Electron. J. Differential Equations, Vol. 2018 (2018), No. 30, pp. 1-17.

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

George L. Karakostas, Konstantina G. Palaska

Abstract:
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 $\lambda$. By this way a Green's kernel can be established and the problem is transformed into an operator equation $u=T_{\lambda}u$. As it can be shown the well known Krasnoselskii's fixed point theorem on cones in the Banach space C[0,1] cannot be applied. More exactly, there is no (positive) value of the parameter $\lambda$ for which the condensing property $\|T_{\lambda}u\|\leq\|u\|$, with $\|u\|=\rho(>0)$ is satisfied. To overcome this fact we enlarge the space $C[0,1]$ and work in ${\mathcal{L}^1[0,1]}$ where, now, Krasnoselskii's fixed point theorem is applicable. Compactness criteria in this space are, certainly, needed.

Submitted July 20, 2017. Published January 19, 2018.
Math Subject Classifications: 34B10, 34B15.
Key Words: Nonlocal boundary value problem; boundary value problems at resonance; second order differential equations; Krasnoselskii's fixed point theorem on cones.

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George L. Karakostas
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: gkarako@uoi.gr
Konstantina G. Palaska
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: cpalaska@cc.uoi.gr

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