# Sign-constancy of Green’s functions for impulsive nonlocal boundary value problems

- 49 Downloads

**Part of the following topical collections:**

## Abstract

In this paper we consider sufficient conditions of nonpositivity of Green’s function for impulsive differential equation with nonlocal boundary conditions.

## Keywords

Second order impulsive differential equations Boundary value problems Sign-constancy of Green’s functions## 1 Introduction

Impulsive differential equations have attracted an attention of a number of recognized mathematicians and have applications in many spheres of science from physics, biology, medicine to economical studies. The following well-known books can be noted in this context [35, 41, 44, 54]. In Ref. [4], the concept of the general theory of functional differential equations was presented. In the frame of this concept finite fundamental system, the Wronskian and Green’s functions can be introduced. On the basis of this concept nonoscillation for the first order impulsive functional differential equations was considered in [10], where positivity of the Cauchy and Green’s functions of the periodic problem was firstly studied. Nonoscillation for the first order impulsive differential equations is also considered in the book [1]. The positivity of Green’s function of one- and two-point boundary value problems for impulsive functional differential equations of the first order was considered in [10] and of the second order in [2, 11, 12, 14, 19, 24, 28, 29, 36, 38].

Various applications with nonlocal problems for ordinary and partial differential equations were presented in the known works by Skubachevskii [45, 46]. Ordinary differential equations with integral boundary conditions arise in the theory of turbulence [47], in the theory of Markov processes [18], in heat flow problems [22, 25, 27, 48, 49], in the study of the response of a spherical cap [3, 5, 40]. In the references in [7], one can find works on applications of nonlocal problems in modeling of thermostats, beams and suspension bridges.

Questions of representation solutions and solvability of nonlocal problems for functional differential equations were considered in [4, 23, 33, 34]. The positivity of solutions for nonlocal boundary value problems for ordinary differential equations was studied in [20, 21, 50, 51, 53]. The method is to reduce nonlocal boundary value problems to the Hammerstein integral equation and then scrupulous analysis of Green’s functions leads researchers to estimates (of the norm or spectral radii in linear case and the fact of a contraction in nonlinear one) of integral operators and conclusions about positivity of solutions. It seems that some of these results can be generalized also on particular cases of delay or functional differential equations, where Green’s functions of ordinary differential equations could be used [1, 16, 17, 20, 21, 23]. For functional differential equations, forms of Green’s functions are essentially more complicated. That is why quite a different approach was proposed for nonlocal problems with functional differential equations [1, 9], where various results on positivity/negativity of Green’s functions were obtained. One of the main ideas is to obtain a connection between sign-constancy of Green’s functions for different problems with functional differential equations. This approach presents a basic method for analysis of the solution’s positivity (see, for example, Theorem 15.3 in [1]). The main results are obtained in the form of theorems about differential inequalities. Choosing the test functions, researchers can get coefficient tests for positivity of Green’s functions. Note that all these works concern positivity of solutions to nonlocal problems only for scalar differential equations and not for systems. There are almost no results on positivity of solutions of nonlocal problems in the case of systems. Among the results we can note results on the existence [7, 26, 33, 34, 43] and results on the positivity of solution-vectors in [26].

*f*, \(a_{j}\), \(b_{j}\): \([0,\omega ] \to \mathbb{R}\) are summable functions and \(\tau _{j}\), \(\theta _{j}\): \([0,\omega ] \to [0,+\infty )\) are measurable functions for \(j=1,2,\ldots,p\),

*p*and

*r*are natural numbers, \(\gamma _{k}\) and \(\delta _{k}\) are real positive numbers.

