In [

1], Ma and Thompson were considered with determining interval of

*μ*, in which there exist nodal solutions for the boundary value problem (BVP)

${u}^{\u2033}\left(t\right)+\mu w\left(t\right)f\left(u\right)=0,\phantom{\rule{1em}{0ex}}t\in \left(0,1\right),\phantom{\rule{1em}{0ex}}u\left(0\right)=u\left(1\right)=0$

(1.1)

under the assumptions:

(C1) *w*(·) ∈ *C*([0, 1], [0, ∞)) and does not vanish identically on any subinterval of [0, 1];

(C2) *f* ∈ *C*(ℝ, ℝ) with *sf*(*s*) > 0 for *s* ≠ 0;

(C3) there exist

*f*_{0},

*f*∞ ∈ (0, ∞) such that

${f}_{0}=\underset{\left|s\right|\to 0}{\text{lim}}\frac{f\left(s\right)}{s},\phantom{\rule{1em}{0ex}}{f}_{\infty}=\underset{\left|s\right|\to \infty}{\text{lim}}\frac{f\left(s\right)}{s}.$

It is well known that under (C1) assumption, the eigenvalue problem

${\phi}^{\u2033}\left(t\right)+\mu w\left(t\right)\phi \left(t\right)=0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}t\in \left(0,1\right),\phantom{\rule{1em}{0ex}}\phi \left(0\right)=\phi \left(1\right)=0$

(1.2)

has a countable number of simple eigenvalues

*μ*_{
k
},

*k* = 1, 2,..., which satisfy

$0<{\mu}_{1}<{\mu}_{2}<\cdots <{\mu}_{k}<\cdots \phantom{\rule{0.3em}{0ex}},\phantom{\rule{1em}{0ex}}\text{and}\underset{k\to \infty}{\text{lim}}{\mu}_{k}=\infty ,$

and let *μ*_{
k
}be the *k* th eigenvalue of (1.2) and *φ*_{
k
}be an eigenfunction corresponding to *μ*_{
k
}, then *φ*_{
k
}has exactly *k* -- 1 simple zeros in (0,1) (see, e.g., [2]).

Using Rabinowitz bifurcation theorem, they established the following interesting results:

**Theorem A** (Ma and Thompson [[1], Theorem 1.1]). *Let (C1)-(C3) hold. Assume that for some k* ∈ ℕ, *either* $\frac{{\mu}_{k}}{{f}_{\infty}}<\mu <\frac{{\mu}_{k}}{{f}_{0}}$ *or* $\frac{{\mu}_{k}}{{f}_{0}}<\mu <\frac{{u}_{k}}{{f}_{\infty}}$. *Then BVP (1.1) has two solutions* ${u}_{k}^{+}$ *and* ${u}_{k}^{-}$ *such that* ${u}_{k}^{+}$ *has exactly k* -- 1 *zeros in (0, 1) and is positive near 0, and* ${u}_{k}^{-}$ *has exactly k* -- 1 *zeros in (0,1) and is negative near 0.*

In [

3], Ma and Thompson studied the existence and multiplicity of nodal solutions for BVP

${u}^{\u2033}\left(t\right)+w\left(t\right)f\left(u\right)=0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}t\in \left(0,1\right),\phantom{\rule{1em}{0ex}}u\left(0\right)=u\left(1\right)=0.$

(1.3)

They gave conditions on the ratio $\frac{f\left(s\right)}{s}$ at infinity and zero that guarantee the existence of solutions with prescribed nodal properties.

Using Rabinowitz bifurcation theorem also, they established the following two main results:

**Theorem B** (Ma and Thompson [[1], Theorem 2]). *Let (C1)-(C3) hold. Assume that either (i) or (ii) holds for some k* ∈ ℕ *and j* ∈ {0} ∪ ℕ;

*(i) f*_{0} < *μ*_{
k
}< ⋯ < *μ*_{k+j}< *f*_{∞};

*(ii) f*_{∞} < *μ*_{
k
}< ⋯ < *μ*_{k+j}< *f*_{0},

*where μ*_{
k
}*denotes the k* th *eigenvalue of (1.2). Then BVP (1.3) has 2*(*j* + 1) *solutions* ${u}_{k+i}^{+},{u}_{k+i}^{-},i=0,\dots ,j$, *such that* ${u}_{k+i}^{+}$ *has exactly k* + *i* -- 1 *zeros in (0, 1) and are positive near 0, and* ${u}_{k+i}^{-}$ *has exactly k* + *i* -- 1 *zeros in (0,1) and are negative near 0.*

**Theorem C** (Ma and Thompson [[

1], Theorem 3]).

*Let (C1)-(C3) hold. Assume that there exists an integer k* ∈ ℕ

*such that*${\mu}_{k-1}<\frac{f\left(s\right)}{s}<{\mu}_{k},$

*where μ*_{
k
}*denotes the k* th *eigenvalue of (1.2). Then BVP (1.3) has no nontrivial solution*.

