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Higher-order p-Laplacian boundary value problems with resonance of dimension two on the half-line

Abstract

We apply the extension of coincidence degree to obtain sufficient conditions for the existence of at least one solution for a class of higher-order p-Laplacian boundary value problems with two-dimensional kernel on the half-line. The result obtained improves and generalizes some of the known results on p-Laplacian boundary value problems in the literature. We also validate our result with an example.

1 Introduction

This paper is concerned with the existence of solution for the following higher-order p-Laplacian boundary value problem:

$$\begin{aligned}& \bigl(\phi _{p}\bigl(y^{(n-1)}(t)\bigr) \bigr)'=h\bigl(t, y(t), y'(t), \ldots , y^{(n-1)}(t)\bigr),\quad t\in (0, \infty ) , n\geq 3, \end{aligned}$$
(1.1)
$$\begin{aligned}& \begin{aligned} &y^{(n-2)}( \infty )=\sum _{i=1}^{m}\alpha _{i}y^{(n-2)}( \xi _{i}), \qquad y^{(n-3)}(0)+y^{(n-2)}(0)=\sum _{j=1}^{m}\beta _{j}y^{(n-3)}( \eta _{j}), \\ &y^{(n-1)}(\infty )=0, \qquad y^{(i)}(0)=0,\quad i=0,1,2, \ldots , n-4, \end{aligned} \end{aligned}$$
(1.2)

where \(\phi _{p}(s)=|s|^{p-2}s\), \(p>1\), \(1/p+1/q=1\), \(\phi _{q}=\phi _{p}^{-1}\), \(h:[0, \infty ) \times \mathbb{R}^{n}\rightarrow \mathbb{R}\) is a Caratheodory’s function, \(0<\xi _{1}<\xi _{2}<\cdots <\xi _{m}<\infty \), \(0<\eta _{1}<\eta _{2}< \cdots <\eta _{m}<\infty ,\alpha _{i}\), \(\beta _{j}\in \mathbb{R}\), \(i= 1,2, \ldots , m\), \(j=1,2, \ldots , m\), \(\sum_{i=1}^{m} \alpha _{i} = \sum_{j=1}^{m} \beta _{j}=\sum_{j=1}^{m} \beta _{j} \eta _{j}=1\).

Our result will be based on the extension of Mawhin’s continuation theorem by Ge and Ren [6]. Higher-order resonant boundary value problems have in recent years become of great interest to various researchers, see for example [1, 35, 7, 8, 12, 13] and the references therein. Some of the results utilized Mawhin’s coincidence degree theory [14] which has continued to play a significant role in the study of boundary value problems when the differential operator is linear. However, when the differential operator is nonlinear, Mawhin’s continuation theorem can no longer be applied directly as was the case in the above references. For some results on the application of the extension of coincidence degree by Ge and Ren, see [10, 11, 13] and the references therein.

p-Laplacian boundary value problems have found applications in diverse areas such as in nonlinear elasticity, blood flow models, non-Newtonian mechanics, glaciology, etc. Although there have been some results on p-Laplacian boundary value problems at resonance with a two-dimensional kernel, see for example [9], to the best of our knowledge this is the first paper on higher-order p-Laplacian boundary value problems with a resonance of dimension two on the half-line. (1.1)–(1.2) is a problem at resonance if \(Ly=(\phi _{p}(y^{(n-1)}(t)))=0\) has nontrivial solutions under the given boundary conditions. Generally, resonance problems can be cast in the abstract form \(Ly=Ny\), where L is not an invertible operator.

The organization of this paper is as follows. In Sect. 2, we recall some technical results such as definitions, theorems, and lemmas. In Sect. 3, we state and prove the main existence result, and in Sect. 4, we provide an example to demonstrate our results.

2 Some technical results

We recall some notations, definitions, lemmas, and theorems.

Definition 2.1

Let Y and Z be two Banach spaces with \(\Vert \cdot \Vert _{Y}\) and \(\Vert \cdot \Vert _{Z}\) respectively. The operator \(L:Y\rightarrow Z\) is quasi-linear if

  1. (i)

    \(\operatorname{Im} L=L(Y\cap \operatorname{dom} L)\) is a closed subset of Z,

  2. (ii)

    \(\ker L =\{y \in Y \cap \operatorname{dom}M: Ly=0\}\) is linearly homeomorphic to \(\mathbb{R}^{n}\).

Let \(P: Y\rightarrow Y_{1}\) and \(Q: Z\rightarrow Z\) be projections such that \(\operatorname{Im}P=\ker L\), \(\ker Q= \operatorname{Im}L\). Let \(Y_{1}=\ker L\), \(Z_{2}=\operatorname{Im}L\) and \(Z_{1}\), \(Y_{2}\) be the complement spaces of \(Z_{2}\) in Z, \(Y_{1}\) in Y. Then

$$ Y=Y_{1}\oplus Y_{2},\qquad Z=Z_{1}\oplus Z_{2}. $$

Definition 2.2

Let Y be a Banach space with \(Y_{1} \subset Y\). The mapping \(Q:Y \to Y_{1}\) is a semi-projector if \(Q^{2}y=Qy\) and \(Q(\sigma y)= \sigma Qy\), \(y \in Y\), \(\sigma \in \mathbb{R}\).

Definition 2.3

Let \(L: Y\cap \operatorname{dom} L\rightarrow Z\) be a quasi-linear operator. Let \(Y_{1}=\operatorname{ker} L\) and \(W\subset Y\) be an open and bounded set with \(0\in W\). Then \(L_{\sigma}: \overline{W}\rightarrow Z\), \(\sigma \in [0,1]\) is said to be L-compact in if \(L_{\sigma}: \overline{W}\rightarrow Z\) is a continuous operator, and there exists an operator \(R: \overline{W}\times [0,1]\rightarrow Y_{2}\) which is continuous and compact such that, for \(\sigma \in [0,1]\),

$$\begin{aligned} (\mathrm{i})&\quad (I-Q)N_{\sigma}(\overline{W})\subset \operatorname{Im}L \subset (I-Q)Z, \end{aligned}$$
(2.1)
$$\begin{aligned} (\mathrm{ii})&\quad QN_{\sigma}y=0,\quad \sigma \in (0,1) \quad \text{iff} \quad QNy=0, \end{aligned}$$
(2.2)
$$\begin{aligned} (\mathrm{iii})&\quad R(\cdot , 0) \text{ is the zero operator}, \end{aligned}$$
(2.3)
$$\begin{aligned} (\mathrm{iv})&\quad R(\cdot , \sigma ) |_{\Omega _{\sigma}} = (I-P)|_{ \Omega _{\sigma}},\quad \text{where } \Omega _{\sigma} = \{ y\in \overline{W}: Ly = N_{\sigma}y \}, \end{aligned}$$
(2.4)
$$\begin{aligned} (\mathrm{v}) &\quad L\bigl[P + R(\cdot , \sigma )\bigr]=(I-Q)N_{\sigma}, \end{aligned}$$
(2.5)

where Q is a semi-projector.

