Positive Solution to a Generalized Lyapunov Equation via a Coupled Fixed Point Theorem in a Metric Space Endowed With a Partial Order
|
|
- Egbert Britton Hopkins
- 6 years ago
- Views:
Transcription
1 Filomat 29:8 2015, DOI /FIL B Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: Positive Solution to a Generalized Lyapunov Equation via a Coupled Fixed Point Theorem in a Metric Space Endowed With a Partial Order Maher Berzig a, Bessem Samet b a Department of Mathematics, Tunis College of Sciences Techniques, 5 ave. Taha Hussein, Tunis, Tunisia b Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia Abstract. We consider the generalized continuous-time Lyapunov equation: A XB + B XA = Q, where Q is an N N Hermitian positive definite matrix A, B are arbitrary N N matrices. Under certain conditions, using a coupled fixed point theorem du to Bhaskar Lakshmikantham combined with the Schauder fixed point theorem, we establish an existence uniqueness result of Hermitian positive definite solution to such equation. Moreover, we provide an iteration method to find convergent sequences which converge to the solution if one exists. Numerical experiments are presented to illustrate our theoretical results. 1. Introduction Consider the generalized continuous-time algebraic Lyapunov equation GCALE A XB + B XA = Q 1 with given matrices Q, A, B unknown matrix X. Such equations play an important role in stability theory [6, 18], optimal control problems [14, 17] balanced model reduction [16]. In [19], it is proved that 1 has a unique Hermitian, positive definite solution X for every Hermitian positive definite matrix Q if only if all eigenvalues of the pencil λa B are finite lie in the open left half-plane. When A = I the identity matrix, 1 becomes the stard Lyapunov equation XB + B X = Q. 2 The classical numerical methods to solve 2 are the Bartels-Stewart method [1], the Hammarling method [10] the Hessenberg-Schur method [9]. An extension of these methods to solve 1 with the assumption A is nonsingular, is given in [5, 7 9, 19]. Other approaches to solve 1 are the sign function method [4, 13, 15], the ADI method [3, 11, 20] Mathematics Subject Classification. Primary 15A24, 15A29; Secondary 47H10 Keywords. Lyapunov equation, positive definite solution, iterative algorithm, coupled fixed point Received: 06 November 2013; Accepted: 12 April 2014 Communicated by Ljubomir Ćirić The second author extends his sincere appreciation to the Deanship of Scientific Research at King Saud University for its funding this Prolific Research group PRG addresses: maher.berzig@gmail.com Maher Berzig, bsamet@ksu.edu.sa Bessem Samet
2 M. Berzig, B. Samet / Filomat 29:8 2015, In this paper, we consider the equation 1, where Q is an N N Hermitian positive definite matrix A, B are arbitrary N N matrices. Using Bhaskar-Lakshmikantham coupled fixed point theorem [2], we provide a sufficient condition that assures the existence uniqueness of a Hermitian positive definite solution to 1. Moreover, we present an algorithm to solve this equation. Numerical experiments are given to illustrate our theoretical result. 2. Notations Preliminaries We shall use the following notations: MN denotes the set of all N N matrices, HN MN is the set of all N N Hermitian matrices PN HN resp. PN is the set of all N N positive definite matrices resp. positive semidefinite matrices. Instead of X PN, we will also write X > 0. Furthermore, X 0 means that X is positive semidefinite. As a different notation for X Y 0 X Y > 0, we will use X Y X > Y. If X, Y HN such that X Y, then [X, Y] will be the set of all Z HN satisfying X Z Y. If X, Y HN such that X < Y, then ]X, Y[ will be the set of all Z HN satisfying X < Z < Y. We denote by the spectral norm, i.e., A = λ + A A, where λ + A A is the largest eigenvalue of A A. The N N identity matrix will be written as I. It turns out that it is convenient here to use the metric induced by the trace norm 1. Recall that this norm is given by A 1 = n j=1 s ja, where s j A, j = 1,..., n are the singular values of A. In fact, we shall use a slight modification of this norm. For Q PN, we define A 1,Q = Q 1/2 AQ 1/2 1. For any U MN, we denote by SpU the spectrum of U, i.e., the set of its eigenvalues. The following lemmas will be useful later. Lemma 2.1 See [21]. Let A 0 B 0 be N N matrices. 0 trab A trb. Here, for T MN, trt denotes the trace of the matrix T. Lemma 2.2 See [12]. Let A Hn such that I < A < I. A < 1. Definition 2.3. Let, be a partially ordered set. We say that a mapping F : has the mixed monotone property if X, Y, J, K, X J, Y K = FX, Y FJ, K. The proof of our main result is based on the following coupled fixed point theorem posed in a metric space endowed with a partial order. Lemma 2.4 Bhaskar Lakshmikantham [2]. Let, be a partially ordered set d be a metric on X such that, d is a complete metric space. Let F : be a continuous mapping. Suppose that i F has the mixed monotone property; ii there exists some δ [0, 1 such that dfx, y, Fu, v δ [dx, u + dy, v], x u, y v; 2 iii there exists x 0, y 0 such that x 0 Fx 0, y 0, y 0 Fy 0, x 0 ; iv for every x, y, there exists z such that x z, y z. there exists a unique x such that Fx, x = x. Moreover, v the sequences x n y n defined by converge to x. x n+1 = Fx n, y n, y n+1 = Fy n, x n
