On a discontinuous beamtype equation with deviating argument in the curvature
 Rubén Figueroa^{1}Email authorView ORCID ID profile
Received: 2 July 2016
Accepted: 21 September 2016
Published: 12 October 2016
Abstract
We deal with a nonlinear fourthorder equation where the linear part is given by the second derivative of an invertible secondorder operator. The equation includes a deviating argument and the boundary conditions provide information as regards the behavior of the solution in an initial and a final interval. Some discontinuities are allowed in all the variables and we prove for this problem the existence of solutions lying between lower and upper solutions. With the extra assumption that the given secondorder operator is inverse positive we prove the existence of extremal solutions. An example is also included in order to show the application of our results.
Keywords
MSC
1 Introduction
The way to study problem (1.1) is via a reduction of order, that is, we will consider an auxiliary secondorder problem and then we will go back to the fourthorder one. To do this we will need some conditions on operator \(\mathscr {L}\), as we will see in Section 3. This technique of reduction of order to look for solutions for fourthorder equations has been used for example in [2], where a functional equation for \(\mathscr {L}=\frac {d^{2}}{dx^{2}}\) with also functional boundary conditions is considered. Coupled with the reduction of order technique we will use a generalized monotone method in the presence of lower and upper solutions. This technique is very typical in the literature devoted to discontinuous differential equations and was developed in depth in [8].
This paper is organized as follows: in Section 2 we consider an auxiliary secondorder problem and so we recall some results on comparison principles and existence of solutions that were published in [9]. Then, in Section 3, we deal with the fourthorder problem (1.1): first we consider the case when the second variable is dropped, and then we deal with the whole problem. We finish our work in Section 4 with an example of application of our results.
2 Secondorder equations
A problem similar to (2.1) was studied in [9], with the difference that in that paper the problem was defined in \([r_{1},L]\) instead of \([r_{1},L+r_{2}]\), so (2.1) is a little more general than that problem. For this reason, and for the sake of completeness, we reformulate now the main results in [9] in order to apply them to problem (2.1), with a little sketch of their proofs. The reader is referred to the original paper for more details.
We will look for solutions for problem (2.1) inside the set \(\hat {\mathscr {X}}=\mathscr {C}(\tilde{I}) \cap W^{2,1}(I)\).
First we include a uniqueness result for the case of Lipschitzian nonlinearities, based on the wellknown Banach contraction principle.
Lemma 2.1
 (i)
For each \(u,v \in \mathbb {R}\), the mapping \(x \in I \longmapsto f(x,u,v)\) is measurable;
 (ii)for each compact subset \(K \subset \mathbb {R}^{2}\) there exists \(\psi_{K} \in L^{1}(I,[0,+\infty))\) such that$$\biglf(x,u,v)\bigr \le\psi_{K}(x) \quad\textit{for all }(u,v) \in K; $$
 (iii)there exist nonnegative functions \(M_{1},M_{2}\) such thatMoreover, the functions \(M_{1},M_{2}\) satisfy one of the following:$$\begin{aligned} &\biglf(x,u,v)f(x,\overline{u},\overline{v})\bigr \le M_{1}(x) u\overline {u} + M_{2}(x) v\overline{v}\\ &\quad \textit{for a.a. }x \in I\textit{ and all }u,v \in \mathbb {R}. \end{aligned}$$
 (C_{1}):

\(M_{1},M_{2} \in L^{\infty}(I)\) and \(\M_{1}+M_{2}\_{\infty} < \frac{1}{L^{2}}\);
 (C_{2}):

\(M_{1},M_{2} \in L^{2}(I)\) and \(\M_{1}+M_{2}\_{2} < (\frac {3}{2L^{3}} )^{1/2}\);
 (C_{3}):

\(M_{1},M_{2} \in L^{1}(I)\) and \(\M_{1}+M_{2}\_{1} < \frac{1}{2L}\).
Sketch of the proof
Conditions (i)(iii) imply that T is a welldefined contractive mapping, and so it has a unique fixed point which corresponds with the unique solution of problem (2.1). □
The following comparison principle will be essential in order to apply the generalized monotone method to the secondorder problem (2.1).
Lemma 2.2
 (i)
\(w''(x) \le M_{1}(x)w(x)+M_{2}(x)w(\tau(x)) \) for a.a. \(x \in[0,L]\);
 (ii)
\(w(0)\ge0\), \(w(L) \ge0\);
 (iii)
\(0 \le w(x) \le w(0)\) for all \(x \in[r_{1},0]\) and \(0 \le w(x) \le w(L)\) for all \(x \in[L,L+r_{2}]\);
 (iv)the functions \(M_{1},M_{2}\) satisfy one of the following conditions:
 \(\hat{(\mathrm{C}_{1})}\) :

