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

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.

Proposition 2.1

The impulsive problem (1.1) is equivalent to Dirichlet problem

with impulsive conditions

Proof

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

Moreover, we have

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

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

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

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

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

Moreover, we have

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

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

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

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).□

Lemma 3.1

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

Then there exist continua C − and C + with

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

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

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

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 − ℒ ( λ 1 ) has odd dimension. Then there exists a continuum

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

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

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

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

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

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

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

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

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

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

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

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

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

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 ℋ 1 ( κ m , u m ) = 0 . This is equivalent to

It follows from (A1) that

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

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

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

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

Combining (4.8) and (4.9), we have

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

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

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

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

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

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.□