Weakly Increasing Solutions of Equations with p-Mean Curvature Operator

Abstract

Globally positive unbounded solutions, with zero derivative at infinity, are here considered for ordinary differential equations involving the generalized Euclidean mean curvature operator. When $p\ge 2$, the results highlight an analogy with an auxiliary equation with the p-Laplacian operator. The results are obtained using some comparison criteria for the principal solutions of a class of associated half-linear equations.

1. Introduction

In this paper we are concerned with the second-order equation

$${\left(a\left(t\right){\mathsf{\Phi }}_C\left(x^{\prime }\right)\right)}^{\prime }+b\left(t\right)F\left(x\right)=0,t\in I=[1,\infty ),$$

(1)

where the functions $a,b$ are continuous and positive on $I,$ the function F is continuous on $\mathbb{R}$, $uF\left(u\right)>0$ for $u\ne 0$ and ${\mathsf{\Phi }}_C$ is the nonhomogeneous operator

$$z={\mathsf{\Phi }}_C\left(u\right)={\displaystyle \frac{{\left|u\right|}^{p-2}u}{{\left(1+{\left|u\right|}^p\right)}^{(p-1)/p}}},p>1.$$

The operator ${\mathsf{\Phi }}_C$ is sometimes called the p-mean curvature operator or generalized Euclidean mean curvature operator. It occurs in studying some nonlinear fluid mechanics problems, in particular capillarity-type phenomena for compressible and incompressible fluids; see, e.g., refs. [1,2,3,4] and references therein. The operator ${\mathsf{\Phi }}_C$ originates from the Euclidean mean curvature operator ${\mathsf{\Phi }}_E$

$${\mathsf{\Phi }}_E\left(u\right)={\displaystyle \frac{u}{\sqrt{1+u^2}}},$$

which corresponds to ${\mathsf{\Phi }}_C$ when $p=2.$

The existence of unbounded nonoscillatory solutions for

$${\left(a\left(t\right){\mathsf{\Phi }}_E\left(x^{\prime }\right)\right)}^{\prime }+b\left(t\right)F\left(x\right)=0$$

(2)

has been considered in the recent paper [5], in which a similarity between (2) and the linear equation

$$\left(a\left(t\right)x^{\prime }\right){)}^{\prime }+b\left(t\right)x=0,$$

(3)

has also been pointed out concerning such solutions. Observe that, roughly speaking, for small values of $\left|u\right|$, the operator ${\mathsf{\Phi }}_E\left(u\right)$ is close to the linear operator $L\left(u\right)=u$, and consequently (2) can be interpreted as in asymptotic similarity with the linear Equation (3). Analogously, for small values of $\left|u\right|$, the operator ${\mathsf{\Phi }}_C$ is close to the p-Laplacian operator

$${\mathsf{\Phi }}_p\left(u\right)={\left|u\right|}^{p-2}u.$$

Hence, a natural question that arises is whether the similarity established in [5] between (2) and (3) continues to hold between (1) and the half-linear equation

$${\left(a\left(t\right){\mathsf{\Phi }}_p\left(x^{\prime }\right)\right)}^{\prime }+b\left(t\right){\mathsf{\Phi }}_p\left(x\right)=0.$$

(4)

It is well known that, if (3) is nonoscillatory, then it has unbounded nonoscillatory solutions x only if $a^{-1}\notin L^1\left(I\right),$ and, in this case, ${lim}_{t\to \infty }x\left(t\right)=\infty ,$${lim}_{t\to \infty }a\left(t\right)x^{\prime }\left(t\right)=0$; see, e.g., ([6] Theorem 4.1.10) with minor changes. Passing to the half-linear case, if (4) is nonoscillatory, then it has unbounded solutions only if

$$J_a={\int }_1^{\infty }{\left({\displaystyle \frac{1}{a\left(t\right)}}\right)}^{1/(p-1)}dt=\infty ,$$

but unbounded solutions x can have a different growth at infinity, since both cases

$$\underset{t\to \infty }{lim}x\left(t\right)=\infty ,\underset{t\to \infty }{lim}a\left(t\right){\mathsf{\Phi }}_p\left(x^{\prime }\left(t\right)\right)=0,$$

(5)

and

$$\underset{t\to \infty }{lim}x\left(t\right)=\infty ,\underset{t\to \infty }{lim}a\left(t\right){\mathsf{\Phi }}_p\left(x^{\prime }\left(t\right)\right)={𝓁}_x,0<{𝓁}_x<\infty ,$$

(6)

are possible; see, e.g., ([7] Theorem 4-(i₁), Corollary 1).

Solutions of the generalized Emden–Fowler equation

$${\left(a\left(t\right){\mathsf{\Phi }}_p\left(x^{\prime }\right)\right)}^{\prime }+b\left(t\right)F\left(x\right)=0$$

(7)

satisfying (5) are called weakly increasing solutions. Sometimes, especially in the special case (4), the terminology intermediate solutions has been used; see, e.g., [7,8]. The existence of weakly increasing solutions is a well-known problem, with a long history, started seventy years ago by Moore and Nehari in [9]. The difficulty in that problem is related to finding sharp upper and lower bounds for solutions x satisfying (5); see, e.g., ([10] p. 241). For the special case (4), the problem has been completely solved in [7].

The aim of this paper is to prove the existence of solutions x of (1) satisfying (5) and which are positive increasing on the whole interval I, that is, $x\left(t\right)>0,x^{\prime }\left(t\right)>0$ on I. Our result also gives an answer to the above question on the similarity between (1) and (4). Moreover, it extends and generalizes analogous ones in [5], in which the existence of unbounded solutions for (2) is proved only for large $t.$ Observe that the continuability at infinity of any solution of (7) holds under very general assumptions, see, e.g., ([11] Appendix), but the positivity of nonoscillatory solutions on the whole interval I is, in general, a nontrivial problem. For instance, it can fail for (7) and also for the special case (4), since nonoscillatory solutions may have an arbitrary large number of zeros. The situation can be very different when the function b takes negative values on a compact subinterval of I. Indeed, in this case, the so-called blow-up solutions can appear, that is, solutions x of (7) such that ${lim}_{t\to T-}\left|x\left(t\right)\right|=\infty$ for some $T>1$; see, e.g., [12,13].

Our approach is based on a fixed-point result for maps defined in a Fréchet space which originates from a result by Cecchi, Furi and Marini; see the monograph ([14] Section 3.1.2) and [15] for more details. In Section 3, we present it in the form that is needed in the sequel. This result simplifies the check of the topological properties of the fixed-point map, since they become an immediate consequence of a priori bounds for an associated half-linear equation. These bounds are obtained in an implicit form by means of the concept of principal solutions for a half-linear equation, whose properties are presented in Section 2. Finally, in Section 4, some comments and suggestions for future research are given.