Definition 2.4

([15])

Let \(\phi _{p}:\mathbb{R}\rightarrow \mathbb{R}\), then \(\phi _{p}\) satisfies the following conditions:

$$\begin{aligned} (\mathrm{i})&\quad \phi _{p}(u+v)\leq \bigl(\phi _{p}(u)+\phi _{p}(v)\bigr),\quad 1< p \leq 2, \end{aligned}$$
(2.6)
$$\begin{aligned} (\mathrm{ii})&\quad \phi _{p}(u+v)\leq 2^{p-2}\bigl(\phi _{p}(u)+\phi _{p}(v)\bigr),\quad p>2. \end{aligned}$$
(2.7)

In what follows, we shall need the following space:

$$\begin{aligned} \begin{aligned} Y={}&\bigl\{ y:[0, \infty )\rightarrow \mathbb{R}: y, \bigl( \phi _{p}\bigl(y^{(n-1)}\bigr)\bigr)\in AC[0, \infty ),\lim_{t \rightarrow \infty}e^{-t}\bigl|y^{(i)}(t)\bigr| \text{ exists}, \\ &0\leq i\leq n-1, \bigl(\phi _{p}\bigl(y^{(n-1)}\bigr) \bigr)'\in L^{1}[0, \infty )\bigr\} \end{aligned} \end{aligned}$$
(2.8)

with the norm

$$ \Vert y \Vert =\max_{0\leq i\leq n-1}\sup_{t \in (0, \infty )} \bigl\vert y^{(i)}(t) \bigr\vert e^{-t}. $$
(2.9)

Then Y is a Banach space.

Definition 2.5

([14])

\(h:[0, \infty ) \times \mathbb{R}^{n}\rightarrow \mathbb{R}\) is \(L^{1}[0, \infty )\) Caratheodory if it satisfies the following conditions:

  1. (i)

    For each \(y\in \mathbb{R}^{n}\), the mapping \(t\rightarrow h(t, y)\) is Lebesgue measurable,

  2. (ii)

    For a.e. \(t\in [0, \infty )\), the mapping \(y\rightarrow h(t, y)\) is continuous on \(\mathbb{R}^{n}\),

  3. (iii)

    For each \(r>0\), there exists \(\alpha _{r}\in L^{1}[0, \infty )\) such that for a.e. \(t\in [0, \infty )\) and every y such that \(\Vert y\Vert \leq r\) we have \(|h(t, y)|<\alpha _{r}\).

Theorem 2.1

([2])

Let X be the space of all continuous and bounded vector-valued functions on \([0, \infty )\) and \(X_{1}\subset X\). Then \(X_{1}\) is relatively compact if

  1. (i)

    \(X_{1}\) is bounded in X,

  2. (ii)

    all functions from \(X_{1}\) are equicontinuous on any compact subinterval of \([0, \infty )\),

  3. (iii)

    all functions from \(X_{1}\) are equiconvergent at infinity.

Let \(L: \operatorname{dom}L \subset Y\rightarrow Z\) where

$$\begin{aligned} \begin{aligned} \operatorname{dom}L ={}& \Biggl\{ y\in Y : \bigl(\phi _{p}\bigl(y^{(n-1)}\bigr)\bigr)' \in L^{1}[0, \infty ), y^{(n-2)}(\infty )= \sum _{i=1}^{m}\alpha _{i}y^{(n-2)}( \xi _{i}) \\ &y^{(n-3)}(0)+y^{(n-2)}(0)=\sum_{j=1}^{m} \beta _{j}y^{(n-3)}(\eta _{j}), y^{(n-1)}(\infty )=0, \\ &y^{(i)}(0)=0, i=0,1,2, \ldots , (n-4) \Biggr\} \end{aligned} \end{aligned}$$
(2.10)

and \(N_{\sigma} : Y\rightarrow Z\) is defined by \(N_{\sigma}y=\sigma h(t, y(t), \ldots , y^{(n-1)}(t))\). Thus (1.1)–(1.2) is of the form

$$ Lu =N _{\sigma}y \quad \text{when } \sigma =1. $$
(2.11)

Theorem 2.2

([6])

Let \(W \subset Y\) be an open and bounded set with \(0 \in W\). Let \(L:Y \cap \operatorname{dom}L \to Z\) be a quasi-linear operator and \(N_{\sigma}:\overline{W} \to Z\), \(\sigma \in [0,1]\) be L-compact. In addition, if the following hold:

  1. (i)

    \(Ly \neq N_{\sigma}y\), \(y \in \partial W \cap \operatorname{dom}L\), \(\sigma \in (0,1)\),

  2. (ii)

    \(\deg (JQN, W \cap \ker L,0)\neq 0\), where \(N=N_{1}\) and \(J:\operatorname{Im}Q \to \ker L\) is the homeomorphism with \(J(0)=0\),

then the abstract equation \(Ly=Ny\) has at least one solution in \(\operatorname{dom}L \cap \overline{W}\).

In what follows we assume the following conditions:

$$\begin{aligned} (A_{1})&\quad \sum_{i=1}^{m} \alpha _{i}=\sum_{j=1}^{m} \beta _{j}=1, \qquad \sum_{j=1}^{m} \beta _{j}\eta _{j}=1, \end{aligned}$$
(2.12)
$$\begin{aligned} (A_{2})&\quad \Delta = \begin{vmatrix} Q_{1}t^{n-3}e^{-t} & Q_{2}t^{n-3}e^{-t} \\ Q_{1}t^{n-2}e^{-t} & Q_{2}t^{n-2}e^{-t} \end{vmatrix}= \begin{vmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{vmatrix}=c_{11}c_{22}-c_{12}c_{21} \neq 0, \end{aligned}$$
(2.13)

where

$$\begin{aligned} & Q_{1}z=\sum_{j=1}^{m} \beta _{j} \int _{0}^{\eta _{j}} \int _{0}^{s} \phi _{q} \biggl( \int _{v}^{\infty}z(\tau )\,d\tau \biggr)\,dv\, ds, \end{aligned}$$
(2.14)
$$\begin{aligned} & Q_{2}z = \sum_{i=1}^{m} \alpha _{i} \int _{\xi _{i}}^{ \infty}\phi _{q} \biggl( \int _{s}^{\infty} z(\tau )\,d\tau \biggr)\,ds . \end{aligned}$$
(2.15)

Lemma 2.1

Suppose that (\(A_{1}\)) and (\(A_{2}\)) hold. Then

  1. (i)

    \(\ker L = \{ y \in \operatorname{dom}L:y(t)=at^{n-3}+bt^{n-2}, a, b \in \mathbb{R}, t \in [0,\infty )\}\);

  2. (ii)

    \(\operatorname{Im}L = \{ z \in Z: Q_{1}z=Q_{2}z=0 \}\).

