# Existence of solution for a resonant p-Laplacian second-order m-point boundary value problem on the half-line with two dimensional kernel

## Abstract

The existence of a solution for a second-order p-Laplacian boundary value problem at resonance with two dimensional kernel will be considered in this paper. A semi-projector, the Ge and Ren extension of Mawhin’s coincidence degree theory, and algebraic processes will be used to establish existence results, while an example will be given to validate our result.

## Introduction

The following second-order p-Laplacian boundary value problem will be considered in this work:

$$\left \{ \textstyle\begin{array}{l} (\varphi_{p}(u'(t)))' + g(t, u(t),u'(t))=0,\quad t \in(0,+\infty), \\ \varphi_{p}(u'(0)) = \int_{0}^{+\infty } v(t)\varphi_{p}(u'(t))\,dt,\qquad \varphi_{p}(u'(+\infty ))= \sum_{j=1}^{m} \beta_{j} \int_{0} ^{\eta_{j}} \varphi_{p}(u'(t))\,dt, \end{array}\displaystyle \right .$$
(1.1)

where $$g:[0,+\infty) \times\mathbb{R}^{2} \to\mathbb{R}$$ is an $$L^{1}$$-Carathéodory function, $$0< \eta_{1}<\eta_{2} < \cdots\leq\eta _{m} < +\infty$$, $$\beta_{j} \in\mathbb{R}$$, $$j=1,2, \ldots, m$$, $$v \in L^{1}[0,+\infty )$$, $$v(t) >0$$ on $$[0,+\infty )$$, and

$$\varphi_{p} (s) = \vert s \vert ^{p-2}s,\quad p \geq2.$$

There are many real life applications of boundary value problems with integral and multi-point boundary conditions on an unbounded domain, for instance, in the study of physical phenomena such as the study of an unsteady flow of fluid through a semi-infinite porous medium and radially symmetric solutions of nonlinear elliptic equations. They also arise in plasma physics and in the study of drain flows; see .

Boundary value problems are said to be at resonance if the solution of the corresponding homogeneous boundary value problem is non-trivial. Many authors in the literature have considered resonant problems. López-Somoza and Minhós  obtained existence results for a resonant multi-point second-order boundary value problem on the half-line, Capitanelli, Fragapane and vivaldi  addressed regularity results for p-Laplacians in pre-fractal domains, while Jiang and Kosmatov  considered resonant p-Laplacian problems with functional boundary conditions. For other work on resonant problems without p-Laplacian operator, see , while for problems with the p-Laplacian operator, see . In , Jiang considered the following p-Laplacian operator:

$$\left \{ \textstyle\begin{array}{l} (\varphi_{p}(u'))' + f(t,u,u') =0, \quad0< t< +\infty ,\\ u(0) =0, \qquad\varphi_{p}(u(+\infty ))=\sum_{i=1}^{n} \alpha_{i} \varphi _{p}(u'(\xi_{i})), \end{array}\displaystyle \right .$$

where $$\alpha_{i} >0$$, $$i=1,2,\dots,n$$, $$\sum_{i=1}^{n} \alpha_{i}=1$$.

To the best of our knowledge p-Laplacian problems with two dimensional kernel on the half-line have not received much attention in the literature.

We will give the required lemmas, theorem and definitions in Sect. 2, Sect. 3 will be dedicated to stating and proving condition for existence of solutions, while an example will be given in Sect. 4 to validate the result obtained.

## Preliminaries

In this section, we will give some definitions and lemmas that will be used in this work.