2. Preliminaries

Consider the equation

$${\left(a\left(t\right){\mathsf{\Phi }}_p\left(x^{\prime }\right)\right)}^{\prime }+\beta \left(t\right){\mathsf{\Phi }}_p\left(x\right)=0,$$

(8)

where $\beta$ is a continuous function for $t\ge 1.$ Equation (8) is called a half-linear equation because the set of solutions of (8) is homogeneous but not generally additive, that is, it has one half of the properties characterizing linearity. When (8) is nonoscillatory, in [16,17], the notion of the principal solution has been introduced by using the associated generalized Riccati equation, namely the equation

$${\xi }^{\prime }+(p-1){\displaystyle \frac{{\left|\xi \right|}^q}{{\mathsf{\Phi }}_q\left(a\left(t\right)\right)}}+\beta \left(t\right)=0,$$

(9)

where q is the conjugate number of $p,$ that is, $p^{-1}+q^{-1}=1.$ More precisely, in [16,17], it is proved that, when (8) is nonoscillatory, among all eventually different-from-zero solutions of (9), there exists one, say ${\xi }_u,$ that is continuable to infinity and is minimal in the sense that any other solution ${\xi }_x$ of (9), that is continuable to infinity, satisfies

$${\xi }_u\left(t\right)<{\xi }_x\left(t\right)\mathrm{for}\mathrm{large}t.$$

(10)

For simplicity of notation, the eventually minimal solution of (9) will be denoted by $\tilde{\xi },$ omitting the dependence on u and, similarly, we denote a generic solution of (9) by $\xi$, instead of ${\xi }_x$. Equation (9) is related with (8) because, for any solution x of (8), such that $x\left(t\right)\ne 0$ for $t\ge T\ge 1$, the function

$${\xi }_x\left(t\right)={\displaystyle \frac{a\left(t\right){\mathsf{\Phi }}_p\left(x^{\prime }\left(t\right)\right)}{{\mathsf{\Phi }}_p\left(x\left(t\right)\right)}}$$

(11)

is a solution of (9). In virtue of (11), inequality (10) becomes

$${\displaystyle \frac{u^{\prime }\left(t\right)}{u\left(t\right)}}<{\displaystyle \frac{x^{\prime }\left(t\right)}{x\left(t\right)}}\mathrm{for}\mathrm{large}t.$$

(12)

Thus, following [16,17], a nontrivial solution u of (8) is said to be the principal solution if, for every nontrivial solution x of (8) such that $x\ne \lambda u$, $\lambda \in \mathbb{R}$, inequality (12) holds. When (8) is nonoscillatory, the set of its principal solutions is nonempty and for any $\mu \ne 0$ there exists a unique principal solution u such that $u\left(1\right)=\mu ,$ i.e., principal solutions are uniquely determined up to a constant factor. Notice that, if x is a nonprincipal solution of (8) and (12) is satisfied for any $t\ge T\ge 1$, with $x\left(T\right)\ge u\left(T\right)\ge 0$, then $u\left(t\right)<x\left(t\right)$ for $t>T$. Hence, roughly speaking, the principal solution is the smallest solution in a neighborhood of infinity.

The characterization of the principal solution in the half-linear case is a harder problem. More precisely, let x be a solution of (8), different from zero for $t\ge T\ge 1.$ Then, a first attempt in finding an integral characterization is given in [18], where the integral

$$Q_x={\int }_T^{\infty }{\displaystyle \frac{x^{\prime }\left(t\right)}{a\left(t\right){\mathsf{\Phi }}_p\left(x^{\prime }\left(t\right)\right)x^2\left(t\right)}}dt$$

was introduced. Observe that, for $p=2$, the integral $Q_x$ reduces to the well-known integral

$${\int }_T^{\infty }{\displaystyle \frac{1}{a\left(t\right)x^2\left(t\right)}}dt,$$

whose divergence characterizes the principal solution in the linear case. Nevertheless, the result ([18] Theorem 3.3) fails and the erratum is due to an incorrect application of ([18] Lemma 2.4) when $1<p<2$, as it is discussed in [19] and references therein. Many efforts to find a universal integral characterization of the principal solution of (8) have continued even recently, see, e.g., [19,20,21], but until now this problem is not completely solved. A partial answer is given by ([19] Theorem A) as the following shows.

Proposition 1. 

Let (8) be nonoscillatory. Suppose that $p\ge 2$, $J_a=\infty$ and $\beta \left(t\right)\ge 0,$ with β not identically zero for large t and $\beta \in L^1\left(I\right)$. Then, a solution x of (8) is the principal solution if and only if $Q_x=\infty$.

Clearly, the principal solution does not have zeros in a neighborhood of infinity. The positiveness of the principal solution on an a priori closed unbounded interval $[T,\infty ),T\ge 1$, is another subtle question. This property plays an important role in the sequel, and can be obtained using some comparison properties between the Riccati minimal solutions.

Consider the half-linear equation

$${\left(a_1\left(t\right){\mathsf{\Phi }}_p\left(y^{\prime }\right)\right)}^{\prime }+{\beta }_1\left(t\right){\mathsf{\Phi }}_p\left(y\right)=0.$$

(13)

If $a_1,{\beta }_1$ are continuous functions for $t\ge 1$ such that

$$0<a_1\left(t\right)\le a\left(t\right),{\beta }_1\left(t\right)\ge \beta \left(t\right),$$

(14)

then Equation (13) is a Sturm majorant of (8). Consider the Riccati equation associated with (13), i.e.,

$${\eta }^{\prime }+(p-1){\displaystyle \frac{{\left|\eta \right|}^q}{{\mathsf{\Phi }}_q\left(a_1\left(t\right)\right)}}+{\beta }_1\left(t\right)=0.$$

(15)

The following comparison result between eventually minimal solutions of (9) and (15) holds.

Theorem 1. 

