Skip to content
BY 4.0 license Open Access Published by De Gruyter Open Access November 22, 2022

Existence and multiplicity of solutions for second-order Dirichlet problems with nonlinear impulses

  • Mantang Ma EMAIL logo and Ruyun Ma
From the journal Open Mathematics

Abstract

We are concerned with Dirichlet problems of impulsive differential equations

u ( x ) λ u ( x ) + g ( x , u ( x ) ) + j = 1 p I j ( u ( x ) ) δ ( x y j ) = f ( x ) for a.e. x ( 0 , π ) , u ( 0 ) = u ( π ) = 0 ,

where λ is a parameter and runs near 1, f L 2 ( 0 , π ) , I j C ( R , R ) , j = 1 , 2 , , p , p N , the nonlinearity g : [ 0 , π ] × R R satisfies the Carathéodory condition, δ = δ ( x ) denote the Dirac delta impulses concentrated at 0, which are applied at given points 0 < y 1 < y 2 < < y p < π . We show the existence and multiplicity of solutions to the aforementioned problem for λ in a neighborhood of 1 by using degree theory and bifurcation theory.

MSC 2010: 34B15; 34B37; 34C23

1 Introduction and main result

In this article, we are concerned with the existence and multiplicity of solutions for Dirichlet problems of impulsive differential equations

(1.1) u ( x ) λ u ( x ) + g ( x , u ( x ) ) + j = 1 p I j ( u ( x ) ) δ ( x y j ) = f ( x ) for a.e. x ( 0 , π ) , u ( 0 ) = u ( π ) = 0 ,

where λ is a parameter and runs near 1, f L 2 ( 0 , π ) , I j : R R is continuous, j = 1 , 2 , , p , p N , the nonlinearity g : [ 0 , π ] × R R satisfies Carathéodory condition, δ = δ ( x ) denote the Dirac delta impulses concentrated at 0, i.e., δ ( x ) = 0 for x 0 , δ ( 0 ) = + , and + δ ( x ) d x = 1 , the Dirac delta impulses δ are applied at given points 0 < y 1 < y 2 < < y p < π .

In recent years, the existence and multiplicity of solutions for problem (1.1) with I j 0 , j = 1 , 2 , , p , have been extensively studied by many authors, see [16] and references therein. In particular, Mawhin and Schmitt [1] studied the Dirichlet problem with the parameter λ near the principal eigenvalue of the form:

(1.2) u ( x ) + λ u ( x ) + g ( x , u ( x ) ) = f ( x ) for a.e. x ( 0 , π ) , u ( 0 ) = u ( π ) = 0 .

Using degree theory together with bifurcation theory, they proved that problem (1.2) had near λ = 1 at least one solution for λ 1 and at least two solutions for λ < 1 provided that

0 π g ( x ) f ( x ) d x < 0 π f ( x ) sin x d x < 0 π g + ( x ) sin x d x ,

where

g ( x ) limsup s g ( x , s ) , g + ( x ) liminf s + g ( x , s ) .

In addition, Chiappinell et al. [2] showed that there exists ν > 0 such that problem (1.2) with λ near 1 had at least one solution for λ 1 and two solutions for 1 < λ < 1 + ν under lim s + g ( x , s ) / s = 0 and a Landesman-Lazer-type condition. Here, we emphasize that the existence and multiplicity of solutions in works [1,2] are obtained without impulsive effects.

As far as we know, the model of the impulsive differential equation describes evolution processes in which their states change abruptly at certain moments in time, see [7,8]. There has recently been increasing interest in studying the impulsive differential equation, see, for instance, [926]. In [9], Drábek and Langerová studied Dirichlet problems of impulsive differential equations

(1.3) u ( x ) μ u ( x ) + g ˜ ( u ( x ) ) = f ( x ) , x ( 0 , π ) { x 1 , x 2 , , x q } , Δ u ( x i ) u ( x i + ) u ( x i ) = I i ( u ( x i ) ) , i = 1 , 2 , , q , u ( 0 ) = u ( π ) = 0 ,

where μ R is a parameter, 0 < x 1 < < x q < π , g ˜ C ( R , R ) , and I i C ( R , R ) . Based on topological degree arguments, they showed that problem (1.3) had at least one solution under the assumption that nonlinearity g ˜ and impulses I i satisfy sublinear growth at ± . Note that the multiplicity of solutions for problem (1.3) in [9] is not studied.

