### NAITO TOSHIKI

Emeritus Professor etc. | Emeritus Professor |

- Profile:

1969-1985 Fundamental theory of differential equations with infinite delay on finite dimensional space

1986-1998 Funcamental theory of differential equations with infinite delay on infinite dimensional space

1999-2002 Study of functional differential equations by using operator analysis

2003-2005 Study of differential equations by using differece equations

Researcher Information

Research Activity Information

### Paper

- Periodic solutions of difference equations

Tetsuo Furumochi; Toshiki Naito

Nonlinear Analysis, Theory, Methods & Applications, 71, 12, 2217-2222, Dec. 2009, Peer-reviwed

Scientific journal, English - Delayed Feedback 方程式とその性質

宮崎倫子; 内藤敏機; 申正善

RIMS Kokyuroku 1637, 数理解析研究所講究録 1637, 1637, 74-86, Apr. 2009

Research institution, Japanese - Stability and robust stability of positive linear Volterra difference equations

Pham Huu Anh Ngoc; Toshiki Naito; Jong Son Shin; Satoru Murakami

INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, JOHN WILEY & SONS LTD, 19, 5, 552-568, Mar. 2009, Peer-reviwed, We first introduce a class of positive linear Volterra difference equations. Then, we offer explicit criteria for uniform asymptotic stability of positive equations. Furthermore, we get a new Perron-Frobenius theorem for positive linear Volterra difference equations. Finally, we study robust stability of positive equations under structured perturbations and affine perturbations. Two explicit stability bounds with respect to these perturbations are given. Copyright (c) 2008 John Wiley & Sons, Ltd.

Scientific journal, English - Lyapunov exponents of solutions to linear differential equations with periodic forcing functions

Toshiki Naito; Pham Huu Anh Ngoc; Jong Son Shin

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, ACADEMIC PRESS INC ELSEVIER SCIENCE, 342, 1, 349-353, Jun. 2008, Peer-reviwed, We give Lyapunov exponents of solutions to linear differential equations of the form x' = Ax + f (t), where A is a complex matrix and f (t) is a tau-periodic continuous function. Notice that f (t) is not "small" as t -> infinity. The proof is essentially based on a representation [J. Kato, T. Naito, J.S. Shin, A characterization of solutions in linear differential equations with periodic forcing functions, J. Difference Equ. Appl. 11 (2005) 1-19] of solutions to the above equation. (C) 2008 Elsevier Inc. All rights reserved.

Scientific journal, English - Representations and asymptotic behavior of solutions to periodic linear difference equations

Toshiki Naito; Pharn Hun Anh Ngoc; Jong Son Shin

FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA, KOBE UNIV, DEPT MATHEMATICS, 51, 1, 55-80, Apr. 2008, Peer-reviwed, We give a new representation of solutions of the periodic linear difference equation of the form x(n + 1) = Bx(n) + b(n), where B is a complex p x p matrix and b(n) is an element of C-p satisfies the condition b(n) = b(n + rho), rho is an element of N, rho >= 2. If B = e(tau A), tau > 0, then the equation has two representations of solutions based on A and B. In particular, the representation of solutions based on A is deduced from the one based on B by using the translation formulae from B to A. Using these representations, we can obtain the complete classification of the set of initial values according to the behavior of solutions. As applications of these results, by the initial values we characterize necessary and sufficient conditions on the existence of a bounded solution and a rho-periodic solution.

Scientific journal, English - Floquet representations and asymptotic behavior of solutions to periodic linear difference equations

Toshiki Naito; Pham Huu Anh Ngoc; Jong Son Shin

HIROSHIMA MATHEMATICAL JOURNAL, HIROSHIMA UNIV, GRAD SCH SCI, 38, 1, 135-154, Mar. 2008, We give new representations of solutions for the periodic linear difference equation of the type x(n + 1) = B(c)x(n) + b(n), where complex nonsingular matrices B(n) and vectors b(n) are rho-periodic. These are based on the Floquet multipliers and the Floquet exponents, respectively. By using these representations, asymptotic behavior of solutions is characterized by initial values. In particular, we can characterize necessary and sufficient conditions that the equation has a bounded solution (or a rho-periodic solution), and the Massera type theorem by initial values.