From above literature, we can see that the existence and multiplicity results are largely based on the assumption that *t* and *u* are separated in nonlinearity term. It is interesting to know what will happen if *t* and *u* are not separated in nonlinearity term? We shall give a confirm answer for this question.

In this article, we consider the existence and multiplicity of nodal solutions for the nonlinear BVP

${u}^{\u2033}+f\left(t,u\right)=0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}t\in \left(0,1\right),\phantom{\rule{1em}{0ex}}u\left(0\right)=u\left(1\right)=0$

(1.4)

under the following assumptions:

(

*H*_{1})

${\lambda}_{k}\le a\left(t\right)\equiv \underset{\left|s\right|\to +\infty}{\text{lim}}\frac{f\left(t,s\right)}{s}$ *uniformly on* [0, 1],

*and the inequality is strict on some subset of positive measure in* (0,1), where

*λ*_{k} denotes the

*k* th eigenvalue of

${u}^{\u2033}\left(t\right)+{\lambda}_{u}\left(t\right)=0,\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}t\in \left(0,1\right),\phantom{\rule{1em}{0ex}}u\left(0\right)=u\left(1\right)=0;$

(1.5)

(*H*_{2}) $0\le \underset{\left|s\right|\to 0}{\text{lim}}\frac{f\left(t,s\right)}{s}\equiv c\left(t\right)\le {\lambda}_{k}$ *uniformly on* [0, 1], *and all the inequalities are strict on some subset of positive measure in* (0, 1), *where λ*_{
k
}*denotes the k* th *eigenvalue of (1.5);*

*(H*_{3}*) f*(*t, s*)*s* > 0 *for t* ∈ (0, 1) *and s* ≠ 0.

**Remark 1.1**. From (*H*_{1})-(*H*_{3}), we can see that there exist a positive constant *ϱ* and a subinterval [*α, β*] of [0, 1] such that *α* < *β* and $\frac{f\left(r,s\right)}{s}\ge \varrho $ for all *r* ∈ [*α, β*] and *s* ≠ 0.

In the celebrated study [4], Rabinowitz established Rabinowitz's global bifurcation theory [[4], Theorems 1.27 and 1.40]. However, as pointed out by Dancer [5, 6] and López-Gómez [7], the proofs of these theorems contain gaps, the original statement of Theorem 1.40 of [4] is not correct, the original statement of Theorem 1.27 of [4] is stronger than what one can actually prove so far. Although there exist some gaps in the proofs of Rabinowitz's Theorems 1.27, 1.40, and 1.27 has been used several times in the literature to analyze the global behavior of the component of nodal solutions emanating from *u* = 0 in wide classes of boundary value problems for equations and systems [1, 2, 8, 9]. Fortunately, López-Gómez gave a corrected version of unilateral bifurcation theorem in [7].

By applying the bifurcation theorem of López-Gómez [[7], Theorem 6.4.3], we shall establish the following:

**Theorem 1.1**. *Suppose that f*(*t, u*) *satisfies (H*_{1}*), (H*_{2}*), and (H*_{3}*), then problem (1.4) possesses two solutions* ${u}_{k}^{+}$ *and* ${u}_{k}^{-}$, *such that* ${u}_{k}^{+}$ *has exactly k* -- 1 *zeros in (0, 1) and is positive near 0, and* ${u}_{k}^{-}$ *has exactly k* -- 1 *zeros in (0,1) and is negative near 0.*

Similarly, we also have the following:

**Theorem 1.2**. *Suppose that f*(*t, u*) *satisfies (H*_{3}*) and*

$\left({H}_{1}^{\prime}\right){\lambda}_{k}\ge a\left(x\right)\equiv \underset{\left|s\right|\to +\infty}{\text{lim}}\frac{f\left(t,s\right)}{s}\ge 0$ *uniformly on* [0, 1], *and all the inequalities are strict on some subset of positive measure in* (0, 1), *where λ*_{k} *denotes the k* th *eigenvalue of (1.5);*

$\left({H}_{2}^{\prime}\right)\underset{\left|s\right|\to 0}{\text{lim}}\frac{f\left(t,s\right)}{s}\equiv c\left(x\right)\ge {\lambda}_{k}$ *uniformly on* [0, 1], *and the inequality is strict on some subset of positive measure in* (0, 1), *where λ*_{
k
}*denotes the k* th *eigenvalue of (1.5), then problem (1.4) possesses two solutions* ${u}_{k}^{+}$ *and* ${u}_{k}^{-}$, *such that* ${u}_{k}^{+}$ has exactly k-- 1 *zeros in (0,1) and is positive near 0, and* ${u}_{k}^{-}$ *has exactly k* -- 1 *zeros in (0,1) and is negative near 0.*

