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Existence and multiplicity of solutions to boundary value problems for fourthorder impulsive differential equations
Boundary Value Problemsvolume 2014, Article number: 105 (2014)
Abstract
In this paper we study the existence and the multiplicity of solutions for an impulsive boundary value problem for fourthorder differential equations. The notions of classical and weak solutions are introduced. Then the existence of at least one and infinitely many nonzero solutions is proved, using the minimization, the mountainpass, and Clarke’s theorems.
MSC: 34B15, 34B37, 58E30.
1 Introduction
The theory of impulsive boundary value problems (IBVPs) became an important area of studies in recent years. IBVPs appear in mathematical models of processes with sudden changes in their states. Such processes arise in population dynamics, optimal control, pharmacology, industrial robotics, etc. For an introduction to theory of IBVPs one is referred to [1]. Some classical tools used in the study of impulsive differential equations are topological methods as fixed point theorems, monotone iterations, upper and lower solutions (see [2–4]). Recently, some authors have studied the existence of solutions of IBVPs using variational methods. The pioneering work in this direction is the paper of Nieto and O’Regan [5], where the secondorder impulsive problem
(with $\mathrm{\Delta}{u}^{\prime}({t}_{j}):={u}^{\prime}({t}_{j}^{+}){u}^{\prime}({t}_{j}^{})$) is studied, using the minimization and the mountainpass theorem. We mention also other papers for secondorder impulsive equations as [6, 7]. In several recent papers [8–10], fourthorder impulsive problems are considered via variational methods.
In this paper, we consider the boundary value problem for fourthorder differential equation with impulsive effects
Here, $0={t}_{0}<{t}_{1}<\cdots <{t}_{n}<{t}_{n+1}=T$, the limits ${u}^{\u2033}({t}_{j}^{\pm})={lim}_{t\to {t}_{j}^{\pm}}{u}^{\u2033}(t)$ and ${u}^{\u2034}({t}_{j}^{\pm})={lim}_{t\to {t}_{j}^{\pm}}{u}^{\u2034}(t)$ exist and $\mathrm{\Delta}{u}^{\u2034}({t}_{j})={u}^{\u2034}({t}_{j}^{+}){u}^{\u2034}({t}_{j}^{})$, $\mathrm{\Delta}{u}^{\u2033}({t}_{j})={u}^{\u2033}({t}_{j}^{+}){u}^{\u2033}({t}_{j}^{})$.
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 ${u}^{\u2033}({t}_{j}^{\pm})={lim}_{t\to {t}_{j}^{\pm}}{u}^{\u2033}(t)$ and ${u}^{\u2034}({t}_{j}^{\pm})={lim}_{t\to {t}_{j}^{\pm}}{u}^{\u2034}(t)$ exist and satisfy the impulsive conditions $\mathrm{\Delta}{u}^{\u2034}({t}_{j})=g({u}^{\prime}({t}_{j}))$, $\mathrm{\Delta}{u}^{\u2033}({t}_{j})=h(u({t}_{j}))$, $j=1,\dots ,n$, and boundary conditions $u(0)=u(T)={u}^{\u2033}(0)={u}^{\u2033}(T)=0$.
Moreover, we introduce, for every $j\in \{1,\dots ,n\}$, the following real functions:
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.
(H2) There exist ${\gamma}_{j},{\sigma}_{j}\in (2,p)$ such that functions ${g}_{j}$, ${h}_{j}$, $j=1,\dots ,n$, satisfy the conditions
A simple example of functions fulfilling the last condition is given by
where ${d}_{j}$ and ${e}_{j}$, $j=1,\dots ,n$, are positive constants.
Note that (2) implies that there exist positive constants ${D}_{j}$, ${E}_{j}$ such that
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:
(H3) There exist positive constants ${A}_{j}$, ${B}_{j}$, $j=1,\dots ,n$ such that the functions ${G}_{j}$, ${H}_{j}$, defined in (3), satisfy the conditions
A simple example of this new situation is given by the functions ${g}_{j}(t)=2{A}_{j}t$ and ${h}_{j}(t)=2{B}_{j}t$.