Recently, Shi and Chen [10] studied Dirichlet problems of impulsive differential equations

(1.4) u ( x ) = h ( x , u ( x ) ) , x ( 0 , π ) { x 1 , x 2 , , x q } , Δ u ( x i ) u ( x i + ) u ( x i ) = I i ( u ( x i ) ) , i = 1 , 2 , , q , u ( 0 ) = u ( π ) = 0

with h C ( [ 0 , π ] × R , R ) . They showed that problem (1.4) had at least three solutions provided that impulses I i satisfy sublinear growth at ± . The method consists in using Morse theory in combination with the minimax arguments. In [11], Liu and Zhao considered a Dirichlet problem of impulsive differential equations. The existence of multiple solutions is obtained by using variational methods combined with a three critical point theorem.

Although papers [10,11] obtained the multiplicity results of solutions for the Dirichlet problems of impulsive differential equations, they did not consider the near resonance problem corresponding to problem (1.3) or (1.4). It is natural to ask whether it is possible to obtain some multiplicity results for Dirichlet problems of impulsive differential equations near resonance. In this article, we show that this program can be developed for problem like (1.1). Specifically, we shall employ degree theory and bifurcation theory to deal with problem (1.1) under the following assumptions:

(A1) I j : R R is continuous and

lim s I j ( s ) s = 0 , j = 1 , , p .

(A2) g : [ 0 , π ] × R R satisfies Carathéodory condition and there exists Γ L 1 ( 0 , π ) with

g ( x , s ) Γ L 1 .

(A3) There exist limits lim s ± g ( x , s ) = g ( x , ± ) for any x ( 0 , π ) and lim s ± I j ( s ) = I j ( ± ) for j = 1 , 2 , , p . Moreover, the following inequality holds:

(1.5) j = 1 p I j ( ) sin y j + 0 π g ( x , ) sin x d x < 0 π f ( x ) sin x d x < j = 1 p I j ( + ) sin y j + 0 π g ( x , + ) sin x d x .

Our main result is the following:

Theorem 1.1

Assume that (A1)–(A3) hold. Then for all λ ( , 4 ) , problem (1.1) has at least one solution. Furthermore, if λ < 1 but close to 1, then problem (1.1) has at least three solutions.

Remark 1.1

For other multiplicity results of solutions of second-order impulsive differential equations, we refer the interested readers to D’Agui et al. [19], Han et al. [20], Henderson and Luca [21], and Lee and Lee [22]. Nevertheless, the difference between these works and our work consists of the fact that we studied the near resonance problem of the impulsive differential equation. Furthermore, the method we use (bifurcation theory) is also different from the methods used in the aforementioned works.

This article is organized as follows. Section 2 is devoted to proving the regularity of a weak solution to problem (1.1). In Section 3, we introduce some preliminary results from Mawhin and Schmitt [6] which are useful for the proof of our main result. Finally, in Section 4, we give a priori bounds of the eventual solutions of problem (1.1) and finish the proof of Theorem 1.1.

2 Regularity of weak solutions

Denote H H 0 1 ( 0 , π ) . We say that u H is a weak solution of problem (1.1) if the integral identity

(2.1) 0 π u ( x ) v ( x ) d x λ 0 π u ( x ) v ( x ) d x + 0 π g ( x , u ( x ) ) v ( x ) d x = 0 π f ( x ) v ( x ) d x j = 1 p I j ( u ( y j ) ) v ( y j )

for any test function v H . For simplicity, we use the following notations:

0 = y 0 < y 1 < y 2 < < y p < y p + 1 = π ; j ( y j 1 , y j ) , j = 1 , 2 , , p + 1 ; ( 0 , π ) { y 1 , y 2 , , y p } = j = 1 p + 1 j .

Let D ( ) , R , be the set of all infinitely differentiable functions on with compact support lying in .

In the sequel, we prove the regularity of a weak solution u H .

Proposition 2.1

The impulsive problem (1.1) is equivalent to Dirichlet problem

(2.2) u ( x ) λ u ( x ) + g ( x , u ( x ) ) = f ( x ) , x j , j = 1 , 2 , , p + 1 , u ( 0 ) = u ( π ) = 0

with impulsive conditions

(2.3) Δ u ( y j ) lim x y j + u ( x ) lim x y j u ( x ) = I j ( u ( y j ) ) , j = 1 , 2 , , p .