Let \(D(t_{1},t_{2},\ldots, t_{r})\) be a space of functions *x*: \([0,\omega ] \to \mathbb{R}\) such that their derivative \(x'(t)\) is absolutely continuous on every interval \(t \in [t_{i}, t_{i+1})\), \(i = 0, 1,\ldots, r\), \(x'' \in L_{\infty }\), we assume also that there exist the finite limits \(x(t_{i}-0) = \lim_{t \to t_{i}^{-}} x(t)\) and \(x'(t_{i}-0) = \lim_{t \to t_{i}^{-}} x'(t)\) and condition (1.3) is satisfied at points \(t_{i}\) (\(i = 0, 1,\ldots,r\)). As a solution *x* we understand a function \(x \in D(t_{1},t_{2},\ldots, t_{r})\) satisfying (1.2)–(1.4).

In this paper we obtain new results on sign-constancy of Green’s functions of nonlocal boundary value problems. Comparing our results with [25, 26, 27, 50, 51], we study an essentially more general object: impulsive differential equations with delays. Note that combining the approach of [25, 26, 27] and our results, nonlinear nonlocal impulsive functional differential boundary value problems can be considered in future research. Thus, the technique we proposed in this paper opens new opportunities in the study of positivity/negativity of solutions for a wide class of impulsive functional differential equations.

## 2 Preliminaries

- \(\nu _{1}(t)\) is a solution of the homogeneous equation$$\begin{aligned}& (Lx) (t)\equiv x''(t)+\sum _{j=1}^{p} a_{j}(t) x'\bigl(t-\tau _{j}(t)\bigr) + \sum _{j=1}^{p} b_{j}(t) x\bigl(t-\theta _{j}(t)\bigr) = 0, \quad t \in [0,\omega ], \end{aligned}$$(2.2)$$\begin{aligned}& \begin{aligned} & x(t_{k})=\gamma _{k} x(t_{k}-0), \quad\quad x'(t_{k})= \delta _{k} x'(t_{k}-0), \quad k=1,2, \ldots,r, \\ & 0 = t_{0} < t_{1} < t_{2} < \cdots < t_{r} < t_{r+1} = \omega , \end{aligned} \end{aligned}$$(2.3)with the initial conditions \(x(0)=1\), \(x'(0)=0\).$$\begin{aligned}& x(\zeta )=0, \quad\quad x'(\zeta )=0, \quad \zeta < 0, \end{aligned}$$(2.4)
- \(C(t,s)\), called the Cauchy function of (2.2)–(2.4), is the solution of the equation$$\begin{aligned}& (L_{s}x) (t)\equiv x''(t)+ \sum_{j=1}^{p} a_{j}(t) x'\bigl(t-\tau _{j}(t)\bigr) + \sum _{j=1}^{p} b_{j}(t) x\bigl(t-\theta _{j}(t)\bigr) = 0, \quad t \in [s,\omega ], \end{aligned}$$(2.5)where$$\begin{aligned}& \begin{aligned} & x(t_{k})=\gamma _{k} x(t_{k}-0), \quad\quad x'(t_{k})= \delta _{k} x'(t_{k}-0), \quad k=m, \ldots,r, \\ & 0 = t_{0} < t_{1} < t_{2} < \cdots < t _{r} < t_{r+1} = \omega , \end{aligned} \end{aligned}$$(2.6)
*m*is a number, such that \(t_{m-1}< s \leq t_{m}\),satisfying the initial conditions \(C(s,s)=0\), \(C_{t}'(s,s)=1\) and \(C(t,s)=0\) for \(t< s\).$$ x(\zeta )=0, \quad\quad x'(\zeta )=0, \quad \zeta < s, $$(2.7)

Below the following definition will be used.

### Definition 2.1

We call \([0,\omega ]\) a semi-nonoscillation interval of (2.2)–(2.4), if every nontrivial solution having a zero of derivative does not have a zero on this interval.

In Ref. [11] for (1.2)–(1.4) the following assertion was proven.

### Lemma 2.1

*If*

- (1)
\(b_{j}(t) \leq 0\)

*for*\(t \in [0,\omega ]\). - (2)
*The Cauchy function*\(C_{1}(t,s)\)*of the first order impulsive equation*$$ \begin{aligned} & y'(t)+\sum _{j=1}^{p} a_{j}(t) y\bigl(t-\tau _{j}(t)\bigr) =0, \quad t \in [0, \omega ], \\ & y(t_{k})=\delta _{k} y(t_{k}-0), \quad k=1,2,\ldots,r, \\ & y(\zeta )=0, \quad \zeta < 0, \end{aligned} $$(2.8)*is positive for*\(0 \leq s \leq t \leq \omega \).

