- Open Access
Positive solutions of second-order non-local boundary value problem with singularities in space variables
© Zima; licensee Springer 2014
- Received: 30 January 2014
- Accepted: 12 August 2014
- Published: 25 September 2014
We discuss a non-local boundary value problem of second-order, where the involved nonlinearity depends on the derivative and may be singular. The boundary conditions are given by Riemann-Stieltjes integrals. We establish sufficient conditions for the existence of positive solutions of the considered problem. Our approach is based on the Krasnoselskii-Guo fixed point theorem on cone expansion and compression.
MSC: 34B10, 34B16, 34B18.
- singular boundary value problem
- positive solution
Throughout the paper we assume that:
(H1) and ,
(H2) f is continuous and nonnegative on ,
(H3) A and B are of bounded variation, and dA and dB are positive measures.
Many interesting results on the existence of solutions for the BVPs singular in the independent and/or the dependent variables can be found in the monographs  and  and in the recent papers; see for example – and . Some of the techniques applied to the singular BVPs are based on the fixed point theorems in cones (see – and ). For other methods, including Leray-Schauder alternative and a priori bounds, see for example , , , ,  and the references therein.
We point out that both regular and singular BVPs under the BCs involving Riemann-Stieltjes integrals are extensively discussed objects. We refer the reader to , , ,  and  for some recent results on this topic.
with f singular in its space variable. This time the fixed point index technique was employed together with the truncation method, that is, the singular nonlinear term f was extended to all of (see also , ).
respectively. Clearly, and . Throughout the paper we work under assumption (see for example )
has only the trivial solution. In order to prove the existence of a fixed point of (3) we make use of the Krasnoselskii-Guo fixed point theorem on cone expansion and compression (see ). It is well known that the key step when one applies the Krasnoselskii-Guo result is to find a suitable cone. We would like to point out here that in our case the choice of a cone is determined not only by the properties of the Green’s function of (4) as it can be frequently found in the literature. The technique we use essentially takes into account the upper bound of the term f on with R being a suitable chosen positive constant. In this way we can deal with f singular in both its space variables.
then and , so is a positive solution of (1).
for . Now we recall some standard facts on cone theory in Banach spaces.
for all and ,
if , then .
Our existence result on positive solutions for (5) is based on the following Krasnoselskii-Guo fixed point theorem on cone expansion and compression.
Let P be a cone in a Banach space E and let, be open bounded subsets of E withand. Ifis a completely continuous operator such that either
1∘: for everyandfor everyor
2∘: for everyandfor every,
then F has a fixed point in.
In addition to (H1)-(H4), we make the following assumptions on the function , the functionals α and β, and the coefficients a and b that appear in (5).
We assume there exist and such that:
(H5) and .
(H6) for .
(H8) for .
Application of Theorem 1 yields the result. □
which provides the lower bound not only for but for as well (see (10) and (11)). Since it is sufficient for our method to work that F defined in (3) maps to P, we would like to emphasize here that we do not need F to be positive on P.
We conclude this section with one numerical example illustrating Theorem 2. Some calculations have been made here with MAPLE.
where the function h is continuous on and on . In this case, , , and . Fix and . For and we have and and we can take and . By Theorem 2, the BVP (17) has a solution u such that , and on .
The sole author personally prepared the manuscript.
This work was partially supported by the Centre for Innovation and Transfer of Natural Science and Engineering Knowledge of University of Rzeszów. The author is grateful to both reviewers for their careful reading of the manuscript and helpful comments.
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