Proof

Choose v D ( j ) and extend v = 0 on ( 0 , π ) j , Then v H . Integrating by parts in (2.1), we obtain

0 π u ( x ) v ( x ) d x λ 0 π v ( x ) d 0 x u ( τ ) d τ + 0 π v ( x ) d 0 x g ( τ , u ( τ ) ) d τ = 0 π v ( x ) d 0 x f ( τ ) d τ j = 1 p I j ( u ( y j ) ) v ( y j ) .

Moreover, we have

(2.4) j u ( x ) + λ y j 1 x u ( τ ) d τ + y j 1 x f ( τ ) d τ y j 1 x g ( τ , u ( τ ) ) d τ v ( x ) d x = 0 .

Since (2.4) holds for arbitrary v D ( j ) , there exists a constant k 1 such that

(2.5) u ( x ) + λ y j 1 x u ( τ ) d τ + y j 1 x f ( τ ) d τ y j 1 x g ( τ , u ( τ ) ) d τ = k 1

for a.e. x j . Therefore, u C 1 ( j ) . Moreover, from (2.5), we infer that

u ( x ) λ u ( x ) + g ( x , u ( x ) ) = f ( x ) , x j .

In particular, the equation in (1.1) holds pointwise in j = 1 p + 1 j .

We have u C 1 [ 0 , π ] by the embedding H C [ 0 , π ] . Set

0 η < min j = 1 , 2 , , p { y j y j 1 , y j + 1 y j } .

Choose v D ( y j η , y j + η ) and extend v = 0 on ( 0 , π ) ( y j η , y j + η ) , i.e., v H . Integration by parts in (2.1) yields

y j η y j + η u ( x ) v ( x ) d x λ y j η y j + η u ( x ) v ( x ) d x + y j η y j + η g ( x , u ( x ) ) v ( x ) d x = y j η y j + η f ( x ) v ( x ) d x y j η y j + η I j ( u ( x ) ) δ ( x y j ) v ( x ) d x .

Moreover, we have

y j η y j + η u ( x ) + λ y j η x u ( τ ) d τ y j η x g ( τ , u ( τ ) ) d τ + y j η x f ( τ ) d τ y j η x I j ( u ( τ ) ) δ ( τ y j ) d τ v ( x ) d x = 0 .

Since v ( y j η , y j + η ) is arbitrary, there are constants k 2 such that

(2.6) u ( x ) + λ y j η x u ( τ ) d τ y j η x g ( τ , u ( τ ) ) d τ + y j η x f ( τ ) d τ y j η x I j ( u ( τ ) ) δ ( τ y j ) d τ = k 2

for a.e. x ( y j η , y j + η ) . Moreover, from (2.6), we conclude

(2.7) lim x y j + u ( x ) = k 2 λ y j η y j u ( s ) d s + y j η y j g ( s , u ( s ) ) d s + I j ( u ( y j ) ) y j η y j f ( s ) d s , lim x y j u ( x ) = k 2 λ y j η y j u ( s ) d s + y j η y j g ( s , u ( s ) ) d s + 0 y j η y j f ( s ) d s .

Therefore, from (2.6) and (2.7), we obtain

Δ u ( y j ) lim x y j + u ( x ) lim x y j u ( x ) = I j ( u ( y j ) ) .

It follows from the aforementioned considerations that u C ( 0 , π ) and that the first derivative u is piecewise continuous with discontinuities of the first kind at the points y 1 , y 2 , , y p .

The boundary conditions u ( 0 ) = u ( π ) = 0 are satisfied automatically due to the fact that every weak solution u belongs to H . Therefore, the impulsive problem (1.1) is equivalent to the Dirichlet problem (2.2) with impulsive conditions (2.3).□

In fact, we have just proved that every weak solution to problem (1.1) is also a classical solution. On the other hand, it is obvious that every classical solution is also a weak solution. With this result at hand, we can look for (classical) solutions as for solutions of certain operator equation induced by (2.1). Therefore, we define the scalar product and the norm of H , respectively, given by

(2.8) u , v 0 π u ( x ) v ( x ) d x , u , v H

and

u 0 π ( u ( x ) ) 2 d x 1 / 2 .

