We devote this section to state some operative criteria which can usefully applied to operators and right-hand sides having the product structure

a(t,x)=\alpha (t)\beta (x)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}f(t,x,y)=b(t,x)c(x,y).

We will highlight how the local behaviors of c(x,\cdot ) at y=0 and of b(\cdot ,x), \alpha (\cdot ) at infinity, related to the maximum and the minimum of the functions *β*, *g* in the interval [{\nu}^{-},{\nu}^{+}], play a relevant role for the existence or non-existence of solutions.

In what follows, we assume that *α*, *β* are continuous positive functions, *b* is a Carathéodory function and *c* is a continuous function such that

c(x,y)>0\phantom{\rule{1em}{0ex}}\text{for every}y\ne 0\text{and}x\in [{\nu}^{-},{\nu}^{+}];\phantom{\rule{2em}{0ex}}c({\nu}^{-},0)=c({\nu}^{+},0)=0.

In this framework, putting \tilde{m}:={min}_{x\in [{\nu}^{-},{\nu}^{+}]}\beta (x) and \tilde{M}:={max}_{x\in [{\nu}^{-},{\nu}^{+}]}\beta (x), we have

m(t)=\tilde{m}\alpha (t)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}M(t)=\tilde{M}\alpha (t)\phantom{\rule{1em}{0ex}}\text{for every}t\in \mathbb{R},

where recall that m(t):={min}_{x\in [{\nu}^{-},{\nu}^{+}]}a(t,x) and M(t):={max}_{x\in [{\nu}^{-},{\nu}^{+}]}a(t,x).

Finally, we put

{m}_{\mathrm{\infty}}:=\underset{t\in \mathbb{R}}{inf}\alpha (t)\ge 0.

(3.1)

### 3.1 Case of Φ growing at most linearly

In this subsection, we deal with differential operators Φ satisfying condition (2.4) that is such that |\mathrm{\Phi}(y)|\le \mathrm{\Lambda}|y| whenever every |y|>H for some H,\mathrm{\Lambda}>0. With this class of operators, we cover differential equations of the type

{(a(t,x(t)){x}^{\prime}(t))}^{\prime}=f(t,x(t),{x}^{\prime}(t)).

The first two existence theorems are an application of Theorem 2.1.

**Proposition 3.1**
*Suppose that*

t\cdot b(t,x)<0\phantom{\rule{1em}{0ex}}\mathit{\text{for a.e.}}t\mathit{\text{such that}}|t|\ge L,\mathit{\text{every}}x\in [{\nu}^{-},{\nu}^{+}]

(3.2)

*for some* L\ge 0, *and there exists a function* \lambda \in {L}_{\mathrm{loc}}^{q}(\mathbb{R}), 1\le q\le +\mathrm{\infty}, *such that*

|b(t,x)|\le \lambda (t)\phantom{\rule{1em}{0ex}}\mathit{\text{for a.e.}}t\in \mathbb{R},\mathit{\text{every}}x\in [{\nu}^{-},{\nu}^{+}].

(3.3)

*Moreover*, *assume that there exists a real constant* *p* (*not necessarily positive*) *such that for every* x\in [{\nu}^{-},{\nu}^{+}], *we have*

{h}_{1}|t{|}^{p}\le \alpha (t)\le {h}_{2}|t{|}^{p}\phantom{\rule{1em}{0ex}}\mathit{\text{a.e.}}|t|L,

(3.4)

{\lambda}_{1}|t{|}^{p-1}\le |b(t,x)|\le {\lambda}_{2}|t{|}^{p-\sigma}\phantom{\rule{1em}{0ex}}\mathit{\text{a.e.}}|t|L,

(3.5)

c(x,y)\ge {k}_{1}\mathrm{\Phi}(|y|)\phantom{\rule{1em}{0ex}}\mathit{\text{for every}}y\in \mathbb{R},

(3.6)

c(x,y)\le {k}_{2}\mathrm{\Phi}{(|y|)}^{\delta}\phantom{\rule{1em}{0ex}}\mathit{\text{whenever}}|y|\rho ,

(3.7)

c(x,y)\le {k}_{2}|\mathrm{\Phi}(y){|}^{2-\frac{1}{q}}\phantom{\rule{1em}{0ex}}\mathit{\text{whenever}}|y|H

(3.8)

*for certain positive constants* {h}_{1}, {h}_{2}, {\lambda}_{1}, {\lambda}_{2}, {k}_{1}, {k}_{2}, *σ*, *δ*, *ρ*, *H* *such that* \delta \le 1,

