Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations

(with $\mathrm{\Delta }{u}^{\prime }(t_j):=u^{\prime }(t_j^{+})-u^{\prime }(t_j^{-})$) is studied, using the minimization and the mountain-pass theorem. We mention also other papers for second-order impulsive equations as [6, 7]. In several recent papers [8–10], fourth-order impulsive problems are considered via variational methods.

Here, $0=t_0<t_1<\cdots <t_n<t_{n+1}=T$, the limits $u^{″}(t_j^{\pm })={lim}_{t\to t_j^{\pm }}{u}^{″}(t)$ and $u^{‴}(t_j^{\pm })={lim}_{t\to t_j^{\pm }}{u}^{‴}(t)$ exist and $\mathrm{\Delta }{u}^{‴}(t_j)=u^{‴}(t_j^{+})-u^{‴}(t_j^{-})$, $\mathrm{\Delta }{u}^{″}(t_j)=u^{″}(t_j^{+})-u^{″}(t_j^{-})$.

We look for solutions in the classical sense, as given in the next definition.

Definition 1 A function $u\in C^1([0,T])$ and $u{|}_{(t_j,t_{j+1})}\in H^2(t_j,t_{j+1})$, $j=0,\dots ,n$ is said to be a classical solution of the problem (P), if u satisfies the equation a.e. on $[0,T]\mathrm{\setminus }\{t_1,t_2,\dots ,t_n\}$, the limits $u^{″}(t_j^{\pm })={lim}_{t\to t_j^{\pm }}{u}^{″}(t)$ and $u^{‴}(t_j^{\pm })={lim}_{t\to t_j^{\pm }}{u}^{‴}(t)$ exist and satisfy the impulsive conditions $\mathrm{\Delta }{u}^{‴}(t_j)=g(u^{\prime }(t_j))$, $\mathrm{\Delta }{u}^{″}(t_j)=-h(u(t_j))$, $j=1,\dots ,n$, and boundary conditions $u(0)=u(T)=u^{″}(0)=u^{″}(T)=0$.

To deduce the existence of solutions, we assume the following conditions:

(H1) The constant a is positive, b and c are continuous functions on $[0,T]$ and there exist positive constants $b_1$, $b_2$, $c_1$, and $c_2$ such that $0<b_1\le b(t)\le b_2$ and $0<c_1\le c(t)\le c_2$. The functions $g_j:\mathbb{R}\to \mathbb{R}$, $h_j:\mathbb{R}\to \mathbb{R}$, $j=1,2,\dots ,n$, are continuous functions.

where $d_j$ and $e_j$, $j=1,\dots ,n$, are positive constants.

In the next section we will prove the following existence result for $p>2$.

Theorem 2 Suppose that $p>2$ and conditions (H1) and (H2) hold. Then the problem (P) has at least one nonzero classical solution.

Having in mind the case $1<p<2$, we introduce the following condition:

A simple example of this new situation is given by the functions $g_j(t)=2A_{j}t$ and $h_j(t)=2B_{j}t$.

The result to be proven is the following.

Theorem 3 Suppose that $1<p<2$, the functions $g_j$, $h_j$, $j=1,\dots ,n$, are odd and conditions (H1) and (H3) hold. Then the problem (P) has infinitely many nonzero classical solutions.

we introduce the following condition.

A simple example now is $g_j(t)={\gamma }_j{d}_j{|t|}^{{\gamma }_j-2}t$, $h_j(t)={\sigma }_j{e}_j{|t|}^{{\sigma }_j-2}t$.

The obtained result is the following.

Theorem 4 Suppose that $p>2$ and conditions (H1) and (H2′) hold. If $0<T\le T_2=\pi \sqrt{\frac{a+\sqrt{a^2+4b_2}}{2b_2}}$, the problem (P₁) has only the zero solution. If $T>T_1=\pi \sqrt{\frac{a+\sqrt{a^2+4b_1}}{2b_1}}$, the problem (P₁) has at least one nonzero classical solution.

The proofs of the main results are given in Section 3.

and the corresponding norm.

where M is a positive constant depending on T, a and $b(t)$.

We have the following compactness embedding, which can be proved in the standard way.

Proposition 5 The inclusion $X\subset C^1([0,T])$ is compact.

In the sequel we introduce the concept of a weak solution of our problem.

As a consequence, the critical points of ϕ are the weak solutions of the problem (P). Let us see that they are, actually, strong solutions too.

Lemma 7 If u is a weak solution of (P) then $u\in X$ is classical solution of (P).

By a standard regularity argument (see [9, 11]) the weak derivative $u_j^{(4)}\in L^2(t_j,t_{j+1})$ and therefore the limits $u^{″}(t_j^{\pm })={lim}_{t\to t_j^{\pm }}{u}^{″}(t)$ and $u^{‴}(t_j^{\pm })={lim}_{t\to t_j^{\pm }}{u}^{‴}(t)$ exist.

Then we obtain $g_j(u^{\prime }(t_j))=\mathrm{\Delta }{u}^{″}(t_j)$ and $u^{″}(0)=u^{″}(T)=0$. Similarly, we prove that $h_j(u^{\prime }(t_j))=-\mathrm{\Delta }{u}^{‴}(t_j)$, which shows that u is a classical solution of the problem (P). The lemma is proved. □

In the proofs of the theorems, we will use three critical point theorems which are the main tools to obtain weak solutions of the considered problems.