Assume (14) and let (13) be nonoscillatory. If the minimal solution $\tilde{\eta }$ of (15) is defined on $I=[1,\infty ),$ then

(i₁) the minimal solution $\tilde{\xi }$ of (9) is defined on $I;$

(i₂) for any $t\in I$ we have

$$\tilde{\xi }\left(t\right)\le \tilde{\eta }\left(t\right).$$

For proving Theorem 1, an auxiliary result is needed. For fixed $k\ge 0$, let $\gamma$ be the continuous function

$$\gamma \left(t\right)=min\left\{\beta \left(t\right),-k\right\},$$

(16)

and consider the half-linear equation

$${\left(a\left(t\right){\mathsf{\Phi }}_p\left(w^{\prime }\right)\right)}^{\prime }+\gamma \left(t\right){\mathsf{\Phi }}_p\left(w\right)=0.$$

(17)

Thus, Equation (8) is a Sturm majorant of (17). Since $\gamma$ is nonpositive, Equation (17) is nonoscillatory; see, e.g., ([6] Section 4.1.2). Moreover, the principal solution $w_0$ of (17) satisfies $w_0\left(t\right)w_0^{\prime }\left(t\right)<0$ for $t\ge 1$; see, e.g., ([6] Theorem 4.2.7.—Part (I)). Consequently, $w_0$ does not have zeros on $[1,\infty )$, and the eventually minimal solution $\tilde{\omega }$ of the Riccati equation associated with (17)

$${\omega }^{\prime }+(p-1){\displaystyle \frac{{\left|\omega \right|}^q}{{\mathsf{\Phi }}_q\left(a\left(t\right)\right)}}+\gamma \left(t\right)=0,$$

(18)

is defined on the whole interval $[1,\infty ).$ The function $\tilde{\omega }$ gives a lower bound for the eventually minimal solution of (9), as the following result shows.

Lemma 1. 

Assume that (8) is nonoscillatory and let $x_0$ be the principal solution of (8) such that $x_0\left(t\right)\ne 0$ on $(T,\infty ),1\le T$. Denote by $\tilde{\xi }$ and $\tilde{\omega }$ the eventually minimal solution of the Riccati Equations (9) and (18), respectively. Then,

$$\tilde{\xi }\left(t\right)\ge \tilde{\omega }\left(t\right)\mathit{on}(T,\infty ),$$

(19)

and

$$\underset{t\to T+}{liminf}\tilde{\xi }\left(t\right)>-\infty .$$

(20)

Proof. 

By contradiction, suppose that there exists $t_1>T$ such that

$$\tilde{\xi }\left(t_1\right)<\tilde{\omega }\left(t_1\right).$$

(21)

Consider the solution $\omega$ of (18) satisfying

$$\omega \left(t_1\right)=\tilde{\xi }\left(t_1\right).$$

(22)

Setting

$$\begin{array}{cc}\hfill X(t,u)& =(1-p){\displaystyle \frac{{\left|u\right|}^q}{{\mathsf{\Phi }}_q\left(a\left(t\right)\right)}}-\beta \left(t\right)\hfill \\ \hfill Y(t,u)& =(1-p){\displaystyle \frac{{\left|u\right|}^q}{{\mathsf{\Phi }}_q\left(a\left(t\right)\right)}}-\gamma \left(t\right),\hfill \end{array}$$

Equations (9) and (18) become

$${\xi }^{\prime }=X(t,\xi )\mathrm{and}{\omega }^{\prime }=Y(t,\omega ),$$

respectively. Moreover, from (16), we have for $(t,u)\in [t_1,\infty )\times \mathbb{R}$

$$X(t,u)-Y(t,u)=\gamma \left(t\right)-\beta \left(t\right)\le 0.$$

Set $I_{\sigma }=[t_1,\sigma ),\sigma \le \infty ,$ the interval of existence of $\omega$ to the right of $t_1.$ From ([22] Chapter III, Theorem 4.1 and Remark 1) with $U=Y$ and $V=X$, we obtain for $t\in I_{\sigma }$

$$\tilde{\xi }\left(t\right)\le \omega \left(t\right),$$

(23)

that is, the solution $\omega$ is bounded from below by $\tilde{\xi }$ on $I_{\sigma }$. In view of the unique solvability of the Riccati equation, the graphs of solutions of (18) cannot intersect and, so, from (21) and (22), we obtain for $t\in I_{\sigma }$

$$\omega \left(t\right)<\tilde{\omega }\left(t\right).$$

(24)

Hence, from (23) and (24), the solution $\omega$ is continuable to infinity and (24) holds for $t\in [t_1,\infty )$, giving a contradiction with the minimality of $\tilde{\omega }.$ Thus, (19) is valid and, since $\tilde{\omega }$ is defined in the whole interval $[1,\infty ),$ the lower bound (20) also follows. □

Proof of Theorem 1. 

First, notice that in the limit case

$$a_1\left(t\right)=a\left(t\right),{\beta }_1\left(t\right)=\beta \left(t\right),t\in I,$$

the assertion is trivially true. Now, suppose that at least one of the inequality (14) is strict on a subinterval of $I.$ Then, in view of the Sturm comparison theorem, Equation (8) is nonoscillatory, too.

Claim $\left(i_1\right).$ By contradiction, suppose that the eventually minimal solution $\tilde{\xi }$ of (9) is defined on $(T,\infty ),T\ge 1,$ and is unbounded in a right neighborhood $I_{\epsilon }=(T,T+\epsilon )$ of $T.$ From Lemma 1, the solution $\tilde{\xi }$ is bounded from below on $I_{\epsilon }$ and so

$$\underset{t\to T+}{limsup}\tilde{\xi }\left(t\right)=\infty .$$

On the other hand, the minimal solution $\tilde{\eta }$ of (15) is defined at $t=T.$ Consequently, there exists $T_1\in I_{\epsilon }$ such that

$$\tilde{\eta }\left(T_1\right)<\tilde{\xi }\left(T_1\right).$$

Consider the solution $\xi$ of (9) starting with the initial value $\xi \left(T_1\right)=\tilde{\eta }\left(T_1\right).$ From this we obtain

$$\xi \left(T_1\right)<\tilde{\xi }\left(T_1\right).$$

(25)

Setting

$$\begin{array}{cc}\hfill X(t,u)& =(1-p){\displaystyle \frac{{\left|u\right|}^q}{{\mathsf{\Phi }}_q\left(a\left(t\right)\right)}}-\beta \left(t\right),\hfill \\ \hfill Y(t,u)& =(1-p){\displaystyle \frac{{\left|u\right|}^q}{{\mathsf{\Phi }}_q\left(a_1\left(t\right)\right)}}-{\beta }_1\left(t\right),\hfill \end{array}$$

Equations (9) and (15) become

$${\xi }^{\prime }=X(t,\xi ),{\eta }^{\prime }=Y(t,\eta ),$$

respectively. Moreover, we obtain for $(t,u)\in [T_1,\infty )\times \mathbb{R}$

$$X(t,u)-Y(t,u)={(p-1)\left|u\right|}^q\left({\displaystyle \frac{1}{{\mathsf{\Phi }}_q\left(a_1\left(t\right)\right)}}-{\displaystyle \frac{1}{{\mathsf{\Phi }}_q\left(a\left(t\right)\right)}}\right)+\left({\beta }_1\left(t\right)-\beta \left(t\right)\right).$$