Scientific journal, English - On stability and robust stability of positive linear Volterra equations

Pham Huu Anh Ngoc; Toshiki Naito; Jong Son Shin; Satoru Murakami

SIAM JOURNAL ON CONTROL AND OPTIMIZATION, SIAM PUBLICATIONS, 47, 2, 975-996, 2008, Peer-reviwed, We first introduce the notion of positive linear Volterra integro differential equations. Then we give some characterizations of these positive equations. An explicit criterion and a Perron-Frobenius-type theorem for positive linear Volterra integro differential equations are given. Then we offer a new criterion for uniformly asymptotic stability of positive equations. Finally, we study stability radii of positive linear Volterra integro differential equations. It is proved that complex, real, and positive stability radii of positive linear Volterra equations under structured perturbations ( or a. ne perturbations) coincide and can be computed by explicit formulae. To the best of our knowledge, most of the results of this paper are new.

Scientific journal, English - Translation formulae and its applications

内藤敏機; 申正善

数理解析研究所講究録, 1582, 108-117, Nov. 2007

Research institution, English - On stability of a class of positive linear functional difference equations

Pham Huu Anh Ngoc; Toshiki Naito; Jong Son Shin

MATHEMATICS OF CONTROL SIGNALS AND SYSTEMS, SPRINGER LONDON LTD, 19, 4, 361-382, Nov. 2007, Peer-reviwed, We first give a sufficient condition for positivity of the solution semigroup of linear functional difference equations. Then, we obtain a Perron-Frobenius theorem for positive linear functional difference equations. Next, we offer a new explicit criterion for exponential stability of a wide class of positive equations. Finally, we study stability radii of positive linear functional difference equations. It is proved that complex, real and positive stability radius of positive equations under structured perturbations (or affine perturbations) coincide and can be computed by explicit formulae.

Scientific journal, English - Characterizations of positive linear Volterra integro-differential systems

Toshiki Naito; Satoru Murakami; Jong Son Shin; Pham Huu Anh Ngoc

INTEGRAL EQUATIONS AND OPERATOR THEORY, BIRKHAUSER VERLAG AG, 58, 2, 255-272, Jun. 2007, Peer-reviwed, We first give a criterion for positivity of the solution semigroup of linear Volterra integro-differential systems. Then, we offer some explicit conditions under which the solution of a positive linear Volterra system is exponentially stable or (robustly) lies in L-2 [0, +infinity).

Scientific journal, English - A representation of solutions for periodic linear difference equations

内藤敏機; 申正善

RIMS Kokyuroku, 1547, 59-67, Apr. 2007

Research institution, English - Stability radius of linear parameter-varying systems and applications

Pham Huu Anh Ngoc; Toshiki Naito

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, ACADEMIC PRESS INC ELSEVIER SCIENCE, 328, 1, 170-191, Apr. 2007, Peer-reviwed, In this paper, we present a unifying approach to the problems of computing of stability radii of positive linear systems. First, we study stability radii of linear time-invariant parameter-varying differential systems. A formula for the complex stability radius under multi perturbations is given. Then, under hypotheses of positivity of the system matrices, we prove that the complex, real and positive stability radii of the system under multi perturbations (or affine perturbations) coincide and they are computed via simple formulae. As applications, we consider problems of computing of (strong) stability radii of linear time-invariant time-delay differential systems and computing of stability radii of positive linear functional differential equations under multi perturbations and affine perturbations. We show that for a class of positive linear time-delay differential systems, the stability radii of the system under multi perturbations (or affine perturbations) are equal to the strong stability radii. Next, we prove that the stability radii of a positive linear functional differential equation under multi perturbations (or affine perturbations) are equal to those of the associated linear time-invariant parameter-varying differential system. In particular, we get back some explicit formulas for these stability radii which are given recently in [P.H.A. Ngoc, Strong stability radii of positive linear time-delay systems, Internat. J. Robust Nonlinear Control 15 (2005) 459-472; P.H.A. Ngoc, N.K. Son, Stability radii of positive linear functional differential equations under multi perturbations, SIAM J. Control Optim. 43 (2005) 2278-2295]. Finally, we give two examples to illustrate the obtained results. (c) 2006 Elsevier Inc. All rights reserved.

