We consider now more general boundary conditions in order to deal with periodic, Neumann, Sturm-Liouville boundary conditions for equation (4), using the result obtained in the previous section for Dirichlet problems, following the idea developed in [1]. To this aim, in what follows we adopt the notation \((D_{c,d})\) to denote the Dirichlet problem \((D)\), when we need to emphasize the values of the boundary conditions.

The following lemma provides a compactness-type result for the solutions of equation (4) obtained by means of Theorem 2.

### Lemma 1

*Let assumption* (9) *be satisfied*, *and let*
\(\alpha, \beta\)
*be a well*-*ordered pair of lower and upper solutions for equation* (4). *Then*, *for every pair of sequences*
\((c_{n})_{n}\)
*and*
\((d_{n})_{n}\)
*of real numbers satisfying*
\(c_{n}\in[\alpha(0), \beta(0)]\)
*and*
\(d_{n}\in[\alpha(T), \beta(T)]\)
*for every*
\(n\in \mathbb {N}\), *and for every sequence*
\((x_{n})_{n}\)
*of solutions of problem*
\(D_{c_{n},d_{n}}\), *equibounded in*
\(C^{1}\)
*and belonging to the functional interval*
\([\alpha,\beta]\), *there exists a subsequence*
\((x_{n_{k}})_{k}\)
*such that*

$$x_{n_{k}}(t)\to x_{0}(t),\qquad x_{n_{k}}'(t)\to x_{0}'(t) \quad\textit{uniformly in } I $$

*for some solution*
\(x_{0}\)
*of equation* (4).

### Proof

Let \((c_{n})_{n}\), \((d_{n})_{n}\) be a pair of sequences of real numbers satisfying \(\alpha(0)\le c_{n}\le\beta(0)\) and \(\alpha (T)\le d_{n}\le\beta(T)\) for every \(n\in \mathbb {N}\), and let \((x_{n})_{n}\) be a sequence of solutions of problem \(D_{c_{n},d_{n}}\), equibounded in \(C^{1}\) and belonging to the functional interval \([\alpha,\beta]\). We can assume without restriction, possibly by passing to subsequences, that \(c_{n}\to c_{0}\), \(d_{n}\to d_{0}\). By the equiboundedness of the sequence \((x_{n})_{n}\) we have \(\vert x_{n}(t) \vert + \vert x'_{n}(t) \vert < r\) for all \(t \in I\), for some \(r>0\). Therefore, by assumption (9) we get

$$\bigl\vert \bigl( a \bigl(t,x_{n}(t) \bigr) \Phi \bigl(x_{n}'(t) \bigr) \bigr)' \bigr\vert = \bigl\vert f \bigl(t,x_{n}(t),x_{n}'(t) \bigr) \bigr\vert \le\gamma_{r}(t) \quad\text{for all } n, \text{ a.e. } t\in I. $$

So, the sequence of functions \(( a(t,x_{n}(t)) \Phi(x_{n}'(t)) )_{n} \) is equicontinuous on *I*. Moreover, since the function \(1/a(t,x)\) is uniformly continuous on \(I\times [-r,r]\), we deduce that also the sequence \((1/a(t,x_{n}(t)))_{n}\) is equicontinuous on *I*. Therefore, the sequence \((\Phi (x_{n}'(t)) )_{n}\) is equicontinuous on *I*, as it is the product of equicontinuous functions. Finally, by the continuity of the function \(\Phi^{-1}\), we deduce that the sequence \((x_{n}')_{n}\) is uniformly continuous on *I*. Recalling that \(\vert x_{n}'(t) \vert \le r\) for every \(n \in \mathbb {N}\) and a.e. \(t\in I\), by the Ascoli-Arzelà theorem, we infer the existence of a subsequence \((x_{n_{k}}')_{k}\) uniformly convergent to a continuous function \(y_{0}\), implying that also \((x_{n_{k}})_{k}\) uniformly converges to the function \(x_{0}(t)=\int_{0}^{t} y_{0}(s) \,\mathrm{ d}s\).

