, 51, 5-6 (2006), 735-745. imar.ro/journals/Revue_Mathematique/home_page.html
Olteanu, O., Analytic solutions of special functional equations, International Journal of Analysis and Applications, 1, 1 (2013), 18-32. etamaths.com/index.php/ijaa/article/view/50
Olteanu, O., Exact geometric solutions and approximating analytic solutions of functional equations, International Journal of Mathematical Analysis, 7, 47 (2013), 2303-2311. www.m-hikari.com/ijma/index.html https://doi.org/10.12988/ijma.2013.36143
Olteanu, O., New Results in Mathematical Analysis, LAMBERT Academic Publishing, Beau Bassin, 2020. www.lap-publishing.com
Olteanu, O., On Newton’s method for convex functions and operators and its relationship with contraction principle, MathLAB Journal, 6 (2020), 53-61. www.purkh.com/index.php/mathlab
Olteanu, O. Recent Results on Markov Moment Problem, Polynomial Approximation, and Related Fields in Analysis, Generis Publishing: Chişinău, 2020. www.generis-publishing.com
Rudin, W., Real, and Complex Analysis. Third Edition, McGraw-Hill Inc., International Edition, 1987. www.bizapedia.com/ny/mcgrawhill-publishing-company-inc.html
How to Cite
Copyright (c) 2020 Octav Olteanu
This work is licensed under a Creative Commons Attribution 4.0 International License.