Scientific journal, English - Characterizations of positive linear functional differential equations

Pham Huu Anh Ngoc; Toshiki Naito; Jong Son Shin

FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA, KOBE UNIV, DEPT MATHEMATICS, 50, 1, 1-17, Apr. 2007, Peer-reviwed, In this paper, we first prove that if a linear neutral functional differential equation is positive then it must degrade into a linear functional differential equation of retarded type. Then, we give some explicit criteria for positive linear functional differential equations. Consequently, we obtain a novel criterion for exponential stability of positive linear functional differential equations.

Scientific journal, English - Stability radii of higher order linear difference systems under multi-perturbations

Pham Huu Anh Ngoc; Thai Bao Tran; Toshiki Naito

JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, TAYLOR & FRANCIS LTD, 13, 1, 15-24, Jan. 2007, Peer-reviwed, We study stability radii of higher order linear difference systems under multi-perturbations. A formula for complex stability radius of higher order linear difference systems under multi-perturbations is given. Then, for the class of positive systems, we prove that the complex stability radius and real stability radius of the system under multi-perturbations coincide and they are computed via a simple formula. These are extensions of corresponding results of Hinrichsen and Son, Hinrichsen et al., Ngoc and Son, and Pappas and Hinrichsen. An example is given to illustrate the obtained results.

Scientific journal, English - On periodicizing functions

Toshiki Naito; Jong Son Shin

Bull. Korean Math. Soc., 43, 2006, Peer-reviwed

Scientific journal, English - Stability radii of positive linear functional differential systems in Banach spaces

Pham Huu Anh Ngoc; Nguyen Van Minh; Toshiki Naito

International J. of Evolution Equations, 2, 1, 2006, Peer-reviwed

Scientific journal, English - A spectral countability condition for almost automorphy of solutions of differential equations

Nguyen Van Minh; Toshiki Naito; Gaston Nguerekata

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, AMER MATHEMATICAL SOC, 134, 11, 3257-3266, 2006, Peer-reviwed, We consider the almost automorphy of bounded mild solutions to equations of the form

(*) dx/dt = A(t)x+f(t)

with (generally unbounded) tau-periodic A(center dot) and almost automorphic f(center dot) in a Banach space X. Under the assumption that X does not contain c(0), the part of the spectrum of the monodromy operator associated with the evolutionary process generated by A(center dot) on the unit circle is countable. We prove that every bounded mild solution of (*) on the real line is almost automorphic.

Scientific journal, English - Global optimization problems in stability analysis of linear dynamical systems

Pham Huu Anh Ngoc; Toshiki Naito; Jong Son Shin

POSITIVE SYSTEMS, PROCEEDINGS, SPRINGER-VERLAG BERLIN, 341, 311-318, 2006, Peer-reviwed, We deal with problems of maximizing the norm of the transfer matrix function of linear dynamical systems. It is shown that if a linear system is positive and (exponentially) asymptotically stable then the norm of its transfer matrix function attains the maximum at a specific point on the boundary of the stable region. For sake of space, exposure is kept to minimum in this paper. Two examples are given to illustrate the obtained results.