3 M. Berzig, B. Samet / Filomat 29:8 2015, Main Results Let U = A B + I, V = A + B + I, W = B I. 2 2 Our first result is the following. Theorem 3.1. Suppose that there exists Q, M PN PN such that a 2U QU + B QB < Q; b 2V QV + W QW < Q; c U MU + B MB < M V MV + W MW; d Q ]U MU + B MB, M V MV + W MW[. i 1 has one only one solution X PN; ii X [Q U MU + B MB, Q + V MV + W MW]. Proof. At first, observe that 1 is equivalent to X = Q + V XV + W XW U XU + B XB. 3 Consider the continuous mapping F : HN HN HN defined by FX, Y = Q + V XV + W XW U YU + B YB, for all X, Y HN. 4 3 is equivalent to X = FX, X. Observe that F has the mixed monotone property. Indeed, for any X, Y, J, K HN such that X J Y K, we have FX, Y = Q + V XV + W XW U YU + B YB Q + V JV + W JW U KU + B KB = FJ, K. Now, let X, Y, J, K HN such that X J Y K. We have FX, Y FJ, K 1, Q = tr Q1/2 FX, Y FJ, K Q 1/2 = tr Q1/2 V X JV + W X JW + U K YU + B K YB Q 1/2 = tr Q1/2 V X JV + W X JW Q 1/2 + tr Q1/2 U K YU + B K YB Q 1/2. On the other h, using Lemma 2.1, we obtain tr Q1/2 V X JV + W X JW Q 1/2 = tr V QV X J + W QW X J = tr V QV X J Q 1/2 Q 1/2 + W QW 1/2 X J Q Q 1/2 = tr Q 1/2 V QV X J Q 1/2 + Q 1/2 W QW X J Q 1/2 = tr Q 1/2 V QV Q 1/2 Q1/2 X J Q 1/2 + Q 1/2 W QW Q 1/2 Q1/2 X J Q 1/2 Q 1/2 V QV Q 1/2 + Q 1/2 W QW Q 1/2 tr Q1/2 X J Q 1/2 = Q 1/2 V QV + W QW Q 1/2 tr Q1/2 X J Q 1/2 = Q 1/2 V QV + W QW Q 1/2 X J 1, Q := E 1. 5
4 M. Berzig, B. Samet / Filomat 29:8 2015, tr Q1/2 V X JV + W X JW Q 1/2 E 1. 6 Similarly, we have where tr Q1/2 U K YU + B K YB Q 1/2 E 2, 7 Now, using 6 7, we get where E 2 := Q 1/2 U QU + B QB Q 1/2 K Y 1, Q. FX, Y FJ, K 1, Q δ 2 X J 1, Q + K Y 1, Q, δ = 2 max Q 1/2 V QV + W QW Q 1/2, Q 1/2 U QU + B QB Q 1/2. Observe that from Lemma 2.2, a b, we have δ 0, 1. On the other h, Taking X 0 = 0 Y 0 = M, from c d, we have X 0 < FX 0, Y 0 Y 0 > FY 0, X 0. 8 we proved that all the conditions of Lemma 2.4 are satisfied. Note that conditions iv of Lemma 2.4 hold in the case = HN. We deduce that there exists a unique X HN solution to 5. Now, to establish our result, we need to prove that X PN. The Schauder fixed point theorem will be useful in this step. Define the mapping G : [F0, M, FM, 0] HN by GX = FX, X, X [F0, M, FM, 0]. Note that from 8 the mixed monotone property of F, we have Let X [F0, M, FM, 0], i.e., Using the mixed monotone property of F, we get On the other h, from 8, we have F0, M FM, 0. F0, M X FM, 0. FF0, M, FM, 0 FX, X = GX FFM, 0, F0, M. 0 < F0, M M > FM, 0. Again, using the mixed monotone property of F, we get FFM, 0, F0, M FM, 0 FF0, M, FM, 0 F0, M. F0, M GX FM, 0. Thus we proved that G[F0, M, FM, 0] [F0, M, FM, 0], i.e., G maps the compact convex set [F0, M, FM, 0] into itself. Since G is continuous, it follows from Schauder s fixed point theorem that G has at least one fixed point in this set. However, fixed points of G are solutions to 1, we proved already that 1 has a unique Hermitian solution. this solution must be in the set [F0, M, FM, 0], i.e., Theorem 3.1 is proved. X [Q U MU + B MB, Q + V MV + W MW] PN.
5 M. Berzig, B. Samet / Filomat 29:8 2015, Remark 3.2. Note that in the proof of the above theorem, we cannot apply the Banach contraction principle to the mapping G. Indeed, the inequalities 6 7 are satisfied only for X J Y K. Recall that the proof of these inequalities is based on Lemma 2.1 which requires that B 0. Theorem 3.3. Suppose that all the assumptions of Theorem 3.1 hold. Let X n Y n be the sequences defined by X 0 = 0 zero matrix, Y 0 = M, Xn+1 = Q + V X n V + W X n W U Y n U + B Y n B, Y n+1 = Q + V Y n V + W Y n W U X n U + B 9 X n B. Proof. It follows immediately from v of Lemma 2.4. lim X n X = lim Y n X = 0. n n Now, we present some consequences following from Theorems , when A, B are Hermitian matrices. Our first consequence is the following. Corollary 3.4. Suppose that a A, B HN; b 2UQU + BQB < Q; c 2VQV + WQW < Q. i 1 has one only one solution X PN; ii X [Q 2UQU + BQB, Q + 2VQV + WQW]. iii Moreover, let X n Y n the sequences defined by X 0 = 0, Y 0 = 2Q, Xn+1 = Q + VX n V + WX n W UY n U + BY n B, Y n+1 = Q + VY n V + WY n W UX n U + BX n B. We have lim X n X = lim Y n X = 0. n n Proof. It follows from Theorems with Q = Q M = 2Q. Corollary 3.5. Suppose that a A, B HN; b 2U 2 + B 2 < I, 2V 2 + W 2 < I; c U 2 + B 2 < Q < I V 2 + W 2. i 1 has one only one solution X PN; ii X [Q U 2 + B 2, Q + V 2 + W 2 ]. iii Moreover, let X n Y n the sequences defined by X 0 = 0, Y 0 = I, Xn+1 = Q + VX n V + WX n W UY n U + BY n B, Y n+1 = Q + VY n V + WY n W UX n U + BX n B. We have lim X n X = lim Y n X = 0. n n Proof. It follows from Theorems with Q = M = I.