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.
If we consider the problem
we introduce the following condition.
(H2′) There exist ${\gamma}_{j},{\sigma}_{j}\in (2,p)$ and positive constants ${d}_{j}$, ${e}_{j}$ such that functions ${g}_{j}$, ${h}_{j}$, ${G}_{j}$ and ${H}_{j}$, $j=1,\dots ,n$, satisfy the conditions
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 $0<T\le {T}_{2}=\pi \sqrt{\frac{a+\sqrt{{a}^{2}+4{b}_{2}}}{2{b}_{2}}}$, the problem (P_{1}) has only the zero solution. If $T>{T}_{1}=\pi \sqrt{\frac{a+\sqrt{{a}^{2}+4{b}_{1}}}{2{b}_{1}}}$, the problem (P_{1}) has at least one nonzero classical solution.
The proofs of the main results are given in Section 3.
2 Preliminaries
Denote by ${L}^{p}(0,T)$ for $p\ge 1$, the Lebesgue space of pintegrable functions over the interval $(0,T)$, endowed with the usual norm ${\parallel u\parallel}_{p}^{p}={\int}_{a}^{b}{u(t)}^{p}\phantom{\rule{0.2em}{0ex}}dt$, and by $\parallel \cdot \parallel $ and ${\parallel \cdot \parallel}_{\mathrm{\infty}}$ the corresponding norms in ${L}^{2}(0,T)$ and $C([0,T])$,
Denote by ${H}_{0}^{1}(0,T)$ and ${H}^{2}(0,T)$ the Sobolev spaces
and
Let $X={H}_{0}^{1}(0,T)\cap {H}^{2}(0,T)$ be the Hilbert space endowed with the usual scalar product
and the corresponding norm.
By assumption (H_{1}) an equivalent scalar product and norm in X are given by
and
It is well known (see [[9], Lemma 2.2], [11]) that the following Poincaré and imbedding inequalities hold for all $u\in X$:
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.
We define the functional $\varphi :X\to \mathbb{R}$, as follows:
By assumption (H1), we find that $\varphi :X\to \mathbb{R}$ is continuously differentiable and, for $v\in X$, the following identity holds:
In the sequel we introduce the concept of a weak solution of our problem.
Definition 6 A function $u\in X$ is said to be a weak solution of the problem (P), if for every $v\in X$, the following identity holds:
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).
Proof Let $u\in X$ be a weak solution of (P), i.e. (11) holds for any $v\in X$. For a fixed $j\in \{0,1,\dots ,n\}$ we take a test function ${w}_{j}$, such that ${w}_{j}(t)=0$ for $t\in [0,{t}_{j}]\cup [{t}_{j+1},T]$. We have by (11)
This means that for every $w\in {X}_{j}={H}^{2}({t}_{j},{t}_{j+1})\cap {H}_{0}^{1}({t}_{j},{t}_{j+1})\subset {C}^{1}([{t}_{j},{t}_{j+1}])$
and ${u}_{j}=u{}_{({t}_{j},{t}_{j+1})}$ satisfies the equation
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 ${u}^{\u2033}({t}_{j}^{\pm})={lim}_{t\to {t}_{j}^{\pm}}{u}^{\u2033}(t)$ and ${u}^{\u2034}({t}_{j}^{\pm})={lim}_{t\to {t}_{j}^{\pm}}{u}^{\u2034}(t)$ exist.