Scientific journal, English - A characterization of solutions in linear differential equations with periodic forcing functions

J Kato; T Naito; JS Shin

JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, TAYLOR & FRANCIS LTD, 11, 1, 1-19, Jan. 2005, Peer-reviwed, We deal with periodic linear inhomogeneous differential equations of the form d x /d t = Ax ( t )+ f ( t ), where A is an m x m matrix and f a tau-periodic continuous function. The solutions of this equation will be characterized as a sum of tau-periodic functions and exponential-like functions in an explicit form. As applications of this result, we can obtain the complete classification of the set of initial values according to the behavior of solutions: bounded solutions on [0, infinity], tau-periodic solutions, quasi-periodic solutions, asymptotically periodic solutions and solutions with the growth order as t -->infinity , etc. The essential part of our method is to give a specific representation of the solutions of difference equations corresponding to the above equations.

Scientific journal, English - Periodic solutions of evolution equations

K Ezzinbi; T Naito; N Minh; J Liu

DYNAMICS OF CONTINUOUS DISCRETE AND IMPULSIVE SYSTEMS-SERIES A-MATHEMATICAL ANALYSIS, WATAM PRESS, 11, 5-6, 601-613, Oct. 2004, Peer-reviwed, Consider the following evolution equations without delay or with finite or infinite delay in a general Banach space X,

u'(t) + A(t)u(t) = f (t, u(t)), t > 0,

u'(t) + A(t)u(t) = f (t, u(t), u(t)), t > 0.

We will analyze some fixed point theorems and then see how they can be applied to derive periodic solutions for the above mentioned equations.

Scientific journal, English - On the Asymptotic Periodic Solutions of Abstract Functional Differential Equations

Takeshi Nishikawa; Nguyen Van Minh; Toshiki Naito

FUNKCIALAJ EKVACIOJ-SERIO INTERNACIA, KOBE UNIV, DEPT MATHEMATICS, 47, 2, 307-327, Aug. 2004, Peer-reviwed, The paper is concerned with conditions for all mild solutions of abstract functional differential equations with finite delay in a Banach space to be periodic and asymptotic periodic, where forcing term is a continuous 1-periodic function. The obtained results extend various recent ones on the subject.

Scientific journal, English - Massera's theorem for almost periodic solutions of functional differential equations

S Murakami; T Naito; N Van Minh

JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, MATH SOC JAPAN, 56, 1, 247-268, Jan. 2004, Peer-reviwed, The Massera Theorem for almost periodic solutions of linear periodic ordinary differential equations of the form (*) x' = A(t)x + f(t), where f is almost periodic, is stated and proved. Furthermore, it is extended to abstract functional differential equations (**) x' = Ax + F(t)x(1) + f(t), where A is the generator of a compact semigroup, F is periodic and f is almost periodic. The main techniques used in the proofs involve a new variation of constants formula in the phase space and a decomposition theorem for almost periodic solutions.

Scientific journal, English - Massera criterion for abstract functional differential equations with advance and delay

Takeshi Nishikawa; Nguyen Van Minh; Toshiki Naito

Applicable Analysis, 83, 11, 1171-1185, 2004, Peer-reviwed

English - Bounded and periodic solutions of infinite delay evolution equations

J Liu; T Naito; N Van Minh

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, ACADEMIC PRESS INC ELSEVIER SCIENCE, 286, 2, 705-712, Oct. 2003, Peer-reviwed, For A (t) and f (t, x, y) T-periodic in t, we consider the following evolution equation with infinite delay in a general Banach space X:

u'(t) + A(t)u(t) = f (t, u(t), u(t)), t > 0, u(s) = phi(s), s less than or equal to 0, (0.1)

where the resolvent of the unbounded operator A(t) is compact, and u(t) (s) = u(t + s), s less than or equal to 0. By utilizing a recent asymptotic fixed point theorem of Hale and Lunel (1993) for condensing operators to a phase space C-g, we prove that if solutions of Eq. (0.1) are ultimate bounded, then Eq. (0.1) has a T-periodic solution. This extends and improves the study of deriving periodic solutions from boundedness and ultimate boundedness of solutions to infinite delay evolution equations in general Banach spaces; it also improves a corresponding result in J. Math. Anal. Appl. 247 (2000) 627-644 where the local strict boundedness is used. (C) 2003 Elsevier Inc. All rights reserved.

