On nonlocal problems for fractional differential equations in Banach spaces

Abstract

Abstract. In this paper, we study the existence and uniqueness of solutions to the nonlocal problems for the fractional differential equation in Banach spaces. New sufficient conditions for the existence and uniqueness of solutions are established by means of fractional calculus and fixed point method under some suitable conditions. Two examples are given to illustrate the results.

Weighted fractional differential equations with infinite delay in Banach spaces

“…It is often applied to the theories of differential and integral equations as well as to the operator theory and geometry of Banach spaces [17][18][19][20][21][22]. One of the most important examples of measure of noncompactness is the Hausdorff's measure of noncompactnessˇY , which is defined by Y .B/ D inffr > 0I B can be covered with a finite number of balls of radius equal to rg for bounded set B in a Banach space Y .…”

Decay integral solutions for a class of impulsive fractional differential equations in Banach spaces

“…In next Section 2, we recall some notions and facts related to fractional calculus, fractional resolvent operators, measures of noncompactness and the fixed point theory for condensing maps. For completeness, Section 3 is devoted to the existence result of integral solutions on compact interval under a general setting via measures of noncompactness, which extends/improves some recent results from [10,12,24,29,30] for the non-impulsive case. Section 4 shows the existence and uniqueness of decay integral solution under Lipschitz conditions imposed on nonlinearities.…”

Nonlocal Cauchy problems for fractional order nonlinear differential systems

“…On the other hand, there are few works on the same topic for systems of nonlinear fractional order differential equations,as seen in [35][36][37][38][39][40]. Recently, Dong et al [18] studied the following nonlocal Cauchy problems for fractional order differential equations in a Banach space X :…”

“…Inspired by the approach in [18,41], we construct an important matrix M α β associated with the fractional order and functionals α β and give some sufficient conditions to guarantee M α β convergent to zero matrix. Then, we can apply some possible fixed point theorems via the techniques that use convergent to zero matrix and vector norm to derive the existence results of solutions for the system (3) under different conditions.…”

“…where [18,41], we discuss nonlocal Cauchy problems for fractional order nonlinear differential systems as follows:…”