Since ${\mathsf{\Phi }}_q$ is increasing, in view of (14), we obtain

$$X(t,u)-Y(t,u)\ge 0.$$

Set $I_{\rho }=[t_1,\rho ),\rho \le \infty ,$ the interval of existence of $\xi$ to the right of $t_1.$ Applying ([22] Chapter III, Theorem 4.1 and Remark 1) with $U=X$ and $V=Y$, we obtain for $t\in I_{\rho }$

$$\tilde{\eta }\left(t\right)\le \xi \left(t\right).$$

Thus, the solution $\xi$ is bounded from below by $\tilde{\eta }$. Moreover, in view of the unique solvability of the Riccati equation (15), the graphs of two solutions cannot intersect. Then, in view of (25), we obtain for $t\in I_{\rho }$

$$\xi \left(t\right)<\tilde{\xi }\left(t\right).$$

(26)

Therefore, $\xi$ is continuable to infinity and (26) holds on $[t_1,\infty )$, giving a contradiction with the minimality of $\tilde{\xi }$.

Claim $\left(i_2\right).$ If, by contradiction, there exists $T_2\ge T$ such that $\tilde{\eta }\left(T_2\right)<\tilde{\xi }\left(T_2\right),$ then, considering the solution $\xi$ of (15) starting with the initial value $\xi \left(T_2\right)=\tilde{\eta }\left(T_2\right)$ and using the same argument as in Claim $\left(i_1\right)$, we obtain a contradiction. The details are left to the reader. □

Theorem 1 can be achieved by using different approaches. Indeed, in ([6] Theorems 4.2.2 and 4.2.3), two results on conjugate points of (8) are given, which, with some modifications and corrections, allow a different proof of Theorem 1 to be given.

From Theorem 1, we immediately obtain the following.

Corollary 1. 

Assume (14) and let (13) be nonoscillatory, with the principal solution positive on the whole interval I. Then, the principal solution $x_0$ of (8), satisfying $x_0\left(1\right)>0$, is positive on I.

Proof. 

Since the principal solution of (13) is positive on the whole interval I, the minimal solution of (15) is defined on the whole interval I. By Theorem 1, the minimal solution of (9) is also defined on I, and therefore $x_0$ cannot vanish, which yields the positivity of $x_0$ on the whole interval I. □

Now, consider the special case of (8) with $\beta$ positive on $I,$ that is, Equation (4). The following result is crucial in our later consideration.

Proposition 2. 

Let (4) be nonoscillatory. If $J_a=\infty ,$ and

$$J_{ab}={\int }_1^{\infty }{\left({\displaystyle \frac{1}{a\left(t\right)}}{\int }_t^{\infty }b\left(r\right)dr\right)}^{1/(p-1)}dt=\infty ,$$

(27)

then the principal solution x of (4) satisfies (5).

Proof. 

The assertion follows by using some results from [7], with minor modifications. Indeed, from Theorems 4, 6 and 7 in [7] and ([7] p. 915 and p. 918-(1)) any eventually positive solution of (4) satisfies either (5) or (6). Moreover, the set of solutions satisfying (5) is nonempty. Now, by contradiction, suppose that the eventually positive principal solution u of (4) satisfies (6). Let x be an eventually positive solution of (4) satisfying (5). Using the l’Hopital rule, we obtain

$$\underset{t\to \infty }{lim}{\displaystyle \frac{u\left(t\right)}{x\left(t\right)}}=\infty ,$$

which yields a contradiction with the definition (12) of the principal solution. □

3. Solvability of the Boundary Value Problem

In this section, we study the existence of solutions x of (1) such that $x\in S,$ where S is the subset

$$S=\left\{x\in C^1\left(I\right):x\left(t\right)>0,x^{\prime }\left(t\right)>0\mathrm{on}I,\underset{t\to \infty }{lim}x\left(t\right)=\infty ,\underset{t\to \infty }{lim}{x}^{\prime }\left(t\right)=0\right\}.$$

To this end, we assume the following:

$${\int }_1^{\infty }b\left(t\right)dt<\infty \mathrm{and}{J}_{ab}=\infty ,$$

(28)

where $J_{ab}$ is given in (27),

$$\underset{u\to \infty }{limsup}{\displaystyle \frac{F\left(u\right)}{{\mathsf{\Phi }}_p\left(u\right)}}<\infty ,$$

(29)

and

$$\underset{u\to \infty }{liminf}{\displaystyle \frac{F\left(u\right)}{{\mathsf{\Phi }}_p\left(u\right)}}>0.$$

(30)

Clearly, (28) implies $J_a=\infty$. For proving the existence of solutions x of (1) such that $x\in S$, as claimed, we use a recent result in ([23] Theorem 2.1). This is useful for solving boundary value problems on the half-line I, associated with second-order equations

$${\left(a\left(t\right)\mathsf{\Phi }\left(x^{\prime }\right)\right)}^{\prime }+b\left(t\right)F\left(x\right)=0,$$

where $\mathsf{\Phi }$ is an invertible operator. The following holds.

Lemma 2. 

Let $S_0$ be a subset of $C(I,{\mathbb{R}}^2).$ Assume that there exists a nonempty closed bounded convex subset $\mathsf{\Omega }\subset$ $C(I,{\mathbb{R}}^2)$ such that

$$u\left(t\right)\ne 0,\left|v\right(t\left)\right|/a\left(t\right)<1\mathit{for}\mathit{all}(u,v)\in \mathsf{\Omega }\mathit{and}t\in I$$

and a nonempty closed subset $S_1$ of $S_0\cap \mathsf{\Omega }$ such that for each $(u,v)\in \mathsf{\Omega }$ the half-linear equation

$${\displaystyle \frac{d}{dt}}\left(A_v\left(t\right){\mathsf{\Phi }}_p\left(y^{\prime }\right)\right)+b\left(t\right){\displaystyle \frac{F\left(u\right(t\left)\right)}{{\mathsf{\Phi }}_p\left(u\left(t\right)\right)}}{\mathsf{\Phi }}_p\left(y\right)=0$$

(31)

has a unique solution $y_{uv}$ satisfying $(y_{uv},A_v\left(t\right){\mathsf{\Phi }}_p\left(y_{uv}^{\prime }\right))\in S_1$, where

$$A_v\left(t\right)=(a^{p/(p-1)}-{\left|v\right|}^{p/(p-1)}{)}^{(p-1)/p}.$$

(32)

Then, (1) has a solution $\widehat{x}$ and $(\widehat{x},a\left(t\right){\mathsf{\Phi }}_C\left({\widehat{x}}^{\prime }\right))\in S_1$.

