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Positive solutions of secondorder nonlocal boundary value problem with singularities in space variables
Boundary Value Problems volume 2014, Article number: 200 (2014)
Abstract
We discuss a nonlocal boundary value problem of secondorder, where the involved nonlinearity depends on the derivative and may be singular. The boundary conditions are given by RiemannStieltjes integrals. We establish sufficient conditions for the existence of positive solutions of the considered problem. Our approach is based on the KrasnoselskiiGuo fixed point theorem on cone expansion and compression.
MSC: 34B10, 34B16, 34B18.
1 Introduction
In the paper we are interested in the existence of positive solutions for the following singular nonlocal boundary value problem (BVP):
Throughout the paper we assume that:
(H1) a>0 and b>0,
(H2) f is continuous and nonnegative on [0,1]\times (0,\mathrm{\infty})\times (0,\mathrm{\infty}),
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 RiemannStieltjes integrals
such that:
(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 [1] and [2] and in the recent papers; see for example [3]–[11] and [12]. Some of the techniques applied to the singular BVPs are based on the fixed point theorems in cones (see [3]–[6] and [12]). For other methods, including LeraySchauder alternative and a priori bounds, see for example [2], [7], [8], [10], [13] and the references therein.
We point out that both regular and singular BVPs under the BCs involving RiemannStieltjes integrals are extensively discussed objects. We refer the reader to [9], [11], [14], [15] and [16] for some recent results on this topic.
A direct inspiration for studying (1) in the present paper were the problems considered in [4] and [5]. In [4], Yan, O’Regan and Agarwal dealt with the following local singular BVP:
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 [5], Infante studied the following nonlocal 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 [0,1]\times [0,\mathrm{\infty}) (see also [17], [18]).
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 nonlocal boundary conditions. The main idea of our method is to restrict the singular nonlinear term f to an appropriately chosen subset [0,1]\times [{\rho}_{1},\mathrm{\infty})\times [{\rho}_{2},\mathrm{\infty}) of [0,1]\times (0,\mathrm{\infty})\times (0,\mathrm{\infty}). Then, following to some extent the approach developed by Webb and Infante in [19], we study the existence of fixed points of a perturbed Hammerstein integral operator of the form
where G(t,s) is the Green’s function of the problem
and γ and δ are the unique solutions of
respectively. Clearly, \gamma (t)=\frac{1}{a} and \delta (t)=t+\frac{b}{a}. Throughout the paper we work under assumption (see for example [19])
(H4) (1\alpha [\gamma ])(1\beta [\delta ])\alpha [\delta ]\beta [\gamma ]\ne 0.
This implies that (1) is nonresonant, 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 KrasnoselskiiGuo fixed point theorem on cone expansion and compression (see [20]). It is well known that the key step when one applies the KrasnoselskiiGuo 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 [0,1]\times [{\rho}_{1},R]\times [{\rho}_{2},R] with R being a suitable chosen positive constant. In this way we can deal with f singular in both its space variables.
2 Preliminaries
Let {\rho}_{1},{\rho}_{2}>0. Denote by \tilde{f} the restriction of f to [0,1]\times [{\rho}_{1},\mathrm{\infty})\times [{\rho}_{2},\mathrm{\infty}). Clearly, \tilde{f} is continuous and nonnegative on [0,1]\times [{\rho}_{1},\mathrm{\infty})\times [{\rho}_{2},\mathrm{\infty}) and if {u}_{0} is a positive solution of the following regular BVP:
then {u}_{0}(t)\ge {\rho}_{1}>0 and {u}_{0}^{\prime}(t)\ge {\rho}_{2}>0, so {u}_{0} is a positive solution of (1).
In what follows we will employ the Green’s function G of the homogeneous BVP (4) corresponding to (5). It is easy to check that, under (H1), G is given by the formula (see [4])
Then
and
for t,s\in [0,1]. Now we recall some standard facts on cone theory in Banach spaces.
Definition 1
A nonempty subset P, P\ne \{0\}, of a real Banach space E is called a cone if P is closed, convex and

(i)
\lambda u\in P for all u\in P and \lambda \ge 0,

(ii)
if u,u\in P, then u=0.
Our existence result on positive solutions for (5) is based on the following KrasnoselskiiGuo fixed point theorem on cone expansion and compression.
Theorem 1
[20]
Let P be a cone in a Banach space E and let{\mathrm{\Omega}}_{1}, {\mathrm{\Omega}}_{2}be open bounded subsets of E with0\in {\mathrm{\Omega}}_{1}and{\overline{\mathrm{\Omega}}}_{1}\subset {\mathrm{\Omega}}_{2}. IfF:P\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1})\to Pis a completely continuous operator such that either
1^{∘}: \parallel Fu\parallel \le \parallel u\parallelfor everyu\in P\cap \partial {\mathrm{\Omega}}_{1}and\parallel Fu\parallel \ge \parallel u\parallelfor everyu\in P\cap \partial {\mathrm{\Omega}}_{2}or
2^{∘}: \parallel Fu\parallel \le \parallel u\parallelfor everyu\in P\cap \partial {\mathrm{\Omega}}_{2}and\parallel Fu\parallel \ge \parallel u\parallelfor everyu\in P\cap \partial {\mathrm{\Omega}}_{1},
then F has a fixed point inP\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1}).