*Then the Cauchy function*\(C(t,s)\)

*of the second order impulsive equation*(1.2)

*–*(1.4)

*and its derivative*\(C_{t}'(t,s)\)

*are positive in*\(0 \leq s \leq t \leq \omega \).

### Lemma 2.2

*Let*\(0<\delta _{j} \leq 1\)

*for*\(j = 1,\ldots,r\)

*and the following inequality be fulfilled*:

*where*\(B(t) = \prod_{j \in D_{t}} \delta _{j}\), \(D_{t} = \{i : t_{i} \in [t- \tau _{1}(t),t]\}\), \(a_{+}(t) = \max \{a_{1}(t),0\}\)

*and*\(m(t) = \max \{t-\tau _{1}(t),0\}\).

*Then the Cauchy function of the first order impulsive differential equation*(2.9)

*is positive in*\(0 \leq s \leq t \leq \omega \).

In the case when the number of terms with delays \(p>1\) the following sufficient condition of nonnegativity of the Cauchy function \(C_{1}(t,s)\), proven in [10], can be used.

### Lemma 2.3

*Let*\(a_{j}(t) \geq 0\)

*for*\(j=1,\ldots,p\), \(0<\delta _{k} \leq 1\)

*for*\(k = 1,\ldots,r\)

*and*

*then the Cauchy function of the first order impulsive differential equation*(2.8)

*is positive in*\(0 \leq s \leq t \leq \omega \).

## 3 Sign-constancy of Green’s function for nonlocal boundary value problem in the case of \(b_{j}(t) \leq 0\)

For the boundary value problem (1.2)–(1.4), (3.2) the following lemma can be obtained.

### Lemma 3.1

*If the conditions*(1)

*and*(2)

*of Lemma*2.1

*are fulfilled and*\(A(t)\)

*for*\(t \in [0,\omega ]\)

*satisfies the conditions*

*and*\(\gamma _{k} \geq 1\), \(\delta _{k} \geq 0\), \(k=1,2,\ldots,r\),

*then the Green’s function*\(G_{1}(t,s)\)

*exists and there exists an interval*\((0,\epsilon _{s})\),

*such that*\(G_{1}(t,s) <0\)

*for*\(t \in (0,\epsilon _{s})\).

### Proof

This means that there exists an interval \((0,\epsilon _{s})\), such that \(G_{1}(t,s) <0\) for \(t \in (0,\epsilon _{s})\). □

### Remark 3.1

*l*is positive, i.e. \(lx \geq 0\) if \(x \geq 0\), and can be presented in the form of Stieltjes integral \(lx = \int _{0}^{\omega } x(s)\,dR(s)\). In this case, a generalization of Lemma 3.1 can be obtained by repeating the proof for problem (1.2)–(1.4), (3.1). For this case, instead of the condition (3.5), the following ones can be used:

### Theorem 3.1

*Assume that the following conditions are fulfilled*:

- (1)
\(a_{j}(t) \geq 0\), \(b_{j}(t) \leq 0\)

*for*\(t \in [0,\omega ]\). - (2)
*The Wronskian*\(W(t)\)*of the fundamental system of solutions of a homogeneous equation*(2.2)*–*(2.4)*satisfies*\(W(t) \neq 0\), \(t \in [0,\omega ]\). - (3)
*The Cauchy function*\(C_{1}(t,s)\)*of the first order equation*(2.8)*is positive for*\(0 \leq s \leq t \leq \omega \). - (4)
\(A(t)\)

*satisfies*(3.5)*for*\(t \in [0,\omega ]\). - (5)
\(\gamma _{k} \geq 1\), \(\delta _{k} \geq 0\), \(k=1,2,\ldots,r\).