Moreover, notice that, for all \(t\in I\), we have

$$a \bigl(t,x_{n_{k}}(t) \bigr)\Phi\bigl(x'_{n_{k}}(t) \bigr)= a \bigl(0,x_{n_{k}}(0) \bigr)\Phi\bigl(x'_{n_{k}}(0) \bigr)+ { \int_{0}^{t} f \bigl(s, x_{n_{k}}(s), x_{n_{k}}'(s) \bigr)\,\mathrm{ d}s}. $$

Hence, by the dominated convergence theorem we obtain

$$a \bigl(t,x_{0}(t) \bigr)\Phi\bigl(x'_{0}(t) \bigr)= a \bigl(0,x_{0}(0) \bigr)\Phi\bigl(x'_{0}(0) \bigr)+ { \int_{0}^{t} f \bigl(s, x_{0}(s), x_{0}'(s) \bigr)\,\mathrm{ d}s}, $$

that is, \(x_{0}\) is a solution of (4). □

In order to handle various types of boundary condition, let us consider the following problem with very general nonlinear boundary conditions:

$$ \textstyle\begin{cases} (a(t,x(t))\Phi(x'(t)))' = f(t,x(t),x'(t)) \quad\mbox{a.e. } t\in I, \\ g(x(0),x(T),x'(0),x'(T))=0, \\ x(T)=h(x(0)), \end{cases} $$

(22)

where \(g:\mathbb {R}^{4}\to \mathbb {R}\) and \(h:\mathbb {R}\to \mathbb {R}\) are continuous functions.

By applying Theorem 2 we are able to prove an existence result also for the general problem (22).

### Theorem 3

*Suppose that there exists a well*-*ordered pair*
*α*, *β*
*of lower and upper solutions for equation* (4) *such that*

$$\textstyle\begin{cases} g(\alpha(0),\alpha(T),\alpha'(0),\alpha'(T))\ge0, \\ \alpha(T)=h(\alpha(0)); \end{cases}\displaystyle \qquad \textstyle\begin{cases} g(\beta(0),\beta(T),\beta'(0),\beta'(T))\le0, \\ \beta(T)=h(\beta(0)). \end{cases} $$

*Let assumptions* (9), (10), (11) *be satisfied*. *Moreover*, *suppose*
*h*
*is increasing and*

$$ g(u,v,\cdot,z)\quad\textit{is increasing; }\qquad g(u,v,w,\cdot)\quad\textit{is decreasing.} $$

(23)

*Then problem* (22) *admits a solution*
*x*, *belonging to the functional interval*
\([\alpha, \beta]\), *such that*
\(\Vert x \Vert _{C^{1}}\le\Lambda\), *where* Λ *is the constant given by Theorem *
2 (*see* (12)), *with respect to*
\(M:=\max\{{\max_{t \in I} \vert \alpha(t) \vert }, {\max_{t \in I} \vert \beta(t) \vert }\}\).

### Proof

Let us fix \(c \in[\alpha(0),\beta(0)]\). By the monotonicity of *h*, we have \(\alpha(T)\le h(c)\le\beta(T)\); so, by Theorem 2, the problem \((D_{c,h(c)})\) admits a solution \(x_{c}\) belonging to the functional interval \([\alpha,\beta]\). Moreover, there exists a constant Λ such that, for every \(c\in[\alpha(0),\beta (0)]\), we have (see (12))

$$ \Vert x_{c} \Vert _{C(I)}\le M \quad\text{and}\quad\bigl\Vert x_{c}' \bigr\Vert _{C(I)}\le\Lambda. $$

(24)

Put

$$\begin{aligned} \Omega:= & \bigl\{ c\in\bigl[\alpha(0),\beta(0)\bigr]: \mbox{ there exists a solution } x_{c} \in[\alpha,\beta] \mbox{ of } (D_{c,h(c)}) \\ & \text{satisfying } (24) \text{ and such that } g \bigl(x_{c}(0),x_{c}(T),x_{c}'(0),x_{c}'(T) \bigr)\ge0\bigr\} . \end{aligned}$$

