On the fractional-order logistic equation

The topic of fractional calculus (derivatives and integrals of arbitrary orders) is enjoying growing interest not only among mathematicians, but also among physicists and engineers (see [E.M. El-Mesiry, A.M.A. El-Sayed, H.A.A. El-Saka, Numerical methods for multi-term fractional (arbitrary) orders differential equations, Appl. Math. Comput. 160 (3) (2005) 683–699; A.M.A. El-Sayed, Fractional differential–difference equations, J. Fract. Calc. 10 (1996) 101–106; A.M.A. El-Sayed, Nonlinear functional differential equations of arbitrary orders, Nonlinear Anal. 33 (2) (1998) 181–186; A.M.A. El-Sayed, F.M. Gaafar, Fractional order differential equations with memory and fractional-order relaxation–oscillation model, (PU.M.A) Pure Math. Appl. 12 (2001); A.M.A. El-Sayed, E.M. El-Mesiry, H.A.A. El-Saka, Numerical solution for multi-term fractional (arbitrary) orders differential equations, Comput. Appl. Math. 23 (1) (2004) 33–54; A.M.A. El-Sayed, F.M. Gaafar, H.H. Hashem, On the maximal and minimal solutions of arbitrary orders nonlinear functional integral and differential equations, Math. Sci. Res. J. 8 (11) (2004) 336–348; R. Gorenflo, F. Mainardi, Fractional calculus: Integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer, Wien, 1997, pp. 223–276; D. Matignon, Stability results for fractional differential equations with applications to control processing, in: Computational Engineering in System Application, vol. 2, Lille, France, 1996, p. 963; I. Podlubny, A.M.A. El-Sayed, On Two Definitions of Fractional Calculus, Solvak Academy of science-institute of experimental phys, ISBN: 80-7099-252-2, 1996. UEF-03-96; I. Podlubny, Fractional Differential Equations, Academic Press, 1999] for example). In this work we are concerned with the fractional-order logistic equation. We study here the stability, existence, uniqueness and numerical solution of the fractional-order logistic equation.     

On finite groups with a reducible permutability-graph

We characterize finite groups in which the permutability-graph has more than one connected component.Research partially supported by G.N.S.A.G.A. of C.N.R. and M.U.R.S.T. of Italy.     

Computational and approximate methods of optimal control

Approximate methods of solving problems of optimal control are classified and analyzed, and their domain of applicability is indicated. Among the special problems the problem of the choice of optimal trajectories for aircraft is considered.Translated from Itogi Nauki i Tekhniki, Matematicheskii Analiz, Vol. 14, pp. 101–166, 1977.The authors of the survey are grateful to N. N. Bolotnik, M. Yu. Borodovskii, G. G. Egiyan, V. A. Korneev, V. M. Mamalyga, A. A. Mironov, Yu. R. Roshchin, and A. P. Seiranyan for their assistance in compiling the bibliography and to R. P. Soldatova and I. S. Kheiker for their help in the shaping of the paper.     

Some common fixed point theorems for weakly compatible mappings

Jungck's [G. Jungck, Compatible mappings and common fixed points, Int. J. Math. Math. Sci. 9 (1986) 771-779] notion of compatible mappings is further extended and used to prove some common fixed point theorems for weakly compatible non-self mappings in complete convex metric spaces. We improve on the method of proof used by Rhoades [B.E. Rhoades, A fixed point theorem for non-self set-valued mappings, Int. J. Math. Math. Sci. 20 (1997) 9-12] and Ahmed and Rhoades [A. Ahmed, B.E. Rhoades, Some common fixed point theorems for compatible mappings, Indian J. Pure Appl. Math. 32 (2001) 1247-1254] and obtain generalization of some known results. In particular, a theorem by Rhoades [B.E. Rhoades, A fixed point theorem for non-self set-valued mappings, Int. J. Math. Math. Sci. 20 (1997) 9-12] is generalized and improved.     