*Then*\(G_{1}(t,s) \leq 0\)

*for*\(t,s \in [0,\omega ]\).

In order to prove Theorem 3.1, we will need the following lemmas from [14].

### Lemma 3.2

*Assume that the following conditions are fulfilled*:

- (1)
\(a_{j}(t) \geq 0\), \(b_{j}(t) \leq 0\), \(j = 1,\ldots,p\), \(t \in [0,\omega ]\).

- (2)
*The Wronskian*\(W(t)\)*of the fundamental system of solutions of homogeneous equation*(2.2)*–*(2.4)*satisfies the inequality*\(W(t) \neq 0\), \(t \in [0,\omega ]\). - (3)
*The Cauchy function*\(C_{1}(t,s)\)*of the first order equation*(2.8)*is positive for*\(0 \leq s \leq t \leq \omega \).

*Then the interval*\([0,\omega ]\)

*is a semi*-

*nonoscillation interval of*(2.2)

*–*(2.4).

### Lemma 3.3

*If*\(a_{j}(t) \geq 0\), \(b_{j}(t) \leq 0\),

*then the following assertions are equivalent*:

- (a)
*The Wronskian*\(W(t)\)*of the fundamental system of solutions of a homogeneous equation*(2.2)*–*(2.4)*satisfies*\(W(t) \neq 0\), \(t \in [0,\omega ]\). - (b)
*The Green’s function*\(G^{\xi }(t,s)\)*with boundary conditions*\(x(\xi )=0\), \(x'(\xi ) =0\)*is nonnegative for*\(t,s \in [0,\xi ]\)*for every*\(0<\xi <\omega \).

### Proof of Theorem 3.1

*t*which are close to 0. Thus, there exist the points

*η*and

*ξ*, such that \(x(\eta )=0\), \(x'(\xi )=0\), \(0<\eta <\xi <\omega \), and \(x(t)>0\) for \(t \in (\eta , \xi )\).

*ξ*follows from the inequalities (3.5). Actually, if there is no such point

*ξ*, the solution is nondecreasing for \(t \in [\eta , \omega ]\), since the condition (5) of Theorem 3.1 is fulfilled, and, in this case, the equality \(x(\omega ) = \int _{0}^{\omega } A(s)x(s)\,ds\) is impossible. The solution \(x(t)\) satisfies the conditions

Consider now the problem (1.2)–(1.4) with the boundary conditions \(x(\xi )=0\), \(x'(\xi )=0\). According to Lemma 3.3, its Green’s function \(G^{\xi }(t,s)\) is nonnegative for \(t,s \in [0, \omega ]\).

The first term \(\int _{0}^{\xi } G^{\xi }(t,s)f(s)\,ds \geq 0\), since \(G^{\xi }(t,s)\geq 0\), \(f(s) \geq 0\), \(t,s \in [0, \xi ]\).

We see that all the conditions of Lemma 3.2 are fulfilled. This means that the interval \([0,\omega ]\) is a semi-nonoscillation interval of (2.2)–(2.4). The solution \(X(t)\) of homogeneous equation (2.2)–(2.4) cannot change its sign. Thus, the solution \(x(t)\) is nonnegative as the sum of two nonnegative terms.

We got the contradiction with the assumption about changing sign of the solution \(x(t)\). This proves that \(G_{1}(t, s)\) should be nonpositive. □

### Remark 3.2

The conditions (2)–(3) of Theorem 3.1 look very difficult for verification, but the known previous results demonstrate that this is not the case. Lemmas 2.2–2.3 give simple inequalities implying positivity of the Cauchy function \(C_{1}(t,s)\). Tests of the nonnegativity of Green’s function \(G^{\xi }(t,s)\) were obtained in [12, 14].

### Remark 3.3

For boundary value problem (1.2)–(1.4), (3.1), an analogue of Theorem 3.1 can be formulated, where instead of the condition (3.5), the functional *l* satisfies the condition (3.8).

