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Article Citation - WoS: 52Citation - Scopus: 62Basics of Riemann Delta and Nabla Integration on Time Scales(Taylor & Francis Ltd, 2002) Guseinov, GS; Kaymakçalan, BIn this paper we introduce and investigate the concepts of Riemann's delta and nabla integrals on time scales. Main theorems of the integral calculus on time scales are proved.Article Citation - WoS: 29Citation - Scopus: 30Boundary Value Problems for Second Order Nonlinear Differential Equations on Infinite Intervals(Academic Press inc Elsevier Science, 2004) Guseinov, GS; Yaslan, IIn this paper, we consider boundary value problems for nonlinear differential equations on the semi-axis (0, infinity) and also on the whole axis (-infinity, infinity), under the assumption that the left-hand side being a second order linear differential expression belongs to the Weyl limit-circle case. The boundary value problems are considered in the Hilbert spaces L-2(0, infinity) and L-2(-infinity, infinity), and include boundary conditions at infinity. The existence and uniqueness results for solutions of the considered boundary value problems are established. (C) 2003 Elsevier Inc. All rights reserved.Article A boundary value problem for second-order nonlinear difference equations on the integers(Cambridge Univ Press, 2005) Dal, F; Guseinov, GSIn this study, we are concerned with a boundary value problem (BVP) for nonlinear difference equations on the set of all integers Z, under the assumption that the left-hand side is a second-order linear difference expression which belongs to the so-called Weyl-Hamburger limit-circle case. The BVP is considered in the Hilbert space l(2) and includes boundary conditions at infinity. Existence and uniqueness results for solution of the considered BVP are established.Article Citation - WoS: 89Citation - Scopus: 99Multiple Integration on Time Scales(Dynamic Publishers, inc, 2005) Bohner, M; Guseinov, GS; MathematicsIn this paper an introduction to integration theory for multivariable functions on time scales is given. Such an integral calculus can be used to develop a theory of partial dynamic equations on time scales in order to unify and extend the usual partial differential equations and partial difference equations.Article Citation - WoS: 44Citation - Scopus: 46Integrable Equations on Time Scales -: Art. No. 113510(Amer inst Physics, 2005) Gürses, M; Guseinov, GS; Silindir, BIntegrable systems are usually given in terms of functions of continuous variables (on R), in terms of functions of discrete variables (on Z), and recently in terms of functions of q-variables (on K-q). We formulate the Gel'fand-Dikii (GD) formalism on time scales by using the delta differentiation operator and find more general integrable nonlinear evolutionary equations. In particular they yield integrable equations over integers (difference equations) and over q-numbers (q-difference equations). We formulate the GD formalism also in terms of shift operators for all regular-discrete time scales. We give a method allowing to construct the recursion operators for integrable systems on time scales. Finally, we give a trace formula on time scales and then construct infinitely many conserved quantities (Casimirs) of the integrable systems on time scales. (c) 2005 American Institute of Physics.Article Citation - WoS: 6Citation - Scopus: 6A Boundary Value Problem for Second Order Nonlinear Difference Equations on the Semi-Infinite Interval(Taylor & Francis Ltd, 2002) Guseinov, GSIn this paper, we consider a boundary value problem (BVP) for nonlinear difference equations on the discrete semi-axis in which the left-hand side being a second order linear difference expression belongs to the so-called Weyl-Hamburger limit-circle case. The BVP is considered in the Hilbert space l(2) and is formed via boundary conditions at a starting point and at infinity. Existence and uniqueness results for solutions of the considered BVP are established.Article Citation - WoS: 283Citation - Scopus: 315Integration on Time Scales(Academic Press inc Elsevier Science, 2003) Guseinov, GS; Hüseyin, Hüseyin Şirin; Hüseyin, Hüseyin Şirin; Mathematics; MathematicsIn this paper we study the process of Riemann and Lebesgue integration oil time scales. The relationship of the Riemann and Lebesgue integrals is considered and a criterion for Riemann integrability is established. (C) 2003 Elsevier Inc. All rights reserved.Article Citation - WoS: 81Citation - Scopus: 93Partial Differentiation on Time Scales(Dynamic Publishers, inc, 2004) Bohner, M; Guseinov, GS; MathematicsIn this paper a differential calculus for multivariable functions on time scales is presented. Such a calculus can be used to develop a theory of partial dynamic equations on time scales in order to unify and extend the usual partial differential and partial difference equations.Article Citation - WoS: 80Citation - Scopus: 86Lyapunov inequalities for discrete linear Hamiltonian systems(Pergamon-elsevier Science Ltd, 2003) Guseinov, GS; Kaymakçalan, BIn this paper, we present some Lyapunov type inequalities for discrete linear scalar Hamiltonian systems when the coefficient c(t) is not necessarily nonnegative valued and when the end-points are not necessarily usual zeros, but rather, generalized zeros. Applying these inequalities, we obtain some disconjugacy and stability criteria for discrete Hamiltonian systems. (C) 2003 Elsevier Science Ltd. All rights reserved.Article Citation - WoS: 3Citation - Scopus: 3Properties of Discrete Composition Operators(Taylor & Francis Ltd, 2005) Dal, F; Guseinov, GSIn this paper, we study the continuity and boundedness properties of the composition operator Fy ( t )= f ( t , y ( t )), where t is a discrete independent variable. A necessary and sufficient condition for the acting of F from a space p 1 ( p 1 greater than or equal to1) into another p 2 ( p 2 greater than or equal to1) is also established.