1-\frac{{\lambda}_{1}{k}_{1}}{{h}_{2}\tilde{M}}\delta +p(1-\delta )<\sigma \le 1,

(3.9)

{\lambda}_{1}{k}_{1}+p{h}_{2}\tilde{M}>0,

(3.10)

*where recall that* \tilde{M}=max\beta (x) *for* x\in [{\nu}^{-},{\nu}^{+}]. *Finally*, *let condition* (2.4) *be satisfied and assume that*

\underset{y\to {0}^{+}}{lim\hspace{0.17em}sup}\frac{\mathrm{\Phi}(y)}{{y}^{\mu}}>0

(3.11)

*for some positive constant*
*μ*
*satisfying*

\mu <\frac{{\lambda}_{1}{k}_{1}}{{h}_{2}\tilde{M}}+p.

(3.12)

*Then problem* (1.1) *admits solutions*.

*Proof* Put \theta (r):={k}_{2}{(\frac{r}{{m}^{\ast}(L)})}^{2-\frac{1}{q}} for r>0. From (3.3) and (3.8), it is immediate to verify the validity of conditions (2.6) and (2.7).

Put

\mathrm{\Lambda}(t):={k}_{1}min\{\underset{x\in [{\nu}^{-},{\nu}^{+}]}{min}b(-t,x),-\underset{x\in [{\nu}^{-},{\nu}^{+}]}{max}b(t,x)\}\phantom{\rule{1em}{0ex}}\mathit{\text{for}}tL

and \mathrm{\Lambda}(t):=0 for 0\le t\le L. By condition (3.3), we have \mathrm{\Lambda}\in {L}_{\mathrm{loc}}^{1}([0,+\mathrm{\infty})) and by (3.2) we have \mathrm{\Lambda}(t)>0 for t>L. Observe that by (3.6) it follows that

f(t,x,y)=b(t,x)c(x,y)\le {k}_{1}b(t,x)\mathrm{\Phi}(|y|)\le -\mathrm{\Lambda}(t)\mathrm{\Phi}(|y|)

and

f(-t,x,y)=b(-t,x)c(x,y)\ge {k}_{1}b(-t,x)\mathrm{\Phi}(|y|)\ge \mathrm{\Lambda}(t)\mathrm{\Phi}(|y|)

for a.e. t\ge L, every x\in [{\nu}^{-},{\nu}^{+}] and every y\in \mathbb{R}. Then condition (2.9) of Theorem 2.1 holds.

Now, from (3.5) it follows that {\lambda}_{1}{k}_{1}|t{|}^{p-1}\le \mathrm{\Lambda}(|t|) for a.e. |t|\ge L and by (3.4), recalling that M(t)=\tilde{M}\alpha (t), we deduce that

\frac{\mathrm{\Lambda}(|t|)}{M(t)}\ge \frac{{\lambda}_{1}{k}_{1}}{{h}_{2}\tilde{M}}\frac{1}{|t|}\phantom{\rule{1em}{0ex}}\text{whenever}|t|L.

So,

\left|{\int}_{0}^{t}\frac{\mathrm{\Lambda}(|\tau |)}{M(\tau )}\phantom{\rule{0.2em}{0ex}}\mathrm{d}\tau \right|\ge \frac{{\lambda}_{1}{k}_{1}}{{h}_{2}\tilde{M}}log\frac{|t|}{L}\phantom{\rule{1em}{0ex}}\text{whenever}|t|L.

Hence, for every fixed C>0, the function {N}_{C}(t) defined in (2.8) satisfies

\mathrm{\Phi}({N}_{C}(t))\le \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.|t{|}^{-(\frac{{\lambda}_{1}{k}_{1}}{{h}_{2}\tilde{M}}+p)}\phantom{\rule{1em}{0ex}}\text{for}|t|\text{large enough}.

(3.13)

By (3.10) we get {N}_{C}(t)\to 0 as |t|\to +\mathrm{\infty}, and therefore by (3.11) we deduce

{N}_{C}(t)\le \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.|t{|}^{-\frac{1}{\mu}(\frac{{\lambda}_{1}{k}_{1}}{{h}_{2}\tilde{M}}+p)}\phantom{\rule{1em}{0ex}}\text{for}|t|\text{large enough},

implying that {N}_{C}(t)\in {L}^{1}(\mathbb{R}) by (3.12). Then (2.8) holds too.