Notice that Ω is not empty because \(\alpha(0)\in\Omega\). Indeed, if \(c=\alpha(0)\), then also \(x_{c}(T)=\alpha(T)\) and, since \(\alpha(t)\le x_{c}(t)\), for all \(t\in I\), we get \(x'_{c}(0)\ge\alpha'(0)\) and \(x'_{c}(T)\le\alpha'(T)\). Thus, from (23), we have

$$g \bigl(x_{c}(0),x_{c}(T),x_{c}'(0),x_{c}'(T) \bigr)\ge0. $$

Let \(c^{*}:=\sup\Omega\), and let us prove that \(c^{*}\in\Omega\). This is trivial if \(c^{*}=\alpha(0)\), whereas, if \(c^{*}>\alpha(0)\), let \((c_{n})_{n}\subset\Omega\) be a sequence converging to \(c_{*}\) from below. Put \(d_{n}=h(c_{n})\), by Theorem 2, for every \(n\in \mathbb {N}\), there exists a solution \(x_{n}\) of problem \((D_{c_{n},d_{n}})\), belonging to the functional interval \([\alpha,\beta]\), such that \(x_{n}\) satisfies (24) for every \(n\in \mathbb {N}\). So, we can apply Lemma 1 to infer the existence of a subsequence \((x_{n_{k}})_{k}\) of solutions with the properties

$$x_{n_{k}}(t)\to x_{*}(t),\qquad x_{n_{k}}'(t)\to x_{*}'(t) \quad\text{for every } t\in I $$

for some solution \(x_{*}\) of equation (4). Of course, \(c^{*}:=x_{*}(0)\) and, by the continuity of *h*, we get that \(x_{*}\) is a solution of problem \((D_{c_{*},h(c_{*})})\). Moreover, since \(c_{n_{k}}\in\Omega\), we have \(g(x_{n_{k}}(0),x_{n_{k}}(T),x_{n_{k}}'(0),x_{n_{k}}'(T))\geq0\) for every \(k \in\mathbb{N}\) and, by the continuity of the function *g*, we obtain

$$ g \bigl(x_{*}(0),x_{*}(T),x_{*}'(0),x_{*}'(T) \bigr)\ge0, $$

(25)

then \(c_{*} \in\Omega\) and so \(c_{*}=\max\Omega\).

If \(c^{*}=\beta(0)\), then \(x_{*}'(0)\le\beta'(0)\). Moreover, \(x_{*}(T)=h(\beta(0))=\beta(T)\) implying \(x_{*}'(T)\ge\beta'(T)\). Hence, by (23) we deduce

$$\begin{aligned} g \bigl(x_{*}(0),x_{*}(T),x_{*}'(0),x_{*}'(T) \bigr)& = g \bigl( \beta(0),\beta(T),x_{*}'(0),x_{*}'(T) \bigr) \\ &\le g\bigl(\beta(0),\beta(T),\beta'(0),\beta'(T) \bigr) \le0. \end{aligned}$$

So, taking account of (25), we get \(g(x_{*}(0),x_{*}(T),x_{*}'(0),x_{*}'(T)) = 0\) and \(x_{*}\) is a solution of problem (22).

Let us now treat the case \(c^{*}<\beta(0)\). In order to do this, let us consider a decreasing sequence \((c_{m})_{m}\) converging to \(c^{*}\). Of course, the functions \(x_{*}\) and *β* can be considered as a well-ordered pair of lower and upper solutions for equation (4), with \(\Vert x^{*} \Vert _{C(I)}\le M\) and \(\Vert x^{*'} \Vert _{C(I)}\le\Lambda \). By applying again Theorem 2, we deduce that, for every \(m\in \mathbb {N}\), there exists a solution \(\xi_{m}\) of problem \((D_{c_{m},h(c_{m})})\) belonging to the functional interval \([x_{*},\beta]\), again with \(\Vert \xi_{m} \Vert _{C(I)}\le M\) and \(\Vert \xi_{m}' \Vert _{C(I)}\le\Lambda\) for every \(m\in \mathbb {N}\). Hence, we can apply Lemma 1 again, obtaining the existence of a subsequence \((\xi_{m_{k}})_{k}\) of solutions such that