\(M_{1},M_{2} \in L^{\infty}(I)\) and \(\M_{1}+M_{2}\_{\infty} < \frac{2}{L^{2}}\);
 \(\hat{(\mathrm{C}_{2})}\) :

\(M_{1},M_{2} \in L^{2}(I)\) and \(\M_{1}+M_{2}\_{2} < \frac {\sqrt{2}}{L}\);
 \(\hat{(\mathrm{C}_{3})}\) :

\(M_{1},M_{2} \in L^{1}(I)\) and \(\M_{1}+M_{2}\_{1} < \frac {1}{L} \).
Proof
It is the same proof as in [9], Lemma 3.1. □
Remark 2.3
Notice that for \(\frac{3}{4} \le L\) each condition (C_{ i }) implies \(\hat{(\mathrm{C}_{i})}\), \(i=1,2,3\), and for \(0< L \le\frac{3}{4}\) we have \((\mathrm{C}_{1}) \Rightarrow \hat{(\mathrm{C}_{1})}\), \((\mathrm{C}_{3}) \Rightarrow \hat{(\mathrm{C}_{3})}\), and \(\hat{(\mathrm{C}_{2})} \Rightarrow(\mathrm{C}_{2})\).
Now we are ready to look for solutions for problem (2.1). We begin by introducing the concept of lower and upper solutions.
Definition 2.4
The main result in this section establishes the existence of extremal solutions for problem (2.1) in the presence of wellordered lower and upper solutions. We recall that two solutions \(u^{*}\), \(u_{*}\) are said to be extremal in a set Y if any solution \(u \in Y\) satisfies \(u_{*} \le u \le u^{*}\).
The proof of this result uses the generalized monotone method and, in particular, the following lemma on the existence of fixed points of nondecreasing operators.
Lemma 2.5
([8], Theorem 1.2.2)
Theorem 2.6
 (H_{1}):

For each \(\gamma\in[\xi,\eta]\) the composition \(x \in I \longmapsto f(x,\gamma(x),\gamma(\tau(x)))\) is measurable;
 (H_{2}):

there exists \(\psi\in L^{1}(I,[0,\infty))\) such that for a.a. \(x \in I\), all \(u \in[\xi(x),\eta(x)]\) and all \(v \in[\xi (\tau (x)),\eta(\tau(x))]\) we have \(f(x,u,v) \le\psi(x)\);
 (H_{3}):

