Existence of solutions for a BVP of a second order FDE at resonance by using Krasnoselskii's fixed point theorem on cones in the L1 space
Karakostas, George L.
Texas State University, Department of Mathematics
We provide sufficient conditions for the existence of positive solutions of a nonlocal boundary value problem at resonance concerning a second order functional differential equation. The method is developed by inserting an exponential factor which depends on a suitable positive parameter λ. By this way a Green's kernel can be established and the problem is transformed into an operator equation u = Tλu. As it can be shown the well known Krasnoselskii's fixed point theorem is no (positive) value of the parameter λ for which the condensing property ∥Tλu∥ ≤ ∥u∥, with ∥u∥ = ρ(> 0) is satisfied. To overcome this face we enlarge the space C[0, 1] and work in L1[0, 1] where, now, Krasnoselskii's fixed point theorem is applicable. Compactness criteria in this space are, certainly, needed.
Nonlocal boundary value problem, Boundary value problems at resonance, Second order differential equations, Krasnoselskii's fixed point theorem on cones
Karakostas, G. L., & Palaska, K. G. (2018). Existence of solutions for a BVP of a second order FDE at resonance by using Krasnoselskii's fixed point theorem on cones in the L1 space. <i>Electronic Journal of Differential Equations, 2018</i>(30), pp. 1-17.