$$\xi_{m_{k}}(t)\to\xi_{*}(t), \qquad\xi_{m_{k}}'(t)\to \xi_{*}'(t)\quad\text{for every } t\in I $$

for some solution \(\xi_{*}\) of equation (4). By the continuity of *h*, the function \(\xi_{*}\) is a solution of problem \((D_{c_{*},h(c_{*})})\). Moreover, since \(c_{m_{k}}>c_{*}=\max\Omega\), we have \(c_{m}\notin\Omega\) for every \(m\in \mathbb {N}\). Since \(\Vert \xi_{m} \Vert _{C^{1}(I)}\le\Lambda\), necessarily we have \(g(\xi_{m}(0),\xi_{m}(T),\xi_{m}'(0),\xi_{m}'(T)) <0 \). Thus, by the continuity of *g*, we have

$$ g \bigl(\xi_{*}(0),\xi_{*}(T),\xi_{*}'(0),\xi_{*}'(T) \bigr) \le0. $$

(26)

Nevertheless, being \(\xi_{*}(t)\ge x_{*}(t)\) for every \(t\in I\), with \(\xi_{*}(0)=x_{*}(0)\) and \(\xi_{*}(t)=x_{*}(T)\), we deduce that \(\xi_{*}'(0)\ge x_{*}'(0)\) and \(\xi_{*}(T)\le x_{*}'(T)\). So, by (23) and (25) we infer

$$\begin{aligned} g \bigl(\xi_{*}(0),\xi_{*}(T),\xi_{*}'(0), \xi_{*}'(T) \bigr)&= g \bigl(x_{*}(0),x_{*}(T),\xi_{*}'(0), \xi_{*}'(T) \bigr) \\ &\ge g \bigl(x_{*}(0),x_{*}(T),x_{*}'(0), x_{*}'(T) \bigr) \ge0 \end{aligned}$$

that jointly with (26) implies

$$g \bigl(\xi_{*}(0),\xi_{*}(T),\xi_{*}'(0), \xi_{*}'(T) \bigr)=0 $$

and \(\xi_{*}\) is a solution of problem (22). □

The general boundary conditions considered in problem (22) include, as a particular case, periodic boundary conditions, that is, the problem

$$ \textstyle\begin{cases} (a(t,x(t))\Phi(x'(t)))' = f(t,x(t),x'(t)) \quad\mbox{a.e. } t\in I, \\ x(0)=x(T),\qquad x'(0)=x'(T) . \end{cases} $$

(27)

As an immediate consequence of Theorem 3, the following existence result holds.

### Theorem 4

*Let*
*α*
*and*
*β*
*be a well*-*ordered pair of lower and upper solutions for equation* (4) *such that*

$$\textstyle\begin{cases} \alpha(0)=\alpha(T) ,\\ \alpha'(0)\ge\alpha'(T), \end{cases}\displaystyle \quad\textit{and}\quad \textstyle\begin{cases} \beta(0)=\beta(T), \\ \beta'(0)\le\beta'(T). \end{cases} $$

*Assume that hypotheses* (9), (10), (11) *are satisfied*. *Then problem* (27) *has a solution belonging to the functional interval*
\([\alpha, \beta]\).

### Proof

The assertion is an immediate consequence of Theorem 3, taking \(g(u,v,w,z):=w-z\) and \(h(r):=r\). □

Let us consider now the following boundary value problem:

$$ \textstyle\begin{cases} (a(t,x(t))\Phi(x'(t)))' = f(t,x(t),x'(t)) \quad\mbox{a.e. } t\in I, \\ p(x(0), x'(0))=0 ,\qquad q(x(T),x'(T))=0, \end{cases} $$

(28)

where \(p,q:\mathbb {R}^{2}\to \mathbb {R}\) are continuous functions. The following existence result for problem (28) holds.