6 M. Berzig, B. Samet / Filomat 29:8 2015, Numerical Experiments In this section, we present some numerical experiments to check the convergence of the proposed algorithm 9. We take X 0 = 0 Y 0 = M PN. For each iteration i, we consider the residual errors E i X = X i Q + V X i V + W X i W U X i U + B X i B, All programs are written in MATLAB version 7.1. First example. We consider 1 with Q = , A = B = E i Y = Y i Q + V Y i V + W Y i W U Y i U + B Y i B. E i = maxe i X, E i Y In this case, we have A, B H3, Sp Q 2UQU + BQB = , , Sp Q 2VQV + WQW = , , , which imply that conditions a-c of Corollary 3.4 are satisfied. Let us consider the iterative method 9 with X 0 = 0 Y 0 = 2Q. After 100 iterations, one gets the following approximation to the positive definite solution X: X X 100 = Y 100 = We obtained E 100 = Second example. We consider 1 with Q = , A = B =
7 In this case, we have A, B H5, Sp I 2U 2 + B 2 = Sp I 2V 2 + W 2 = Sp Q U 2 + B 2 = Sp I V 2 + W 2 Q = M. Berzig, B. Samet / Filomat 29:8 2015, , , , , , , , , , , , , , , , , , , , , which imply that conditions a-c of Corollary 3.5 are satisfied. Considering the iterative method 9 with X 0 = 0 Y 0 = I, after 82 iterations one gets an approximation to the positive definite solution X given by X X 82 = Y 82 = We obtained E 82 = References [1] R.H. Bartels, G.W. Stewart, Solution of the equation AX + XB = C, Comm. ACM [2] T.G. Bhaskar, V. Lakshmikantham, Fixed point theory in partially ordered metric spaces applications, Nonlinear Anal [3] P. Benner, J.R. Li, T. Penzl, Numerical solution of large Lyapunov equations, Riccati equations, linear-quadratic control problems, Numer. Linear Algebra Appl [4] P. Benner, E.S. Quintana-Ortí, Solving stable generalized Lyapunov equations with the matrix sign function, Numer. Algorithms [5] K.E. Chu, The solution of the matrix equations AXB CXD = E YA DZ, YC BZ = E, F, Linear Algebra Appl [6] F.R. Gantmacher, Theory of Matrices, Chelsea, New York, [7] J.D. Gardiner, A.J. Laub, J.J. Amato, C.B. Moler, Solution of the Sylvester matrix equation AXB T + CXD T = E, ACM Trans. Math. Software [8] J.D. Gardiner, M.R. Wette, A.J. Laub, J.J. Amato, C.B. Moler, Algorithm 705: A Fortran-77 software package for solving the Sylvester matrix equation AXB T + CXD T = E, ACM Trans. Math. Software [9] G.H. Golub, S. Nash, C. Van Loan, A Hessenberg-Schur method for the problem AX + XB = C, IEEE Trans. Automat. Control [10] S.J. Hammarling, Numerical solution of the stable non-negative definite Lyapunov equation, IMA J. Numer. Anal [11] J.R. Li, J. White, Low rank solution of Lyapunov equations, SIAM J. Matrix Anal. Appl [12] J.H. Long, X.Y. Hu, L. Zhang, On the Hermitian positive definite solution of the nonlinear matrix equation X+A X 1 A+B X 1 B = I, Bull Braz Math Soc. New Series [13] C.S. Kenney, A.J. Laub, The matrix sign function, IEEE Trans. Automat. Control [14] P. Lancaster, L. Rodman, The Algebraic Riccati Equation, Oxford University Press, Oxford, [15] V.B. Larin, F.A. Aliev, Generalized Lyapunov equation factorization of matrix polynomials, Systems Control Lett [16] A.J. Laub, M.T. Heath, C.C. Paige, R.C. Ward, Computation of system balancing transformations other applications of simultaneous diagonalization algorithms, IEEE Trans. Automat. Control [17] V. Mehrmann, The Autonomous Linear Quadratic Control Problem, Theory Numerical Solution, Lecture Notes in Control Information Sciences, 163. Springer-Verlag, Heidelberg, [18] P.C. Müller, Stability of linear mechanical systems with holonomic constraints, Appl. Mech. Rev [19] T. Penzl, Numerical solution of generalized Lyapunov equations, Adv. Comput. Math [20] T. Penzl, A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM J. Sci. Comp [21] A.C.M. Ran, M.C.B. Reurings, A fixed point theorem in partially ordered sets some applications to matrix equations, Proc. Amer. Math. Soc
Improved Newton s method with exact line searches to solve quadratic matrix equation
Journal of Computational and Applied Mathematics 222 (2008) 645 654 wwwelseviercom/locate/cam Improved Newton s method with exact line searches to solve quadratic matrix equation Jian-hui Long, Xi-yan
More informationApproximate Low Rank Solution of Generalized Lyapunov Matrix Equations via Proper Orthogonal Decomposition
Applied Mathematical Sciences, Vol. 4, 2010, no. 1, 21-30 Approximate Low Rank Solution of Generalized Lyapunov Matrix Equations via Proper Orthogonal Decomposition Amer Kaabi Department of Basic Science
More informationSingular Value Inequalities for Real and Imaginary Parts of Matrices
Filomat 3:1 16, 63 69 DOI 1.98/FIL16163C Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Singular Value Inequalities for Real Imaginary
More informationA Continuation Approach to a Quadratic Matrix Equation
A Continuation Approach to a Quadratic Matrix Equation Nils Wagner nwagner@mecha.uni-stuttgart.de Institut A für Mechanik, Universität Stuttgart GAMM Workshop Applied and Numerical Linear Algebra September
More informationStability and Inertia Theorems for Generalized Lyapunov Equations
Published in Linear Algebra and its Applications, 355(1-3, 2002, pp. 297-314. Stability and Inertia Theorems for Generalized Lyapunov Equations Tatjana Stykel Abstract We study generalized Lyapunov equations
More informationIterative Solution of a Matrix Riccati Equation Arising in Stochastic Control
Iterative Solution of a Matrix Riccati Equation Arising in Stochastic Control Chun-Hua Guo Dedicated to Peter Lancaster on the occasion of his 70th birthday We consider iterative methods for finding the
More informationA fixed point problem under two constraint inequalities