Proof. 

Let us show that the assumptions in ([23] Theorem 2.1) are verified. From $z={\mathsf{\Phi }}_c\left(u\right)$, we obtain $1-{\left|z\right|}^{p/(p-1)}={\left(1+{\left|u\right|}^p\right)}^{-1}$, from which we have

$$\left|u\right|={\left|z\right|}^{1/(p-1)}{\left(1-{\left|z\right|}^{p/(p-1)}\right)}^{-1/p}.$$

(33)

Denoting by q the conjugate number of p, that is,

$$q={\displaystyle \frac{p}{p-1}},$$

(34)

it holds

$$p={\displaystyle \frac{q}{q-1}}\mathrm{and}p-1={\displaystyle \frac{1}{q-1}}.$$

Thus, from (33) and taking into account that $\mathrm{sgn}u=\mathrm{sgn}z$ we have

$$u={\displaystyle \frac{{\left|z\right|}^{q-2}z}{{\left(1-{\left|z\right|}^q\right)}^{(q-1)/q}}},q>1.$$

Then, the operator ${\mathsf{\Phi }}_C$ is invertible and its inverse ${\mathsf{\Phi }}^{*}$ verifies condition $\left(i_0\right)$ in ([23] Theorem 2.1), i.e., ${\mathsf{\Phi }}^{*}\left(z\right)=\mathsf{\Psi }\left(z\right){\left|z\right|}^{1/\beta }\mathrm{sgn}z$, with

$$\mathsf{\Psi }\left(z\right)={\displaystyle \frac{1}{{\left(1-{\left|z\right|}^q\right)}^{(q-1)/q}}}\mathrm{and}\beta ={(q-1)}^{-1}.$$

Moreover, we have

$${\left(\mathsf{\Psi }\left({\displaystyle \frac{v\left(t\right)}{a\left(t\right)}}\right)\right)}^{-1/(q-1)}={\left(1-{\left({\displaystyle \frac{v\left(t\right)}{a\left(t\right)}}\right)}^q\right)}^{1/q}$$

and so the function H in ([23] (2.8)) reads as

$$H(t,v\left(t\right))=a\left(t\right){\left(1-{\left({\displaystyle \frac{v\left(t\right)}{a\left(t\right)}}\right)}^{p/(p-1)}\right)}^{(p-1)/p}=A_v\left(t\right).$$

It is easy to verify that the remaining assumptions of Theorem 2.1 in [23] are satisfied and so the assertion follows. □

Let

$$d<{\displaystyle \frac{p}{p-1}},d>0$$

(35)

be fixed and set

$$M={\left(1-{\left({\displaystyle \frac{d(p-1)}{p}}\right)}^p\right)}^{(p-1)/p},$$

(36)

$$F_1=\underset{u\ge d}{sup}{\displaystyle \frac{F\left(u\right)}{{\mathsf{\Phi }}_p\left(u\right)}}.$$

(37)

The following holds.

Theorem 2. 

Let (28), (29) and (30) be satisfied. If $p\ge 2$ and $\mathsf{\Lambda }>0$ exists such that

$$a\left(t\right)\ge \mathsf{\Lambda },b\left(t\right)\le {\displaystyle \frac{M\mathsf{\Lambda }}{F_1}}{\left({\displaystyle \frac{p-1}{p}}\right)}^p{t}^{-p}\mathit{for}\mathit{any}t\ge 1,$$

(38)

then (1) has infinitely many solutions x$\in S.$ In particular, for any d satisfying (35), Equation (1) has a solution x$\in S$ with $x\left(1\right)=d.$

Proof. 

Consider the half-linear Euler equation

$${\left(M\mathsf{\Lambda }{\mathsf{\Phi }}_p\left(x^{\prime }\right)\right)}^{\prime }+M\mathsf{\Lambda }{\left({\displaystyle \frac{p-1}{p}}\right)}^p{\displaystyle \frac{1}{t^p}}{\mathsf{\Phi }}_p\left(x\right)=0.$$

(39)

Equation (39) is nonoscillatory and its principal solution ${\epsilon }_0,$ satisfying ${\epsilon }_0\left(1\right)=d,$ is

$${\epsilon }_0\left(t\right)=d{t}^{(p-1)/p},$$

see, e.g., ([6] pp. 146–147, point (iii)). In view of (38), Equation (39) is a majorant of

$${\left(Ma\left(t\right){\mathsf{\Phi }}_p\left(w^{\prime }\right)\right)}^{\prime }+F_{1}b\left(t\right){\mathsf{\Phi }}_p\left(w\right)=0.$$

(40)

Thus, from Corollary 1 and Proposition 2, the principal solution $w_0$ of (40), starting at $w_0\left(1\right)=d,$ is positive on the whole interval I and

$$\underset{t\to \infty }{lim}{w}_0\left(t\right)=\infty ,\underset{t\to \infty }{lim}a\left(t\right){\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)=0.$$

(41)

Moreover, from (40), the function $a{\mathsf{\Phi }}_p\left(w_0^{\prime }\right)$ is decreasing on I and, so, in view of (41), we have $a\left(t\right){\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)>0$ for any $t\in I$, that is, $w_0$ is increasing on I.

Since $a\left(t\right)\ge \mathsf{\Lambda }$, from (41), we obtain ${lim}_{t\to \infty }{w}_0^{\prime }\left(t\right)=0.$ Further, from Theorem 1 and (11), a standard calculation yields for any $t\ge 1$

$${\displaystyle \frac{a\left(t\right){\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)}{{\mathsf{\Phi }}_p\left(w_0\left(t\right)\right)}}\le \mathsf{\Lambda }{\displaystyle \frac{{\mathsf{\Phi }}_p\left({\epsilon }_0^{\prime }\left(t\right)\right)}{{\mathsf{\Phi }}_p\left({\epsilon }_0\left(t\right)\right)}}=\mathsf{\Lambda }{\left({\displaystyle \frac{p-1}{pt}}\right)}^{p-1}<\mathsf{\Lambda }.$$

(42)

Hence,

$$c=a\left(1\right){\mathsf{\Phi }}_p\left(w_0^{\prime }\left(1\right)\right)\le \mathsf{\Lambda }{\left({\displaystyle \frac{p-1}{p}}\right)}^{p-1}{d}^{p-1}<\mathsf{\Lambda }.$$