Scientific journal, English - Asymptotic properties and initial values of solutions to periodic linear equations

Junji Kato; Toshiki Naito; Jong Son Shin

数理解析研究所講究録 1309, 京都大学数理解析研究所, 1309, 100-107, 2003

Research institution, English - Boundedness and almost periodicity of solutions of partial functional differential equations

T Furumochi; T Naito; N Van Minh

JOURNAL OF DIFFERENTIAL EQUATIONS, ACADEMIC PRESS INC ELSEVIER SCIENCE, 180, 1, 125-152, Mar. 2002, Peer-reviwed, We study necessary and sufficient conditions for the abstract functional differential equation (x) over dot = Ax + Fx(i) + f (t) to have almost periodic, quasi periodic solutions with the same structure of spectrum as f. The main conditions are stated in terms of the imaginary solutions of the associated characteristic equations and the spectrum of the forcing term f. The obtained results extend recent results to abstract functional differential equations. (C) 2002 Elsevier Science (USA).

Scientific journal, English - A variation-of-constants formula for abstract functional differential equations in the phase space

Y Hino; S Murakami; T Naito; N Van Minh

JOURNAL OF DIFFERENTIAL EQUATIONS, ACADEMIC PRESS INC ELSEVIER SCIENCE, 179, 1, 336-355, Feb. 2002, Peer-reviwed, For linear functional differential equations with infinite delay in a Banach space, a variation-of-constants formula is established in the phase space. As an application one applies it to study the admissibility of some spaces of functions whose spectra are contained in a closed subset of the real line, (C) 2002 Elsevier Science.

Scientific journal, English - A Massera type theorem for functional differential equations with infinite delay

Toshiki Naito; Nguyen Van Minh; Jong Son Shin

Japanese Journal of Mathematics, 28, 1, 31-49, 2002, Peer-reviwed

Scientific journal, English - Existence of bounded solutions to lindear differential equations, (I)

Tohiki Naito; Jong Son Shin

数理解析研究所講究録, 1254, 64-72, 2002

Research institution, English - Spectrum and (almost) periodic solutions of functional differential equations

Toshiki Naito; Nguyen Van Minh; Jong Son Shin

Springer, Vietnam Journal of Mathematics, 30, SI, 577-589, 2002, Peer-reviwed

International conference proceedings, English - Bounded solutions and periodic solutions to linear differential equations in Banach spaces

Junji Kato; Toshik Naito; Jong Son Shin

Springer, Vietnam Journal of Mathematics, 30, SI, 561-575, 2002, Peer-reviwed

International conference proceedings, English - Periodic and almost periodic solutions of functional differential equations with finite and infinite delay

T Naito; N Van Minh; JS Shin

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, PERGAMON-ELSEVIER SCIENCE LTD, 47, 6, 3989-3999, Aug. 2001, Peer-reviwed, We survey some new methods and results on periodic and almost periodic solutions of differential equations in Banach spaces.

Scientific journal, English - New spectral criteria for almost periodic solutions of evolution equations

T Naito; N Van Minh; JS Shin

STUDIA MATHEMATICA, POLISH ACAD SCIENCES INST MATHEMATICS, 145, 2, 97-111, 2001, We present a general spectral decomposition technique for bounded solutions to inhomogeneous linear periodic evolution equations of the form (x) over dot = A(t)x + f(t) (*), with f having precompact range, which is then applied to find new spectral criteria for the existence of almost periodic solutions with specific spectral properties in the resonant case where e(i sp(f)) may intersect the spectrum of the monodromy operator P of (*) there sp(f) denotes the Carleman spectrum of (f). We show that if (*) has a bounded uniformly continuous mild solution u and sigma (Gamma)(P)\e(isp(f)) is closed, where sigma (Gamma)(P) denotes the part of sigma (P) on the unit circle, then (*) has a bounded uniformly continuous mild solution w such that e(i sp(w)) = e(i sp(f)). Moreover, w is a "spectral component" of u. This allows us to solve the general Massera-type problem for almost periodic solutions. Various spectral criteria for the existence of almost periodic and quasi-periodic mild solutions to (*) are given.