Moreover, we define operators J , S , G , I : H H and an element f as follows:

J u , v 0 π u ( x ) v ( x ) d x , S u , v 0 π u ( x ) v ( x ) d x , G ( u ) , v 0 π g ( x , u ( x ) ) v ( x ) d x , I ( u ) , v j = 1 p I j ( u ( y j ) ) v ( y j ) , f , v 0 π f ( x ) v ( x ) d x , u , v H .

Then problem (2.2), (2.3) is then equivalent to operator equation

(2.9) J u λ S u + G u + I ( u ) = f .

It follows from the definitions of operators J , S , G , I and the compact embedding H C [ 0 , π ] that J is just identity, S is a linear compact operator, and G is a nonlinear compact operator. In addition, the element f is also well defined because f L 2 ( 0 , π ) .

3 Bifurcation and continuation

Let E be a real Banach space with the norm E and let A : E × R E be a completely continuous operator. Consider the operator equation

(3.1) u A ( u , λ ) = 0 .

It follows from Mawhin and Schmitt [6] that the following results.

Lemma 3.1

[6, Theorem 1] Let there exists a bounded open set Ω in E such that

(3.2) deg ( I d A ( , a ) , Ω , 0 ) 0 .

Then there exist continua C and C + with

C E × ( , a ] ( I d A ) 1 ( 0 ) , C + E × [ a , + ) ( I d A ) 1 ( 0 ) ,

and for both C = C and C = C + the following are valid:

  1. C Ω × { a } .

  2. Either C is unbounded or else C E Ω ¯ × { a } .

Lemma 3.2

[6, Corollary 2] Assume the conditions of Lemma 3.1, where Ω is given by

Ω B R ( 0 ) = { u E : u E < R } .

Moreover, assume that there exists b > a such that for any λ , a λ b , we have that u E < R , where ( u , λ ) is a solution of operator equation (2.9). Then there exists a constant α > 0 , such that for b λ b + α , there exists ( u , λ ) C + with u E 2 R .

Remark 3.1

A similar statement holds for values of λ to the left of a .

As a further tool, we need a result that guarantees bifurcation from infinity. For this, we shall assume a particular form for the completely continuous mapping A , namely

(3.3) A ( u , λ ) ( λ ) u + ( u , λ ) ,

where ( λ ) is a family of compact linear operators and the perturbation term satisfies

(3.4) ( u , λ ) u E 0 as u E .

Lemma 3.3

[6, Theorem 3] Assume (3.3) and (3.4) hold and assume that for λ = λ 1 the generalized nullspace of I d ( λ 1 ) has odd dimension. Then there exists a continuum

C ( I d A ) 1 ( 0 )

which bifurcates from infinity at λ = λ 1 . That is, for any ε > 0 , there exists ( u , λ ) C with

λ λ 1 < ε and u E > 1 ε .

4 Proof of Theorem 1.1

Theorem 4.1

Assume that (A1)–(A3) hold. Then for given γ , 1 < γ < 4 , there exists a constant R 0 > 0 , such that, if 1 λ γ , then any solution u of problems (2.2) and (2.3) satisfies

(4.1) u max 0 x π u ( x ) < R 0 .

Proof

We break the proof into two parts, according to λ = 1 or 1 < λ γ .

Case 1. 1 < λ γ . Assume there exists a constant R 1 > 0 such that

J u λ S u + G u + I ( u ) f 0

for all u = R 1 , then the properties of operators J , S , G , I , and f imply that the Leray-Schauder degree

deg ( J λ S + G + I f , B R 1 , 0 )

is well-defined, where B R 1 { u H : u < R 1 } . If we find R 1 > 0 such that

(4.2) deg ( J λ S + G + I f , B R 1 , 0 ) 0 ,

then there exists u B R 1 satisfying operator equation (2.9). In order to search R 1 > 0 such that (4.2) holds, we use the homotopy invariance property of the Leray-Schauder degree. Specifically, we introduce the homotopy

λ ( κ , u ) J u λ S u ( 1 κ ) θ S u + κ G u + κ I κ f ,

where κ [ 0 , 1 ] is a homotopy parameter and θ > 0 satisfies

λ + ( 1 κ ) θ 1 for any κ [ 0 , 1 ] .