there exist nonnegative functions \(M_{1},M_{2}\) such that for a.a. \(t \in I\) we havewhenever \(\xi(x) \le u\le\overline{u} \le\eta(x)\), \(\xi(\tau(x)) \le v \le\overline{v} \le\eta(\tau(x))\).$$f(x,\overline{u},\overline{v})f(x,u,v) \le M_{1}(x) (\overline{u}u) + M_{2}(x) (\overline{v}v) $$
Moreover, if \(L \ge\frac{3}{4}\) then functions \(M_{1},M_{2}\) satisfy one of the conditions (C_{1}), (C_{2}), (C_{3}) and if \(0< L<\frac{3}{4}\) then they satisfy one of the following: (C_{1}), \(\hat{(\mathrm{C}_{2})}\), \((\mathrm{C}_{3})\).
Sketch of the proof
Operator T is well defined by application of Lemma 2.1. Moreover, we can show by application of Lemma 2.2, in a similar way to [9], Theorem 3.5, that T is a nondecreasing operator which maps interval \([\xi,\eta]\) into itself and which, moreover, maps monotone sequences into convergent ones in \(\mathscr {C}(\tilde{I})\). Then Lemma 2.5 guarantees that T has the extremal fixed points in \([\xi,\eta]\), which correspond with the extremal solutions of problem (2.1) in \([\xi,\eta]\). □
Remark 2.7
Condition (H_{3}) states, roughly speaking, that the functions \(f_{1}(u)=f(x,u,v)M_{1}(x) u\) and \(f_{2}(v)=f(x,u,v)M_{2}(x) v\) must be nonincreasing. In Section 4 we will prove a lemma that can be useful in order to check this condition in practice.
3 Fourth order equations
We define lower and upper solutions for problem (3.1).
Definition 3.1
Now we prove the existence of solutions for problem (3.1). We recall that we say that the operator \(\mathscr {L}:W^{2,1}\longrightarrow \mathscr {C}\) is invertible in a certain set \(Z\subset W^{2,1}\) if there exists \(\mathscr {L}^{1}:\mathscr {C} \longrightarrow Z\) such that \(v=\mathscr {L}^{1}[\mathscr {L}[v]]\) for all \(v \in Z\). This is equivalent to the fact that there exists an associated Green’s function in Z (see [10] for more details).
Theorem 3.2
Proof
Remark 3.3
Notice that equation (3.2) does not imply the existence of \(\mathscr {L}[u](0)\) nor \(\mathscr {L}[u](L)\). However, the fact of \(\mathscr {L}[u]=v\) almost everywhere, joined with condition (1.2), is enough to guarantee that (3.4) makes sense. On the other hand, we are assuming throughout this paper that \(r_{1},r_{2}>0\). We can relax this condition and require these constants to be only nonnegative. Therefore, if \(r_{1}=0\) we could actually provide some additional condition about the value of \(\mathscr {L}[u](0)\), and analogously if \(r_{2}=0\).
The following result states the existence of extremal solutions under the additional hypothesis that the operator \(\mathscr {L}\) is inverse positive, that is, \(u\ge0 \Rightarrow \mathscr {L}^{1}[u] \ge0\), in a certain set \(Z'\). This condition is equivalent to saying that \(\mathscr {L}\) satisfies the antimaximum principle in \(Z'\), that is, \(\mathscr {L}[u] \ge0 \Rightarrow u\ge0\). This provides a certain monotonicity condition that will be essential in order to apply the monotone iterative technique. To see more examples of application of this technique in the presence of inverse positive or inverse negative operators the reader is referred to [11] or [10].
Theorem 3.4
Proof
We will divide the proof in three steps.
Claim 1: Let \(v \in[\mathscr {L}[\alpha],\mathscr {L}[\beta]]\) be a solution of problem ( 2.1 ) and u a solution of ( 3.1 ) obtained from v by equation ( 3.2 ). Then \(\alpha\le u \le\beta\) on Ĩ. It is clear by construction of u and the conditions on \(\alpha, \beta\) that \(\alpha(x) \le u(x) \le\beta(x)\) on \(I_{} \cup I_{+}\), so we only have to prove that this inequality is true in I.
In a same way we prove that \(u \le\beta\) on Ĩ.
Claim 3: If v̂ is the greatest solution in \([\mathscr {L}[\alpha],\mathscr {L}[\beta]]\) of problem ( 2.1 ) then û obtained from v̂ by equation ( 3.2 ) provides the greatest solution of problem ( 3.1 ) in \([\alpha,\beta]\) . Let \(u \in [\alpha ,\beta]\) be an arbitrary solution of problem (3.1) and \(v \in [\mathscr {L}[\alpha],\mathscr {L}[\beta]]\) provided by Claim 2. Then \(v(0)=\hat{v}(0)\), \(v(L)=\hat{v}(L)\), and \(v \le\hat{v}\) on I. As \(\mathscr {L}\) is inverse positive in Z we obtain \(u=\mathscr {L}^{1}[v] \le \mathscr {L}^{1}[\hat{v}]=\hat{u}\) on I. Taking into account that \(u(x) = \hat{u}(x)\) for all \(x \in I_{} \cup I_{+}\) we conclude that \(u \le\hat{u}\) in Ĩ and so û is the greatest solution of problem (3.1) in \([\alpha,\beta]\). In an analogous way we obtain the least solution of the problem in \([\alpha ,\beta]\). □
Now we are going to develop a monotone method in order to look for solutions to the whole problem (1.1).
Definition 3.5
The following is the main result in this paper.
Theorem 3.6
 (H_{4}):