Jleli and Samet Fixed Point Theory and Applications 2016) 2016:18 DOI 10.1186/s13663-016-0504-9 RESEARCH Open Access A fixed point problem under two constraint inequalities Mohamed Jleli and Bessem Samet*
More informationISOLATED SEMIDEFINITE SOLUTIONS OF THE CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATION
ISOLATED SEMIDEFINITE SOLUTIONS OF THE CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATION Harald K. Wimmer 1 The set of all negative-semidefinite solutions of the CARE A X + XA + XBB X C C = 0 is homeomorphic
More informationThe iterative methods for solving nonlinear matrix equation X + A X 1 A + B X 1 B = Q
Vaezzadeh et al. Advances in Difference Equations 2013, 2013:229 R E S E A R C H Open Access The iterative methods for solving nonlinear matrix equation X + A X A + B X B = Q Sarah Vaezzadeh 1, Seyyed
More informationThe Solvability Conditions for the Inverse Eigenvalue Problem of Hermitian and Generalized Skew-Hamiltonian Matrices and Its Approximation
The Solvability Conditions for the Inverse Eigenvalue Problem of Hermitian and Generalized Skew-Hamiltonian Matrices and Its Approximation Zheng-jian Bai Abstract In this paper, we first consider the inverse
More informationOn Multivalued G-Monotone Ćirić and Reich Contraction Mappings
Filomat 31:11 (2017), 3285 3290 https://doi.org/10.2298/fil1711285a Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On Multivalued
More informationTrace Inequalities for a Block Hadamard Product
Filomat 32:1 2018), 285 292 https://doiorg/102298/fil1801285p Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat Trace Inequalities for
More informationOn the effect of α-admissibility and θ-contractivity to the existence of fixed points of multivalued mappings
Nonlinear Analysis: Modelling and Control, Vol. 21, No. 5, 673 686 ISSN 1392-5113 http://dx.doi.org/10.15388/na.2016.5.7 On the effect of α-admissibility and θ-contractivity to the existence of fixed points
More informationOn the Existence of Bounded Solutions to a Class of Nonlinear Initial Value Problems with Delay
Filomat : (27, 25 5 https://doi.org/.2298/fil725a Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat On the Existence of Bounded Solutions
More informationSome notes on a second-order random boundary value problem
ISSN 1392-5113 Nonlinear Analysis: Modelling and Control, 217, Vol. 22, No. 6, 88 82 https://doi.org/1.15388/na.217.6.6 Some notes on a second-order random boundary value problem Fairouz Tchier a, Calogero
More informationFactorized Solution of Sylvester Equations with Applications in Control
Factorized Solution of Sylvester Equations with Applications in Control Peter Benner Abstract Sylvester equations play a central role in many areas of applied mathematics and in particular in systems and
More informationCommon fixed point of multivalued mappings in ordered generalized metric spaces
Filomat 6:5 (01), 1045 1053 DOI 10.98/FIL105045A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Common fixed point of multivalued
More information2 Computing complex square roots of a real matrix
On computing complex square roots of real matrices Zhongyun Liu a,, Yulin Zhang b, Jorge Santos c and Rui Ralha b a School of Math., Changsha University of Science & Technology, Hunan, 410076, China b
More informationGAUSS-SIDEL AND SUCCESSIVE OVER RELAXATION ITERATIVE METHODS FOR SOLVING SYSTEM OF FUZZY SYLVESTER EQUATIONS
GAUSS-SIDEL AND SUCCESSIVE OVER RELAXATION ITERATIVE METHODS FOR SOLVING SYSTEM OF FUZZY SYLVESTER EQUATIONS AZIM RIVAZ 1 AND FATEMEH SALARY POUR SHARIF ABAD 2 1,2 DEPARTMENT OF MATHEMATICS, SHAHID BAHONAR
More informationAn error estimate for matrix equations
Applied Numerical Mathematics 50 (2004) 395 407 www.elsevier.com/locate/apnum An error estimate for matrix equations Yang Cao, Linda Petzold 1 Department of Computer Science, University of California,
More informationMultiplicative Perturbation Bounds of the Group Inverse and Oblique Projection
Filomat 30: 06, 37 375 DOI 0.98/FIL67M Published by Faculty of Sciences Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Multiplicative Perturbation Bounds of the Group
More informationAn Accelerated Jacobi-gradient Based Iterative Algorithm for Solving Sylvester Matrix Equations
Filomat 31:8 (2017), 2381 2390 DOI 10.2298/FIL1708381T Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat An Accelerated Jacobi-gradient
More informationFixed Point Theorems for -Monotone Maps on Partially Ordered S-Metric Spaces
Filomat 28:9 (2014), 1885 1898 DOI 10.2298/FIL1409885D Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Fixed Point Theorems for
More informationThe symmetric linear matrix equation
Electronic Journal of Linear Algebra Volume 9 Volume 9 (00) Article 8 00 The symmetric linear matrix equation Andre CM Ran Martine CB Reurings mcreurin@csvunl Follow this and additional works at: http://repositoryuwyoedu/ela
More informationOn Solving Large Algebraic. Riccati Matrix Equations
International Mathematical Forum, 5, 2010, no. 33, 1637-1644 On Solving Large Algebraic Riccati Matrix Equations Amer Kaabi Department of Basic Science Khoramshahr Marine Science and Technology University
More informationDownloaded 05/27/14 to Redistribution subject to SIAM license or copyright; see
SIAM J. MATRIX ANAL. APPL. Vol. 31, No. 2, pp. 272 288 c 2009 Society for Industrial and Applied Mathematics INEXACT KLEINMAN NEWTON METHOD FOR RICCATI EQUATIONS F. FEITZINGER, T. HYLLA, AND E. W. SACHS
More informationAbstract. In this article, several matrix norm inequalities are proved by making use of the Hiroshima 2003 result on majorization relations.
