Existence of mild solutions for fractional evolution equations

Abstract

In this paper, we study the nonlocal Cauchy problems of fractional evolution equations with Riemann-Liouville derivative by considering an integral equation which is given in terms of probability density. By using the theory of Hausdorff measure of noncompactness, we establish various existence theorems of mild solutions for the Cauchy problems in the cases C 0 semigroup is compact or noncompact. 2010 AMS Mathematics subject classification. Primary 26A33, 34A08, 35R11. Keywords and phrases. Fractional evolution equations, integral equations, Riemann-Liouville derivative, mild solutions, C 0 semigroup, measure of noncompactness.

Cauchy problem for nonlinear fractional differential equations with positive constant coefficient

“…For more history and basic results on fractional calculus theory, one can see monograph [1][2][3][4][5][6][7][8] and the references therein. Many researchers studied Cauchy problem and long time behavior for nonlinear fractional differential equations and obtained many interesting results by using all kinds of fixed point theorems (see [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]). …”

Existence of mild solution for evolution equation with Hilfer fractional derivative

“…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.…”

Existence and Attractivity for Fractional Evolution Equations

“…On the other hand, the existence theory of solutions for time fractional evolution equations has been investigated intensively by many authors; for example, see Kim et al [16], Bazhlekova [22], Wang et al [23], Zacher [24], and Zhou et al [25]. However, to the best of our knowledge, there are no results on the attractivity of solutions for fractional evolution equations in the literature.…”