We prove that there exists R 1 > 0 such that for all u H , u = R 1 , and all κ [ 0 , 1 ] we have

(4.3) λ ( κ , u ) 0 .

We prove (4.3) via contradiction. Assume that there exist u m H , u m , κ m [ 0 , 1 ] such that λ ( κ m , u m ) = 0 for 1 < λ γ . Then, setting v m u m / u m , this is equivalent to

(4.4) J v m λ S v m ( 1 κ m ) θ S v m + κ m G ( u m ) u m + κ m I ( u m ) u m κ m f u m = 0 .

The last three terms in (4.4) tend to zero due to assumptions (A1), (A2), and the fact that f is a fixed element. Passing to subsequences if necessary, we may assume v m v (weakly) in H for some v H and κ m κ [ 0 , 1 ] . Since S is compact, S v m S v (strongly) in H . Now the strong convergence J v m J v follows from (4.4). In particular, v = 1 and

(4.5) J v ( λ + ( 1 κ ) θ ) S v = 0 .

Since λ + ( 1 κ ) θ 1 , equation (4.5) has only trivial solution, a contradiction. Hence, there exists a positive constant R 1 which is independent of the parameters κ m and κ such that u R 1 .

Case 2. λ = 1 . In this case, our goal is to prove

deg ( J S + G + I f ; B R 2 , 0 ) 0

for some R 2 > 0 . To this end, we introduce homotopy

1 ( κ , u ) J u S u ( 1 κ ) θ S u + κ G u + κ I κ f ,

where κ [ 0 , 1 ] is a homotopy parameter and 0 < θ < 3 . We prove that there exists R 2 > 0 such that

(4.6) 1 ( κ , u ) 0

holds for all u H , u = R 2 , and all κ [ 0 , 1 ] . Then the result follows from (4.6) and the homotopy invariance of the Leary-Schauder degree as in the proof of Case 1.

We also prove (4.6) via contradiction. Let u m H , u m , κ m [ 0 , 1 ] be such that 1 ( κ m , u m ) = 0 . This is equivalent to

J v m S v m ( 1 κ m ) θ S v m + κ G v m + κ m I ( u m ) u m κ m f u m = 0 .

It follows from (A1) that

κ m I ( u m ) u m 0 , κ m f u m 0 .

Similarly as in the proof of above, we arrive at the limit equation

(4.7) J v ( 1 + ( 1 κ ) θ ) S v = 0 .

Since 1 + ( 1 κ ) θ < 4 due to the choice of θ , (4.7) may occur only if κ = 1 and v ( x ) = ± 2 π sin x . First, we assume

v ( x ) = 2 π sin x .

Taking the inner product of 1 ( κ m , u m ) = 0 with sin x , we obtain

(4.8) ( 1 κ m ) θ 0 π u m ( x ) sin x d x + κ m j = 1 p I j ( u m ( y j ) ) sin y j κ m 0 π ( f ( x ) g ( x , u m ) ) sin x d x = 0 .

It follows from the embedding H C [ 0 , π ] that v m ( x ) 2 π sin x (uniformly) on [ 0 , π ] . Therefore, we have

(4.9) 0 π u m ( x ) sin x d x > 0 , m 1 .

Combining (4.8) and (4.9), we have

(4.10) j = 1 p I j ( u m ( y j ) ) sin y j + 0 π g ( x , u m ( x ) ) sin x d x 0 π f ( x ) sin x d x .

Passing to the limit for m in (4.10), we obtain

j = 1 p I j ( + ) sin y j + 0 π g ( x , + ) sin x d x 0 π f ( x ) sin x d x .

However, this contradicts the second inequality in (1.5). If

v ( x ) = 2 π sin x ,

we derive similarly a contradiction with the first inequality in (1.5). Hence, there exists a positive constant R 2 , which is independent of the parameters κ m and κ such that u R 2 .

Consequently, set R 3 max { R 1 , R 2 } , we obtain

(4.11) u R 3 .

