Existence of mild solution for evolution equation with Hilfer fractional derivative

The paper is concerned with existence of mild solution of evolution equation with Hilfer fractional derivative which generalized the famous Riemann–Liouville fractional derivative. By noncompact measure method, we obtain some sufficient conditions to ensure the existence of mild solution. Our results are new and more general to known results. Nowadays, fractional calculus receives increasing attention in the scientific community, with a growing number of applications in physics, electrochemistry, biophysics, viscoelasticity, biomedicine, control theory, signal processing, etc(see [22,29] and the references therein). Fractional differential equations also have been proved to be useful tools in the modeling of many phenomena in various fields of science and engineering. There has been a significant development in fractional differential equations in recent years, see the monographs of Kilbas et al. A strong motivation for investigating fractional evolution equations comes from physics. Fractional diffusion equations are abstract partial differential equations that involve fractional derivative in space and time. For example, EI-Sayed [10] discussed fractional order diffusion-wave equation. Eidelman and Kochubei [11] investigated the Cauchy problem for fractional diffusion equation. As stated in [11], fractional diffusion equations describe anomalous diffusion on fractals. Physical objects of fractional dimension, like some amorphous semiconductors or strongly porous materials. This class of equations can provide a nice instrument for the description of memory and hereditary properties of various materials and processes. Some recent papers investigated the problem of the existence of mild solution for abstract differential equations with fractional derivative [2,8,15,25]. Since the mild solution definition in integer order abstract differential equations obtained by variation of constant formulas can not be generalized directly to fractional order abstract differential equations, Zhou and Jiao [30] gave a suit concept on mild solutions by applying laplace transform and probability density functions for evolution equation with Caputo fractional derivative. Using the same method, Zhou et al. [31] gave a suit concept on mild solutions for evolution equation with Riemann–Liouville fractional derivative. By using sectorial operator, Su et al. [25] gave a definition of mild solution for fractional differential equation with order 1 < a < 2 and investigated it's existence. Agarwal et al. [2] studied the existence and dimension of the set for mild solutions of semilinear fractional differential equations inclusions. Wang [26] researched the abstract fractional Cauchy problem with almost sectorial operators. On the other hand, Hilfer [14]