To this end, we introduce classical notations and results. Let E be a reflexive real Banach space. Recall that a functional $I:E\to \mathbb{R}$ is lower semi-continuous (resp. weakly lower semi-continuous (w.l.s.c.)) if $u_k\to u$ (resp. $u_k\rightharpoonup u$) in E implies $lim{inf}_{k\to \mathrm{\infty }}I(u_k)\ge I(u)$ (see [12], pp.3-5).

We have the following well-known minimization result.

Theorem 8 Let I be a weakly lower semi-continuous operator that has a bounded minimizing sequence on a reflexive real Banach space E. Then I has a minimum $c={min}_{u\in E}I(u)=I(u_0)$. If $I:E\to \mathbb{R}$ is a differentiable functional, $u_0$ is a critical point of I.

Note that a functional $I:E\to \mathbb{R}$ is w.l.s.c. on I if $I(u)=I_1(u)+I_2(u)$, $I_1$ is convex and continuous and $I_2$ is sequentially weakly continuous (i.e. $u_k\rightharpoonup u$ in E implies ${lim}_{k\to \mathrm{\infty }}$ $I_2(u_k)=I_2(u)$) (see [13], pp.301-303). The existence of a bounded minimizing sequence appears, when the functional I is coercive, i.e. $I(u)\to +\mathrm{\infty }$ as $\parallel u\parallel \to +\mathrm{\infty }$.

Next, recall the notion of the Palais-Smale (PS) condition, the mountain-pass theorem and Clarke’s theorem.

We say that I satisfies condition (PS) if any sequence $(u_k)\subset E$ for which $I(u_k)$ is bounded and $I^{\prime }(u_k)\to 0$ as $k\to \mathrm{\infty }$ possesses a convergent subsequence.

Theorem 9 ([[14], p.4])

Then I possesses a critical value $c\ge \alpha$. Moreover, c can be characterized as $c=inf\{max\{I(u):u\in \gamma ([0,1])\}:\gamma \in \mathrm{\Gamma }\}$ where $\mathrm{\Gamma }=\{\gamma \in C([0,1],E):\gamma (0)=0,\gamma (1)=e\}$.

Theorem 10 ([[14], p.53])

Let E be a real Banach space and $I\in C^1(E,\mathbb{R})$ with I even, bounded from below, and satisfying condition (PS). Suppose that $I(0)=0$, there is a set $K\subset E$ such that K is homeomorphic to ${\mathbb{S}}^{m-1}$ by an odd map, and $sup\{I(u):u\in K\}<0$.

Then I possesses, at least, m distinct pairs of critical points.

This section is devoted to the proof of the three theorems enunciated in the introduction of this work.

First consider the case $p>2$ for which we prove that the functional ϕ satisfies the Palais-Smale condition.

Lemma 11 Suppose that $p>2$ and conditions (H1) and (H2) hold. Then the functional $\varphi :X\to \mathbb{R}$ satisfies condition (PS).

which implies that $(u_k)$ is a bounded sequence in X.

i.e., $u_k\to u$ strongly in X, which completes the proof. □

Now, we are in a position to prove the main results of this paper.

where $C=max\{C_j,K_j:1\le j\le n\}$.

Since $p>2$, we conclude that $\varphi (u_{\lambda })<0$ for sufficiently large λ. According to the mountain-pass Theorem 9, together with Lemmas 11 and 7, we deduce that there exists a nonzero classical solution of the problem (P). □

Now consider the case $1<p<2$. In the next result we prove that the Palais-Smale condition is also valid.

Lemma 12 Suppose that $1<p<2$ and conditions (H1) and (H3) hold. Then the functional $\varphi :X\to \mathbb{R}$ is bounded from below and satisfies condition (PS).

Further, if $(u_k)$ is a (PS) sequence, by (17) it follows that $(u_k)$ is a bounded sequence in X. Then, as in Lemma 11, we conclude that $(u_k)$ has a convergent subsequence. □

Now we are in a position to prove the next existence result for the problem (P).

where ${\parallel \cdot \parallel }_{X_m}$ is the induced norm of ${\parallel \cdot \parallel }_X$ on $X_m$.

where $K=\frac{1}{2}+\frac{4nC}{T{b}_1}$.

we obtain $\varphi (w)<0$ for any $w\in K_{\rho }^m$.

By Clarke’s Theorem 10, there exist at least m pairs of different critical points of the functional ϕ. Since m is arbitrary, there exist infinitely many solutions of the problem (P), which concludes the proof. □

Proof of Theorem 4 By the Poincaré inequalities (7) we find that $⦀u{⦀}^2={\int }_0^T(u^{\prime \prime 2}+a{u}^{\prime 2})dt$ is an equivalent norm to ${\parallel \cdot \parallel }_X$ in X and the functional $I_1(u)=\frac{1}{2}⦀u{⦀}^2$ is convex.

is sequentially weakly continuous, from the fact that the inclusion $X\subset C^1([0,T])$ is compact, we deduce that the functional ${\varphi }_1:X\to \mathbb{R}$ is weakly lower semi-continuous.

Then, by Theorem 8, there exists a minimizer of ${\varphi }_1$, which is a critical point of ${\varphi }_1$.