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
and
Observe that (H1), (H2), and (H3) imply
In addition to (H1)(H4), we make the following assumptions on the function \tilde{f}, the functionals α and β, and the coefficients a and b that appear in (5).
We assume there exist 0<r<R and M,m>0 such that:
(H5) \frac{b}{a+b}min\{1,\frac{b}{a}\}r\ge {\rho}_{1} and cmin\{1,\frac{b}{a}\}r\ge {\rho}_{2}.
(H6) \tilde{f}(t,u,v)\le RM for (t,u,v)\in [0,1]\times [R\frac{b}{a+b},R]\times [Rcmin\{1,\frac{b}{a}\},R].
(H7) \frac{1}{a}{\int}_{0}^{1}dA(s)+(1+\frac{b}{a}){\int}_{0}^{1}dB(s)+Mmax\{\frac{1}{2}+\frac{b}{a},1\}\le 1.
(H8) \tilde{f}(t,u,v)\ge rm for (t,u,v)\in [0,1]\times [r\frac{b}{a+b},r]\times [rcmin\{1,\frac{b}{a}\},r].
(H9) \frac{b}{a+b}min\{1,\frac{b}{a}\}[\frac{1}{a}{\int}_{0}^{1}dA(s)+(1+\frac{b}{a}){\int}_{0}^{1}dB(s)]+\frac{3}{2}m\ge 1.
Theorem 2
Under the assumptions (H1)(H9), the regular BVP (5) has a solution u, positive on[0,1], with
and
Proof
Let {C}^{1}[0,1] denote a Banach space of continuously differentiable functions with the norm
where
Let
Then P is a cone in {C}^{1}[0,1]. Observe that the constant c that appears in P involves the maximum {M}_{R} of \tilde{f} on the set [0,1]\times [{\rho}_{1},R]\times [{\rho}_{2},R] (see (7) and (8)). Moreover, if u\in P, then u is increasing on [0,1] and
Hence
We also have
Hence
Let
For u\in P\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1}) consider the operator (3)
that is,
It is clear that every fixed point of F is a solution of (5) (see for example [5] and [19]). We will show that F fulfills the assumptions of Theorem 1. First we prove that F:P\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1})\to P. If u\in P\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1}) then Fu\in {C}^{1}[0,1] and by (6) we have
Since
we get
To show that {(Fu)}^{\prime}(t)\ge c{\parallel Fu\parallel}_{\mathrm{\infty}}, we observe first that for t\in [0,1] we have
On the other hand, (6) and (7) give
By (H3) and (10) we obtain
and
Moreover, we can rewrite (8) as
Then (9) combined with (12), (13), and (14) implies
and therefore
which gives {(Fu)}^{\prime}(t)\ge c{\parallel Fu\parallel}_{\mathrm{\infty}}. Thus F maps P\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1}) to P. By standard arguments we can show that F is completely continuous on P\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1}). Let u\in P\cap \partial {\mathrm{\Omega}}_{2}. Then, in particular, \parallel u\parallel =R, R\frac{b}{a+b}\le u(t)\le R and Rcmin\{1,\frac{b}{a}\}\le {u}^{\prime}(t)\le R. Since dA and dB are positive measures, we get by (H6)
and
Thus, (H7), (15), and (16) imply
For u\in P\cap \partial {\mathrm{\Omega}}_{1} we have \parallel u\parallel =r, r\frac{b}{a+b}\le u(t)\le r and rcmin\{1,\frac{b}{a}\}\le {u}^{\prime}(t)\le r. Hence, from (H8) and (H9), we obtain
Application of Theorem 1 yields the result. □
Remark 1
In [4] 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 u(t) but for {u}^{\prime}(t) as well (see (10) and (11)). Since it is sufficient for our method to work that F defined in (3) maps P\cap ({\overline{\mathrm{\Omega}}}_{2}\setminus {\mathrm{\Omega}}_{1}) to P, we would like to emphasize here that we do not need F to be positive on P.
Remark 2
Observe that Theorem 2 implies the existence of a positive solution of (1). Indeed, by (H5), a solution u of (5) evidently satisfies u(t)\ge \frac{b}{a+b}min\{1,\frac{b}{a}\}r\ge {\rho}_{1} and {u}^{\prime}(t)\ge cmin\{1,\frac{b}{a}\}r\ge {\rho}_{2} on [0,1].
We conclude this section with one numerical example illustrating Theorem 2. Some calculations have been made here with MAPLE.
