This paper treats the approximate controllability of fractional differential systems of Sobolev type in Banach spaces. We first characterize the properties on the norm continuity and compactness of some resolvent operators (also called solution operators). And then via the obtained properties on resolvent operators and fixed point technique, we give some approximate controllability results for Sobolev type fractional differential systems in the Caputo and Riemann-Liouville fractional derivatives with order 1 <

For f and g polynomials in p variables, we relate the special value at a non-positive integer s = −N , obtained by analytic continuation of the Dirichlet series ζ(s; f, g) = ∞ k1=0

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In this article, we investigate the existence and uniqueness of solutions of linear and semilinear second–order equations involving time scales. To obtain such results, we make use of exponential dichotomy and fixed point results. Also, we present some examples and applications to illustrate our main results.

cosine and sine functions defined on a Banach space are useful tools in the study of wide classes of abstract evolution equations. In this paper, we introduce a definition of cosine and sine functions on time scales, which unify the continuous, discrete and the cases "in between." Our definition includes several types of time scales such as real numbers set, integers numbers set, quantum scales, among others. We investigate the relationship between the cosine function on time scales and its infinitesimal generator, proving several properties concerning it. Also, we investigate the sine functions on time scales, presenting their main properties. Finally, we apply our theory to study the homogeneous and inhomogeneous abstract Cauchy problem on time scales in Banach spaces.