Since {lim}_{|t|\to +\mathrm{\infty}}{N}_{C}(t)=0, a constant {L}_{C}^{\ast}>L exists such that {N}_{C}(t)\le \rho for every |t|\ge {L}_{C}^{\ast}. Let us define \stackrel{\u02c6}{C}:={max}_{|t|\le {L}_{C}^{\ast}}{N}_{C}(t) and

{\eta}_{C}(t):=\{\begin{array}{ll}{max}_{x\in [{\nu}^{-},{\nu}^{+}]}|b(t,x)|\cdot {max}_{(x,y)\in [{\nu}^{-},{\nu}^{+}]\times [-\stackrel{\u02c6}{C},\stackrel{\u02c6}{C}]}c(x,y)& \text{if}|t|\le {L}_{C}^{\ast},\\ {\lambda}_{2}{k}_{2}|t{|}^{p-\sigma}\mathrm{\Phi}{({N}_{C}(t))}^{\delta}& \text{if}|t|{L}_{C}^{\ast}.\end{array}

By (3.5) and (3.7), for a.e. t\in \mathbb{R}, for every x\in [{\nu}^{-},{\nu}^{+}] and every y\in \mathbb{R} such that |y|\le {N}_{C}(t), we have

|f(t,x,y)|=|b(t,x)|c(x,y)\le {\eta}_{C}(t),

that is, condition (2.10). It remains to prove that {\eta}_{C}\in {L}^{1}(\mathbb{R}).

By (3.3) and the continuity of the function *c*, we have {\eta}_{C}\in {L}^{1}([-{L}_{C}^{\ast},{L}_{C}^{\ast}]). Moreover, when |t|>{L}_{C}^{\ast}, by (3.13), we have

{\eta}_{c}(t)\le \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.|t{|}^{-\sigma -\frac{{\lambda}_{1}{k}_{1}}{{h}_{2}\tilde{M}}\delta +p(1-\delta )}

implying that {\eta}_{c}(t)\in {L}^{1}(\mathbb{R}\setminus [-{L}_{C}^{\ast},{L}_{C}^{\ast}]) by assumption (3.9).

Therefore, Theorem 2.1 applies and guarantees the assertion of the present result. □

**Remark 3.2** The introduction of the constants *σ* and *δ* serves to state the result in the most general form, but often they can be taken both equal to 1, in such a way that assumption (3.9) is trivially verified.

If {m}_{\mathrm{\infty}}>0 (see (3.1)), condition (3.6) can be weakened, requiring that it holds only for |y| small enough, as the following result states.

**Proposition 3.3** *Let all the assumptions of Proposition * 3.1 *be satisfied*, *with the exception of* (3.6) *replaced by*

c(x,y)\ge {k}_{1}\mathrm{\Phi}(|y|)\phantom{\rule{1em}{0ex}}\mathit{\text{whenever}}|y|\rho .

(3.14)

*Moreover*, *assume that* {m}_{\mathrm{\infty}}>0. *Then problem* (1.1) *admits solutions*.

*Proof* For every C>0, put

\begin{array}{c}{\mathrm{\Gamma}}_{C}:=max\{\rho ,{\mathrm{\Phi}}^{-1}(\frac{{M}^{\ast}(L)}{{m}_{\mathrm{\infty}}}\mathrm{\Phi}(C))\},\phantom{\rule{2em}{0ex}}{\stackrel{\u02c6}{m}}_{C}:=\underset{(x,y)\in [{\nu}^{-},{\nu}^{+}]\times [\rho ,{\mathrm{\Gamma}}_{C}]}{min}c(x,y),\hfill \\ {h}_{C}:=min\{{k}_{1},\frac{{\stackrel{\u02c6}{m}}_{C}}{\mathrm{\Phi}({\mathrm{\Gamma}}_{C})}\}\hfill \end{array}

and finally put

{\mathrm{\Lambda}}_{C}(t):={h}_{C}min\{\underset{x\in [{\nu}^{-},{\nu}^{+}]}{min}b(-t,x),-\underset{x\in [{\nu}^{-},{\nu}^{+}]}{max}b(t,x)\}

for t>L and {\mathrm{\Lambda}}_{C}(t):=0 for t\in [0,L].

As it is immediate to verify, c(x,y)\ge {h}_{C}\mathrm{\Phi}(|y|) whenever {\nu}^{-}\le x\le {\nu}^{+} and |y|\le {\mathrm{\Gamma}}_{C}. So, (2.9) holds since {N}_{C}(t)\le {\mathrm{\Gamma}}_{C} for every t>L.