Moreover, (4.11) together with the compact embedding H C [ 0 , π ] imply that

u R 0 ,

where R 0 is a positive constant. This completes the proof.□

Proof of Theorem 1.1

From assumption (A2), g is bounded in R . Assume that λ ( , 1 ) ( 1 , γ ) . Then we may apply the Schauder fixed point theorem. To prove the other parts of the result, we shall employ Lemmas 3.2 and 3.3.

The necessary L 1 ( 0 , π ) bound of course follows immediately. Moreover, since the a priori bounds (4.1) of Theorem 4.1 are independent of the parameter κ , we may, for 1 < λ γ , compute the Leray-Schauder degree (3.2) as that of a linear homeomorphism, which is nonzero (in our case -1). Therefore, we may apply Lemma 3.2 to deduce the existence of solutions for 1 λ < 4 .

On the other hand, λ = 1 is the principle eigenvalue of the linear eigenvalue problem

u ( x ) = λ u ( x ) , x ( 0 , π ) , u ( 0 ) = u ( π ) = 0 ,

and which is of multiplicity one. Hence, Lemma 3.3 also is applied and we may conclude that there is bifurcation from infinity at λ = 1 . However, we have established a priori bounds for the eventual solutions of problem (1.1) with 1 λ γ < 4 , the continua bifurcating from infinity, must do so for λ < 1 but close to 1, there must exist large positive and a large negative solution. This coupled with Lemma 3.2 and what has been proved above yields the existence of three solutions for λ < 1 close to 1.□

Acknowledgements

The authors are very grateful to the anonymous referees for their valuable suggestions.

  1. Funding information: This research was supported by the NSFC (No. 12061064).

  2. Author contributions: The authors claim that the research was realized in collaboration with the same responsibility. All authors read and approved the last of the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: Data sharing not applicable to this article as no datasets were generated.

References

[1] J. Mawhin and K. Schmitt, Landesman-Lazer type problems at an eigenvalue of odd multiplicity, Result. Math. 14 (1988), no. 1–2, 138–146. 10.1007/BF03323221Search in Google Scholar

[2] R. Chiappinelli, J. Mawhin, and R. Nugari, Bifurcation from infinity and multiple solutions for some Dirichlet problems with unbounded nonlinearities, Nonlinear Anal. 18 (1992), no. 12, 1099–1112. 10.1016/0362-546X(92)90155-8Search in Google Scholar

[3] A. Ambrosetti and G. Mancini, Existence and multiplicity results for nonlinear elliptic problems with linear part at resonance the case of the simple eigenvalue, J. Differential Equations 28 (1978), no. 2, 220–245. 10.1016/0022-0396(78)90068-2Search in Google Scholar

[4] N. P. Các, On an elliptic boundary value problem at double resonance, J. Math. Anal. Appl. 132 (1988), no. 2, 473–483. 10.1016/0022-247X(88)90075-3Search in Google Scholar

[5] D. G. Costa and J. V. A. Goncalves, Existence and multiplicity results for a class of nonlinear elliptic boundary value problems at resonance, J. Math. Anal. Appl. 84 (1981), no. 2, 328–337. 10.1016/0022-247X(81)90171-2Search in Google Scholar

[6] J. Mawhin and K. Schmitt, Nonlinear eigenvalue problems with the parameter near resonance, Ann. Polon. Math. 51 (1990), 241–248. 10.4064/ap-51-1-241-248Search in Google Scholar

[7] M. Benchohra, J. Henderson, and S. Ntouyas, Theory of Impulsive Differential Equations, Contemporary Mathematics and Its Applications, Vol. 2, Hindawi Publishing Corporation, New York, 2006. 10.1155/9789775945501Search in Google Scholar

[8] V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific Press, Singapore, 1989. 10.1142/0906Search in Google Scholar

[9] P. Drábek and M. Langerová, On the second order equations with nonlinear impulses. Fredholm alternative type results, Topol. Methods Nonlinear Anal. 44 (2014), no. 1, 249–261. 10.12775/TMNA.2014.046Search in Google Scholar

[10] H. X. Shi and H. B. Chen, Multiplicity results for a class of boundary value problems with impulsive effects, Math. Nachr. 289 (2016), no. 5–6, 718–726. 10.1002/mana.201400341Search in Google Scholar

[11] J. Liu and Z. Q. Zhao, Multiple solutions for impulsive problems with non-autonomous perturbations, Appl. Math. Lett. 64 (2017), 143–149. 10.1016/j.aml.2016.08.020Search in Google Scholar

