A Variational Approach towards Solving an Ill-Posed Cauchy Problem for a Functional-Differential Equation

MR Yu. F. DOLGII and P. G. SURKOV Yu. F. DOLGII and P. G. SURKOV A Variational Approach towards Solving an Ill-Posed Cauchy Problem for a Functional-Differential Equation Funkcialaj Ekvacioj. Serio Internacia 59 2016 157--183 http://fe.math.kobe-u.ac.jp/FE/FullPapers/59-2/59_157.pdf http://www.ams.org/mathscinet-getitem?mr=MR We investigate an ill-posed Cauchy problem for a linear functional-differential equation of retarded type on the negative half-line. Using a step-by-step procedure this problem is replaced by an inverse problem for an operator equation of the first kind in a functional space. The Tikhonov's method is then used for finding solutions. We also construct special boundary value problems for the functional-differential equations. Solutions of these boundary value problems determine regularized solutions of the ill-posed Cauchy problem by the step-by-step procedure. Linear functional differential equation, Ill-posed problems. 34K06, 47A52. 59-157 2016 A Variational Approach towards Solving an Ill-Posed Cauchy Problem for a Functional-Differential Equation Yu. F. DOLGII and P. G. SURKOV Yu. F. DOLGII and P. G. SURKOV 1 Hale, J. K.; Verduyn Lunel, S. M. Introduction to Functional-Differential Equations Applied Mathematical Sciences, 99, Springer-Verlag, New York 1993 MR1243878

2 Dolgii, Yu. F.; Putilova, E. N. Backward of Solutions of a Linear Differential Equation with Delay as an Ill-Posed Problem Differentsial'nye Uravneniya 29 1993 1317-1323 MR1281254

3 Dolgii, Yu. F.; Surkov, P. G. Asymptotics of Regularized Solutions of a Linear Nonautonomous System of Advanced Differential Equations Differ. Equ. 46 2010 470-488 MR2796522

4 Tikhonov, A. N.; Arsenin, V. Ya. Solutions of ill-posed problems Scripta Series in Mathematics, John Wiley & Sons, New York-Toronto, Ont.-London 1977 MR0455365

5 Vasin, V. V. Some tendencies in the Tikhonov regularization of ill-posed problems J. Inverse Ill-Posed Problems 14 2006 813-840 MR2270702

6 Dolgii, Yu. F. Characteristic Equation in the Problem of Asymptotic Stability of Periodic System with Aftereffect Proc. Steklov Inst. Math. 2005, Dynamical Systems and Control Problems, suppl. 1 2005 82-94 MR2156251

7 Naimark, M. A. Linear differential operators, Part I: Elementary theory of linear differential operators Frederick Ungar Publishing Co., New York 1967 MR0216050 8 Yosida, K. Functional Analysis Grundlehren der Mathematischen Wissenschaften, 123, Springer-Verlag, Berlin-New York 1980 MR0617913 9 Azbelev, N.; Maksimov, V.; Rakhmatullina, L. Introduction to the theory of linear functional differential equations Advanced Series in Mathematical Science and Engineering, 3, World Federation Publishers Company, Atlanta, GA 1995 MR1422013

10 Maksimov, V. P.; Rumyantsev, A. N.; Shishkin, V. A. On constructing solutions of functional-differential systems with a guaranteed precision Funct. Differ. Equ. 3 1995 135-144 MR1418981

11 Neverova, D. A.; Skubachevskii, A. L. On the Classical and Generalized Solutions of Boundary-Value Problems for Difference-Differential Equations with Variable Coeffcients Math. Notes 94 2013 653-667 MR3227012

12 Miller, J. J.; O'Riordan, E.; Shishkin, G. I. Fitted numerical methods for singular perturbation problems. Error estimates in the maximum error for linear problems in one and two dimensions World Scientific Publishing Co., Inc., River Edge, NJ 1996 MR1439750