### Example 3.1

Thus, the conditions (1)–(3), (5) of Theorem 3.1 are fulfilled. According to Theorem 3.1, for each \(A(t)\), satisfying the condition (3.5), the Green’s function \(G_{1}(t,s)\) is nonpositive.

## 4 Sign-constancy of Green’s function for nonlocal boundary value problem in the case when \(b_{j}(t)\) can change sign

We prove the assertion about the nonpositivity of the Green’s function of (1.2)–(1.4), (3.2) without the assumption about the sign-constancy of \(b_{j}(t)\).

### Theorem 4.1

*Assume that the following conditions are fulfilled*:

- (1)
\(a_{j}(t) \geq 0\), \(j=1,\ldots,p\), \(t \in [0,\omega ]\).

- (2)
*The Wronskian*\(W(t)\)*of the fundamental system of solutions of a homogeneous equation*(2.2)*–*(2.4)*satisfies*\(W(t) \neq 0\), \(t \in [0,\omega ]\). - (3)
*The Cauchy function*\(C_{1}(t,s)\)*of the first order equation*(2.8)*is positive for*\(0 \leq s \leq t \leq \omega \). - (4)
\(A(t)\)

*satisfies*(3.5)*for*\(t \in [0,\omega ]\). - (5)
\(\gamma _{k} \geq 1\), \(\delta _{k} \geq 0\), \(k=1,2,\ldots,r\).

- (6)
*The spectral radius*\(\rho (K)\)*of the operator*\(K : L_{\infty }\to L _{\infty }\),*defined by*(4.9),*satisfies the inequality*\(\rho (K) < 1\).

*Then Green’s function*\(G_{1}(t,s)\)

*of the nonlocal problem*(1.2)

*–*(1.4), (3.2)

*is nonpositive for*\(t,s \in [0, \omega ]\).

### Proof

The conditions (1)–(5) of Theorem 4.1 correspond to all the conditions of Theorem 3.1, so they imply that the Green’s function \(G_{1}^{-}(t,s)\) of the auxiliary boundary value problem (4.1)–(4.3), (3.2) is nonpositive for \(t,s \in [0, \omega ]\).

We have noted above, in Eq. (4.7), that \(b_{j}^{+}(t) \geq 0\). Together with the fact that \(G_{1}^{-}(t,s) \leq 0\), this implies that the operator *K* is positive.

It follows from the inequality \(G_{1}^{-}(t,s) \leq 0\) that all operators \(K^{j}\) are positive and, consequently, for this case, the operator \(\sum_{j=0}^{\infty } K^{j}\) is positive.

### Example 4.1

Let us verify, whether the condition \(\rho (K)<1\) is satisfied. For our example, after substitution into (4.13), this inequality has the form \(0.965<1\). Thus, the spectral radius \(\rho (K)<1\).

Thus, the conditions (1)–(3), (5)–(6) of Theorem 4.1 are fulfilled. According to Theorem 4.1, for each \(A(t)\), satisfying the condition (3.5), the Green’s function \(G_{1}(t,s)\) is nonpositive.

## Notes

### Acknowledgements

This paper is part of the second author’s Ph.D. thesis, which is being carried out in the Department of Mathematics at Ariel University.

### Availability of data and materials

Not applicable.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

### Funding

Not applicable.

### Competing interests

The authors declare that they have no competing interests.