HIROSHIMA S THEOREM AND MATRIX NORM INEQUALITIES MINGHUA LIN AND HENRY WOLKOWICZ Abstract. In this article, several matrix norm inequalities are proved by making use of the Hiroshima 2003 result on majorization
More informationOn intermediate value theorem in ordered Banach spaces for noncompact and discontinuous mappings
Int. J. Nonlinear Anal. Appl. 7 (2016) No. 1, 295-300 ISSN: 2008-6822 (electronic) http://dx.doi.org/10.22075/ijnaa.2015.341 On intermediate value theorem in ordered Banach spaces for noncompact and discontinuous
More informationAn Exact Line Search Method for Solving Generalized Continuous-Time Algebraic Riccati Equations
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO., JANUARY 998 0 An Exact Line Search Method for Solving Generalized Continuous-Time Algebraic Riccati Equations Peter Benner and Ralph Byers Abstract
More informationConvergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations
Convergence Analysis of Structure-Preserving Doubling Algorithms for Riccati-Type Matrix Equations Wen-Wei Lin Shu-Fang u Abstract In this paper a structure-preserving transformation of a symplectic pencil
More informationTwo Results About The Matrix Exponential
Two Results About The Matrix Exponential Hongguo Xu Abstract Two results about the matrix exponential are given. One is to characterize the matrices A which satisfy e A e AH = e AH e A, another is about
More informationSolving projected generalized Lyapunov equations using SLICOT
Solving projected generalized Lyapunov equations using SLICOT Tatjana Styel Abstract We discuss the numerical solution of projected generalized Lyapunov equations. Such equations arise in many control
More informationFixed Points of Multivalued Non-Linear F -Contractions with Application to Solution of Matrix Equations
Filomat 31:11 (017), 3319 3333 https://doi.org/10.98/fil1711319i Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Fixed Points of
More informationH 2 optimal model reduction - Wilson s conditions for the cross-gramian
H 2 optimal model reduction - Wilson s conditions for the cross-gramian Ha Binh Minh a, Carles Batlle b a School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Dai
More informationCommon fixed point of -approximative multivalued mapping in partially ordered metric space
Filomat 7:7 (013), 1173 118 DOI 1098/FIL1307173A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://wwwpmfniacrs/filomat Common fixed point of -approximative
More informationPERTURBATION ANAYSIS FOR THE MATRIX EQUATION X = I A X 1 A + B X 1 B. Hosoo Lee
Korean J. Math. 22 (214), No. 1, pp. 123 131 http://dx.doi.org/1.11568/jm.214.22.1.123 PERTRBATION ANAYSIS FOR THE MATRIX EQATION X = I A X 1 A + B X 1 B Hosoo Lee Abstract. The purpose of this paper is
More informationBOUNDS OF MODULUS OF EIGENVALUES BASED ON STEIN EQUATION
K Y BERNETIKA VOLUM E 46 ( 2010), NUMBER 4, P AGES 655 664 BOUNDS OF MODULUS OF EIGENVALUES BASED ON STEIN EQUATION Guang-Da Hu and Qiao Zhu This paper is concerned with bounds of eigenvalues of a complex
More informationA Fixed Point Theorem for JS-contraction Type Mappings with Applications to Polynomial Approximations
Filomat 31:15 (2017, 4969 4978 https://doi.org/10.2298/fil1715969a Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Fixed Point
More informationNew Coupled Common Fixed Point for Four Mappings satisfying Rational Contractive Expression
New Coupled Common Fixed Point for Four Mappings satisfying Rational Contractive Expression H K Nashine 1, A Gupta 2 1 Department of Mathematics, Amity University, Manth Kharora, Raipur-(Chhattisgarh),
More informationFixed points of Ćirić quasi-contractive operators in normed spaces
Mathematical Communications 11(006), 115-10 115 Fixed points of Ćirić quasi-contractive operators in normed spaces Arif Rafiq Abstract. We establish a general theorem to approximate fixed points of Ćirić
More informationELA
POSITIVE DEFINITE SOLUTION OF THE MATRIX EQUATION X = Q+A H (I X C) δ A GUOZHU YAO, ANPING LIAO, AND XUEFENG DUAN Abstract. We consider the nonlinear matrix equation X = Q+A H (I X C) δ A (0 < δ 1), where
More informationSome inequalities for sum and product of positive semide nite matrices
Linear Algebra and its Applications 293 (1999) 39±49 www.elsevier.com/locate/laa Some inequalities for sum and product of positive semide nite matrices Bo-Ying Wang a,1,2, Bo-Yan Xi a, Fuzhen Zhang b,
More informationYONGDO LIM. 1. Introduction
THE INVERSE MEAN PROBLEM OF GEOMETRIC AND CONTRAHARMONIC MEANS YONGDO LIM Abstract. In this paper we solve the inverse mean problem of contraharmonic and geometric means of positive definite matrices (proposed
More informationKrylov Subspace Type Methods for Solving Projected Generalized Continuous-Time Lyapunov Equations
Krylov Subspace Type Methods for Solving Proected Generalized Continuous-Time Lyapunov Equations YUIAN ZHOU YIQIN LIN Hunan University of Science and Engineering Institute of Computational Mathematics
More informationSOME FIXED POINT THEOREMS FOR ORDERED REICH TYPE CONTRACTIONS IN CONE RECTANGULAR METRIC SPACES
Acta Math. Univ. Comenianae Vol. LXXXII, 2 (2013), pp. 165 175 165 SOME FIXED POINT THEOREMS FOR ORDERED REICH TYPE CONTRACTIONS IN CONE RECTANGULAR METRIC SPACES S. K. MALHOTRA, S. SHUKLA and R. SEN Abstract.