Scientific journal, English - Existence and uniqueness of periodic solutions to periodic linear functional differential equations with finite delay

Jong Son Shin; Toshiki Naito; Nguyen Van Minh

Funkcialaj Ekvacioj, 44, 53-71, 2001

English - Periodic solutions of linear differential equatiions

Toshiki Naito; Jong Song Shin; Nguyen Van Minh

数理解析研究所講究録, 1216, 78-89, 2001

Research institution, English - Boundedness and almost periodicity in dynamical systems

T Naito; NV Minh; R Miyazaki; Y Hamaya

JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, TAYLOR & FRANCIS LTD, 7, 4, 507-527, 2001, Peer-reviwed, We consider spectral criteria for the existence of bounded solutions to difference equations of the form x(n+1) = Ax(n) + f(n) with specific spectral properties. The results will be then applied to find periodic, almost periodic solutions to dx/dt = A(t)x +f(t)(*) and dx/dt = Ax + F(t)x(i) + f(t)(**) with (in general, unbounded) tau -periodic A(.), tau -periodic F(t),f(.). This provides a new and simple approach to find spectral criteria for the existence of periodic, almost periodic solutions to differential equations (*), (**).

Scientific journal, English - Evolution semigroups and sums of commuting operators: A new approach to the admissibility theory of function spaces

S Murakami; T Naito; N Van Minh

JOURNAL OF DIFFERENTIAL EQUATIONS, ACADEMIC PRESS INC, 164, 2, 240-285, Jul. 2000, This payer is concerned with conditions for the admissibility of a translation invariant function space M with respect to a well posed linear evolution equation du/dt = Au + f(t), t is an element of R (*). We propose a new approach to this problem by considering the sum of two commuting operators - d/dt : = - D-M and the operator of multiplication by A on M. On the one hand, the closure of this operator is the infinitesimal generator of the so-called evolution semigroup associated with (*). On the other hand, the generator G of this semigroup relates a mild solution u of (*) to the forcing term f by the rule Gu = -f. Consequently, various spectral criteria of the type sigma(D-M) boolean AND sigma(A) = circle divide for the admissibility of the function space M with respect to (*) can be proved in an elegant manner. Moreover, they can be naturally extended to general classes of differential equations, including higher order and abstract functional differential equations. Applications and examples are provided to illustrate the obtained results. (C) 2000 Academic Press.

Scientific journal, English - Spectral analysis of an operator associated with linear functional differential equations and its applications

Satoru Murakami; Toshiki Naito; Nguyen Van Minh

EJQTDE, Prod. 6th coll. QTDE, 20, 1-15, 2000

English - A decomposition theorem for bounded solutions and the existence of periodic solutions of periodic differential equations

T Naito; N Van Minh; R Miyazaki; JS Shin

JOURNAL OF DIFFERENTIAL EQUATIONS, ACADEMIC PRESS INC ELSEVIER SCIENCE, 160, 1, 263-282, Jan. 2000, We prove a decomposition theorem fbr bounded uniformly continuous mild solutions to tau-periodic evolution equations of the form dx/dt = A(t) x + f(t) (*) with (in general, unbounded) tau-periodic A(.), tau-periodic f(.), and compact monodromy operator. By this theorem, every bounded uniformly continuous mild solution to (*) is a sum of a tau-periodic solution to (*) and a quasi periodic solution to its homogeneous equation. An analog of this for bounded solutions has been proved for abstract functional differential equations dx/dt = Ax + F(t) x(t) + f(t) with finite delay, where A generates a compact semigroup, As an immediate consequence, the existence of such a solution implies the existence of a tau-periodic solution to the inhomogeneous equation as well as a formula for its Fourier coefficients. This, even for the classical case of equations, improves considerably the previous results on the subject. (C) 2000 Academic Press.