## References

- 1.Agarwal, R.P., Berezansky, L., Braverman, E., Domoshnitsky, A.: Nonoscillation Theory of Functional Differential Equations with Applications. Springer, New York (2012) zbMATHCrossRefGoogle Scholar
- 2.Agarwal, R.P., O’Regan, D.: Multiple nonnegative solutions for second order impulsive differential equations. Appl. Math. Comput.
**114**(1), 51–59 (2000) MathSciNetzbMATHGoogle Scholar - 3.Agarwal, R.P., O’Regan, D.: Upper, lower solutions for singular problems with nonlinear boundary data. NoDEA Nonlinear Differ. Equ. Appl.
**9**, 419–440 (2002) MathSciNetzbMATHCrossRefGoogle Scholar - 4.Azbelev, N.V., Maksimov, V.P., Rakhmatullina, L.F.: Introduction to the Theory of Functional Differential Equations: Methods and Applications. Hindawi, New York (2007) zbMATHCrossRefGoogle Scholar
- 5.Baxley, J.: A singular nonlinear boundary value problem: membrane response of a spherical cap. SIAM J. Appl. Math.
**48**, 497–505 (1988) MathSciNetzbMATHCrossRefGoogle Scholar - 6.Bolojan-Nica, O., Infante, G., Pietramala, P.: Existence results for impulsive systems with initial nonlocal conditions. Math. Model. Anal.
**18**(5), 599–611 (2013) MathSciNetzbMATHCrossRefGoogle Scholar - 7.Bolojan-Nica, O., Infante, G., Precup, R.: Existence results for systems with coupled nonlocal initial conditions. Nonlinear Anal.
**94**, 231–242 (2014) MathSciNetzbMATHCrossRefGoogle Scholar - 8.Calamai, A., Infante, G.: Nontrivial solutions of boundary value problems for second order functional differential equations. arXiv:1406.7508v1 [math. CA] (2014)
- 9.Domoshnitsky, A.: Maximum principles and nonoscillation intervals for first order Volterra functional differential equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal.
**15**, 769–814 (2008) MathSciNetzbMATHGoogle Scholar - 10.Domoshnitsky, A., Drakhlin, M.: Nonoscillation of first order impulsive differential equations with delay. J. Math. Anal. Appl.
**206**(1), 254–269 (1997) MathSciNetzbMATHCrossRefGoogle Scholar - 11.Domoshnitsky, A., Landsman, G.: Semi-nonoscillation intervals in the analysis of sign constancy of Green’s functions of Dirichlet, Neumann and focal impulsive problems. Adv. Differ. Equ.
**2017**, 81 (2017) MathSciNetzbMATHCrossRefGoogle Scholar - 12.Domoshnitsky, A., Landsman, G., Yanetz, S.: About sign-constancy of Green’s functions of one-point problem for impulsive second order delay equations. Funct. Differ. Equ.
**21**(1–2), 3–15 (2014) MathSciNetzbMATHGoogle Scholar - 13.Domoshnitsky, A., Mizgireva, Iu.: Sign-constancy of Green’s functions for two-point impulsive boundary value problems. Miskolc Math. Notes
**20**, 193–208 (2019) MathSciNetzbMATHCrossRefGoogle Scholar - 14.Domoshnitsky, A., Mizgireva, Iu., Raichik, V.: Semi-nonoscillation intervals and sign-constancy of Green’s functions of two-point impulsive boundary value problems. Ukr. Math. J. (accepted) (2019) Google Scholar
- 15.Domoshnitsky, A., Volinsky, I.: About positivity of Green’s functions for nonlocal boundary value problems with impulsive delay equations. Sci. World J.
