Xuping Zhang, Youhui Su, and Yibo Kong.
Positive Solutions for Boundary Value Problem of Fractional Differential Equation in Banach Spaces.
Dynamic Systems and Applications 30 (2021) No. 9, 1449-1461

https://doi.org/10.46719/dsa202130.09.04

ABSTRACT.
In this paper, we consider the existence of positive solutions for the following boundary
value problem of fractional differential equations in Banach space E

D^α u(t) = f(t, u(t)), t ϵ I,

lim t^{1 – α} u(t) = w^{1 – α} u(w),
t → 0⁺

where 0 < α ≤ 1 is real number, I = (0; w], D^α is the Riemann-Liouville fractional derivative, f : I x K → K is continuous, K is a normal cone in Banach space E. Under more general growth and noncompactness measure conditions about nonlinearity f, we obtained the existence of positive solutions by applying the fixed point index theory of condensing mapping.

AMS (MOS) Subject Classification. 34B15, 34B18, 47H08.
Key Words and Phrases. Fractional differential equation, Positive solutions, Cone, Measure of noncompactness, Condensing mapping.