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Positive solutions of second-order non-local boundary value problem with singularities in space variables
Boundary Value Problems volume 2014, Article number: 200 (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.
In the paper we are interested in the existence of positive solutions for the following singular non-local boundary value problem (BVP):
Throughout the paper we assume that:
(H1) and ,
(H2) f is continuous and nonnegative on ,
and we consider f to be singular at the value 0 of its space variables, that is, f may be singular in its second and third variable. The boundary conditions (BCs) involve linear functionals given by Riemann-Stieltjes integrals
(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.
They established the existence of multiple positive solutions using the fixed point index technique combined with the approximation of the singular BVP (2) by an appropriate sequence of regular BVPs. The nonlinearity f in (2) allowed to be singular in its second and third variable. In , Infante studied the following non-local singular BVP:
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 , ).
The aim of our paper is to establish sufficient conditions for the existence of positive solutions for (1), that is, for the singular BVP with the derivative dependence and non-local boundary conditions. The main idea of our method is to restrict the singular nonlinear term f to an appropriately chosen subset of . Then, following to some extent the approach developed by Webb and Infante in , we study the existence of fixed points of a perturbed Hammerstein integral operator of the form
where is the Green’s function of the problem
and γ and δ are the unique solutions of
respectively. Clearly, and . Throughout the paper we work under assumption (see for example )
This implies that (1) is non-resonant, that is, the following BVP:
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.
Let . Denote by the restriction of f to . Clearly, is continuous and nonnegative on and if is a positive solution of the following regular BVP:
then and , so is a positive solution of (1).
for . Now we recall some standard facts on cone theory in Banach spaces.
A nonempty subset P, , of a real Banach space E is called a cone if P is closed, convex and
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.
3 Existence result for the regular BVP
In this section we state a result for the existence of a positive solution of (5). For positive numbers r and R we set
Observe that (H1), (H2), and (H3) imply
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 .
Under the assumptions (H1)-(H9), the regular BVP (5) has a solution u, positive on, with
Let denote a Banach space of continuously differentiable functions with the norm
We also have
For consider the operator (3)
It is clear that every fixed point of F is a solution of (5) (see for example  and ). We will show that F fulfills the assumptions of Theorem 1. First we prove that . If then and by (6) we have
To show that , we observe first that for we have
By (H3) and (10) we obtain
Moreover, we can rewrite (8) as
which gives . Thus F maps to P. By standard arguments we can show that F is completely continuous on . Let . Then, in particular, , and . Since dA and dB are positive measures, we get by (H6)
For we have , and . Hence, from (H8) and (H9), we obtain
Application of Theorem 1 yields the result. □
In  the authors used the cone
The cone we consider in the proof of Theorem 2 is of the form
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.
Consider the following four-point boundary value problem:
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.
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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.
The author declares that she has no competing interests.