Proof

Obviously, (i) holds. Hence kerL is homeomorphic to \(\mathbb{R}^{2}\). Thus \(\dim \ker L=2\). To prove (ii), let \(z \in \operatorname{Im}L\) and consider the equation

$$ \bigl(\phi _{p}\bigl(y^{(n-1)}(t)\bigr) \bigr)'=z(t) $$
(2.16)

with boundary conditions (1.2). Then

$$\begin{aligned}& y^{(n-3)}(t)=- \int _{0}^{t} \int _{0}^{s} \phi _{q} \biggl( \int _{v}^{ \infty}z(\tau )\,d\tau \biggr)\,dv\, ds + y^{(n-2)}(0)t+y^{(n-3)}(0), \\& y^{(n-2)}(t)=- \int _{0}^{t} \phi _{q} \biggl( \int _{s}^{\infty} z( \tau )\,d\tau \biggr)\,ds + y^{(n-2)}(0). \end{aligned}$$

Hence from the boundary conditions we derive

$$\begin{aligned} y^{(n-3)}(0)+ y^{(n-2)}(0)&= -\sum_{j=1}^{m} \beta _{j} \int _{0}^{ \eta _{j}} \int _{0}^{s}\phi _{q} \biggl( \int _{v}^{\infty}z(\tau )\,d\tau \biggr)\,dv\, ds \\ &\quad {} + \sum _{j=1}^{m} \beta _{j} \eta _{j} y^{(n-2)}(0)+ \sum_{j=1}^{m} \beta _{j} y^{(n-3)}(0). \end{aligned}$$

Since \(\sum_{j=1}^{m}\beta _{j}=\sum_{j=1}^{m}\beta _{j}\eta _{j}=1\), we obtain

$$ \sum_{j=1}^{m} \beta _{j} \int _{0}^{\eta _{j}} \int _{0}^{s} \phi _{q} \biggl( \int _{v}^{\infty}z(\tau )\,d\tau \biggr)\,dv\, ds =Q_{1}z =0. $$

Similarly,

$$\begin{aligned} y^{(n-2)}(\infty )&=- \int _{0}^{\infty} \phi _{q} \biggl( \int _{s} ^{ \infty} z(\tau )\,d\tau \biggr) \,ds + y^{(n-2)}(0) \\ &=-\sum_{i=1}^{m}\alpha _{i} \int _{0}^{\xi _{i}} \phi _{q} \biggl( \int _{s}^{\infty}z(\tau )\,d\tau \biggr)\,ds + \sum _{i=1}^{m} \alpha _{i} y^{(n-2)}(0), \end{aligned}$$

which implies

$$ \sum_{i=1}^{m}\alpha _{i} \int _{\xi _{i}}^{\infty} \phi _{q} \biggl( \int _{s}^{\infty}z(\tau )\,d\tau \biggr)\,ds =Q_{2}z=0.$$

Thus L is a quasi-linear operator.

On the other hand, if \(z \in Z\) satisfies \(Q_{1}z=Q_{2}z=0\), we take

$$ y(t)=at^{n-3}+ bt^{n-2}-\frac{1}{(n-2)!} \int _{0}^{t} (t-s)^{n-2} \phi _{q} \biggl( \int _{s}^{\infty}z(\tau )\,d\tau \biggr)\,ds , $$
(2.17)

where a, b are arbitrary constants. Then, for \(y \in Y\), \((\phi _{p}(y^{(n-1)}(t)))'=z(t)\) satisfies (1.2). Thus \(y \in \operatorname{dom}L\), that is, \(z \in \operatorname{Im}L\). □

We define the projector \(P:Y \to \ker L\) by

$$ Py(t)=\frac{y^{(n-3)}(0)t^{n-3}}{(n-3)!} + \frac{y^{(n-2)}(0)}{(n-2)!}t^{n-2}, $$
(2.18)

and the operator \(T_{1}, T_{2} : Z \to Z_{1}\) by

$$\begin{aligned}& T_{1}z=\frac{e^{-t}}{\Delta}[c_{22}Q_{1}z - c_{21}Q_{2}z], \end{aligned}$$
(2.19)
$$\begin{aligned}& T_{2}z=\frac{e^{-t}}{\Delta}[-c_{12}Q_{1}z + c_{11}Q_{2}z]. \end{aligned}$$
(2.20)

Define the operator \(Q:Z \to Z\) by

$$ Qz=T_{1}z(t)t^{n-3} + T_{2}z(t)t^{n-2}. $$

Then we can calculate and obtain \(T_{1}((T_{1}z)t^{n-3})=T_{1}z\), \(T_{1}((T_{2}z)t^{n-2})=0\), \(T_{2}((T_{1}z)t^{n-3})=0\), \(T_{2}((T_{2}z)t^{n-2})=T_{2}z\). Hence, \(Q^{2}z=Qz\) and \(Q(\sigma z) = \sigma Qz\). Thus Q is a semi-projector.

Lemma 2.2

If h is an \(L^{1}[0,\infty )\) Caratheodory’s function, then \(N_{\sigma} : \overline{W} \to Z\) is L-compact in for \(W \subset Y\) an open and bounded subset with \(0 \in W\).

Proof

To prove (2.1) we have

$$ Q(I-Q)N_{\sigma}(\overline{W})=QN_{\sigma}(\overline{W})-Q^{2}N_{ \sigma}( \overline{W})=QN_{\sigma}(\overline{W})-QN_{\sigma}( \overline{W})=0. $$

Thus, \((I-Q)N_{\sigma}(\overline{W})\subset \operatorname{Im}L\). Also, for \(z \in \operatorname{Im}L\), we have \(Qz=0\). Hence \(z \in \ker Q\) i.e. \(z \in (I-Q)z\). Hence, \(\operatorname{Im}L \subset (I-Q)z\). Therefore,

$$ (I-Q)N_{\sigma}(\overline{W}) \subset \operatorname{Im}L \subset (I-Q)Z. $$

To prove (2.2), suppose \(QN_{\sigma}y=0\) for \(\sigma \in (0,1)\). Then

$$ 0=QN_{\sigma}y = Q\bigl(\sigma h\bigl(t,y(t), \dots , y^{(n-1)}(t)\bigr)\bigr)=\sigma Qh\bigl(t,y(t), \dots , y^{(n-1)}(t)\bigr)=\sigma QNy. $$

Thus, \(QNy=0\). On the other hand, if \(QNy=0\), we have

$$\begin{aligned} 0&=QNy=T_{1}(QN_{\sigma}y)t^{n-3}-T_{2}(QN_{\sigma}y)t^{n-2} \\ &=\frac{e^{-t}}{\Delta} \bigl[c_{22}Q_{1}(QN_{\sigma}y)t^{n-3} - c_{21}Q_{2}(QN_{ \sigma}y)t^{n-3} \\ &\quad {} - c_{12}Q_{1}(QN_{\sigma}y)t^{n-2} + c_{11}Q_{2}(QN_{\sigma}y)t^{n-2} \bigr] \\ &=\frac{1}{\Delta}\bigl[(c_{11}c_{22}-c_{21}c_{12})+ (-c_{21}c_{12} + c_{11}c_{22}) \bigr](QN_{ \sigma}y) \\ &=2QN_{\sigma}y. \end{aligned}$$

Accordingly, \(QN_{\sigma}y=0\). To establish (2.3), (2.4), and (2.5) we define

$$ R(y,\sigma ) (t)=-\frac{1}{(n-2)!} \int _{0}^{t} (t-s)^{n-2}\phi _{q} \biggl( \int _{s}^{\infty}(I-Q)N_{\sigma}y(\tau )\,d\tau \biggr)\,ds . $$
(2.21)