More informationThe Nearest Doubly Stochastic Matrix to a Real Matrix with the same First Moment
he Nearest Doubly Stochastic Matrix to a Real Matrix with the same First Moment William Glunt 1, homas L. Hayden 2 and Robert Reams 2 1 Department of Mathematics and Computer Science, Austin Peay State
More informationAvailable online at J. Nonlinear Sci. Appl., 10 (2017), Research Article
Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 10 (2017), 2719 2726 Research Article Journal Homepage: www.tjnsa.com - www.isr-publications.com/jnsa An affirmative answer to
More informationOn a quadratic matrix equation associated with an M-matrix
Article Submitted to IMA Journal of Numerical Analysis On a quadratic matrix equation associated with an M-matrix Chun-Hua Guo Department of Mathematics and Statistics, University of Regina, Regina, SK
More informationOn an iterative algorithm for variational inequalities in. Banach space
MATHEMATICAL COMMUNICATIONS 95 Math. Commun. 16(2011), 95 104. On an iterative algorithm for variational inequalities in Banach spaces Yonghong Yao 1, Muhammad Aslam Noor 2,, Khalida Inayat Noor 3 and
More informationSome Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph
J o u r n a l of Mathematics and Applications JMA No 39, pp 81-90 (2016) Some Fixed Point Theorems for G-Nonexpansive Mappings on Ultrametric Spaces and Non-Archimedean Normed Spaces with a Graph Hamid
More informationAn iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation AXB =C
Journal of Computational and Applied Mathematics 1 008) 31 44 www.elsevier.com/locate/cam An iterative method for the skew-symmetric solution and the optimal approximate solution of the matrix equation
More informationFixed point theorems for Ćirić type generalized contractions defined on cyclic representations
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 8 (2015), 1257 1264 Research Article Fixed point theorems for Ćirić type generalized contractions defined on cyclic representations Adrian Magdaş
More informationMultiplicative metric spaces and contractions of rational type
Advances in the Theory of Nonlinear Analysis and its Applications 2 (2018) No. 4, 195 201. https://doi.org/10.31197/atnaa.481995 Available online at www.atnaa.org Research Article Multiplicative metric
More informationA Fixed Point Theorem for G-Monotone Multivalued Mapping with Application to Nonlinear Integral Equations
Filomat 31:7 (217), 245 252 DOI 1.2298/FIL17745K Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat A Fixed Point Theorem for G-Monotone
More informationMiskolc Mathematical Notes HU e-issn Some common xed point theorems for a class of fuzzy contractive mappings. M. A.
Miskolc Mathematical Notes HU e-issn 1787-2413 Vol. 8 (2007), No 2, pp. 109-115 DOI: 10.18514/MMN.2007.110 Some common xed point theorems for a class of fuzzy contractive mappings M. A. Ahmed Miskolc Mathematical
More informationA FIXED POINT THEOREM FOR UNIFORMLY LOCALLY CONTRACTIVE MAPPINGS IN A C-CHAINABLE CONE RECTANGULAR METRIC SPACE
Surveys in Mathematics and its Applications A FIXED POINT THEOREM FOR UNIFORMLY LOCALLY CONTRACTIVE MAPPINGS IN A C-CHAINABLE CONE RECTANGULAR METRIC SPACE Bessem Samet and Calogero Vetro Abstract. Recently,
More informationAn SVD-Like Matrix Decomposition and Its Applications
An SVD-Like Matrix Decomposition and Its Applications Hongguo Xu Abstract [ A matrix S C m m is symplectic if SJS = J, where J = I m I m. Symplectic matrices play an important role in the analysis and
More informationInvariant Approximation Results of Generalized Contractive Mappings
Filomat 30:14 (016), 3875 3883 DOI 10.98/FIL1614875C Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Invariant Approximation Results
More informationBest proximity points of Kannan type cyclic weak ϕ-contractions in ordered metric spaces
An. Şt. Univ. Ovidius Constanţa Vol. 20(3), 2012, 51 64 Best proximity points of Kannan type cyclic weak ϕ-contractions in ordered metric spaces Erdal Karapınar Abstract In this manuscript, the existence
More informationAPPLICATIONS OF THE HYPER-POWER METHOD FOR COMPUTING MATRIX PRODUCTS
Univ. Beograd. Publ. Eletrotehn. Fa. Ser. Mat. 15 (2004), 13 25. Available electronically at http: //matematia.etf.bg.ac.yu APPLICATIONS OF THE HYPER-POWER METHOD FOR COMPUTING MATRIX PRODUCTS Predrag
More informationarxiv: v1 [math.fa] 1 Oct 2015
SOME RESULTS ON SINGULAR VALUE INEQUALITIES OF COMPACT OPERATORS IN HILBERT SPACE arxiv:1510.00114v1 math.fa 1 Oct 2015 A. TAGHAVI, V. DARVISH, H. M. NAZARI, S. S. DRAGOMIR Abstract. We prove several singular
More informationAnalytical formulas for calculating extremal ranks and inertias of quadratic matrix-valued functions and their applications
Analytical formulas for calculating extremal ranks and inertias of quadratic matrix-valued functions and their applications Yongge Tian CEMA, Central University of Finance and Economics, Beijing 100081,
More informationMoore Penrose inverses and commuting elements of C -algebras
Moore Penrose inverses and commuting elements of C -algebras Julio Benítez Abstract Let a be an element of a C -algebra A satisfying aa = a a, where a is the Moore Penrose inverse of a and let b A. We
More informationThe Kalman-Yakubovich-Popov Lemma for Differential-Algebraic Equations with Applications
MAX PLANCK INSTITUTE Elgersburg Workshop Elgersburg February 11-14, 2013 The Kalman-Yakubovich-Popov Lemma for Differential-Algebraic Equations with Applications Timo Reis 1 Matthias Voigt 2 1 Department
More informationON THE HÖLDER CONTINUITY OF MATRIX FUNCTIONS FOR NORMAL MATRICES
Volume 10 (2009), Issue 4, Article 91, 5 pp. ON THE HÖLDER CONTINUITY O MATRIX UNCTIONS OR NORMAL MATRICES THOMAS P. WIHLER MATHEMATICS INSTITUTE UNIVERSITY O BERN SIDLERSTRASSE 5, CH-3012 BERN SWITZERLAND.