for a.a. \(x \in I\), all \(u \in[\mathscr {L}[\alpha ](x),\mathscr {L}[\beta](x)]\) and all \(v \in[\mathscr {L}[\alpha ](\tau (x)),\mathscr {L}[\beta](\tau(x))]\) function \(f(x,\cdot,u,v)\) is nonincreasing in \([\alpha,\beta]\).
Proof
Finally, we will show that T maps monotone sequences into convergent ones. So, let \(\{\gamma_{n}\}_{n \in \mathbb {N}}\) be a monotone sequence in \([\alpha,\beta]\). As T is nondecreasing and \(T([\alpha,\beta]) \subset[\alpha,\beta]\) then \(\{T\gamma_{n}\}_{n \in \mathbb {N}}\) is a monotone sequence which has its pointwise limit in \([\alpha,\beta]\), say Γ. We have to prove that \(T\gamma_{n} \to\Gamma\) in \(\mathscr {C}(I)\). As \(\{T\gamma_{n}\}_{n \in \mathbb {N}}\) is constant in \(I_{}\cup I_{+}\), we only need to check that the convergence is uniform in I. To see this, put \(g_{n}=\mathscr {L}[T\gamma_{n}]\) and notice that \(g_{n} \le \mathscr {L}[\beta ]\) for all \(n \in \mathbb {N}\), and so \(g_{n}\) is uniformly bounded. As the operator \(\mathscr {L}^{1}\) is compact we conclude that \(\{T\gamma_{n}\} _{n \in \mathbb {N}}\) is precompact and so it has a convergent subsequence that coincides with \(\{T\gamma_{n}\}_{n \in \mathbb {N}}\) because of monotonicity.
We conclude by application of Lemma 2.5 that T has the greatest fixed point, \(u^{*}\) in \([\alpha,\beta]\) and we claim that \(u^{*}\) is the greatest solution of our problem in \([\alpha,\beta]\). Indeed, it is obvious that \(u^{*}\) is a solution. To see that it is the greatest one, let \(u \in[\alpha,\beta]\) another solution. Then \(Tu=u\) and so characterization (2.2) implies that \(u^{*} \ge u\).
To obtain the least solution of problem (1.1) in \([\alpha,\beta]\) we must redefine operator T, considering the least solution of auxiliar problem \((P_{\gamma})\) instead of the greatest one, and reasoning like above. □
4 An example of application
In this section we provide an example to show how our results can be applied in practice. First we begin by introducing a lemma that will be useful for us in order to check condition (H_{3}).
Lemma 4.1
 (i)$$\lim_{x \to x_{k}^{}}f(x) \ge f(x_{k}) \ge\lim _{x \to x_{k}^{+}} f(x); $$
 (ii)
there exist \(M_{k}=\sup\{f'(x) : x \in (x_{k1},x_{k})\}\) and \(M=\sup\{M_{k} : k \in \mathbb {N}\}\).
Under these conditions, for all \(\tilde{M} \ge M\) the function \(g:x \in \mathbb {R}\longmapsto g(x)=f(x)  \tilde{M} x\) is nonincreasing.
Proof
Fixing \(k \in \mathbb {N}\) we see that the function g is differentiable in \((x_{k1},x_{k})\) and \(g'(x) \le M_{k}  M \le0\) for all \(x \in(x_{k1}, x_{k})\), so g is nonincreasing in that interval. On the other hand, condition (i) provides that f is nonincreasing at point \(x_{k}\), and so g. □
Our main goal in this section is to show that problem (4.1) has the extremal solutions between suitable lower and upper solutions. We will do it in several steps and we begin by checking the measurability conditions.
Lemma 4.2
Proof
On the other hand, the function \(f_{1}\) is continuous almost everywhere and so the composition \(f_{1} \circ\lambda\) is measurable. Therefore (4.2) is a measurable function as a linear combination of measurable ones. □
Now we show the existence of wellordered lower and upper solutions.
Proposition 4.3
Functions \(\alpha\equiv0\) and \(\beta(x)=\cos(x)\) are, respectively, a lower and an upper solution for problem (4.1).
Proof
First it is trivially satisfied that \(\alpha\le \beta\) on Ĩ. On the other hand, as we have \(\mathscr {L}[\alpha ](x)=0\) and \(\mathscr {L}[\beta](x)=(M+1)\cos(x) \ge0\), we also obtain \(\mathscr {L}[\alpha](x) \le \mathscr {L}[\beta](x)\) for all \(x \in \tilde{I}\).
Finally, we prove that problem (4.1) has the extremal solutions between α and β.
Proposition 4.4
If \(n \le7\) then problem (4.1) has the extremal solutions between the lower solution \(\alpha\equiv0\) and the upper solution \(\beta(x)=\cos(x)\).
We will prove the statement by application of Theorem 3.6.
Declarations
Acknowledgements
Partially supported by Xunta de Galicia (Spain), project EM2014/032.
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.
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