Clearly, \(R(y,0)=0\). For \(y \in \Omega _{\sigma}=\{y \in \overline{W}:Ly=N_{\sigma}y\}\),

$$ \bigl(\phi _{p}\bigl(y^{(n-1)}(t)\bigr) \bigr)'=\sigma h\bigl(t,y(t),y'(t),\dots , y^{(n-1)}(t)\bigr) \in \operatorname{Im}L \subset \ker Q. $$

Hence

$$ \begin{aligned} R(y,\sigma ) (t)&=-\frac{1}{(n-2)!} \int _{0}^{t} (t-s)^{n-2}\phi _{q} \biggl( \int _{s}^{\infty}(I-Q)N_{\sigma}y(\tau )\,d\tau \biggr)\,ds \\ &= \int _{0}^{t} (t-s)^{n-2}y^{(n-1)}(s)\,ds \\ &=y(t)-\frac{y^{(n-2)}(0)t^{n-2}}{(n-2)!}- \frac{y^{(n-3)}(0)t^{n-3}}{(n-3)!} \\ &=(I-P)y(t). \end{aligned} $$
(2.22)

Similarly,

$$\begin{aligned} L\bigl[P+R(y,\sigma )\bigr](t)&= \biggl\lbrace \phi _{p} \biggl[ \frac{y^{(n-3)}(0)t^{n-3}}{(n-3)!}+\frac{y^{(n-2)}(0)t^{n-2}}{(n-2)!} \\ &\quad {} -\frac{1}{(n-2)!} \int _{0}^{t} (t-s)^{n-2}\phi _{q} \biggl( \int _{s}^{\infty}(I-Q)N_{\sigma}y(\tau )\,d\tau \biggr)\,ds ) \biggr]^{(n-1)} \biggr\rbrace ' \\ &= \biggl\lbrace -\phi _{p} \biggl[\phi _{q} \biggl( \int _{t}^{\infty}(I-Q)N_{ \sigma}y(\tau )\,d\tau \biggr) \biggr] \biggr\rbrace ' \\ &=(I-Q)N_{\sigma}y(t). \end{aligned}$$
(2.23)

This verifies (2.3) and (2.4). Next we show that R is relatively compact for \(\sigma \in [0,1]\).

Let \(W\subset Y\) be a bounded set, that is, there exists \(r>0\) such that \(r=\sup \{ \Vert y \Vert : y \in W\}\). Since L is \(L^{1}[0,\infty )\) Caratheodory, there exists \(\alpha _{r} \in L^{1}[0,\infty )\) such that for \(y\in W\) and a.e. \(t \in [0,\infty )\)

$$ \bigl\vert h\bigl(t,y(t),y'(t),\dots ,y^{(n-1)}(t) \bigr) \bigr\vert \leq \alpha (t). $$
(2.24)

Therefore, for \(y \in W\),

$$ \int _{0}^{\infty} \bigl\vert N_{\sigma}y( \tau ) \bigr\vert \,d\tau + \int _{0}^{\infty} \bigl\vert QN_{ \sigma}y( \tau ) \bigr\vert \,d\tau \leq \Vert \alpha _{r} \Vert _{1} + \Vert QN_{ \sigma} \Vert _{1}, $$
(2.25)

where \(\Vert z \Vert _{1}= \int _{0}^{\infty}|z(t)|\,dt \), \(z \in Z\).

For \(y \in W\) and setting

$$ E_{n}=\max_{0\leq i \leq n-2} \Bigl(\sup_{t \in [0,\infty )}e^{-t}t^{n-2-i} \Bigr), $$
(2.26)

we have for \(0\leq i \leq n-2\)

$$\begin{aligned} e^{-t} \bigl\vert R^{(i)}(y,\sigma ) (t) \bigr\vert &=e^{-t} \biggl\vert -\frac{1}{(n-2-i)!} \int _{0}^{t} (t-s)^{n-2-i}\phi _{q} \biggl( \int _{s}^{\infty}(I-Q)N_{\sigma}y(\tau )\,d\tau \biggr)\,ds \biggr\vert \\ &\leq \max_{0\leq i \leq n-2} \Bigl(\sup_{t \in [0,\infty )}e^{-t}t^{n-2-i} \Bigr)\phi _{q} \bigl( \Vert \alpha _{r} \Vert _{1} + \Vert QN_{ \sigma} \Vert _{1} \bigr) \\ &= E_{n} \phi _{q} \bigl( \Vert \alpha _{r} \Vert _{1} + \Vert QN_{ \sigma} \Vert _{1} \bigr). \end{aligned}$$
(2.27)

For \(i=n-1\),

$$ \begin{aligned} e^{-t} \bigl\vert R^{(n-1)}(y,\sigma ) (t) \bigr\vert &=e^{-t} \biggl\vert \phi _{q} \biggl( \int _{s}^{\infty}(I-Q)N_{\sigma}y(\tau )\,d\tau \biggr) \biggr\vert \\ &\leq \phi _{q} \bigl( \Vert \alpha _{r} \Vert _{1} + \Vert QN_{ \sigma} \Vert _{1} \bigr). \end{aligned} $$
(2.28)

Therefore from (2.27) and (2.28) we obtain

$$ \bigl\Vert R(y,\sigma ) \bigr\Vert \leq \max (E_{n},1)\phi _{q} \bigl( \Vert \alpha _{r} \Vert _{1} + \Vert QN_{\sigma} \Vert _{1} \bigr)=C. $$
(2.29)

Thus \(R(y,\sigma )\) is uniformly bounded in Y. For \(t_{1}, t_{2} \in [0,D]\), \(D\in (0,\infty )\) with \(t_{1} < t_{2}\), \(y\in W\) and \(0\leq i \leq n-2\), we have

$$ \begin{aligned} \bigl\vert e^{-t_{2}}R^{(i)}(y,\sigma ) (t_{2})-e^{-t_{1}}R^{(i)}(y, \sigma ) (t_{1}) \bigr\vert &= \biggl\vert \int _{t_{1}}^{t_{2}} \bigl[e^{-\tau}R^{(i)}(y, \sigma ) (\tau )\bigr]'\,d\tau \biggr\vert \\ &= \biggl\vert \int _{t_{1}} ^{t_{2}}\bigl[-e^{-\tau}R^{(i)}(y, \sigma ) (\tau )+e^{- \tau}R^{(i+1)}(y,\sigma ) (\tau )\bigr]\,d\tau \biggr\vert \\ &\leq 2(t_{2}-t_{1}) \bigl\Vert R(y,\sigma ) \bigr\Vert \to 0 \quad \text{as } t_{1} \to t_{2}. \end{aligned} $$