[12] T. Jankowski, Positive solutions for second order impulsive differential equations involving Stieltjes integral conditions, Nonlinear Anal. 74 (2011), no. 11, 3775–3785. 10.1016/j.na.2011.03.022Search in Google Scholar

[13] P. Chen and X. H. Tang, Existence and multiplicity of solutions for second-order impulsive differential equations with Dirichlet problems, Appl. Math. Comput. 218 (2012), no. 24, 11775–11789. 10.1016/j.amc.2012.05.027Search in Google Scholar

[14] X. N. Lin and D. Q. Jiang, Multiple positive solutions of Dirichlet boundary value problems for second order impulsive differential equations, J. Math. Anal. Appl. 321 (2006), no. 2, 501–514. 10.1016/j.jmaa.2005.07.076Search in Google Scholar

[15] X. N. Hao, L. S. Liu, and Y. H. Wu, Positive solutions for second order impulsive differential equations with integral boundary conditions, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 1, 101–111. 10.1016/j.cnsns.2010.04.007Search in Google Scholar

[16] Y. S. Liu and D. O’Regan, Multiplicity results using bifurcation techniques for a class of boundary value problems of impulsive differential equations, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 4, 1769–1775. 10.1016/j.cnsns.2010.09.001Search in Google Scholar

[17] H. W. Chen and Z. M. He, Variational approach to some damped Dirichlet problems with impulses, Math. Methods Appl. Sci. 36 (2013), no. 18, 2564–2575. 10.1002/mma.2777Search in Google Scholar

[18] J. H. Chen, C. C. Tisdell, and R. Yuan, On the solvability of periodic boundary value problems with impulse, J. Math. Anal. Appl. 331 (2007), no. 2, 902–912. 10.1016/j.jmaa.2006.09.021Search in Google Scholar

[19] G. D’Agui, B. Di Bella, and S. Tersian, Multiplicity results for superlinear boundary value problems with impulsive effects, Math. Methods Appl. Sci. 39 (2016), no. 5, 1060–1068. 10.1002/mma.3545Search in Google Scholar

[20] X. N. Hao, M. Y. Zuo, and L. S. Liu, Multiple positive solutions for a system of impulsive integral boundary value problems with sign-changing nonlinearities, Appl. Math. Lett. 82 (2018), 24–31. 10.1016/j.aml.2018.02.015Search in Google Scholar

[21] J. Henderson and R. Luca, Positive solutions for an impulsive second-order nonlinear boundary value problem, Mediterr. J. Math. 14 (2017), no. 2, 1–16. 10.1007/s00009-017-0897-7Search in Google Scholar

[22] E.-L. Lee and Y.-H. Lee, Multiple positive solutions of two point boundary value problems for second order impulsive differential equations, Appl. Math. Comput. 158 (2004), no. 3, 745–759. 10.1016/j.amc.2003.10.013Search in Google Scholar

[23] Y.-H. Lee and X. Z. Liu, Study of singular boundary value problems for second order impulsive differential equation, J. Math. Anal. Appl. 331 (2007), no. 1, 159–176. 10.1016/j.jmaa.2006.07.106Search in Google Scholar

[24] Z. G. Luo and J. J. Nieto, New result for the periodic boundary value problem for impulsive integro-differential equations, Nonlinear Anal. 70 (2009), no. 6, 2248–2260. 10.1016/j.na.2008.03.004Search in Google Scholar

[25] D. B. Qian and X. Y. Li, Periodic solutions for ordinary differential equations with sublinear impulsive effects, J. Math. Anal. Appl. 303 (2005), no. 1, 288–303. 10.1016/j.jmaa.2004.08.034Search in Google Scholar

[26] Z. H. Zhang and R. Yuan, An application of variational methods to Dirichlet boundary value problem with impulses, Nonlinear Anal. Real World Appl. 11 (2010), no. 1, 155–162. 10.1016/j.nonrwa.2008.10.044Search in Google Scholar

Received: 2022-06-25
Revised: 2022-10-06
Accepted: 2022-10-07
Published Online: 2022-11-22

© 2022 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Downloaded on 2.3.2024 from https://www.degruyter.com/document/doi/10.1515/math-2022-0519/html
Scroll to top button