From now on, the proof proceeds as that of Proposition 3.1. □

We state now two non-existence results, obtained applying Theorem 2.3.

**Proposition 3.4**
*Suppose that*

t\cdot b(t,x)\le 0\phantom{\rule{1em}{0ex}}\mathit{\text{for a.e.}}t\in \mathbb{R}\mathit{\text{and every}}x\in [{\nu}^{-},{\nu}^{+}],

(3.15)

*and let there exist a constant* p\in \mathbb{R}, *a constant* L>0 *and a positive function* \ell (t)\in {L}^{1}([0,L]) *such that*

|b(t,x)|\le {\lambda}_{2}|t{|}^{p-1}\phantom{\rule{1em}{0ex}}\mathit{\text{for every}}x\in [{\nu}^{-},{\nu}^{+}]\mathit{\text{a.e.}}|t|L,

(3.16)

|b(t,x)|\le \ell (|t|)\phantom{\rule{1em}{0ex}}\mathit{\text{for a.e.}}|t|\le L,x\in [{\nu}^{-},{\nu}^{+}],

(3.17)

c(x,y)\le {k}_{2}\mathrm{\Phi}(y)\phantom{\rule{1em}{0ex}}\mathit{\text{for every}}x\in [{\nu}^{-},{\nu}^{+}],0y\rho

(3.18)

*for some positive constants* {\lambda}_{2}, {k}_{2}, *ρ*. *Moreover*, *assume that* (3.4) *holds for some constants* {h}_{1}, {h}_{2}, *p* *such that*

{\lambda}_{2}{k}_{2}+{h}_{1}p\tilde{m}>0,

(3.19)

*where recall that* \tilde{m}=min\beta (x) *for* x\in [{\nu}^{-},{\nu}^{+}].

*Furthermore*, *suppose that*

\underset{y\to 0}{lim\hspace{0.17em}sup}\frac{\mathrm{\Phi}(y)}{{y}^{\mu}}<+\mathrm{\infty}

(3.20)

*for some positive constant*
*μ*
*satisfying*

\mu \ge \frac{{\lambda}_{2}{k}_{2}}{{h}_{1}\tilde{m}}+p.

(3.21)

*Finally*, *suppose that there exist two constants* \u03f5,H>0 *such that*

\alpha (t)\le H\alpha (t+r)\phantom{\rule{1em}{0ex}}\mathit{\text{for every}}t0\mathit{\text{and}}0r\u03f5,

(3.22)

\alpha (t+r)\le H\alpha (t)\phantom{\rule{1em}{0ex}}\mathit{\text{for every}}t0\mathit{\text{and}}0r\u03f5.

(3.23)

*Then problem* (1.1) *does not admit solutions*.

*Proof* First of all, notice that assumption (3.15) implies condition (2.16) and assumptions (3.22) and (3.23) respectively imply conditions (2.17), (2.18).

Putting

\mathrm{\Lambda}(t):=\{\begin{array}{ll}{k}_{2}\ell (t)& \text{for}t\in [0,L],\\ {\lambda}_{2}{k}_{2}{t}^{p-1}& \text{for}tL,\end{array}

we have that Λ is a positive function belonging to {L}_{\mathrm{loc}}^{1}([0,+\mathrm{\infty})) and one can easily verify that conditions (3.16), (3.17) and (3.18) guarantee the validity of (2.13) and (2.14). Moreover, by (3.4) we get

\frac{\mathrm{\Lambda}(|t|)}{\alpha (t)}\le \frac{{\lambda}_{2}{k}_{2}}{{h}_{1}}\frac{1}{|t|}\phantom{\rule{1em}{0ex}}\text{for}|t|\text{large enough.}

Hence, if {N}_{C}(t) is the function defined in (2.15), we have

\mathrm{\Phi}({N}_{C}(t))\ge \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.|t{|}^{-\frac{{\lambda}_{2}{k}_{2}}{{h}_{1}\tilde{m}}-p}\phantom{\rule{1em}{0ex}}\text{for}|t|\text{large enough}.

(3.24)

By assumption (3.19), we get {N}_{C}(t)\to 0 as |t|\to +\mathrm{\infty} and by virtue of (3.20), we obtain

{N}_{C}(t)\ge \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.|t{|}^{-\frac{1}{\mu}(\frac{{\lambda}_{2}{k}_{2}}{{h}_{1}\tilde{m}}+p)}>0\phantom{\rule{1em}{0ex}}\text{for}|t|\text{large enough}.