Example 1
Consider the following fourpoint boundary value problem:
where the function h is continuous on [0,1] and 1\le h(t)\le 2 on [0,1]. In this case, f(t,u,v)=h(t)(\frac{0.1}{u}+\frac{0.0001}{v}), a=b=1, \alpha [u]=\frac{1}{10}u(\frac{1}{4}) and \beta [u]=\frac{2}{5}u(\frac{1}{2}). Fix {\rho}_{1}=0.1 and {\rho}_{2}=0.0001. For r=\frac{1}{5} and R=3 we have {M}_{R}=4 and c=\frac{2}{319}\approx 0.0063 and we can take M=\frac{1}{20} and m=2. By Theorem 2, the BVP (17) has a solution u such that \frac{1}{5}\le \parallel u\parallel \le 3, \frac{1}{10}\le u(t)\le 3 and \frac{6}{319}\le {u}^{\prime}(t)\le 3 on [0,1].
Author’s contributions
The sole author personally prepared the manuscript.
References
O’Regan D, Agarwal RP: Singular Differential and Integral Equations with Applications. Kluwer, Dordrecht; 2003.
Rachůnková I, Staněk S, Tvrdý M: Solvability of Nonlinear Singular Problems for Ordinary Differential Equations. Hindawi, New York; 2008.
Liu Z, Ume JS, Anderson DR, Kang SM: Twin monotone positive solutions to a singular nonlinear thirdorder differential equation. J. Math. Anal. Appl. 2007, 334: 299313. 10.1016/j.jmaa.2006.12.067
Yan B, O’Regan D, Agarwal RP: Multiple positive solutions of singular second order boundary value problems with derivative dependence. Aequ. Math. 2007, 74: 6289. 10.1007/s000100062850x
Infante, G: Positive solutions of nonlocal boundary value problems with singularities. Discrete Contin. Dyn. Syst. suppl., 377384 (2009)
Suna Y, Liu L, Zhanga J, Agarwal RP: Positive solutions of singular threepoint boundary value problems for secondorder differential equations. J. Comput. Appl. Math. 2009, 230: 738750. 10.1016/j.cam.2009.01.003
Chu J, Fan N, Torres PJ: Periodic solutions for second order singular damped differential equations. J. Math. Anal. Appl. 2012, 388: 665675. 10.1016/j.jmaa.2011.09.061
FewsterYoung N, Tisdell CC: The existence of solutions to secondorder singular boundary value problems. Nonlinear Anal. 2012, 75: 47984806. 10.1016/j.na.2012.03.029
Webb JRL: Existence of positive solutions for a thermostat model. Nonlinear Anal., Real World Appl. 2012, 13: 923938. 10.1016/j.nonrwa.2011.08.027
Rachůnková I, Spielauer A, Staněk S, Weinmüller EB: Positive solutions of nonlinear Dirichlet BVPs in ODEs with time and space singularities. Bound. Value Probl. 2013., 2013: 10.1186/1687277020136
Jankowski T: Positive solutions to SturmLiouville problems with nonlocal boundary conditions. Proc. R. Soc. Edinb., Sect. A 2014, 144: 119138. 10.1017/S0308210512000960
Yao Q: Triple positive periodic solutions of nonlinear singular secondorder boundary value problems. Acta Math. Sin. 2014, 30: 361370. 10.1007/s1011401312914
Kiguradze IT, Shekhter BL: Singular boundary value problems for secondorder ordinary differential equations. J. Sov. Math. 1988, 43: 23402417. 10.1007/BF01100361
Jankowski T: Existence of positive solutions to third order differential equations with advanced arguments and nonlocal boundary conditions. Nonlinear Anal. 2012, 75: 913923. 10.1016/j.na.2011.09.025
Webb JRL, Zima M: Multiple positive solutions of resonant and nonresonant nonlocal fourthorder boundary value problems. Glasg. Math. J. 2012, 54: 225240. 10.1017/S0017089511000590
Infante G, Pietramala P, Venuta M: Existence and localization of positive solutions for a nonlocal BVP arising in chemical reactor theory. Commun. Nonlinear Sci. Numer. Simul. 2014, 19: 22452251. 10.1016/j.cnsns.2013.11.009
Lan KQ: Multiple positive solutions of Hammerstein integral equations and applications to periodic boundary value problems. Appl. Math. Comput. 2004, 154: 531542. 10.1016/S00963003(03)007331
Guo Y, Ge W: Positive solutions for threepoint boundary value problems with dependence on the first order derivative. J. Math. Anal. Appl. 2004, 290: 291301. 10.1016/j.jmaa.2003.09.061
Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc. 2006, 74: 673693. 10.1112/S0024610706023179
Guo D, Lakshmikantham V: Nonlinear Problems in Abstract Cones. Academic Press, San Diego; 1988.
Acknowledgements
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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Zima, M. Positive solutions of secondorder nonlocal boundary value problem with singularities in space variables. Bound Value Probl 2014, 200 (2014). https://doi.org/10.1186/s1366101402009
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DOI: https://doi.org/10.1186/s1366101402009