**2014**, Article ID 978519 (2014) Google Scholar - 16.Domoshnitsky, A., Volinsky, I.: About differential inequalities for nonlocal boundary value problems with impulsive delay equations. Math. Bohem.
**140**(2), 121–128 (2015) MathSciNetzbMATHGoogle Scholar - 17.Domoshnitsky, A., Volinsky, I., Shklyar, R.: About Green’s functions for impulsive differential equations. Funct. Differ. Equ.
**20**, 55–81 (2013) MathSciNetzbMATHGoogle Scholar - 18.Feller, W.: Diffusion processes in one dimension. Trans. Am. Math. Soc.
**77**, 1–31 (1954) MathSciNetzbMATHCrossRefGoogle Scholar - 19.Feng, M., Xie, D.: Multiple positive solutions of multi-point boundary value problem for second-order impulsive differential equations. J. Math. Anal. Appl.
**223**(1), 438–448 (2009) MathSciNetzbMATHGoogle Scholar - 20.Graef, J.R., Henderson, J., Yang, B.: Existence and nonexistence of positive solutions of an n-th order nonlocal boundary value problem. Proc. Dyn. Syst. Appl.
**5**, 86–191 (2007) Google Scholar - 21.Graef, J.R., Yang, B.: Positive solutions of a third order nonlocal boundary value problem. Discrete Contin. Dyn. Syst., Ser. S
**1**, 89–97 (2008) MathSciNetzbMATHGoogle Scholar - 22.Guidotti, P., Merino, S.: Gradial loss of positivity and hidden invariant cones in a scalar heat equation. Differ. Integral Equ.
**13**, 1551–1568 (2000) zbMATHGoogle Scholar - 23.Hakl, R., Lomtatidze, A., Sremr, J.: Some boundary value problems for first order scalar functional differential equations. FOLIA, Masaryk University, Brno, Czech Republic (2002) Google Scholar
- 24.Hu, L., Liu, L., Wu, Y.: Positive solutions of nonlinear singular two-point boundary value problems for second-order impulsive differential equations. Nonlinear Anal., Theory Methods Appl.
**69**(11), 3774–3789 (2008) zbMATHCrossRefGoogle Scholar - 25.Infante, G.: Positive solutions of some nonlinear BVPs involving singularities and integral BCs. Discrete Contin. Dyn. Syst., Ser. S
**1**(1), 99–106 (2008) MathSciNetzbMATHCrossRefGoogle Scholar - 26.Infante, G., Minhos, F., Pietramala, P.: Non-negative solutions of systems of ODEs with coupled boundary conditions. Commun. Nonlinear Sci. Numer. Simul.
**17**(12), 4952–4960 (2012) MathSciNetzbMATHCrossRefGoogle Scholar - 27.Infante, G., Webb, J.: Nonlinear non-local boundary-value problems and perturbed Hammerstein integral equations. Proc. Edinb. Math. Soc.
**49**(3), 637–656 (2006) MathSciNetzbMATHCrossRefGoogle Scholar - 28.Jankowsky, T.: Positive solutions to second order four-point boundary value problems for impulsive differential equations. Appl. Math. Comput.
**202**(2), 550–561 (2008) MathSciNetGoogle Scholar - 29.Jiang, D., Lin, X.: Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations. J. Math. Anal. Appl.
**321**(2), 501–514 (2006) MathSciNetzbMATHCrossRefGoogle Scholar - 30.Karaca, I.Y.: On positive solutions for second-order boundary value problems of functional differential equations. Appl. Math. Comput.
**219**, 5433–5439 (2013) MathSciNetzbMATHGoogle Scholar - 31.Karakostas, I.Y., Mavridis, K.G., Tsamatos, P.Ch.: Triple solutions for a nonlocal functional boundary value problem by Leggett–Williams theorem. Appl. Anal.
**83**, 957–970 (2004) MathSciNetzbMATHCrossRefGoogle Scholar - 32.Karakostas, I.Y., Tsamatos, P.Ch.: Existence of multiple positive solutions for a nonlocal boundary value problem. Topol. Methods Nonlinear Anal.
**19**, 109–121 (2002) MathSciNetzbMATHCrossRefGoogle Scholar - 33.Kiguradze, I., Puza, B.: On boundary value problems for systems of linear functional differential equations. Czechoslov. Math. J.
**47**(2), 341–373 (1997) MathSciNetzbMATHCrossRefGoogle Scholar - 34.Kiguradze, I., Puza, B.: Boundary value problems for systems of linear functional differential equations. FOLIA, Brno, Czech Republic (2002) Google Scholar
- 35.Lakshmikantham, V., Bainov, D.D., Simeonov, P.S.: Theory of Impulsive Differential Equations. World Scientific, Singapore (1989) CrossRefGoogle Scholar