Finally, assumption (3.21) implies that {N}_{C}(t) is not summable in ℝ and the assertion follows as an application of Theorem 2.3. □

**Remark 3.5** As for the validity of conditions (3.22), (3.23) in the previous non-existence theorem, notice that when dealing with autonomous operators, that is, for \alpha (t)\equiv 1, they are trivially satisfied. However, also in the non-autonomous case, they hold in many relevant situations. For instance, they are satisfied if one the following conditions is satisfied:

\alpha (t) is decreasing in (-\mathrm{\infty},0) and increasing in (0,+\mathrm{\infty});

*α* is uniformly continuous in ℝ and {inf}_{t\in \mathbb{R}}\alpha (t)>0;

\alpha (t)\sim |t{|}^{-p} as |t|\to +\mathrm{\infty} for some p>0.

When condition (3.19) does not hold, we can use the following non-existence result.

**Proposition 3.6** *Let all the assumptions of Proposition * 3.4 *be satisfied with the exception of* (3.20), (3.21) *and with assumption* (3.19) *replaced by the opposite one*,

{\lambda}_{2}{k}_{2}+{h}_{1}p\tilde{m}\le 0,

(3.25)

*then problem* (1.1) *does not admit solutions*.

*Proof* With the same notations of the proof of Proposition 3.4, notice that under condition (3.25), by (3.24), we have {N}_{C}(t)\ge \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}.>0 for |t| large enough, implying that {N}_{C} is not summable and the assertion follows from Theorem 2.3. □

Let us now provide some examples of applications of the previous results.

**Example 3.7** Let us consider the differential equation

{(|t{|}^{p}\beta (x){x}^{\prime}(t))}^{\prime}=-\frac{t}{1+{t}^{2}}|t{|}^{p}g(x)|{x}^{\prime}(t)|,

with *β*, *g* positive continuous functions.

It is easy to show that all the assumptions of Proposition 3.1 are satisfied with p>-2, q=1, {h}_{1}={h}_{2}={k}_{1}={k}_{2}=1, {\lambda}_{1}<{min}_{x\in [{\nu}^{-},{\nu}^{+}]}g(x):={m}_{g}, {\lambda}_{2}={max}_{x\in [{\nu}^{-},{\nu}^{+}]}g(x):={M}_{g}, p>-\frac{{m}_{g}}{\tilde{M}}, \sigma =\delta =\mu =1, and *L* large enough (depending on {m}_{g}-{\lambda}_{1}).

By applying Propositions 3.1 and 3.4, we deduce that if p>1-\frac{{m}_{g}}{\tilde{M}}, then problem (1.1) admits solutions, whereas if p\le 1-\frac{{M}_{g}}{\tilde{m}}, then (1.1) does not admit solutions. Recalling that \tilde{M}=max\beta (x) (\tilde{m}=min\beta (x)) for x\in [{\nu}^{-},{\nu}^{+}], the existence or non-existence of solutions depends on the boundary data {\nu}^{-}, {\nu}^{+}. For instance, if \beta (x):=(1+{x}^{2}) and g(x):={e}^{-|x|} and the boundary data are symmetric, that is, {\nu}^{+}=-{\nu}^{-}=\nu, then {M}_{g}=\tilde{m}=1, \tilde{M}=1+{\nu}^{2} and {m}_{g}={e}^{-\nu}. So, if p>1-\frac{1}{{e}^{\nu}(1+{\nu}^{2})}, problem (1.1) admits solutions, whereas if p\le 0, it does not admit solutions. Notice that for every p>0, problem (1.1) is solvable for *ν* small enough.

**Example 3.8** Let us consider the differential equation

{(\frac{\beta (x)}{1+{t}^{2}}{x}^{\prime}(t)arctan{x}^{\prime}(t))}^{\prime}=-\frac{t}{1+{t}^{4}}g(x){x}^{\prime}{(t)}^{2},

with *β*, *g* positive continuous functions.

As one can immediately verify, assumptions (3.2)-(3.9) and (3.11) of Proposition 3.1 hold with q=+\mathrm{\infty}, p=-2, {h}_{1}<1, {h}_{2}=1, {\lambda}_{1}<{min}_{x\in [{\nu}^{-},{\nu}^{+}]}g(x):={m}_{g}, {\lambda}_{2}={max}_{x\in [{\nu}^{-},{\nu}^{+}]}g(x):={M}_{g}, {k}_{1}=1, {k}_{2}>1, \sigma =\delta =1, \mu =2 and *L*, *H* large enough. Therefore, if {m}_{g}>4\tilde{M}, both conditions (3.10) and (3.12) are satisfied and problem (1.1) admits solutions. Instead, if {M}_{g}\le 4\tilde{m}, then (1.1) has no solutions as a consequence of Proposition 3.4.