(43)

In view of (40), the function $a{\mathsf{\Phi }}_p\left(w_0^{\prime }\right)$ is decreasing on I. Hence, we obtain

$$0<a\left(t\right){\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)\le a\left(1\right){\mathsf{\Phi }}_p\left(w_0^{\prime }\left(1\right)\right)=c\mathrm{and}d=w_0\left(1\right)\le w_0\left(t\right).$$

(44)

Let us prove that (1) admits a solution $x\in S.$ Let $\mathsf{\Omega }$ be the subset of $C(I,{\mathbb{R}}^2),$ given by

$$\mathsf{\Omega }=\left\{(u,v)\in C(I,{\mathbb{R}}^2):d\le u\left(t\right)\le w_0\left(t\right),0\le v\left(t\right)\le a\left(t\right){\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)\right\},$$

and let $S_0=S_1=$$\mathsf{\Omega }.$ We start by showing that, for each $(u,v)\in \mathsf{\Omega }$, Equation (31) has a unique solution $y_{uv}$ with $(y_{uv},A_v{\mathsf{\Phi }}_p\left(y_{uv}^{\prime }\right))\in \mathsf{\Omega },$ where $A_v$ is given by (32).

Step 1: existence. For any $(u,v)\in \mathsf{\Omega },$ since $a\left(t\right)\ge \mathsf{\Lambda },$ using (43) and (44), we have

$$\begin{array}{cc}\hfill {\left({\displaystyle \frac{v\left(t\right)}{a\left(t\right)}}\right)}^{p/(p-1)}& \le {\displaystyle \frac{{\left(a\left(t\right){\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)\right)}^{p/(p-1)}}{{\left(a\left(t\right)\right)}^{p/(p-1)}}}\le {\displaystyle \frac{c^{p/(p-1)}}{{\left(a\left(t\right)\right)}^{p/(p-1)}}}\le {\left({\displaystyle \frac{c}{\mathsf{\Lambda }}}\right)}^{p/(p-1)}\hfill \\ \hfill & \le {\left({\displaystyle \frac{d(p-1)}{p}}\right)}^p.\hfill \end{array}$$

(45)

Hence, from (36), (45) and (37), we obtain

$$\begin{array}{cc}\hfill A_v\left(t\right)& =a\left(t\right){\left(1-{\left({\displaystyle \frac{v\left(t\right)}{a\left(t\right)}}\right)}^{p/(p-1)}\right)}^{(p-1)/p}\hfill \\ \hfill & \ge a\left(t\right){\left(1-{\left({\displaystyle \frac{d(p-1)}{p}}\right)}^p\right)}^{(p-1)/p}=Ma\left(t\right)\hfill \end{array}$$

(46)

and

$$b\left(t\right){\displaystyle \frac{F\left(u\right(t\left)\right)}{{\mathsf{\Phi }}_p\left(u\left(t\right)\right)}}\le F_{1}b\left(t\right).$$

Hence, (31) is a minorant of (40) and so (31) is nonoscillatory. For any $(u,v)\in \mathsf{\Omega }$, let $y_{uv}$ be the principal solution of (31) such that

$$y_{uv}\left(1\right)=w_0\left(1\right)=d.$$

(47)

Then, from Corollary 1, the solution $y_{uv}$ is positive for any $t\ge 1.$ Moreover, from Theorem 1 and (11), we obtain

$${\displaystyle \frac{A_v\left(t\right){\mathsf{\Phi }}_p\left(y_{uv}^{\prime }\left(t\right)\right)}{{\mathsf{\Phi }}_p\left(y_{uv}\left(t\right)\right)}}\le Ma\left(t\right){\displaystyle \frac{{\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)}{{\mathsf{\Phi }}_p\left(w_0\left(t\right)\right)}}$$

(48)

or, from (46),

$${\displaystyle \frac{{\mathsf{\Phi }}_p\left(y_{uv}^{\prime }\left(t\right)\right)}{{\mathsf{\Phi }}_p\left(y_{uv}\left(t\right)\right)}}\le {\displaystyle \frac{{\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)}{{\mathsf{\Phi }}_p\left(w_0\left(t\right)\right)}}.$$

(49)

Since ${\mathsf{\Phi }}_p$ is increasing, we obtain for $t\ge 1$

$${\displaystyle \frac{y_{uv}^{\prime }\left(t\right)}{y_{uv}\left(t\right)}}\le {\displaystyle \frac{w_0^{\prime }\left(t\right)}{w_0\left(t\right)}},$$

(50)

which, in view of (47), yields $d\le y_{uv}\left(t\right)\le w_0\left(t\right).$ Moreover, since $M<1,$ from (48), we obtain

$$\begin{array}{cc}\hfill A_v\left(t\right){\mathsf{\Phi }}_p\left(y_{uv}^{\prime }\left(t\right)\right)& \le M{\displaystyle \frac{{\mathsf{\Phi }}_p\left(y_{uv}\left(t\right)\right)}{{\mathsf{\Phi }}_p\left(w_0\left(t\right)\right)}}a\left(t\right){\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)\le Ma\left(t\right){\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)\hfill \\ \hfill & <a\left(t\right){\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)\hfill \end{array}$$

(51)

and so $(y_{uv},A_v\left(t\right){\mathsf{\Phi }}_p\left(y_{uv}^{\prime }\right))\in \mathsf{\Omega }$.

Step 2: uniqueness. For fixed $(u,v)\in \mathsf{\Omega }$, by contradiction, let ${\xi }_{uv}$ be another solution of (31) with $({\xi }_{uv},A_v\left(t\right){\mathsf{\Phi }}_p\left({\xi }_{uv}^{\prime }\right))\in \mathsf{\Omega }$ and ${\xi }_{uv}\neg \equiv y_{uv}.$ Thus, ${\xi }_{uv}\left(1\right)=y_{uv}\left(1\right)=d$ and so ${\xi }_{uv}$ is a nonprincipal solution of (31). From the definition of the set $\mathsf{\Omega }$, we have

$$A_v\left(t\right){\mathsf{\Phi }}_p\left({\xi }_{uv}^{\prime }\left(t\right)\right)\le a\left(t\right){\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)$$

or, in view of (46),

$${\mathsf{\Phi }}_p\left({\xi }_{uv}^{\prime }\left(t\right)\right)\le {\displaystyle \frac{a\left(t\right)}{A_v\left(t\right)}}{\mathsf{\Phi }}_p\left(w_0^{\prime }\left(t\right)\right)\le {\displaystyle \frac{1}{M}}{\mathsf{\Phi }}_p(w_0^{\prime }\left(t\right).$$