For \(i=n-1\),

$$\begin{aligned} &\bigl\vert e^{-t_{2}}\phi _{p}(R^{(n-1)}(y,\sigma ) (t_{2})-e^{-t_{1}}\phi _{p}(R^{(n-1)}(y, \sigma ) (t_{1}) \bigr\vert \\ &\quad = \biggl\vert e^{-t_{2}} \int _{t_{2}}^{\infty}(I-Q)N_{\sigma}y(\tau )\,d\tau - e^{-t_{1}} \int _{t_{1}}^{\infty}(I-Q)N_{\sigma}y(\tau )\,d\tau \biggr\vert \\ &\quad \leq \bigl\vert e^{-t_{2}}-e^{-t_{1}} \bigr\vert \int _{t_{2}}^{\infty} \bigl\vert (I-Q)N_{\sigma}y( \tau ) \bigr\vert \,d\tau + e^{-t_{1}} \int _{t_{2}}^{t_{1}} \bigl\vert (I-Q)N_{\sigma}y( \tau ) \bigr\vert \,d\tau \\ &\quad \leq \bigl\vert e^{-t_{2}}-e^{-t_{1}} \bigr\vert \bigl[ \Vert \alpha _{r} \Vert _{1} + \Vert QN_{ \sigma} \Vert _{1}\bigr] + e^{-t_{1}} \int _{t_{2}}^{t_{1}}\bigl[ \vert \alpha _{r} \vert + \vert QN_{\sigma} \vert \bigr]\,d\tau \\ &\quad \to 0 \quad \text{as } t_{1} \to t_{2}. \end{aligned}$$

Thus

$$ \bigl\vert e^{-t_{2}}R^{(n-1)}(y,\sigma ) (t_{2})-e^{-t_{1}}R^{(n-1)}(y, \sigma ) (t_{1}) \bigr\vert \to 0 \quad \text{as } t_{1} \to t_{2}. $$

We therefore conclude that \(R(y,\sigma )\) is equicontinuous on every compact subset of \([0,\infty )\). We next show that \(R(y,\sigma )(W)\) is equiconvergent a infinity.

For \(y \in W\) and \(0 \leq i \leq n-2\), we have

$$\begin{aligned} e^{-t} \bigl\vert R^{(i)}(y,\sigma ) (t) \bigr\vert &=e^{-t} \biggl\vert \frac{1}{(n-2-i)!} \int _{0}^{t}(t-s)^{n-2-i} \phi _{q} \biggl( \int _{s}^{\infty}(I-Q)N_{\sigma}y(\tau )\,d\tau \biggr)\,ds \biggr\vert \\ &\leq e^{-t}t^{n-2-i}\phi _{q}\bigl( \Vert \alpha _{r} \Vert _{1} + \Vert QN_{ \sigma} \Vert _{1}\bigr) \to 0 \quad \text{as } t \to \infty . \end{aligned}$$

For \(i=n-1\),

$$\begin{aligned} e^{-t} \bigl\vert R^{(n-1)}(y,\sigma ) (t) \bigr\vert &=e^{-t} \biggl\vert \phi _{q} \biggl( \int _{t}^{ \infty}(I-Q)N_{\sigma}y(\tau ) \,d\tau \biggr) \biggr\vert \\ &\leq \phi _{q} \biggl( \int _{t}^{\infty}\bigl( \bigl\vert \alpha _{r}(\tau ) \bigr\vert + \bigl\vert QN_{ \sigma}y(\tau ) \bigr\vert \bigr)\,d\tau \biggr) \\ & \to 0\quad \text{as } t \to \infty . \end{aligned}$$

Therefore \(R(y,\sigma )(W)\) is equiconvergent at infinity. Thus all the conditions of Theorem 2.1 are satisfied. The continuity of \(R(y,\sigma )\) follows from the Lebesque convergence theorem. Hence, \(N_{\sigma}\) is compact in . □

3 Main result

We assume the following conditions:

(\(H_{1}\)):

\(\sum_{i=1}^{m} \alpha _{i}=\sum_{j=1}^{m}\beta _{j}= \sum_{j=1}^{m}\beta _{j} \eta _{j}=1\).

(\(H_{2}\)):

There exist functions \(a_{i},r \in L^{1}[0,\infty )\) such that for a.e. \(t \in [0,\infty )\)

$$ \bigl\vert h(t,y_{1},y_{2},\dots , y_{n}) \bigr\vert \leq \phi _{p}\bigl(e^{-t} \bigr)\Biggl[\sum_{i=1}^{n}a_{i}(t) \bigl\vert y_{i}(t) \bigr\vert ^{p-1}\Biggr]+r(t). $$
(3.1)
(\(H_{3}\)):

For \(y \in \operatorname{dom}L\), there exist constants \(D>0\), \(B_{n}>0\) such that if \(|y^{(n-3)}(t)|>B_{n}\) for \(t \in [0,D]\) or \(|y^{(n-2)}(t)|>B_{n}\) for every \(t \in [0,\infty )\), then either

$$ Q_{1}Ny(t) \neq 0\quad \text{or}\quad Q_{2}Ny(t) \neq 0. $$
(\(H_{4}\)):

There exists a constant \(D_{n}>0\) such that for \(|y^{(n-3)}(0)|>D_{n}\) or \(y^{(n-2)}(0)>D_{n}\) either

$$ Q_{1}N\bigl(at^{n-3}+bt^{n-2}\bigr) + Q_{2}\bigl(at^{n-3} + bt^{n-2}\bigr)< 0, \quad t \in (0,\infty ) $$

or

$$ Q_{1}N\bigl(at^{n-3}+bt^{n-2}\bigr) + Q_{2}\bigl(at^{n-3} + bt^{n-2}\bigr)>0, \quad t \in (0,\infty ).$$

Theorem 3.1

If conditions (\(H_{1}\))(\(H_{4}\)) are fulfilled, then boundary value problem (1.1)(1.2) has at least one solution provided

$$ 2^{q-2} \Biggl(\sum_{i=1}^{n} \Vert a_{i} \Vert _{i} \Biggr)^{q-1}E_{n}(1+D)< 1\quad \textit{if } 1< p\leq 2 $$
(3.2)

or

$$ \Biggl(\sum_{i=1}^{n} \Vert a_{i} \Vert _{1} \Biggr)^{q-1}E_{n}(1+D) < 1\quad \textit{if } p>2. $$
(3.3)

Proof

We construct an open bounded set \(W\subset Y\) that satisfies the assumptions of Theorem 2.1. Let \(U_{1}=\{y \in \operatorname{dom}L:Ly=N_{\sigma}y, \sigma \in (0,1) \}\). For \(y \in U_{1}\), then \(QN_{\sigma}y=0\). Therefore from (\(H_{3}\)) there exist \(t_{1} \in [0,D]\), \(t_{2} \in [0,\infty )\) such that \(y^{(n-3)}(t_{1}) < B_{n}\), \(y^{(n-3)}(t_{2}) < B_{n}\),

$$ \bigl\vert y^{(n-2)}(t) \bigr\vert = \biggl\vert y^{(n-2)}(t_{2})- \int _{t} ^{t_{2}}y^{(n-1)}(s)\,ds \biggr\vert \leq B_{n} + \bigl\Vert y^{(n-1)} \bigr\Vert _{1}. $$
(3.4)

Hence

$$\begin{aligned}& \bigl\Vert y^{(n-2)} \bigr\Vert _{\infty} \leq B_{n} + \bigl\Vert y^{(n-1)} \bigr\Vert _{1}, \end{aligned}$$
(3.5)
$$\begin{aligned}& \bigl\vert y^{(n-3)}(0) \bigr\vert = \biggl\vert y^{(n-3)}(t_{1}) - \int _{0}^{t_{1}}y^{(n-2)}(s)\,ds \biggr\vert \leq B_{n} + \bigl\Vert y^{(n-2)} \bigr\Vert _{\infty}D. \end{aligned}$$
(3.6)