Book reviews

L.N. TREFETHEN and D. BAU, III,Numerical Linear Algebra,SIAM, Philadelphia, 1997G.-C. ROTA,Indiscrete Thoughts,Birkhäuser, Boston, 1997D.E. KEYES, A. SAMEH and V. VENKATAKRISHNAN, eds.Parallel Numerical Algorithms,Kluwer, Dordrecht, 1997A. KIRSCH,An Introduction to the Mathematical Theory of Inverse Problems,Springer, New York, 1996L.F. SHAMPINE, R.C. ALLEN, Jr. and S. PRUESS,Fundamentals of Numerical Computing,Wiley, New York, 1997C.W. UEBERHUBERNumerical Computation, 2 vols.Springer, Berlin, 1997W.G. McCALLUM et al.Multivariate Calculus,Wiley, New York, 1997ZHI-QUAN LUO, JONG-SHI PANG and D. RALPH,Mathematical Programs with Equilibrium Constraints,Cambridge University Press, Cambridge, 1996P.R. POPIVANOV and D.K. PALAGACHEV,The Degenerate Oblique Derivative Problem for Elliptic and Parabolic Equations,Akademie Verlag, Berlin, 1997     

A modified halpern-type iteration algorithm for totally quasi-?-asymptotically nonexpansive mappings with applications

The purpose of this article is to modify the Halpern-type iteration algorithm for total quasi-?-asymptotically nonexpansive mapping to have the strong convergence under a limit condition only in the framework of Banach spaces. The results presented in the paper improve and extend the corresponding results of [X.L. Qin, Y.J. Cho, S.M. Kang, H. Y. Zhou, Convergence of a modified Halpern-type iterative algorithm for quasi-?-nonexpansive mappings, Appl. Math. Lett. 22 (2009) 1051-1055], [Z.M. Wang, Y.F. Su, D.X. Wang, Y.C. Dong, A modified Halpern-type iteration algorithm for a family of hemi-relative nonexpansive mappings and systems of equilibrium problems in Banach spaces, J. Comput. Appl. Math. 235 (2011) 2364-2371], [Y.F. Su, H.K. Xu, X. Zhang, Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Anal. 73 (2010) 3890-3906], [C. Martinez-Yanes, H.K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006) 2400-2411] and others.     

某些加权复合算子之不变子空间的存在性

本文研究某些加权复合算子之非平凡不变子空间的存在性。特别地,证明了每个亚正规加权复合算子均有非平凡的不变子空间并且提出了一个新概念,称其为本性可逆变换。对于概率空间上本性可逆变换所确定的加权复合算子,给出其非平凡不变子空间存在性的一个等价刻画。     

On second-order directional derivatives of value functions

Directional derivatives of value functions play an essential role in the sensitivity and stability analysis of parametric optimization problems, in studying bi-level and min–max problems, in quasi-differentiable calculus. Their calculation is studied in numerous works by A.V. Fiacco, V.F. Demyanov and A.M. Rubinov, R.T. Rockafellar, A. Shapiro, J.F. Bonnans, A.D. Ioffe, A. Auslender and R. Cominetti, and many other authors. This article is devoted to the existence of the second order directional derivatives of value functions in parametric problems with non-single-valued solutions. The main idea of the investigation approach is based on the development of the method of the first-order approximations by V.F. Demyanov and A.M. Rubinov.     

Book Reviews

Linear Algebra, 2nd edition by Serge Lang (1971). Addison-Wesley Publishing Company.

Linear Algebra with Applications by Hugh G. Campbell. Appleton-Century-Crofts, New York, 1971. xiii + 396 pp. + A45.An Engineering Approach to Linear Algebra, by W. W. Sawyer. Cambridge University Press, Cambridge, 1972. 304 pp. ($11.50)

An Engineering Approach to Linear Algebra, By W. W. Sawyer. Cambridge University Press, Cambridge, 1972. 304 pp. ($11.50) Sparse Matrices, by R. P. Tewarson. ($11.95)

Sparse Matrices, by R. P. Tewarson. ($11.95)     

An Efficient Algorithm to Generate Official Statistical Reporting Areas: The Case of the 1984 Travel-to-Work Areas Revision in Britain

Travel-to-work areas (T.T.W.A.s) are used by the Department of Employment for the reporting of monthly local unemployment statistics. T.T.W.A.s are also used to demarcate those parts of Britain to benefit from the public expenditure on industry under regional policy. The 1984 revision of T.T.W.A. boundaries provided academics with a rare opportunity to help rationalize official statistical areas. This involved the specification of zone-design criteria and the implementation of these in a regionalization methodology. It was found that methods based on the model of local labour market areas yielded the most reasonable set of boundaries. The final methodology is detailed here, together with the actual parameter values used in the analysis that produced the new T.T.W.A.s. The paper ends by considering the implementation of the results, and evaluates the new areas against the old T.T.W.A.s.     