**Remark 1.2**. We would like to point out that the assumptions (*H*_{1}) and (*H*_{2}) are weaker than the corresponding conditions of Theorem A. In fact, if we let *f*(*t, s*) ≡ *μw*(*t*)*f*(*s*), then we can get $\underset{\left|s\right|\to +\infty}{\text{lim}}\frac{f\left(t,s\right)}{s}\equiv \mu w\left(t\right){f}_{\infty}:=a\left(t\right)$ and $\underset{\left|s\right|\to 0}{\text{lim}}\frac{f\left(t,s\right)}{s}\equiv \mu w\left(t\right){f}_{0}:=c\left(t\right)$. By the strict decreasing of *μ*_{
k
}(*f*) with respect to weight function *f* (see [10]), where *μ*_{
k
}(*f*) denotes the *k* th eigenvalue of (1.2) corresponding to weight function *f*, we can show that our condition *c*(*t*) ≤ *λ*_{
k
}≤ *a*(*t*) is equivalent to the condition $\frac{{\mu}_{k}}{{f}_{\infty}}<\mu <\frac{{\mu}_{k}}{{f}_{0}}$. Similarly, our condition *c* (*t*) ≥ *λ*_{k} ≥ *a* (*t*) is equivalent to the condition $\frac{{\mu}_{k}}{{f}_{0}}<\mu <\frac{{u}_{k}}{{f}_{\infty}}$. Therefore, Theorem A is the corollary of Theorems 1.1 and 1.2.

Using the similar proof with the proof Theorems 1.1 and 1.2, we can obtain the more general results as follows.

**Theorem 1.3**. *Suppose that* (*H*_{3}) *holds, and either (i) or (ii) holds for some k* ∈ ℕ *and j* ∈ {0} ∪ ℕ:

*(i)* $0\le c\left(t\right)\equiv \underset{\left|s\right|\to 0}{\text{lim}}\frac{f\left(t,s\right)}{s}\le {\lambda}_{k}<\cdots <{\lambda}_{k+j}\le a\left(t\right)\equiv \underset{\left|s\right|\to +\infty}{\text{lim}}\frac{f\left(t,s\right)}{s}$ *uniformly on* [0, 1], *and the inequalities are strict on some subset of positive measure in* (0,1), *where λ*_{
k
}*denotes the k* th *eigenvalue of (1.5);*

*(ii)* $0\le a\left(t\right)\equiv \underset{\left|s\right|\to +\infty}{\text{lim}}\frac{f\left(t,s\right)}{s}\le {\lambda}_{k}<\cdots <{\lambda}_{k+j}\le c\left(t\right)\equiv \underset{\left|s\right|\to +\infty}{\text{lim}}\frac{f\left(t,s\right)}{s}$ *uniformly on* [0, 1], *and the inequality is strict on some subset of positive measure in* (0, 1), *where λ*_{
k
}*denotes the k* th *eigenvalue of (1.5).*

*Then BVP (1.4) has* 2(*j* + 1) *solutions* ${u}_{k+i}^{+},{u}_{k+i}^{-},i=0,\dots ,j$, *such that* ${u}_{k+i}^{+}$ *has exactly k* + *i* -- 1 *zeros in (0,1) and are positive near 0, and* ${u}_{k+i}^{-}$ *has exactly k* + *i* -- 1 *zeros in (0,1) and are negative near 0.*

Using Sturm Comparison Theorem, we also can get a non-existence result when *f* satisfies a non-resonance condition.

**Theorem 1.4**.

*Let* (

*H*_{3})

*hold. Assume that there exists an integer k* ∈ ℕ

*such that*${\lambda}_{k-1}<\frac{f\left(t,u\right)}{u}<{\lambda}_{k}$

(1.6)

*for any t* ∈ [0, 1], *where λ*_{
k
}*denotes the k* th *eigenvalue of (1.5). Then BVP (1.4) has no nontrivial solution.*

**Remark 1.3**. Similarly to Remark 1.2, we note that the assumptions (i) and (ii) are weaker than the corresponding conditions of Theorem B. In fact, if we let *f*(*t, s*) ≡ *w*(*t*) *f*(*s*), then we can get $\underset{\left|s\right|\to +\infty}{\text{lim}}\frac{f\left(t,s\right)}{s}\equiv w\left(t\right){f}_{\infty}:=a\left(t\right)$ and $\underset{\left|s\right|\to 0}{\text{lim}}\frac{f\left(t,s\right)}{s}\equiv w\left(t\right){f}_{0}:=c\left(t\right)$. By the strict decreasing of *μ*_{
k
}(*f*) with respect to weight function *f* (see [11]), where *μ*_{
k
}(*f*) denotes the *k* th eigenvalue of (1.2) corresponding to weight function *f*, we can show that our condition *c*(*t*) ≤ *λ*_{
k
}< ⋯ < *λ*_{k+j}≤ *a*(*t*) is equivalent to the condition *f*_{0} < *μ*_{
k
}< · · · < *μ*_{k+j}< *f*_{∞}. Similarly, our condition *a*(*t*) ≤ *λ*_{
k
}< · · · < *λ*_{k+j}≤ *c*(*t*) is equivalent to the condition *f*_{∞} < *μ*_{
k
}< ⋯ < *μ*_{k+j}< *f*_{0}. Therefore, Theorem B is the corollary of Theorem 1.3. Similar, we get Theorem C is also the corollary of Theorem 1.4.