Electron. J. Differential Equations,
Vol. 2018 (2018), No. 30, pp. 117.
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
.
By this way a Green's kernel can be established and the problem is transformed
into an operator equation
.
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
for which the condensing property
, with
is satisfied. To overcome this fact we enlarge the space
and work in
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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