Nonzero-sum differential games (the noncooperative variant)

One presents the fundamental results of the theory of noncooperative differential games: necessary and sufficient equilibrium conditions, existence theorems, properties of equilibrium solutions, numerical methods. One indicates applications to concrete problems in economics, the mechanics of controlled motions, and to strategic games.Translated from Itogi Nauki i Tekhniki, Matematicheskii Analiz, Vol. 15, pp. 199–266, 1977.The authors are grateful to É. M. Vaisbord, A. F. Koaonenko, V. S. Molostvov, V. P. Patsyukov, V. A. Plotnikov, V. V. Podinovskii, R. A. Polyak, and R. T. Yanushevskii for the problems in Sec. 11 and for the remarks which have been taken into account in the final editing of the survey.     

Resonance theorems and functional series

Self-referred dissertation in competition for the academic doctor's degree in physico-mathematical sciences (printed with elisions). The dissertation was defended on May 7, 1971 at the teaching council of the mechanics-mathematics faculty of the Moscow State University. The official opponents were: Dr. Physico-Mathematical Sciences, Prof. N. P. Kuptsov, Dr. Physico-Mathematical Sciences, Prof. A. M. Olevskii, and Corresponding Member of the Academy of Sciences of the Armenian SSR, Dr. Physico-Mathematical Sciences, Prof. A. A. Talalyan.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 583–595, November, 1971.The author extends hearty thanks to Prof. P. L. Ul'yanov for purposeful scientific conferences on the results of the dissertation.     

Convergence theorems of common fixed points for a finite family of Lipschitz pseudocontractions in Banach spaces

In this paper, we continue to study weak convergence problems for the implicit iteration process for a finite family of Lipschitzian continuous pseudocontractions in general Banach spaces. The results presented in this paper improve and extend the corresponding ones of Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim. 22 (2001) 767–773], Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73–81], Chen et al. [R. Chen, Y.S. Song, H.Y. Zhou, Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings, J. Math. Anal. Appl. 314 (2006) 701–709] and others.     

Classes of sequences of real numbers,games and selection properties

This is a continuation of our investigation of classes of sequences of positive real numbers satisfying some selection principles as well as having certain game-theoretic properties. We improve main results from [D. Djurčić, Lj.D.R. Kočinac, M.R. Žižović, Some properties of rapidly varying sequences, J. Math. Anal. Appl. 327 (2007) 1297–1306] and [D. Djurčić, Lj.D.R. Kočinac, M.R. Žižović, Rapidly varying sequences and rapid convergence, Topology Appl. (2008), doi: 10.1016/j.topol.2007.05.026, in press].     

Formal asymptotic integration for one class of weakly nonlinear infinite-dimensional Hamiltonian systems

An asymptotic integration technique is described for a class of infinite-dimensional Hamiltonian systems. The method will generate a solution with any desired accuracy. The solutions are constructed using the theory of generalized functions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 148, pp. 42–51, 1985.In conclusion, I would like to acknowledge the helpful commments of V. B. Babich, I. A. Molotkov, V. F. Lazutkin, M. M. Popov and other participants of the LOMI wave diffraction seminar.     

The fundamental theorem of algebra for Hamilton and Cayley numbers

In this paper we prove the fundamental theorem of algebra for polynomials with coefficients in the skew field of Hamilton numbers (quaternions) and in the division algebra of Cayley numbers (octonions). The proof, inspired by recent definitions and results on regular functions of a quaternionic and of a octonionic variable, follows the guidelines of the classical topological argument due to Gauss. G. Gentili and F. Vlacci are partially supported by G.N.S.A.G.A. of the I.N.D.A.M. and by M.I.U.R.     