So, as in the previous example, the above conditions for the existence and non-existence of solutions become conditions on the boundary data {\nu}^{-}, {\nu}^{+}.

### 3.2 Case of Φ having superlinear growth

We handle now operators Φ having possibly superlinear growth at infinity, that is, we now remove condition (2.4). The non-existence Propositions 3.4 and 3.6 hold also in this case, since they do not require condition (2.4). As for the existence results, we now use Theorem 2.2 instead of Theorem 2.1 by assuming (2.12). As it will be clear after the proof of the next result, condition (2.12) is not satisfied when {m}_{\mathrm{\infty}}=0, so from now on we assume {m}_{\mathrm{\infty}}>0.

**Proposition 3.9** *Let all the assumptions of Proposition * 3.3 *hold true*, *with the exception of* (3.8) *replaced by*

\underset{|y|\to +\mathrm{\infty}}{lim}\frac{{max}_{x\in [{\nu}^{-},{\nu}^{+}]}c(x,y)}{|\mathrm{\Phi}(y)|}=0.

(3.26)

*Then problem* (1.1) *admits solutions*.

*Proof* Put

\theta (r):=\underset{(t,x)\in [-L,L]\times [{\nu}^{-},{\nu}^{+}]}{max}(max\{c(x,{\mathrm{\Phi}}^{-1}\left(\frac{r}{a(t,x)}\right)),c(x,{\mathrm{\Phi}}^{-1}(-\frac{r}{a(t,x)}))\}),

it is immediate to check that *θ* is a continuous function on [0,+\mathrm{\infty}) such that

\theta (a(t,x)|\mathrm{\Phi}(y)\left|\right)\ge c(x,y)\phantom{\rule{1em}{0ex}}\text{for every}t\in [-L,L],x\in [{\nu}^{-},{\nu}^{+}],y\in \mathbb{R},

hence (2.6) holds. Moreover, by (3.26), for every \u03f5>0, there exists a real {c}_{\u03f5} such that

c(x,y)\le \u03f5|\mathrm{\Phi}(y)|\phantom{\rule{1em}{0ex}}\text{for every}x\in [{\nu}^{-},{\nu}^{+}],|y|\ge {c}_{\u03f5}.

Hence, for every s\ge {M}^{\ast}(L)max\{\mathrm{\Phi}({c}_{\u03f5}),-\mathrm{\Phi}(-{c}_{\u03f5})\}, we have \theta (s)\le \frac{\u03f5}{{m}^{\ast}(L)}s, that is,

\underset{s\to +\mathrm{\infty}}{lim}\frac{\theta (s)}{s}=0.

Hence, condition (2.12) holds and the proof proceeds as that of Proposition 3.3, applying Theorem 2.2 instead of Theorem 2.1. □

Note that condition (3.26) is not compatible with (3.6). For this reason, in the case of superlinear growth, we only treat the case {m}_{\mathrm{\infty}}>0.

**Example 3.10** Let us consider the following differential equation:

{((1+|t|)\beta (x){x}^{\prime}(t){e}^{|{x}^{\prime}(t)|})}^{\prime}=-\frac{t}{|t|}g(x)|{x}^{\prime}(t)|(1+{x}^{\prime}{(t)}^{2})

with *β*, *g* positive continuous functions.

In this case, we can apply Proposition 3.9 since {m}_{\mathrm{\infty}}>0 and condition (3.26) is trivially satisfied. Moreover, all the other assumptions of Proposition 3.9 hold with q=+\mathrm{\infty}, {h}_{1}=1, {h}_{2}>1, p=1, {\lambda}_{1}={min}_{x\in [{\nu}^{-},{\nu}^{+}]}g(x), {\lambda}_{2}={max}_{x\in [{\nu}^{-},{\nu}^{+}]}g(x), {k}_{1}<1<{k}_{2}, \mu =1. Hence, since (3.12) is satisfied whatever {\lambda}_{1},\tilde{M}>0 may be, problem (1.1) admits solutions for every boundary data {\nu}^{-}, {\nu}^{+}.