Gaston M. N'Guérékata

Topics in Fractional Differential Equations

Preface.- 1. Preliminary Background.- 2. Partial Hyperbolic Functional Differential Equations.- 3. Partial Hyperbolic Functional Differential Inclusions.- 4. Impulsive Partial Hyperbolic Functional Differential Equations.- 5. Impulsive Partial Hyperbolic Functional Differential Inclusions.- 6. Implicit Partial Hyperbolic Functional Differential Equations.- 7. Fractional Order Riemann-Liouville Integral Equations.- References.- Index.

Almost automorphic solutions of evolution equations

This paper is concerned with the existence of almost automorphic mild solutions to equations of the form (*) u(t) = Au(t) + f(t), where A generates a holomorphic semigroup and f is an almost automorphic function. Since almost automorphic functions may not be uniformly continuous, we introduce the notion of the uniform spectrum of a function. By modifying the method of sums of commuting operators used in previous works for the case of bounded uniformly continuous solutions, we obtain sufficient conditions for the existence of almost automorphic mild solutions to (*) in terms of the imaginary spectrum of A and the uniform spectrum of f.

A spectral countability condition for almost automorphy of solutions of differential equations

We consider the almost automorphy of bounded mild solutions to equations of the form (*) dx/dt = A(t)x+f(t) with (generally unbounded) r-periodic A(·) and almost automorphic f(·) 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(·) on the unit circle is countable. We prove that every bounded mild solution of (*) on the real line is almost automorphic.

Optimal control of a fractional diffusion equation with state constraints

This paper is concerned with the state constrained optimal control problems of a fractional diffusion equation in a bounded domain. The fractional time derivative is considered in the Riemann-Liouville sense. Under a Slater type condition we prove the existence a Lagrange multiplier and a decoupled optimality system.

Square-mean almost automorphic mild solutions to non-autonomous stochastic differential equations in Hilbert spaces

In this paper, we first refine the definition of square-mean almost automorphic functions introduced in Fu and Liu (2010) [11], then we prove the existence and uniqueness of square-mean almost automorphic mild solutions for a class of non-autonomous stochastic differential equations in a real separable Hilbert space. Some additional properties of square-mean almost automorphic functions are also provided. To prove our main result, we use the Banach contraction mapping principle.

Composition of pseudo almost periodic and pseudo almost automorphic functions and applications to evolution equations

We establish new theorems for the composition of pseudo almost periodic and pseudo almost automorphic functions in Banach spaces. Our results extend the recent ones [H. Li, F. Huang and J. Li, Composition of pseudo almost-periodic functions and semilinear differential equations, J. Math. Anal. Appl. 255 (2001), pp. 436–446; J. Liang, J. Zhang, T.J. Xiao, Composition of pseudo almost automorphic and asymptotically almost automorphic functions, J. Math. Anal. Appl. 340 (2001), pp. 1493–1499]. We also study some sufficient conditions for the continuity of the superposition operator. As an application to the abstract results, we give some existence theorems of pseudo almost periodic/automorphic solutions for some semilinear evolution equations and examples with the heat equation.

Topics on stability and periodicity in abstract differential equations

Stability and Exponential Dichotomy Almost Periodic Solutions Almost Automorphic Solutions Evolution Semigroup Method Sums of Commuting Operators Decomposition Theorem Nonlinear Equations with Finite Delay Nonlinear Equations with Infinite Delay Non-Densely Defined Equations Evolution Semigroups and Semilinear Equations Comments and Further Reading Guide.

On fractional integro-differential inclusions with state-dependent delay in Banach spaces

In this article we investigate the existence of solutions on a compact interval for the fractional integro-differential inclusions with state-dependent delay in Banach spaces when the delay is infinite. We consider the cases when the multivalued nonlinear term takes convex values as well as nonconvex values.

A note on a semilinear fractional differential equation of neutral type with infinite delay.

We deal in this paper with the mild solution for the semilinear fractional differential equation of neutral type with infinite delay: , , , , with and . We prove the existence (and uniqueness) of solutions, assuming that is a linear closed operator which generates an analytic semigroup on a Banach space by means of the Banachs fixed point theorem. This generalizes some recent results.

Mild solutions for abstract fractional differential equations

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 behaviour of mild solutions to abstract composite fractional relaxation equations. We include in our analysis the Basset and Bagley–Torvik equations.