From (3.4) we obtain

$$ \bigl\vert y^{(n-2)}(0) \bigr\vert \leq B_{n} + \bigl\Vert y^{(n-1)} \bigr\Vert _{1}. $$
(3.7)

From \(y \in U_{1}\), \((I-P)y \in \operatorname{dom}L \cap \ker P\). Hence, from (2.22) and (2.29), we derive

$$ \bigl\Vert (I-P) \bigr\Vert = \bigl\Vert R(y,\sigma ) \bigr\Vert \leq C. $$
(3.8)

From the definition of P in (2.18) we obtain

$$\begin{aligned}& (Py)^{(i)}(t)=\frac{y^{(n-3)}(0)t^{n-3-i}}{(n-3-i)!}\ 0\leq i \leq n-3 + \frac{y^{(n-2)}(0)t^{n-2-i}}{(n-2-i)!}\ 0\leq i \leq n-2, \\& \begin{aligned} \Vert Py \Vert &\leq \max \Bigl[ \max_{0\leq i \leq n-3} \Bigl( \bigl\vert y^{(n-3)}(0) \bigr\vert \sup_{t \in [0,\infty )}e^{-t}t^{n-3-i} + \bigl\vert y^{(n-2)}(0) \bigr\vert \sup_{t \in [0,\infty )}e^{-t}t^{n-2-i} \Bigr), \\ &\quad {} \sup_{t \in [0,\infty )}e^{-t} \bigl\vert y^{(n-2)}(0) \bigr\vert \Bigr] \\ &\leq A_{n}\bigl[ \bigl\vert y^{(n-3)}(0) \bigr\vert + \bigl\vert y^{(n-2)}(0) \bigr\vert \bigr], \end{aligned} \end{aligned}$$
(3.9)

where

$$ A_{n}=\max \Bigl[\max_{0\leq i \leq n-3} \Bigl(\sup _{t \in [0, \infty )}e^{-t}t^{n-3-i}+\sup _{t \in [0,\infty )}e^{-t}t^{n-2-i} \Bigr),1 \Bigr]. $$
(3.10)

Hence, from (3.6) and (3.7), we get

$$\begin{aligned}& \begin{aligned} \Vert Py \Vert &\leq A_{n}\bigl(B_{n} + \bigl\Vert y^{(n-1)} \bigr\Vert _{1} + B_{n} + \bigl\Vert y^{(n-2)} \bigr\Vert _{\infty}D\bigr) \\ & \leq A_{n} \bigl[B_{n} + \bigl\Vert y^{(n-1)} \bigr\Vert _{1} + B_{n} +D \bigl(B_{n} + \bigl\Vert y^{(n-1)} \bigr\Vert _{1}\bigr)\bigr] \\ &=2B_{n}A_{n} +A_{n}B_{n}D +A_{n} \bigl\Vert y^{(n-1)} \bigr\Vert _{1}+ A_{n}D \bigl\Vert y^{(n-1)} \bigr\Vert _{1} \\ &=B_{n}A_{n}(2+D) + \bigl\Vert y^{(n-1)} \bigr\Vert _{1}(A_{n}+A_{n}D), \end{aligned} \end{aligned}$$
(3.11)
$$\begin{aligned}& \begin{aligned} \Vert y \Vert &= \bigl\Vert Py +(I-P)y \bigr\Vert \leq \Vert Py \Vert + \bigl\Vert (I-P)y \bigr\Vert \\ &\leq B_{n}A_{n}(2+D) + \bigl\Vert y^{(n-1)} \bigr\Vert _{1} (A_{n}+A_{n}D) +C. \end{aligned} \end{aligned}$$
(3.12)

If \(p \leq 2\), then from (2.6), (2.17), and (3.1), we obtain

$$ \begin{aligned} \bigl\Vert y^{(n-1)} \bigr\Vert _{1}&= \int _{0}^{\infty} \biggl\vert \phi _{q} \biggl( \int _{t}^{\infty}N_{\sigma}y(\tau )\,d\tau \biggr) \biggr\vert \,dt \\ &\leq \phi _{q} \Biggl( \sum_{i=1}^{n} \Vert a_{i} \Vert _{1} \Vert y \Vert ^{p-1} + \Vert r \Vert _{1} \Biggr) \\ &\leq 2^{q-2} \Biggl[ \Biggl(\sum_{i=1}^{n} \Vert a_{i} \Vert _{1} \Biggr)^{q-1} \Vert y \Vert + \Vert r \Vert _{1}^{q-1} \Biggr]. \end{aligned} $$
(3.13)

Using (3.2) in (3.13), we derive

$$ \bigl\Vert y^{(n-1)} \bigr\Vert _{1}\leq 2^{q-2} \Biggl\lbrace \Biggl(\sum_{i=1}^{n} \Vert a_{i} \Vert _{1} \Biggr)^{q-1} \bigl[B_{n}A_{n}(2+D)+ \bigl\Vert y^{(n-1)} \bigr\Vert _{1}(A_{n}+A_{n}D) \bigr]+C_{n}+ \Vert r \Vert _{1}^{q-1} \Biggr\rbrace $$

or

$$\begin{aligned}& \begin{aligned}[b] &\Biggl[1-2^{q-2} \Biggl(\sum_{i=1}^{n} \Vert a_{i} \Vert _{1} \Biggr)^{q-1}A_{n}(1+D) \Biggr] \bigl\Vert y^{(n-1)} \bigr\Vert _{1} \\ &\quad \leq 2^{q-2} \Biggl(\sum_{i=1}^{n} \Vert a_{i} \Vert _{1} \Biggr)^{q-1} \bigl[B_{n}A_{n}(2+D)+C_{n}\bigr] +2^{q-2} \Vert r \Vert _{1}^{q-1}, \end{aligned} \\& \bigl\Vert y^{(n-1)} \bigr\Vert _{1} \leq \frac{2^{q-2} (\sum_{i=1}^{n} \Vert a_{i} \Vert _{1} )^{q-1}[B_{n}A_{n}(2+D)+C]+2^{q-2} \Vert r \Vert _{1}^{q-1}}{1-2^{q-2} (\sum_{i=1}^{n} \Vert a_{i} \Vert _{1} )^{q-1}A_{n}(1+D)}. \end{aligned}$$
(3.14)

From (3.12) and (3.14), we obtain \(C_{n}^{*}>0\) such that \(\Vert y \Vert \leq C_{n}^{*}\). So \(U_{1}\) is bounded.