More informationNorm inequalities related to the matrix geometric mean
isid/ms/2012/07 April 20, 2012 http://www.isid.ac.in/ statmath/eprints Norm inequalities related to the matrix geometric mean RAJENDRA BHATIA PRIYANKA GROVER Indian Statistical Institute, Delhi Centre
More information1 Introduction and preliminaries
Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Filomat 5: (0), 5 3 DOI: 0.98/FIL05D MULTIVALUED GENERALIZATIONS OF THE KANNAN FIXED POINT THEOREM
More informationarxiv: v1 [math.na] 21 Oct 2014
Computing Symmetric Positive Definite Solutions of Three Types of Nonlinear Matrix Equations arxiv:1410.5559v1 [math.na] 21 Oct 2014 Negin Bagherpour a, Nezam Mahdavi-Amiri a, a Department of Mathematical
More informationFixed points of almost quadratic Geraghty contractions and property(p) in partially ordered metric spaces
Journal of Advanced Research in Applied Mathematics Online ISSN: 1942-9649 Vol. 7, Issue., 2015, pp. 1-9 doi: 10.5373/jaram. 1 2 3 4 5 6 7 Fixed points of almost quadratic Geraghty contractions and property(p)
More informationCOUPLED FIXED POINT THEOREMS FOR MONOTONE MAPPINGS IN PARTIALLY ORDERED METRIC SPACES
Kragujevac Journal of Mathematics Volume 382 2014, Pages 249 257. COUPLED FIXED POINT THEOREMS FOR MONOTONE MAPPINGS IN PARTIALLY ORDERED METRIC SPACES STOJAN RADENOVIĆ Abstract. In this paper, by reducing
More informationOn matrix equations X ± A X 2 A = I
Linear Algebra and its Applications 326 21 27 44 www.elsevier.com/locate/laa On matrix equations X ± A X 2 A = I I.G. Ivanov,V.I.Hasanov,B.V.Minchev Faculty of Mathematics and Informatics, Shoumen University,
More informationFixed point theorem for generalized weak contractions satisfying rational expressions in ordered metric spaces
RESEARCH Open Access Fixed point theorem for generalized weak contractions satisfying rational expressions in ordered metric spaces Nguyen Van Luong * and Nguyen Xuan Thuan * Correspondence: luonghdu@gmail.com
More informationYuqing Chen, Yeol Je Cho, and Li Yang
Bull. Korean Math. Soc. 39 (2002), No. 4, pp. 535 541 NOTE ON THE RESULTS WITH LOWER SEMI-CONTINUITY Yuqing Chen, Yeol Je Cho, and Li Yang Abstract. In this paper, we introduce the concept of lower semicontinuous
More informationS N. hochdimensionaler Lyapunov- und Sylvestergleichungen. Peter Benner. Mathematik in Industrie und Technik Fakultät für Mathematik TU Chemnitz
Ansätze zur numerischen Lösung hochdimensionaler Lyapunov- und Sylvestergleichungen Peter Benner Mathematik in Industrie und Technik Fakultät für Mathematik TU Chemnitz S N SIMULATION www.tu-chemnitz.de/~benner
More informationChebfun2: Exploring Constant Coefficient PDEs on Rectangles
Chebfun2: Exploring Constant Coefficient PDEs on Rectangles Alex Townsend Oxford University with Sheehan Olver, Sydney University FUN13 workshop, 11th of April 2013 Alex Townsend Constant coefficient PDEs
More informationPerov s fixed point theorem for multivalued mappings in generalized Kasahara spaces
Stud. Univ. Babeş-Bolyai Math. 56(2011), No. 3, 19 28 Perov s fixed point theorem for multivalued mappings in generalized Kasahara spaces Alexandru-Darius Filip Abstract. In this paper we give some corresponding
More informationw T 1 w T 2. w T n 0 if i j 1 if i = j
Lyapunov Operator Let A F n n be given, and define a linear operator L A : C n n C n n as L A (X) := A X + XA Suppose A is diagonalizable (what follows can be generalized even if this is not possible -
More informationTheorem Let X be a symmetric solution of DR(X) = 0. Let X b be a symmetric approximation to Xand set V := X? X. b If R b = R + B T XB b and I + V B R
Initializing Newton's method for discrete-time algebraic Riccati equations using the buttery SZ algorithm Heike Fabender Zentrum fur Technomathematik Fachbereich - Mathematik und Informatik Universitat
More informationOn coupled generalised Banach and Kannan type contractions
Available online at www.tjnsa.com J. Nonlinear Sci. Appl. 5 (01, 59 70 Research Article On coupled generalised Banach Kannan type contractions B.S.Choudhury a,, Amaresh Kundu b a Department of Mathematics,
More informationEE/ACM Applications of Convex Optimization in Signal Processing and Communications Lecture 4
EE/ACM 150 - Applications of Convex Optimization in Signal Processing and Communications Lecture 4 Andre Tkacenko Signal Processing Research Group Jet Propulsion Laboratory April 12, 2012 Andre Tkacenko
More informationAn Iterative Procedure for Solving the Riccati Equation A 2 R RA 1 = A 3 + RA 4 R. M.THAMBAN NAIR (I.I.T. Madras)
An Iterative Procedure for Solving the Riccati Equation A 2 R RA 1 = A 3 + RA 4 R M.THAMBAN NAIR (I.I.T. Madras) Abstract Let X 1 and X 2 be complex Banach spaces, and let A 1 BL(X 1 ), A 2 BL(X 2 ), A
More informationWEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES
Fixed Point Theory, 12(2011), No. 2, 309-320 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html WEAK CONVERGENCE THEOREMS FOR EQUILIBRIUM PROBLEMS WITH NONLINEAR OPERATORS IN HILBERT SPACES S. DHOMPONGSA,
