Research Article Spectrums of Solvable Pantograph Differential-Operators for First Order Z. I. Ismailov 1 and P. Ipek 2 1 Department of Mathematics, Faculty of Arts and Sciences, Karadeniz Technical University, 61080 Trabzon, Turkey 2 Institute of Fundamental Sciences, Karadeniz Technical University, 61080 Trabzon, Turkey Correspondence should be addressed to Z. I. Ismailov; zameddin@yahoo.com Received 23 May 2014; Revised 17 July 2014; Accepted 25 July 2014; Published 14 August 2014 Academic Editor: Abdullah S. Erdo˘ gan Copyright © 2014 Z. I. Ismailov and P. Ipek. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. By using the methods of operator theory, all solvable extensions of minimal operator generated by frst order pantograph-type delay diferential-operator expression in the Hilbert space of vector-functions on fnite interval have been considered. As a result, the exact formula for the spectrums of these extensions is presented. Applications of obtained results to the concrete models are illustrated. 1. Introduction Te quantitative and qualitative theory of linear pantograph diferential equations, sometimes known as pantograph-type delay diferential equations, was frst studied in detail by T. Kato and J. B. McLeod [1], L. Fox et al. [2], and A. Iserles [3] in the nineteen seventies. Tese equations arose as a mathematical model of an industrial problem involving wave motion in the overhead supply line to an electrifed railway system, so they are ofen called pantograph equations. In industrial applications in works [2, 4] and in studies on biology and economics, control and electrodynamics in works [5–7] have been researched (for more comprehensive list of features see [3]). Since an analytical computation of solutions, eigenval- ues, and eigenfunctions of corresponding problems is very difcult theoretically and technically, then in this theory methods of numerical analysis play a signifcant role (for more information see [8–13]). Let us remember that an operator  : () ⊂  →  in Hilbert space  is called solvable, if  is one-to-one, () = , and  −1 ∈ (). In this work, by using methods of operator theory all solv- able extensions of minimal operator generated by panto- graph-type delay diferential-operator expression for frst order in the Hilbert space of vector-functions at a fnite interval have been described in terms of boundary values in Section 2. Consequently, the resolvent operators of these extensions can be written clearly. Te exact formula for the spectrums of these extensions has been given in Section 3. Applications of obtained results to the concrete models have been illustrated in Section 4. 2. Description of Solvable Extensions In the Hilbert space  2 (,(0,1)) of vector-functions consider a linear pantograph diferential-operator expression for frst order in the form ()=  ()+  ∑ =1   () (  ), (1) where (1)  is a separable Hilbert space with inner product (⋅,⋅)  and norm ‖⋅‖  , (2) operator-function   (⋅) : [0,1] → (), = 1,2,3,...,, is continuous on the uniform operator topology, (3) =1,2,3,...,−1, 0<  <1,   =1. Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 837565, 8 pages http://dx.doi.org/10.1155/2014/837565