Let \(U_{2}=\{y \in \ker L:N_{\sigma}y \in \operatorname{Im}L \}\). For \(y \in U_{2} = \{y \in \ker L:y(t) =at^{n-3}+bt^{n-2}, a,b \in \mathbb{R}, t \in (0,\infty )\}\), \(Ny \in \operatorname{Im}L\) implies that \(QNy=0\), and hence

$$ Q_{1}N\bigl(at^{n-3}+bt^{n-2} \bigr)=Q_{2}N\bigl(at^{n-3} +bt^{n-2}\bigr)=0. $$

From (\(H_{4}\)) we get

$$ \vert a \vert + \vert b \vert < 2D_{n}. $$
(3.15)

Thus \(U_{2}\) is bounded. We choose \(W_{0}>0\) large enough such that

$$ W =\bigl\{ y \in W: \Vert y \Vert < W_{0}\bigr\} \supset \overline{U}_{1}\cup \overline{U}_{2}.$$

Then, from the above computations, \(Ly \neq Ny\) for \(y\in \partial W \cap \operatorname{dom}L\). Thus, the first part of Theorem 2.2 is verified. Let

$$ H(y,\lambda )=-\lambda Jy + (1-\lambda )QNy,\quad \lambda \in [0,1], $$
(3.16)

where \(J:\ker L \to \operatorname{Im}Q\) is the homeomorphism

$$ J\bigl(at^{n-3} +bt^{n-2}\bigr)=\frac{e^{-t}}{\Delta} \bigl[\bigl(c_{11} \vert a \vert + C_{12} \vert b \vert \bigr)t^{n-3}+\bigl(c_{21} \vert a \vert +c_{22} \vert b \vert \bigr)t^{n-2}\bigr]. $$
(3.17)

For \(y \in W \cap \ker L\), \(y(t)=at^{n-3}+bt^{n-2} \neq 0\) and \(H(y,0)=QNy\neq 0\) since \(Ny \notin \operatorname{Im}L\). Hence, for \(\lambda =0\), \(\lambda =1\), \(H(y,\lambda ) \neq 0\). Assume \(H(y,\lambda )=0\) for \(0<\lambda <1\), where \(y(t)=at^{n-3}+bt^{n-2} \in \partial W \cap \ker L\). Then from (3.16), (3.17) we obtain

$$\begin{aligned}& \lambda \bigl[c_{11} \vert a \vert +c_{12} \vert b \vert \bigr]=(1-\lambda )\bigl[c_{11}Q_{1}N \bigl(at^{n-3}+bt^{n-2}\bigr)+c_{12}Q_{2}N \bigl(at^{n-3}+bt^{n-2}\bigr)\bigr], \\& \lambda \bigl[c_{21} \vert a \vert +c_{22} \vert b \vert \bigr]=(1-\lambda )\bigl[c_{21}Q_{1}N \bigl(at^{n-3}+bt^{n-2}\bigr)+c_{22}Q_{2}N \bigl(at^{n-3}+bt^{n-2}\bigr)\bigr], \end{aligned}$$

or

$$\begin{aligned}& c_{11}\bigl[\lambda \vert a \vert -(1-\lambda )Q_{1}N\bigl(at^{n-3}+bt^{n-2}\bigr) \bigr]+c_{12}\bigl[ \lambda \vert b \vert -(1-\lambda )Q_{2}N\bigl(at^{n-3}+bt^{n-2}\bigr)\bigr]=0, \\& c_{21}\bigl[\lambda \vert a \vert -(1-\lambda )Q_{1}N\bigl(at^{n-3}+bt^{n-2}\bigr) \bigr]+c_{22}\bigl[ \lambda \vert b \vert -(1-\lambda )Q_{2}N\bigl(at^{n-3}+bt^{n-2}\bigr)\bigr]=0. \end{aligned}$$

Since Δ=| c 11 c 12 c 21 c 22 |= c 22 c 11 c 21 c 22 0, then

$$\begin{aligned}& \lambda \vert a \vert =(1-\lambda )Q_{1}N\bigl(at^{n-3}+bt^{n-2} \bigr), \\& \lambda \vert b \vert = (1-\lambda )Q_{2}N9at^{n-3} +bt^{n-2}). \end{aligned}$$

If \(|a|>D_{n}\), \(|b|>D_{n}\), then from (\(H_{4}\)) we obtain

$$ \lambda \bigl( \vert a \vert + \vert b \vert \bigr)=(1-\lambda ) \bigl[Q_{1}N\bigl(at^{n-3}+bt^{n-2} \bigr)+Q_{2}N\bigl(at^{n-3}+bt^{n-2}\bigr) \bigr]< 0, $$

which is a contradiction. If the second part of (\(H_{4}\)) holds, let

$$ H(y,\lambda )=\lambda Jy +(1-\lambda )Qy,\quad \lambda \in [0,1]. $$

Then, using a similar argument as above, we obtain a contradiction. Hence, \(H(y,\lambda ) \neq 0\) for \(y \in \partial W \cap \ker L\), \(\lambda \in [0,1]\). Therefore, by the invariance of the degree under a homotopy, we obtain

$$\begin{aligned} \deg (QN|_{\ker L},W\cap \ker L, 0)&=\deg \bigl((H\cdot ,0),W\cap \ker L,0\bigr) \\ &=\deg \bigl(H(\cdot ,1),W\cap \ker L,0\bigr) \\ &=\deg (\pm J,W\cap \ker L,0) \\ &=\operatorname{sgn} \left \lbrace \pm \begin{vmatrix} \frac{c_{11}}{\Delta} & \frac{c_{21}}{\Delta} \\ \frac{c_{21}}{\Delta} & \frac{c_{22}}{\Delta} \end{vmatrix} \right \rbrace \\ &=\operatorname{sgn} \biggl(\frac{\pm 1}{\Delta} \biggr)=\pm 1 \neq 0. \end{aligned}$$

Thus from Theorem 2.2 we conclude that \(Ly=Ny\) has at least one solution in \(\operatorname{dom}L \cap W\), which in turn implies that (1.1)–(1.2) has at least one solution in Y. □

4 Example

Consider the third order boundary value problem

$$\begin{aligned}& \bigl(\phi _{p}\bigl(y''(t)\bigr) \bigr)'=h\bigl(t,y(t),y'(t),y''(t) \bigr), \quad t \in (0,\infty ), \end{aligned}$$
(4.1)
$$\begin{aligned}& y'(\infty )=\sum_{i=1}^{2} \alpha _{i} y'(\xi _{i}),\qquad y(0)+y'(0) = \sum_{j=1}^{2} \beta _{j} y(\xi _{j}),\qquad y''( \infty )=0 \end{aligned}$$
(4.2)

corresponding to problem (1.1)–(1.2), we have \(m=2\), \(n=3\), \(\beta _{1}=-1\), \(\beta _{2}=2\), \(\eta _{1}=1/2\), \(\eta _{2}=3/4\), \(\alpha _{1} = \alpha _{2}=1/2\), \(\xi _{1}=1\) \(\xi _{2}=2\), \(p=4/3\), \(q=4\). Then \(\sum_{j=1}^{2}\beta _{j} \eta _{j}= \sum_{i=1}^{2} \alpha _{i} = \sum_{j=1}^{2} \beta _{j}=1\). Hence condition (\(H_{1}\)) is satisfied.