More informationContour integral solutions of Sylvester-type matrix equations
Contour integral solutions of Sylvester-type matrix equations Harald K. Wimmer Mathematisches Institut, Universität Würzburg, 97074 Würzburg, Germany Abstract The linear matrix equations AXB CXD = E, AX
More informationPROXIMAL, DISTAL AND ASYMPTOTIC POINTS IN COMPACT CONE METRIC SPACES. P. Raja (Received 30 June, 2015)
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 46 (2016), 135-140 PROXIMAL, DISTAL AND ASYMPTOTIC POINTS IN COMPACT CONE METRIC SPACES P. Raja (Received 30 June, 2015) Abstract. In this paper, proximal points,
More informationIntegral type Fixed point Theorems for α-admissible Mappings Satisfying α-ψ-φ-contractive Inequality
Filomat 3:12 (216, 3227 3234 DOI 1.2298/FIL1612227B Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Integral type Fixed point Theorems
More informationTHE NEARLY ADDITIVE MAPS
Bull. Korean Math. Soc. 46 (009), No., pp. 199 07 DOI 10.4134/BKMS.009.46..199 THE NEARLY ADDITIVE MAPS Esmaeeil Ansari-Piri and Nasrin Eghbali Abstract. This note is a verification on the relations between
More informationOn the Convergence of Ishikawa Iterates to a Common Fixed Point for a Pair of Nonexpansive Mappings in Banach Spaces
Mathematica Moravica Vol. 14-1 (2010), 113 119 On the Convergence of Ishikawa Iterates to a Common Fixed Point for a Pair of Nonexpansive Mappings in Banach Spaces Amit Singh and R.C. Dimri Abstract. In
More informationNONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS
Fixed Point Theory, Volume 9, No. 1, 28, 3-16 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html NONTRIVIAL SOLUTIONS TO INTEGRAL AND DIFFERENTIAL EQUATIONS GIOVANNI ANELLO Department of Mathematics University
More informationON THE MATRIX EQUATION XA AX = X P
ON THE MATRIX EQUATION XA AX = X P DIETRICH BURDE Abstract We study the matrix equation XA AX = X p in M n (K) for 1 < p < n It is shown that every matrix solution X is nilpotent and that the generalized
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J. Math. Anal. 6 (2012), no. 1, 139 146 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) www.emis.de/journals/bjma/ AN EXTENSION OF KY FAN S DOMINANCE THEOREM RAHIM ALIZADEH
More informationSome unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces
An. Şt. Univ. Ovidius Constanţa Vol. 19(1), 211, 331 346 Some unified algorithms for finding minimum norm fixed point of nonexpansive semigroups in Hilbert spaces Yonghong Yao, Yeong-Cheng Liou Abstract
More informationETNA Kent State University
Electronic Transactions on Numerical Analysis Volume 2 pp 138-153 September 1994 Copyright 1994 ISSN 1068-9613 ETNA etna@mcskentedu ON THE PERIODIC QUOTIENT SINGULAR VALUE DECOMPOSITION JJ HENCH Abstract
More informationNonlinear equations. Norms for R n. Convergence orders for iterative methods
Nonlinear equations Norms for R n Assume that X is a vector space. A norm is a mapping X R with x such that for all x, y X, α R x = = x = αx = α x x + y x + y We define the following norms on the vector
More informationMONOTONE GENERALIZED WEAK CONTRACTIONS IN PARTIALLY ORDERED METRIC SPACES
Fixed Point Theory, 11(2010), No. 2, 375-382 http://www.math.ubbcluj.ro/ nodeacj/sfptcj.html MONOTONE GENERALIZED WEAK CONTRACTIONS IN PARTIALLY ORDERED METRIC SPACES R. SAADATI AND S.M. VAEZPOUR Department
More informationMAXIMALITY OF SUMS OF TWO MAXIMAL MONOTONE OPERATORS
MAXIMALITY OF SUMS OF TWO MAXIMAL MONOTONE OPERATORS JONATHAN M. BORWEIN, FRSC Abstract. We use methods from convex analysis convex, relying on an ingenious function of Simon Fitzpatrick, to prove maximality
More informationEfficient algorithms for finding the minimal polynomials and the
Efficient algorithms for finding the minimal polynomials and the inverses of level- FLS r 1 r -circulant matrices Linyi University Department of mathematics Linyi Shandong 76005 China jzh108@sina.com Abstract:
More informationStanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon Measures
2 1 Borel Regular Measures We now state and prove an important regularity property of Borel regular outer measures: Stanford Mathematics Department Math 205A Lecture Supplement #4 Borel Regular & Radon
More information2 Two-Point Boundary Value Problems
2 Two-Point Boundary Value Problems Another fundamental equation, in addition to the heat eq. and the wave eq., is Poisson s equation: n j=1 2 u x 2 j The unknown is the function u = u(x 1, x 2,..., x
More informationTopology and its Applications
Topology and its Applications 156 (2009) 997 1001 Contents lists available at ScienceDirect Topology and its Applications www.elsevier.com/locate/topol Fixed point results for multimaps in CAT(0) spaces
More informationApplication of Measure of Noncompactness for the System of Functional Integral Equations
Filomat 3:11 (216), 363 373 DOI 1.2298/FIL161163A Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Application of Measure of Noncompactness
More information