Scientific journal, English - Uniqueness of periodic solutions to periodic functional differential equations with finite delay

Toshiki Naito; Jong Son Shin

RIMS kokyuroku,1128, 1128, 28-36, 2000

Research institution, English - On the spectrum of some functional differential equations

T Naito; N Van Minh; JS Shin

SEMIGROUPS OF OPERATORS: THEORY AND APPLICATIONS, BIRKHAUSER VERLAG AG, 42, 222-228, 2000

International conference proceedings, English - On stability of linear autonomous functional differential equations

Jong Son Shin; Toshi Naito; Nguyen Van Minh

Funkcialaj Ekvacioj,43, 323-337, 2000

English - Periodic solutions of linear differential equations

Toshiki Naito; Jong; Song Shin; Nguyen Van Minh

2000

English - Existence and continuous dependence of mild solutions to semilinear functional differential equations in banach spaces

JS Shin; T Naito

TOHOKU MATHEMATICAL JOURNAL, TOHOKU UNIVERSITY, 51, 4, 555-583, Dec. 1999, This paper is concerned with a general existence and continuous dependence of mild solutions to semilinear functional differential equations with infinite delay in Banach spaces. In particular, our results are applicable to the equations whose Cg-semigroups and nonlinear operators, defined on an open set, are noncompact.

Scientific journal, English - Semi-Fredholm operators and periodic solutions for linear functional differential equations

JS Shin; T Naito

JOURNAL OF DIFFERENTIAL EQUATIONS, ACADEMIC PRESS INC ELSEVIER SCIENCE, 153, 2, 407-441, Apr. 1999, We deal with the inhomogeneous linear periodic equation with infinite delay of the form dx/dt = Ax(t) + B(t, x(t)) + F(t), where A is the generator of a C-0-semigroup on a Banach space. Assuming that it has a bounded solution, we obtain several criteria on the existence and the uniqueness of periodic solutions for the equation in the general phase space B and in the concrete phase space B = UCg. The key of our approach is the employment of the perturbation theory of semi-Fredholm operators to show that the period map satisfies the condition of the fixed point theorem by Chow and Hair (Funkcial. Ekvac. 17 (1974), 31-38). (C) 1999 Academic Press.

Scientific journal, English - Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations

T Naito; N Van Minh

JOURNAL OF DIFFERENTIAL EQUATIONS, ACADEMIC PRESS INC, 152, 2, 358-376, Mar. 1999, We investigate spectral criteria for the existence of(almost) periodic solutions to linear I-periodic evolution equations of the Form dx/dt = A (t) x + f(t) with (in general, unbounded) A(t) and (almost) periodic f. Using the evolution semigroup associated with the evolutionary process generated by the equation under consideration we show that if the spectrum of the monodromy operator does not intersect the set <(e(isp(f)))over bar>, then the above equation has an almost periodic (mild) solution x(f) which is unique if one requires sp(x(f)) subset of <(lambda+2 pi k, k is an element of Z, lambda is an element of sp(f)})over bar>. We emphasize that our method allows us to treat the equations without assumption on the existence of Floquet representation. This improves recent results on the subject. In addition we discuss some particular cases, in which the spectrum of monodromy operator does not intersect the unit circle, and apply the obtained results to study the asymptotic behavior of solutions. Finally, an application to parabolic equations is considered. (C) 1999 Academic Press.