### Theorem 5

*Let*
*α*
*and*
*β*
*be a well*-*ordered pair of lower and upper solutions for equation* (4) *such that*

$$\textstyle\begin{cases} p(\alpha(0), \alpha'(0))\ge0, \\ q(\alpha(T), \alpha'(T))\ge0; \end{cases}\displaystyle \quad\textit{and}\quad \textstyle\begin{cases} p(\beta(0),\beta'(0))\le0, \\ q(\beta(T), \beta'(T))\le0. \end{cases} $$

*Assume that hypotheses* (9), (10), (11) *are satisfied*. *Moreover*, *assume that*, *for every*
\(s\in \mathbb {R}\), *we have*

$$ p(s,\cdot) \quad\textit{is increasing}\quad\textit{and}\quad q(s,\cdot) \quad\textit{is decreasing}. $$

(29)

*Then problem* (28) *has a solution belonging to the functional interval*
\([\alpha, \beta]\).

### Proof

For every \(c\in[\alpha(0), \beta(0)]\) and \(d\in[\alpha(T), \beta(T)]\), let \(x_{c,d}\) denote a solution of problem \((D_{c,d})\) belonging to the functional interval \([\alpha,\beta]\), whose existence is ensured by Theorem 2 satisfying (12).

For every fixed \(d\in[\alpha(T),\beta(T)]\), let us consider now the following auxiliary boundary value problem:

So, by virtue of Theorem 3, we have that, for every \(d\in[\alpha(T),\beta(T)]\), there exists a solution \(y_{d}\) of problem (\(Q_{d}\)), belonging to the functional interval \([\alpha,\beta]\), such that \(\Vert y \Vert _{C(I)}\le M \) and \(\Vert y' \Vert _{C(I)}\le\Lambda\).

Observe now that, taking account of (29), when \(d=\alpha (T)\), we have \(q(y(T),y'(T))\ge q(\alpha(T),\alpha'(T))\ge0\) for every solution *y* of problem (\(Q_{d}\)); whereas for \(d=\beta(T)\), we have \(q(y(T),y'(T))\le q(\beta(T),\beta'(T))\le0\) for every solution *y* of problem (\(Q_{d}\)). Our goal is to show that there exists a value \(d^{*}\in[\alpha (T),\beta(T)]\) and a solution \(y_{*}\) of problem \((Q_{d^{*}})\) such that \(q(y_{d^{*}}(T),y_{d^{*}}'(T))= 0\).

To this aim, put

$$\begin{aligned} {\Upsilon}:= {}& \bigl\{ d\in\bigl[\alpha(T),\beta(T)\bigr]: \text{ there exists a solution } \xi_{d} \text{ of problem } (Q_{d}) \\ & \text{ such that } \Vert \xi_{d} \Vert _{C(I)}\le M , \bigl\Vert \xi_{d}' \bigr\Vert _{C(I)}\le \Lambda\text{ and } q\bigl(\xi_{d}(T),\xi_{d}'(T) \bigr)\ge0\bigr\} . \end{aligned}$$

We have that \(\alpha(T)\in\Upsilon\), so \(\Upsilon\ne\emptyset\). Let \(d^{*}:=\sup\Upsilon\), and let us prove that \(d^{*}=\max\Upsilon \), that is, \(q(\xi_{*}(T),\xi_{*}'(T))\ge0 \) for some solution \(\xi_{*}\) of problem \((Q_{d^{*}})\). This is trivial if \(d^{*}=\alpha(T)\), whereas if \(d^{*}>\alpha(T)\), let \((d_{n})_{n}\subset\Upsilon\) be an increasing sequence converging to \(d^{*}\), and let \((\xi_{d_{n}})_{n}\) be a sequence of solutions of problem \((Q_{d_{n}})\) satisfying \(q( \xi_{d_{n}}(T), \xi_{d_{n}}'(T))\ge 0\). By virtue of Lemma 1 we get the existence of a subsequence \((\xi_{d_{n_{k}}})_{k}\) such that

$$\xi_{d_{n_{k}}}(t) \to\xi_{*}(t), \qquad\xi_{d_{n_{k}}}'(t)\to \xi_{*}'(t)\quad\text{for every } t\in I, $$

for some solution \(\xi_{*}\) of equation (4). By the continuity of *p* and *q*, we get \(\xi_{*}\) is a solution of problem \((Q_{d^{*}})\) such that \(q(\xi_{*}(T), \xi_{*}'(T))\ge0\), hence \(d^{*}\in\Upsilon\).