Hence,

$${\displaystyle \frac{1}{{\xi }_{uv}^{\prime }\left(t\right)}}\ge {\displaystyle \frac{M^{1/(p-1)}}{w_0^{\prime }\left(t\right)}}.$$

From this, since $p\ge 2$, we obtain

$${\left({\displaystyle \frac{1}{{\xi }_{uv}^{\prime }\left(t\right)}}\right)}^{p-2}\ge M^{(p-2)/(p-1)}{\left({\displaystyle \frac{1}{w_0^{\prime }\left(t\right)}}\right)}^{p-2}.$$

(52)

Moreover, since $({\xi }_{uv},A_v{\mathsf{\Phi }}_p\left({\xi }_{uv}^{\prime }\right))\in \mathsf{\Omega },$ we have

$${\left({\xi }_{uv}\left(t\right)\right)}^{-2}\ge {\left(w_0\left(t\right)\right)}^{-2}.$$

(53)

Using (52) and (53), we obtain

$$\begin{array}{cc}\hfill Q_{\xi }& ={\int }_1^{\infty }{\displaystyle \frac{{\xi }_{uv}^{\prime }\left(t\right)}{A_v\left(t\right){\mathsf{\Phi }}_p\left({\xi }_{uv}^{\prime }\right){\left({\xi }_{uv}\left(t\right)\right)}^2}}dt={\int }_1^{\infty }{\displaystyle \frac{1}{A_v\left(t\right){\left({\xi }_{uv}\left(t\right)\right)}^2{\left({\xi }_{uv}^{\prime }\left(t\right)\right)}^{p-2}}}dt\ge \hfill \\ & \ge M^{(p-2)/(p-1)}{\int }_1^{\infty }{\displaystyle \frac{1}{A_v\left(t\right)}}{\displaystyle \frac{1}{{\left(w_0\left(t\right)\right)}^2{\left(w_0^{\prime }\left(t\right)\right)}^{p-2}}}dt,\phantom{|}\hfill \end{array}$$

or, since $A_v\left(t\right)\le a\left(t\right),$

$$Q_{\xi }\ge M^{(p-2)/(p-1)}{\int }_1^{\infty }{\displaystyle \frac{1}{a\left(t\right){\left(w_0\left(t\right)\right)}^2{\left(w_0^{\prime }\left(t\right)\right)}^{p-2}}}dt=M{M}^{(p-2)/(p-1)}{Q}_w=\infty ,$$

which contradicts Proposition 1, since ${\xi }_{uv}$ is a nonprincipal solution of (31).

Hence, for each $(u,v)\in \mathsf{\Omega }$, the principal solution $y_{uv}$ is the unique solution of (31) such that $(y_{uv},A_v{\mathsf{\Phi }}_p\left(y_{uv}^{\prime }\right))\in \mathsf{\Omega }.$ Moreover, using $A_v\left(t\right)\le a\left(t\right)$, (30) and Proposition 2, we have

$$\underset{t\to \infty }{lim}{y}_{uv}\left(t\right)=\infty ,\underset{t\to \infty }{lim}{A}_v\left(t\right){\mathsf{\Phi }}_p\left(y_{uv}^{\prime }\left(t\right)\right)=0.$$

(54)

Since $A_v\left(t\right)$ is bounded away from zero, from (54), we also obtain

$$\underset{t\to \infty }{lim}{y}_{uv}^{\prime }\left(t\right)=0.$$

(55)

Let $\mathcal{T}$ be the operator $\mathcal{T}:\mathsf{\Omega }\to \mathsf{\Omega }$ which associates to any $(u,v)\in \mathsf{\Omega }$ the couple $(y_{uv},A_v{\mathsf{\Phi }}_p\left(y_{uv}^{\prime }\right)).$ Using Lemma 2, $\mathcal{T}$ has a fixed point, namely $(\widehat{x},{\widehat{x}}^{\prime })$, which belongs to $\mathsf{\Omega }$ and $\widehat{x}$ is a solution of (1), with $\widehat{x}\left(1\right)=d$. In view of (54) and (55), we have $\widehat{x}\in S$ and the proof is complete. □

When $p=2$, Theorem 2 extends ([5] Theorem 3.1), in which the existence of positive unbounded solutions x satisfying ${lim}_{t\to \infty }a\left(t\right)x^{\prime }\left(t\right)=0$ is proved for any large t.

The following example illustrates Theorem 2.

Example 1. 

Consider the differential equation

$${\left({\displaystyle \frac{|x^{\prime }{|}^2{x}^{\prime }}{{\left(1+|x^{\prime }{|}^4\right)}^{3/4}}}\right)}^{\prime }+\mu b\left(t\right)F\left(x\right)=0,t\in I=[1,\infty ),$$

(56)

where

$$b\left(t\right)={\displaystyle \frac{1}{t^4{\left(1+logt\right)}^{1/2}}},F\left(u\right)={\displaystyle \frac{{\left|u\right|}^{3}u}{1+\left|u\right|}}$$

and μ is a positive real parameter. We have

$${\int }_1^{\infty }b\left(t\right)dt<\infty .$$

Moreover, from the inequality $t^4{\left(1+logt\right)}^{1/2}\le t^4\left(1+logt\right),$ a standard calculation gives

$${\int }_1^{\infty }{\left({\int }_t^{\infty }b\left(r\right)dr\right)}^{1/3}dt=\infty .$$

Thus, (28) is verified. Clearly, (29) and (30) are also valid and, for fixed $d\in (0,4/3)$, see (35), we obtain $F_1=1,$ where $F_1$ is given in (37). We have for any $t\in I$

$$\mu t^{4}b\left(t\right)={\displaystyle \frac{\mu }{{\left(1+logt\right)}^{1/2}}}\le \mu .$$

Hence, (38) is valid for any μ such that

$$0<\mu \le M{\left({\displaystyle \frac{3}{4}}\right)}^4,$$

(57)

where M is defined in (36). All the assumptions of Theorem 2 are therefore verified and (56) has a solution x which belongs to set S for any μ satisfying (57).