Scientific journal, English - On stability of solutions in linear autonomous functional differenteal equations

John Son Shin; Toshiki Naito; Nguyen Van Minh

RIMS kokyuroku,1083, 231-242, 1999

Research institution, English - On a spectral criterion for almost periodicity of solutions to periodic evolutions equatons

Toshiki Naito; Nguyen Van Minh

Electric Journal of Qualitative Theory of Differential Equations,1, 1-28, 1999

Scientific journal, English - Evokution semigroups and harmonic analysis of bounded solutions of evolution equations

Toshiki Naito; Nguyen Van Minh; Jong Son Shin

Surikaisekikenkyusho Kokyuroku,1083, 166-168, 1999

English - Spectral criteria for almost periodicity of solutions of periodic evolution equations(共著)

T. Naito; Nguyen V.M

RIMS Kokyuroku,1034, Kyoto University, 1034, 49-51, 1998

Research institution, English - Existence of periodic solutions for periodic linear functional differential equations in Banach Spaces(共著)

J.S. Shin; T. Naito

RIMS Kokyuroku,1034, 120-129, 1998

Research institution, English - The generator of the solution semigroup for the general ilnear functional differential equation

Jong Son Shin; Toshiki Naito; Nguyen Van Minh

Bulletin of the University of Electro-Communications,11-1, 29-38, 1998

Research institution, English - On solution semigroups of general functional differential equations

T Naito; JS Shin; S Murakami

NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS, PERGAMON-ELSEVIER SCIENCE LTD, 30, 7, 4565-4576, Dec. 1997

Scientific journal, English - Evolution Equations with Infinite Delay

RIMS Kokyuroku, 984, 147-160, 1997

Research institution, English - On solution semigroups of furctional differential equations

数理解析研究所講究録, 940, 161-175, 1996

Research institution, English - Fading memory spaces and stability properties for functional differential equations with infinite delay

Funkcialaj Ekvacioj, 32, 1989

Scientific journal, English - Adjoint semigroups associated with linear functional differential equations with infinite delay

Toshiki Naito

Finite and Infinite Dimensional Dynamics, Lecture Notes in Numerical and Applied Analysis, Vol. 15, Kinokuniya, Tokyo, 165-185, Aug. 1988, Peer-reviwed

International conference proceedings, English - A modified form of the variation-of-constants formula for equations with infinite delay

Tohoku Mathematical Journal, 36, 1984

Scientific journal, English - On linear autonomous retarded equations with an abstract phase space for infinite delay

Journal of Differential Equations, 33, 1979

Scientific journal, English - On autonomous linear functional differential equations with infinite retardations

Journal of Differential Equations, 21, 1976

Scientific journal, English

### MISC

### Books and other publications

- 初等常微分方程式の解法

内藤敏機; 申正善

Japanese, Joint work, 牧野書店, Nov. 2005 - 理工系基礎数学演習

石田晴久; 伊藤裕也; 大野真裕; 木田雅成; 田吉隆夫; 内藤敏機; 山口耕平; 山田裕一

Japanese, Joint work, 昭晃堂, Apr. 2005 - タイムラグをもつ微分方程式

内藤敏機; 原惟行; 日野義之; 宮崎倫子

Japanese, Joint work, 第５章 線形関数微分方程式, 牧野書店, Nov. 2002 - Almost Periodic Solutions of Differential Equations in Banach Spaces

Yoshiyuki Hino; Toshiki Naito; Nguyen Van Minh; Jong Son Shin

English, Joint work, Taylor and Francis,, 2002 - Dual Semigroups Associated with Linear Functional Differential Equations with Infinite Delay

English, International Conference on Differential Equations Barcelona 1991, World Scientific, 1993 - Functional Differential Equations with Infinite Delay

Yoshiyuki Hino; Satoru Murakami; Toshiki Naito

English, Joint work, Springer-Verlag, 1991 - Asymptotic stability of linear fanctional differential equations with the fadeng memory space

English, Proc International symposium Functional Differential Equations, Kyoto 1990, World Scientific, 1991