We have for $v\in X$
Summing the last identities for $j=0,\dots ,n$ we obtain
Therefore, by (11) and (12), we have
Now, take a test function $v={v}_{j}$, $j=0,\dots ,n+1$, such that
Then we obtain ${g}_{j}({u}^{\prime}({t}_{j}))=\mathrm{\Delta}{u}^{\u2033}({t}_{j})$ and ${u}^{\u2033}(0)={u}^{\u2033}(T)=0$. Similarly, we prove that ${h}_{j}({u}^{\prime}({t}_{j}))=\mathrm{\Delta}{u}^{\u2034}({t}_{j})$, 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 semicontinuous (resp. weakly lower semicontinuous (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.35).
We have the following wellknown minimization result.
Theorem 8 Let I be a weakly lower semicontinuous 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.301303). 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 PalaisSmale (PS) condition, the mountainpass 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])
Let E be a real Banach space and $I\in {C}^{1}(E,\mathbb{R})$ satisfying condition (PS). Suppose $I(0)=0$ and

(i)
there are constants $\rho ,\alpha >0$ such that $I(u)\ge \alpha $ if $\parallel u\parallel =\rho $,

(ii)
there is an $e\in E$, $\parallel e\parallel >\rho $ such that $I(e)\le 0$.
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 ${\mathbb{S}}^{m1}$ by an odd map, and $sup\{I(u):u\in K\}<0$.
Then I possesses, at least, m distinct pairs of critical points.
3 Proofs of main results
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 PalaisSmale condition.
Lemma 11 Suppose that $p>2$ and conditions (H1) and (H2) hold. Then the functional $\varphi :X\to \mathbb{R}$ satisfies condition (PS).
Proof Let $({u}_{k})\subset X$ and $C>0$ be such that
Then we have
and
for all sufficiently large k, $k>N$. Taking $v={u}_{k}$ in (15), we have for $k>N$
In particular,
Adding the last inequality with (14), by assumption (H2), we obtain
which implies that $({u}_{k})$ is a bounded sequence in X.
Then, by the compact inclusion $X\subset {C}^{1}([0,T])$, it follows that, up to a subsequence, ${u}_{k}\rightharpoonup u$ weakly in X and ${u}_{k}\to u$ strongly in ${C}^{1}([0,T])$. As a consequence, from the inequality
it follows that
and
Then by (16) it follows that
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.
Proof of Theorem 2 We find by (H1) and (8) that the following inequalities are valid for every $u\in X$:
It is evident that this last expression is strictly positive when ${\parallel u\parallel}_{X}=\rho $, with ρ small enough. Next, let ${u}_{0}(t)=sin(\frac{\pi t}{T})\in X$ and ${u}_{\lambda}(t)=\lambda {u}_{0}(t)$, with $\lambda >0$. Then, by (H2) and (3), we have
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 mountainpass 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 PalaisSmale 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).
Proof By $1<p<2$, conditions (H1), (H3), and inequality (8), it follows that the functional ϕ is bounded from below:
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).
Proof of Theorem 3 By assumption, we know that ${g}_{j}$ and ${h}_{j}$ are odd functions. So ${G}_{j}$ are ${H}_{j}$ are even functions and the functional ϕ is even. By Lemma 12 we know that ϕ is bounded from below and satisfies condition (PS). Let $m\in \mathbb{N}$, $m\ge 3$ be a natural number and define, for any $\rho >0$ fixed, the set
${K}_{\rho}^{m}$ is homeomorphic to ${\mathbb{S}}^{m1}$ by the odd mapping defined as $H:{K}_{\rho}^{m}\to {\mathbb{S}}^{m1}$
Moreover, for $w={\sum}_{j=1}^{m}{\lambda}_{j}sin(\frac{j\pi t}{T})\in {K}_{\rho}^{m}$, the following inequalities hold:
Clearly ${K}_{\rho}^{m}$ is a subset of the mdimensional subspace
and there exist positive constants ${C}_{1}(m)$ and ${C}_{2}(m)$, such that
where ${\parallel \cdot \parallel}_{{X}_{m}}$ is the induced norm of ${\parallel \cdot \parallel}_{X}$ on ${X}_{m}$.