If \(d_{*}=\beta(T)\), then \(\xi_{*}'(T)\ge\beta'(T)\) and by (29) we get

$$q \bigl(\xi_{*}(t), \xi_{*}'(T) \bigr)= q \bigl(\beta(T), \xi_{*}'(T) \bigr)\le q \bigl(\beta(T),\beta'(T) \bigr) \le0. $$

So, \(\xi_{*}\) is a solution of problem \((Q)\).

Finally, let us treat the case \(d^{*}<\beta(T)\). Let \((d_{m})_{m}\) be a decreasing sequence converging to \(d^{*}\). Notice that the functions \(\xi_{*}\) and *β* are a well-ordered pair of lower and upper solutions for equation (4) satisfying the assumptions of Theorem 3 (for \(h(c):=d\) and \(g(u,v,w,z):=p(u,w)\)). So, for every \(m\in \mathbb {N}\), there exists a solution \(\zeta_{m}\) of problem *G* such that

$$\xi_{*}(t)\le\zeta_{m}(t)\le\beta(t)\quad\text{for every } t \in I. $$

Since \(d_{m}\notin\Upsilon\), we have

$$ q \bigl(\zeta_{m}(T),\zeta_{m}'(T) \bigr) < 0 \quad\text{for every } m\in \mathbb {N}. $$

(30)

We can apply again Lemma 1 to deduce the existence of a subsequence \((\zeta_{m_{k}})_{k}\) such that

$$\zeta_{m_{k}} \to\zeta_{*}(t),\qquad\zeta_{m_{k}}'(t) \to \zeta_{*}'(t) \quad\text{for every } t \in I $$

and

$$ \xi_{*}(t)\le\zeta_{*}(t)\le\beta(t) \quad\text{for every } t\in I, $$

(31)

for some \(\zeta_{*}\) solution of equation (4). By the continuity of the function *p*, we have \(p(\zeta_{*}(0),\zeta_{*}'(0))=0\), and by (30) and the continuity of *q*, we get

$$ q \bigl(\zeta_{*}(0),\zeta_{*}'(0) \bigr)\le0. $$

(32)

On the other hand, since \(\xi_{*}(T)=d^{*}= \zeta_{*}(T)\), by (31) we get \(\zeta_{*}'(T)\le\xi_{*}'(T)\). Therefore, from (29) we infer

$$q(\zeta_{*}(T), \zeta_{*}'(T)=q \bigl(\xi_{*}(T),\zeta_{*}'(T) \bigr)\ge q \bigl(\xi_{*}(T),\xi_{*}'(T) \bigr) \ge0 $$

that jointly with (32) implies \(q(\zeta_{*}(0),\zeta _{*}'(0))= 0\). So, \(\zeta_{*}\) is a solution of problem \((Q)\). □

### Remark 2

The boundary conditions in problem (28) cover, as particular cases, both the Sturm-Liouville and the Neumann problems. In fact, for the former problem, one takes \(p(s,t):=\ell _{0}s+m_{0}t-\nu_{0}\) and \(q(s,t)=\ell_{1}s-m_{1}t-\nu_{1}\), with \(m_{0},m_{1}\ge0\). With this choice, one gets the boundary conditions

$$\ell_{0}x(0)+m_{0}x'(0)= \nu_{0} , \qquad\ell_{1}x(T)-m_{1}x'(T)= \nu_{1}. $$

For the latter problem, one takes \(p(s,t):=t-\nu_{0}\) and \(q(s,t):=\nu _{1}-t\), and problem (28) becomes the Neumann problem

$$ \textstyle\begin{cases} (a(t,x(t))\Phi(x'(t)))' = f(t,x(t),x'(t)) \quad\mbox{a.e. } t\in I, \\ x'(0)=\nu_{0},\qquad x'(T)=\nu_{1}. \end{cases} $$

(P)

Thus, Theorem 5 gives a condition for the existence of solutions for both these problems.