4. Concluding Remarks

(1) It is well known that the half-linear case presents some crucial differences from the corresponding linear case, especially as it concerns the asymptotic classification of nonoscillatory solutions; see, e.g., [7] for more details. In particular, the study of proximity between Equations (1) and (4) presents new situations with respect to the comparison between (2) and (3). Indeed, when $J_a=\infty ,$ the classification of nonoscillatory solutions of (4) depends on the divergence of integrals $J_{ab}$ and

$$J_{ba}={\int }_1^{\infty }b\left(t\right){\int }_1^t{\left({\displaystyle \frac{1}{a\left(s\right)}}\right)}^{1/(p-1)}dsdt.$$

It is easy to verify that, in the linear case, namely for $p=2,$ the integrals $J_{ab}$ and $J_{ba}$ coincide, that is, either $J_{ab}=J_{ba}=\infty$ or $J_{ab}<\infty$, $J_{ba}<\infty .$ This fact does not occur for $p\ne 2,$ in which the cases

$$\left(i_1\right):J_{ab}=\infty ,J_{ba}<\infty ;\left(i_2\right):J_{ab}<\infty ,J_{ba}=\infty$$

are also possible. More precisely, the case $\left(i_1\right)$ occurs when $p>2$ and the case $\left(i_2\right)$ occurs when $1<p<2$; see ([7] Lemma 2). Moreover, Equation (4) has weakly increasing solutions only in the cases $\left(i_1\right),\left(i_2\right)$ and

$$\left(i_0\right):J_{ab}=J_{ba}=\infty \mathrm{and}\left(4\right)\mathrm{nonoscillatory},$$

see ([7] Theorems 6 and 7).

Concerning the existence of weakly increasing solutions to (1), Theorem 2 covers the cases $\left(i_0\right)$ with $p\ge 2$ and $\left(i_1\right)$. Thus, it is an open problem if (1) admits weakly increasing solutions in the cases $\left(i_0\right)$ with $p<2$ and $\left(i_2\right).$ Observe that the argument given in the proof of Theorem 2 does not work in the case $\left(i_0\right)$ with $p<2,$ because, due to the erratum in ([18] Lemma 2.4), until now it is not known whether Proposition 1 is also valid when $1<p<2$ and $J_a=J_{ab}=\infty .$ Further, in the case $\left(i_2\right)$, the argument given in the proof of Theorem 2 cannot be used, because the principal solutions of (31) are bounded and weakly increasing solutions of (31) are nonprincipal solutions; see, e.g., [7]. Then, for proving the existence of weakly increasing solutions of (1) when the case $\left(i_2\right)$ occurs, it is necessary to use a different approach to the one here employed. This will be the object of a forthcoming paper.

(2) Assumption (30) is used in Theorem 2 to obtain that the principal solution of (31) is unbounded; see (54). Without this assumption, the proof of Theorem 2 yields the existence of solutions of (1), which are positive increasing on the whole half-line and with zero derivative at infinity. Nevertheless, they can be bounded or unbounded. The study of the boundedness at infinity of these solutions may be the subject of future investigations.

(3) In the proof of Theorem 2, an important role is played by the half-linear Euler Equation (39), whose principal solution ${\epsilon }_0$ is known and satisfies for any $t\ge 1$

$${\displaystyle \frac{{\mathsf{\Phi }}_p\left({\epsilon }_0^{\prime }\left(t\right)\right)}{{\mathsf{\Phi }}_p\left({\epsilon }_0\left(t\right)\right)}}<1.$$

Further, the assumption (38) also depends on the Equation (39), since (38) is needed in order to have that Equation (40) is a minorant of the nonoscillatory Equation (39).

Clearly, the role of the half-linear Euler Equation (39) can be replaced by any other nonoscillatory half-linear equation

$${\left(\widehat{a}\left(t\right){\mathsf{\Phi }}_p\left(z^{\prime }\right)\right)}^{\prime }+\widehat{b}\left(t\right){\mathsf{\Phi }}_p\left(z\right)=0,$$

where $\widehat{a},\widehat{b}$ are positive continuous functions on I, and whose principal solution $z_0$ satisfies $z_0\left(1\right)>0$, ${lim}_{t\to \infty }{z}_0\left(t\right)=\infty$, ${lim}_{t\to \infty }\widehat{a}\left(t\right){\mathsf{\Phi }}_p\left(z_0^{\prime }\right)=0$, and for any $t\ge 1$

$${\displaystyle \frac{\widehat{a}\left(t\right){\mathsf{\Phi }}_p\left(z_0^{\prime }\left(t\right)\right)}{{\mathsf{\Phi }}_p\left(z_0\left(t\right)\right)}}<1.$$

For instance, consider the generalized Euler equation

$${\left(t^n{\mathsf{\Phi }}_p\left(z^{\prime }\right)\right)}^{\prime }+{\left({\displaystyle \frac{p-1-n}{p}}\right)}^p{t}^{n-p}{\mathsf{\Phi }}_p\left(z\right)=0,$$

(58)

where $p\ge 2$ and n is a real number such that

$$0\le n<p-1.$$

Setting $\delta =(p-1-n)/p,$ it is easy to verify that

$$z_0\left(t\right)=t^{\delta }$$

is a solution of (58). Hence, (58) is nonoscillatory. From Proposition 1, the solution $z_0$ is the principal solution and a direct computation shows that for any $t\ge 1$

$${\displaystyle \frac{t^n{\mathsf{\Phi }}_p\left(z_0^{\prime }\left(t\right)\right)}{{\mathsf{\Phi }}_p\left(z_0\left(t\right)\right)}}<1.$$

Let $d_1>0$ be such that

$$d_1<{\displaystyle \frac{p}{p-1-n}}={\displaystyle \frac{1}{\delta }},$$

(59)

and put

$$M_1={\left(1-{\left(d_1\delta \right)}^p\right)}^{(p-1)/p}.$$

Theorem 2 can be reformulated as follows.

Theorem 3. 

Let (28) and (29) be satisfied. If $p\ge 2$ and $\mathsf{\Lambda }>0$ exists such that

$$a\left(t\right)\ge \mathsf{\Lambda }{t}^n,b\left(t\right)\le {\displaystyle \frac{M_1\mathsf{\Lambda }}{F_1}}{\delta }^p{t}^{n-p}\mathit{for}\mathit{any}t\ge 1,$$

then (1) has infinitely many solutions x$\in S.$ In particular, for any $d_1$ satisfying (59), Equation (1) has a solution x $\in S$ with $x\left(1\right)=d_1.$

The proof is analogous to the one of Theorem 2, taking as the majorant equation, in place of (39), the equation

$${\left(M_1\mathsf{\Lambda }{t}^n{\mathsf{\Phi }}_p\left(z^{\prime }\right)\right)}^{\prime }+M_1\mathsf{\Lambda }{\left({\displaystyle \frac{p-1-n}{p}}\right)}^p{t}^{n-p}{\mathsf{\Phi }}_p\left(z\right)=0.$$

The details are left to the reader.