Arguing as in [[15], pp.1618], one can prove that there exists $\epsilon =\epsilon (m)>0$, such that
Denote
By (H3) we see that for every $w\in {K}_{\rho}^{m}$, $w={\sum}_{k=1}^{m}{\lambda}_{k}sin(\frac{k\pi t}{T})$, the following inequalities are fulfilled:
and
Denote $C=max\{{A}_{j}{m}^{3}{(\frac{\pi}{T})}^{2},{B}_{j}m:1\le j\le n\}$. Then by (18)(21) we have
where $K=\frac{1}{2}+\frac{4nC}{T{b}_{1}}$.
By the last inequality, it follows that $\varphi (w)<0$ if ${\parallel w\parallel}_{{X}_{m}}<{(\frac{{\epsilon}^{2}{C}_{1}^{p}(m)}{pK})}^{1/(2p)}$. Then, by (18), choosing
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. □
Concerning the problem (P_{1}), one can introduce similarly the notions of classical and weak solutions. In this case it is not difficult to verify that the weak solutions are critical points of the functional ${\varphi}_{1}:X\to \mathbb{R}$ defined as
Proof of Theorem 4 By the Poincaré inequalities (7) we find that $\u2980u{\u2980}^{2}={\int}_{0}^{T}({u}^{\prime \prime 2}+a{u}^{\prime 2})\phantom{\rule{0.2em}{0ex}}dt$ is an equivalent norm to ${\parallel \cdot \parallel}_{X}$ in X and the functional ${I}_{1}(u)=\frac{1}{2}\u2980u{\u2980}^{2}$ is convex.
Since the functional
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 semicontinuous.
Next, let us see that ${\varphi}_{1}:X\to \mathbb{R}$ is bounded from below:
where
Then, by Theorem 8, there exists a minimizer of ${\varphi}_{1}$, which is a critical point of ${\varphi}_{1}$.
Let u be a weak solution of (P_{1}), i.e., a critical point of ${\varphi}_{1}$. Then
If $0<T\le {T}_{2}=\pi \sqrt{\frac{a+\sqrt{{a}^{2}+4{b}_{2}}}{2{b}_{2}}}$ then ${(\frac{\pi}{T})}^{4}+a{(\frac{\pi}{T})}^{2}{b}_{2}\ge 0$. Suppose that u is a nonzero solution and $0<T\le {T}_{2}$. By (H2′), (7), and (24) it follows that
which is a contradiction. Then, for $0<T\le {T}_{2}$, the problem (P_{1}) has only the zero solution.
Suppose now that $T\ge {T}_{1}=\pi \sqrt{\frac{a+\sqrt{{a}^{2}+4{b}_{1}}}{2{b}_{1}}}$.
Take ${u}_{\epsilon}(t)=\epsilon sin(\frac{\pi t}{T})\in X$, $\epsilon >0$. Then
where $D=max\{{d}_{j},{e}_{j}:1\le j\le n\}$. For $T>{T}_{1}=\pi \sqrt{\frac{a+\sqrt{{a}^{2}+4{b}_{1}}}{2{b}_{1}}}$ it follows that ${(\frac{\pi}{T})}^{4}+a{(\frac{\pi}{T})}^{2}{b}_{1}<0$. Then, since ${\gamma}_{j},{\sigma}_{j}\in (2,p)$, by (25) it follows that ${\varphi}_{1}({u}_{\epsilon})<0$ for sufficiently small $\epsilon >0$. In consequence we show that $min\{{\varphi}_{1}(u):u\in X\}<0$. So we ensure the existence of a nonzero minimizer of ${\varphi}_{1}$, which completes the proof of Theorem 4. □
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The authors are thankful to the anonymous referees for the careful reading of the manuscript and suggestions.
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Keywords
 fourthorder differential equations
 impulsive conditions
 weak solution
 classical solution
 PalaisSmale condition
 mountainpass theorem
 Clarke’s theorem