Papers by Gaston Nguerekata
Nonlinear Analysis, 2004
The main purpose of this paper is to deal with almost automorphic and asymptotically almost autom... more The main purpose of this paper is to deal with almost automorphic and asymptotically almost automorphic solutions of the initial value problem as well as the nonlinear Volterra integral equation in Banach spaces. We obtain a collection of existence results of such solutions to these equations. We investigate also a topological structure of such solution sets. Moreover, we prove Aronszajn-type theorems for solutions of the initial value problem as well as the nonlinear Volterra integral equation, defined on the whole real line.
This paper deals with the existence of asymptotic almost automorphic solution of fractional integ... more This paper deals with the existence of asymptotic almost automorphic solution of fractional integro differential equation. We prove the result by using fixed point theorems. We show the result with Lipschitz condition and without Lipschitz condition on the forcing term. At the end examples have been given to illustrate the analytical findings.
Considering the Green's operator $(\G f)(t)=\int_0^t X(t,s)f(s)ds$ we prove that if the following... more Considering the Green's operator $(\G f)(t)=\int_0^t X(t,s)f(s)ds$ we prove that if the following conditions hold \bullet \quad the map $\G f$ lies in $L^q(\R_+,\X)$ for all $f\in L^{p}(\R_+,\X)$, and \bullet \quad $\G:L^{p}(\R_+,\X)\to L^{q}(\R_+,\X)$ is Lipschitz continuous, i.e. there exists $K>0$ such that $$|\G f-\G g|_{q} \leq K\|f-g\|_{p}, for all f,g\in L^p(\R_+,\X),$$ then the above mild solution will have an exponential decay.
Http Dx Doi Org 10 1080 00036811 2012 698271, Jul 12, 2013
We propose a unified functional analytic approach to derive a variation of constants formula for ... more We propose a unified functional analytic approach to derive a variation of constants formula for a wide class of fractional differential equations using results on (a, k)-regularized families of bounded and linear operators, which covers as particular cases the theories of C 0 -semigroups and cosine families. Using this approach we study the existence of mild solutions to fractional differential equation with nonlocal conditions. We also investigate the asymptotic behavior of mild solutions to abstract composite fractional relaxation equations. We include in our analysis the Basset and Bagley-Torvik equations. D α t u(t) = Au(t) + D α−1 t f (t, u(t)), t ∈ [0, T ], u(0) + g(u) = u 0 , u (0) = 0, Key words and phrases. Linear and semilinear evolution equations; regularized operator families; mild solutions.
Almost automorphic solutions to some semilinear fuzzy differential equations
ABSTRACT In this paper the authors study the existence and uniqueness of an almost automorphic mi... more ABSTRACT In this paper the authors study the existence and uniqueness of an almost automorphic mild solution to the following fuzzy differential equation: x′(t)=Ax(t)⊕f(t,x(t)),t∈R, where A is the infinitesimal generator of an exponentially stable C0-semigroup on the fuzzy number type spaces (X,⊕,⊙,d).
Alomst automorphic solutions of semilinear evolution equations
Proceedings of the American Mathematical Society, 2005
The Space of Continuous Periodic Functions Is a Set of First Category in AP(X)
Journal of function spaces and applications
We prove that the space of continuous periodic functions is a set of first category in the space ... more We prove that the space of continuous periodic functions is a set of first category in the space of almost periodic functions, and we also show that the space of almost periodic functions is a set of first category in the space of almost automorphic functions.
Measure of Noncompactness and Nondensely Defined Semilinear Functional Differential Equations with Fractional Order
Cubo (Temuco), 2010
Nonlinear Analysis: Theory, Methods & Applications, 2010
In this paper, we establish a composition theorem for weighted pseudo-almost automorphic function... more In this paper, we establish a composition theorem for weighted pseudo-almost automorphic functions under a weaker Lipschitz condition. Our composition theorem generalizes some known results. Moreover, the existence and uniqueness of pseudo-almost automorphic solutions for abstract semilinear evolution equations are studied.
Journal of Differential Equations, 2009
We consider the existence and uniqueness of bounded solutions of periodic evolution equations of ... more We consider the existence and uniqueness of bounded solutions of periodic evolution equations of the form u ′ = A(t)u + ǫH(t, u) + f (t), where A(t) is, in general, an unbounded operator depending 1-periodically on t, H is 1-periodic in t, ǫ is small, and f is a bounded and continuous function that is not necessarily uniformly continuous. We propose a new approach to the spectral theory of functions via the concept of "circular spectrum" and then apply it to study the linear equations u ′ = A(t)u+f (t) with general conditions on f . For small ǫ we show that the perturbed equation inherits some properties of the linear unperturbed one. The main results extend recent results in the direction, saying that if the unitary spectrum of the monodromy operator does not intersect the circular spectrum of f , then the evolution equation has a unique mild solution with its circular spectrum contained in the circular spectrum of f .
International Journal of Mathematics and Mathematical Sciences, 1984
Almost periodic functions with values in Banach spaces, and even more generally in Fréchet spaces... more Almost periodic functions with values in Banach spaces, and even more generally in Fréchet spaces and p-Fréchet spaces, have been investigated by many authors. The purpose of this paper is to investigate the extent to which some results of these authors also hold in the setting of topological vector spaces, not necessarily locally convex or p-locally convex. In fact, most of our results (Theorems 2-6) do not require even completeness or metrizability of the range space. We thus extend and unify several known results.
International Journal of Mathematics and Mathematical Sciences, 2001
We prove almost periodicity of solutions of the equation x (t) = Ax(t) when the linear operator A... more We prove almost periodicity of solutions of the equation x (t) = Ax(t) when the linear operator A satisfies an inequality of the form Re(Ax, x) ≥ 0. 2000 Mathematics Subject Classification. 34G10, 34K14.
Electronic Journal of Differential Equations, 2010
In this work, we give sufficient conditions for the existence and uniqueness of a weighted pseudo... more In this work, we give sufficient conditions for the existence and uniqueness of a weighted pseudo-almost periodic solutions for some neutral partial functional differential equations. Our working tools are based on the variation of constant formula and the spectral decomposition of the phase space. To illustrate our main result, we propose to study the existence and uniqueness of a weighted pseudo-almost periodic solution for some neutral model arising in physical systems.
Nonlinear Analysis, 2010
In this paper we introduce the concept of weighted pseudo-almost periodicity in the sense of Step... more In this paper we introduce the concept of weighted pseudo-almost periodicity in the sense of Stepanov, also called S p -weighted pseudo-almost periodicity and study its properties. Next, we present a result on the existence of weighted pseudo-almost periodic solutions to the N-dimensional heat equation with S p -weighted pseudo-almost periodic coefficients.
Almost automorphic solutions to semilinear evolution equations
Advanced fractional differential and integral equations
Topics in fractional differential equations
We study some properties of bounded and C (1) -almost automorphic solutions of the following Lién... more We study some properties of bounded and C (1) -almost automorphic solutions of the following Liénard equation:
Electronic Journal of Qualitative Theory of Differential Equations, 2010
This paper deals with the existence of a mild solution for some fractional semilinear differentia... more This paper deals with the existence of a mild solution for some fractional semilinear differential equations with non local conditions. Using a more appropriate definition of a mild solution than the one given in [12], we prove the existence and uniqueness of such solutions, assuming that the linear part is the infinitesimal generator of an analytic semigroup that is compact for t > 0 and the nonlinear part is a Lipschitz continuous function with respect to the norm of a certain interpolation space. An example is provided.
Asymptotically typed solutions to a semilinear integral equation
Journal of Integral Equations and Applications, 2014
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Papers by Gaston Nguerekata