Explicit formulas for Hankel norm approximations of infinite-dimensional systems

Recently Hankel norm approximation for finite dimensional systems has been studied extensively, both for continuous time systems and discrete time systems. One of the approaches is to combine the work of J.A. Ball and J.W. Helton with results on Wiener-Hopf factorization to produce explicit formulas for the solutions. This has been done by J.A. Ball and A.C.M. Ran for the finite-dimensional case and for a class of infinite-dimensional systems by A.C.M. Ran. Here we show that this approach can be extended to a much larger class of infinite-dimensional systems; those with an exponentially stable semigroup and unbounded input and output operators which satisfy a smoothness condition. This class is larger than the nuclear class for which this problem was solved by K. Glover, R.F. Curtain and J.R. Partington by an approximation approach.     

Peripheral acyclicity and homology manifolds

In this paper we study various concepts of peripheral acyclicity and local homotopical properties, generalizing and extending earlier work due to R. C. Lacher, D. R. McMillan jr. and the authors. Then we obtain some applications for cell-like decompositions of manifolds and construct exotic factors (i.e. homology manifolds) of certain Euclidean spaces.Work performed under the auspices of the G.N.S.A.G.A of Italian C.N.R. and during the permanence of the second author at the Dept. of Math. (Univ. of Modena) as a visiting professor.Supported in part by the Ministero della Università e della Ricerca Scientifica e Tecnologica within the project «Geometria Reale e Complessa».Supported in part by the Commission of the European Communities Go West Fellowship No. 792 and by the Ministry for Science and Technology of the Republic of Slovenia Research Grant No. P1-0214-101-94.     

A Recursive Algorithm for Adaptive Estimation and Parameter Change Detection of Time Series Models

The discounted recursive least-squares concept of R. G. Brown's adaptive smoothing is extended to the Box-Jenkins models. The D.R.L.S. algorithm is derived for the autoregressive moving-average models. Two important applications of the D.R.L.S. for parameter change detection and adaptive forecasting are presented.     

On the semilocal convergence of inexact Newton methods in Banach spaces

We provide two types of semilocal convergence theorems for approximating a solution of an equation in a Banach space setting using an inexact Newton method [I.K. Argyros, Relation between forcing sequences and inexact Newton iterates in Banach spaces, Computing 63 (2) (1999) 134–144; I.K. Argyros, A new convergence theorem for the inexact Newton method based on assumptions involving the second Fréchet-derivative, Comput. Appl. Math. 37 (7) (1999) 109–115; I.K. Argyros, Forcing sequences and inexact Newton iterates in Banach space, Appl. Math. Lett. 13 (1) (2000) 77–80; I.K. Argyros, Local convergence of inexact Newton-like iterative methods and applications, Comput. Math. Appl. 39 (2000) 69–75; I.K. Argyros, Computational Theory of Iterative Methods, in: C.K. Chui, L. Wuytack (Eds.), in: Studies in Computational Mathematics, vol. 15, Elsevier Publ. Co., New York, USA, 2007; X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242]. By using more precise majorizing sequences than before [X. Guo, On semilocal convergence of inexact Newton methods, J. Comput. Math. 25 (2) (2007) 231–242; Z.D. Huang, On the convergence of inexact Newton method, J. Zheijiang University, Nat. Sci. Ed. 30 (4) (2003) 393–396; L.V. Kantorovich, G.P. Akilov, Functional Analysis, Pergamon Press, Oxford, 1982; X.H. Wang, Convergence on the iteration of Halley family in weak condition, Chinese Sci. Bull. 42 (7) (1997) 552–555; T.J. Ypma, Local convergence of inexact Newton methods, SIAM J. Numer. Anal. 21 (3) (1984) 583–590], we provide (under the same computational cost) under the same or weaker hypotheses: finer error bounds on the distances involved; an at least as precise information on the location of the solution. Moreover if the splitting method is used, we show that a smaller number of inner/outer iterations can be obtained.