Results Math 72 (2017), 369–383
2016 Springer International Publishing
published online October 24, 2016
Results in Mathematics
Generalized Hammerstein Equations
, Lingju Kong , and Feliz Minh´os
Abstract. In this paper the authors study the Hammerstein generalized
k(t, s) g(s) f (s, u(s),u
where k :[0, 1]
→ R are kernel functions, m ≥ 1, g :[0, 1] → [0, ∞), and
f :[0, 1] ×R
→ [0, ∞)isaL
−Carath´eodory function. The existence
of solutions of integral equations has been studied in concrete and abstract
cases, by diﬀerent methods and techniques. However, in the existing liter-
ature, the nonlinearity depends only on the unknown function. This paper
is one of a very few to consider equations having discontinuous nonlinear-
ities that depend on the derivatives of the unknown function and having
discontinuous kernels functions that have discontinuities in the partial
derivatives with respect to their ﬁrst variable. Our approach is based on
the Krasnosel’ski˘ı–Guo compression/expansion theorem on cones and it
can be applied to boundary value problems of arbitrary order n>m.
The last two sections of the paper contain an application to a third order
nonlinear boundary value problem and a concrete example.
Mathematics Subject Classiﬁcation. Primary 45G10; Secondary 34B15,
Keywords. Hammerstein integral equation, Krasnosel’ski˘ı–Guo theorem,
Boundary value problems, Discontinuous kernels, Nonlinearities depend-
ing on derivatives.
F. Minh´os was supported by National Founds through FCT-Funda¸c˜ao para a Ciˆencia e a
Tecnologia, project SFRH/BSAB/114246/2016.