$$\begin{aligned}& h\bigl(t,y,y',y'' \bigr)=e^{-t} \biggl[\frac{\sin ^{\frac{1}{3}}}{24}+ \frac{y^{\prime \,\frac{1}{3}}}{24}+ \frac{\sin ^{\frac{1}{3}}y'}{24}+ \frac{y^{\prime\prime \,\frac{1}{3}}}{48}+\frac{\sin ^{\frac{1}{3}}y''}{48}- \frac{1}{24} \biggr], \\& \bigl\vert h\bigl(t,y,y',y'' \bigr) \bigr\vert \leq e^{-\frac{t}{3}} \biggl[ \frac{e^{-\frac{2}{3}t} \vert y \vert ^{\frac{1}{3}}}{24}+ \frac{e^{-\frac{2}{3}t} \vert y' \vert ^{\frac{1}{3}}}{2}+ \frac{e^{-\frac{2}{3}t} \vert y'' \vert ^{\frac{1}{3}}}{24} \biggr]- \frac{e^{-t}}{24}. \end{aligned}$$

Thus condition (\(H_{2}\)) is verified. To verify conditions (\(H_{3}\)) and (\(H_{4}\)), we have

$$ \Delta = c_{11}c_{22}-c_{12}c_{22}=072(0.076)-0.018(0622)=0.497 \neq 0. $$

\(a_{1}(t)=\frac{e^{-\frac{2}{3}t}}{24}\), \(a_{2}(t)=\frac{e^{-\frac{2}{3}t}}{12}\), \(a_{3}(t)=\frac{e^{-\frac{2}{3}t}}{24}\), \(r(t)=-\frac{e^{-t}}{24}\). We set \(B_{n}=5^{3}\). Let \(|y'(t)|>B_{n}\), then \(y'(t) >B_{n}\) or \(y'(t) <-B_{n}\). If \(y'(t) >B_{n}\), then

$$\begin{aligned} Q_{2}Ny&=\frac{1}{2} \int _{\frac{1}{2}}^{\infty} \biggl( \int _{s}^{ \infty}e^{-t} \biggl[ \frac{\sin ^{\frac{1}{3}}}{24}+ \frac{y^{\prime \,\frac{1}{3}}}{24}+\frac{\sin ^{\frac{1}{3}}y'}{24}+ \frac{y^{\prime\prime \,\frac{1}{3}}}{48}+\frac{\sin ^{\frac{1}{3}}y''}{48}- \frac{1}{24} \biggr]\,dt \biggr)^{3}\,ds \\ &\quad {} +\frac{1}{2} \int _{\frac{3}{4}}^{\infty} \biggl( \int _{s}^{ \infty}e^{-t} \biggl[ \frac{\sin ^{\frac{1}{3}}}{24}+ \frac{y^{\prime \,\frac{1}{3}}}{24}+\frac{\sin ^{\frac{1}{3}}y'}{24}+ \frac{y^{\prime\prime \,\frac{1}{3}}}{48}+\frac{\sin ^{\frac{1}{3}}y''}{46}- \frac{1}{24} \biggr]\,dt \biggr)^{3}\,ds \\ &>\frac{1}{2} \int _{\frac{1}{2}}^{\infty} \biggl( \int _{s}^{\infty}e^{-t} \biggl[- \frac{1}{24}+\frac{B_{n}^{\frac{1}{3}}}{24}-\frac{1}{24}- \frac{1}{48}-\frac{1}{24} \biggr]\,dt \biggr)^{3}\,ds \\ &\quad {} + \frac{1}{2} \int _{\frac{3}{4}}^{\infty} \biggl( \int _{s}^{ \infty}e^{-t} \biggl[- \frac{1}{24}+\frac{B_{n}^{\frac{1}{3}}}{24}- \frac{1}{24}- \frac{1}{48}-\frac{1}{24} \biggr]\,dt \biggr)^{3}\,ds \\ &=\frac{1}{2} \biggl(\frac{2B_{n}^{\frac{1}{3}}-7}{48} \biggr)^{3} \int _{\frac{1}{2}}^{\infty} \biggl( \int _{5}^{\infty}e^{-t} \biggr)^{3}\,dt\, ds + \frac{1}{2} \biggl(\frac{2B_{n}^{\frac{1}{3}}-7}{48} \biggr)^{3} \int _{\frac{3}{4}}^{\infty} \biggl( \int _{5}^{\infty}e^{-t} \biggr)^{3}\,dt\, ds \\ &>0. \end{aligned}$$

If \(y'(t) < -B_{n}\), then

$$\begin{aligned} Q_{2}Ny &\leq \frac{1}{2} \int _{\frac{1}{2}}^{\infty} \biggl( \int _{s}^{ \infty}e^{-t} \biggl[ \frac{1}{24}-\frac{B_{n}^{\frac{1}{3}}}{24}+ \frac{1}{24}+ \frac{1}{48}-\frac{1}{24} \biggr]\,dt \biggr)^{3}\,ds \\ &\quad {} + \frac{1}{2} \int _{\frac{3}{4}}^{\infty} \biggl( \int _{s}^{ \infty}e^{-t} \biggl[ \frac{1}{24}-\frac{B_{n}^{\frac{1}{3}}}{24}+ \frac{1}{24}+ \frac{1}{48}-\frac{1}{24} \biggr]\,dt \biggr)^{3}\,ds \\ &=\frac{1}{2} \biggl( \frac{3-2B_{n}^{\frac{1}{3}}}{48} \biggr)^{3} \int _{\frac{1}{2}}^{\infty} \biggl( \int _{s}^{\infty}e^{-t} \biggr)^{3}\,dt\, ds < 0. \end{aligned}$$

Thus condition (\(H_{3}\)) is verified. Taking \(D_{n}=6^{3}\) then for \(|b|>D_{n}\), that is, \(b>D_{n}\) or \(b<-D_{n}\). If \(b>D_{n}\), then we can verify that

$$ Q_{1}(a+bt) +Q_{2}(a+bt) >0. $$

Similarly, if \(b <-D_{n}\), then

$$ Q_{1}(a+bt) +Q_{2}(a+bt) < 0, $$

which verifies (\(H_{4}\)). Finally, \(\Vert a _{1} \Vert _{1} =\frac{1}{16}\), \(\Vert a_{2} \Vert _{1} =\frac{1}{8}\), \(\Vert a_{3} \Vert _{1}=\frac{1}{16}\),

$$\begin{aligned} A_{n}&=\max \Bigl\{ \sup_{t \in [0,\infty )}e^{-t}+ \sup_{t \in [0, \infty )}te^{-t},1\Bigr\} \\ &=\max \bigl[1+e^{-1},1\bigr]=1+e^{-1}. \end{aligned}$$

Taking \(D=1\), we have for \(P \leq 2\)

$$ 2^{q-2} \Biggl( \sum_{i=1}^{3} \Vert a_{i} \Vert _{1} \Biggr)^{q-1}A_{n}(1+D)=2^{2} \biggl(\frac{1}{4} \biggr)^{3}2\bigl(1+e^{-1} \bigr)=\frac{1+e^{-1}}{8}< 1. $$

Hence, all the conditions of Theorem 3.1 are verified. Thus (4.1)–(4.2) has at least one solution.

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Iyase, S.A., Imaga, O.F. Higher-order p-Laplacian boundary value problems with resonance of dimension two on the half-line. Bound Value Probl 2022, 47 (2022). https://doi.org/10.1186/s13661-022-01629-7

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