On a connection between powers of operators and fractional Cauchy problems

Many phenomena in mathematical physics and in the theory of stochastic processes are recently described through fractional evolution equations. We investigate a general framework for connections between ordinary non-homogeneous equations in Banach spaces and fractional Cauchy problems. When the underlying operator generates a strongly continuous semigroup, it is known, using a subordination argument, that the fractional evolution equation is well posed. In this case, we provide an explicit form of the solution involving special functions, one example being the Airy function.     

Existence and uniqueness of holomorphic solutions for fractional Cauchy problem

By employing majorant functions, the existence and uniqueness of holomorphic solutions to nonlinear fractional partial differential equations (the Cauchy problems) are introduced. Furthermore, the analytic continuation of solutions is studied.     

Existence and uniqueness results for Cauchy problem of variable-order fractional differential equations

In this paper, the existence and uniqueness results of variable-order fractional differential equations (VOFDEs) are studied. The variable-order fractional derivative is defined in the Caputo sense, and the fractional order is a bounded function. The existence result of Cauchy problem of VOFDEs is obtained by constructing an iteration series which converges to the analytical solution. The uniqueness result is obtained by employing the contraction mapping principle. Since the variable-order fractional derivatives contain classical and fractional derivatives as special cases, many existence and uniqueness results of references are significantly generalized. Finally, we draw some conclusions of variable-order fractional calculus, and two examples are given for demonstrating the theoretical analysis.     

Growth properties of semigroups generated by fractional powers of certain linear operators

Certain semigroups are generated by powers ?(?A)^a, for closed operators A in Banach space and 0 < a < 1. Properties of extent of the resolvent set and size of the resolvent operator of A correspond to properties relating to the sectors of holomorphy of the semigroups, and their growth near the origin and infinity. In this paper, we deal with semigroups having two different types of growth properties. In the first instance, the semigroup grows near the origin as r^?t, 0 < t < 1. We show that such semigroups are fractional-power semi-groups of operators A, whose resolvents decay as r^?s, 0 < s < 1, in subsectors of the right-hand half-plane. In the second instance, the semigroups are bounded near the origin, and admit special estimates on growth at the periphery of their sectors of definition. We show that for the corresponding A, the resolvent is defined and admits special growth estimates in a region which contains every subsector of the right half-plane; and in these subsectors, the resolvent decays as r^?1.     

Purely imaginary fractional powers of operators

Hydromechanics Institute, Academy of Sciences of the Ukrainian SSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 2, pp. 78–80, April–June, 1991.     

A note on fractional derivatives and fractional powers of operators

Definitions of fractional derivatives and fractional powers of positive operators are considered. The connection of fractional derivatives with fractional powers of positive operators is presented. The formula for fractional difference derivative is obtained.     

On the uniqueness of solutions of a certain characteristic Cauchy problem

The question of uniqueness of solutions of the global Cauchy problem (1)–(2) below is discussed. We assume that there exists a complex constant c such that the modified equation $$\frac{{\partial ^2 u}}{{\partial t^2 }} = c_{\left| \alpha \right| \leqq 2} \sum a_\alpha (x) D_x^\alpha u$$ becomes hyperbolic. Under this and some other additional conditions (See Condition A in §2) we prove the uniqueness of solutions of the Cauchy problem within the class of functions u(t, x) such that $$|u(t,x)| \leqq C exp(a|x|^2 ) ,$$ C and a being positive constants     

Continuous random walks and fractional powers of operators

We derive a probabilistic representation for the Fourier symbols of the generators of some stable processes. This short paper represents a bridge between probabilists and researchers working in PDE?s.     

On the Cauchy problem for a semilinear fractional elliptic equation

We study, for the first time in the literature on the subject, the Cauchy problem for a semilinear fractional elliptic equation. Under an a priori assumption on the solution, we propose the Fourier truncation method for stabilizing the ill-posed problem. A stability estimate of logarithmic type is established.     

A formula for the powers of fractional order operators

We derive none some explicit formula for the power of fractional order (differential and integral) operators.     

On the Cauchy problem for integro-differential operators in Sobolev classes and the martingale problem

The existence and uniqueness in Sobolev spaces of solutions of the Cauchy problem to parabolic integro-differential equation with variable coefficients of the order α∈(0,2)$\alpha \in (0,2)$ is investigated. The principal part of the operator has kernel m(t,x,y)/|y|^d+α$m(t,x,y)/{|y|}^{d+\alpha }$ with a bounded nondegenerate m, Hölder in x and measurable in y. The lower order part has bounded and measurable coefficients. The result is applied to prove the existence and uniqueness of the corresponding martingale problem.     

On the Cauchy problem for linear hyperbolic functional-differential equations

We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing literature.     

On the nonlocal Cauchy problem for semilinear fractional order evolution equations

In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O’Regan fixed point theorem.