- 36.Lee, E.K., Lee, Y.H.: Multiple positive solutions of singular two point boundary value problems for second order impulsive differential equations. Appl. Math. Comput.
**158**(3), 745–759 (2004) MathSciNetzbMATHGoogle Scholar - 37.Ma, R.: A survey on nonlocal boundary value problems. Appl. Math. E-Notes
**7**, 257–279 (2001) MathSciNetzbMATHGoogle Scholar - 38.Mizgireva, Iu.: On positivity of Green’s functions of two-point impulsive problems. Funct. Differ. Equ.
**25**(3–4), 189–200 (2018) MathSciNetGoogle Scholar - 39.Ntouyas, S.K.: Nonlocal Initial and Boundary Value Problems: A Survey, Handbook of Differential Equations: Ordinary Differential Equations, Volume II. Elsevier, Amsterdam (2005) Google Scholar
- 40.O’Regan, D.: Upper and lower solutions for singular problems arising in the theory of membrane response of a spherical cap. Nonlinear Anal.
**47**, 1163–1174 (2001) MathSciNetzbMATHCrossRefGoogle Scholar - 41.Pandit, S.G., Deo, S.G.: Differential Systems Involving Impulses. Springer, Berlin (1982) zbMATHCrossRefGoogle Scholar
- 42.Picone, M.: Su un problema al contorno nelle equazioni differenziali lineari ordinarie del secondo ordine. Ann. Sc. Norm. Super. Pisa, Cl. Sci.
**10**, 1–95 (1908) MathSciNetzbMATHGoogle Scholar - 43.Precup, R., Trif, D.: Multiple positive solutions of non-local initial value problems for first order differential systems. Nonlinear Anal.
**75**, 5961–5970 (2012) MathSciNetzbMATHCrossRefGoogle Scholar - 44.Samoilenko, A.M., Perestyuk, N.A.: Impulsive Differential Equations. World Scientific, Singapore (1995) zbMATHCrossRefGoogle Scholar
- 45.Skubachevskii, A.L.: Elliptic Functional Differential Equations and Applications. Birkhäuser, Basel (1997) zbMATHGoogle Scholar
- 46.Skubachevskii, A.L.: Nonclassical boundary value problems. J. Math. Sci.
**155**(2), 199–334 (2008) MathSciNetzbMATHGoogle Scholar - 47.Sommerfeld, S.: Ein beitrag zur hydrodynamiscen Erklarung der turbulenten Flussigkeitsbewegungen. In: Atti del IV Congr. Intern dei Matem. Rome, vol. 3, pp. 116–124 (1909) Google Scholar
- 48.Webb, J.: Multiple positive solutions of some nonlinear heat flow problems. Discrete Contin. Dyn. Syst.
**2005**(suppl.), 895–903 (2005) MathSciNetzbMATHGoogle Scholar - 49.Webb, J.: Optimal constants in a nonlocal boundary value problem. Nonlinear Anal.
**63**, 672–685 (2005) MathSciNetzbMATHCrossRefGoogle Scholar - 50.Webb, J., Infante, G.: Positive solutions of nonlocal boundary value problems: a unified approach. J. Lond. Math. Soc.
**74**, 673–693 (2006) MathSciNetzbMATHCrossRefGoogle Scholar - 51.Webb, J., Infante, G.: Positive solutions of nonlocal boundary value problems involving integral conditions. NoDEA Nonlinear Differ. Equ. Appl.
**15**, 45–67 (2008) MathSciNetzbMATHCrossRefGoogle Scholar - 52.Whyburn, W.M.: Differential equations with general boundary conditions. Bull. Am. Math. Soc.
**48**, 692–704 (1942) MathSciNetzbMATHCrossRefGoogle Scholar - 53.Yang, Z.: Positive solutions of a second-order integral boundary value problem. J. Math. Anal. Appl.
**321**, 751–765 (2006) MathSciNetzbMATHCrossRefGoogle Scholar - 54.Zavalishchin, S.G., Sesekin, A.N.: Dynamic Impulse Systems: Theory and Applications. Springer, Dordrecht (1997) zbMATHCrossRefGoogle Scholar

## Copyright information

**Open Access** This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.