### Definition 2.1

()

A map $$w:[0,+\infty) \times\mathbb{R}^{2} \to\mathbb{R}$$ is $$L^{1}[0,+\infty )$$-Carathéodory, if the following conditions are satisfied:

1. (i)

for each $$(d,e) \in\mathbb{R}^{2}$$, the mapping $$t \to w(t,d,e)$$ is Lebesgue measurable;

2. (ii)

for a.e. $$t\in[0,\infty)$$, the mapping $$(d,e) \to w(t,d,e)$$ is continuous on $$\mathbb{R}^{2}$$;

3. (iii)

for each $$k>0$$, there exists $$\varphi_{k}(t) \in L_{1}[0,+\infty)$$ such that, for a.e. $$t \in[0,\infty)$$ and every $$(d,e) \in[-k,k]$$, we have

$$\bigl\vert w(t,d,e) \bigr\vert \leq\varphi_{k}(t).$$

### Definition 2.2

()

Let $$(U, \Vert\cdot\Vert_{U})$$ and $$(Z, \Vert\cdot\Vert_{Z})$$ be two Banach spaces. The continuous operator $$M:U \cap \operatorname {dom}M \to Z$$, is quasi-linear if the following hold:

1. (i)

$$\operatorname {Im}M = M(U\cap \operatorname {dom}M)$$ is a closed subset of Z;

2. (ii)

$$\ker M = \{ u \in U \cap \operatorname {dom}M :Mu=0\}$$ is linearly homeomorphic to $$\mathbb{R}^{n}$$, $$n < +\infty$$.

### Definition 2.3

()

Let U be a Banach space and $$U_{1} \subset U$$ a subspace. Let $$P, Q:U \to U_{1}$$ be operators, then P is a projector if

1. (i)

$$P^{2} =P$$;

2. (ii)

$$P(\lambda_{1}u_{1} + \lambda_{2}u_{2})=\lambda_{1}Pu_{1} + \lambda _{2}Pu_{2}$$ where $$u_{1}, u_{2} \in U$$, $$\lambda_{1}, \lambda_{2} \in\mathbb{R}$$,

and Q is a semi-projector if

1. (i)

$$Q^{2} = Q$$;

2. (ii)

$$Q(\lambda u) = \lambda Qu$$ where $$u \in U$$, $$\lambda\in \mathbb{R}$$.

Let $$U_{1} = \ker M$$ and $$U_{2}$$ be the complement space of $$U_{1}$$ in U, then $$U=U_{1} \oplus U_{2}$$. Similarly, if $$Z_{1}$$ is a subspace of Z and $$Z_{2}$$ is the complement space of $$Z_{1}$$ in Z, then $$Z = Z_{1} \oplus Z_{2}$$. Let $$P: U \to U_{1}$$ be a projector, $$Q:Z \to Z_{1}$$ be a semi-projector and $$\varOmega\subset U$$ an open bounded set with $$\theta\in\varOmega$$ the origin. Also, let $$N_{1}$$ be denoted by N, let $$N_{\lambda}: \overline{\varOmega} \to Z$$, where $$\lambda\in [0,1]$$ is a continuous operator and $$\varSigma_{\lambda} =\{ u \in \overline{\varOmega}:Mu=N_{\lambda}u \}$$.

### Definition 2.4

()

Let U be the space of all continuous and bounded vector-valued functions on $$[0,+\infty )$$ and $$X \subset U$$. Then X is said to be relatively compact if the following statements hold:

1. (i)

X is bounded in U;

2. (ii)

all functions from X are equicontinuous on any compact subinterval of $$[0,+\infty )$$;

3. (iii)

all functions from X are equiconvergent at ∞, i.e. $$\forall \epsilon>0$$, ∃ a $$T = T(\epsilon)$$ such that $$\Vert A(t) - A(+\infty )\Vert_{R^{n}}<\epsilon$$$$\forall t >T$$ and $$A \in X$$.

### Definition 2.5

()

Let $$N_{\lambda}: \overline{\varOmega} \to Z$$, $$\lambda\in[0,1]$$ be a continuous operator. The operator $$N_{\lambda}$$ is said to be M-compact in Ω̅ if there exist a vector subspace $$Z_{1} \in Z$$ such that $$\dim Z_{1} = \dim U_{1}$$ and a compact and continuous operator $$R:\overline{\varOmega} \times[0,1] \to U_{2}$$ such that, for $$\lambda\in[0,1]$$, the following holds:

1. (i)

$$(I - Q)N_{\lambda}(\overline{\varOmega}) \subset \operatorname {Im}M \subset(I-B)Z$$,

2. (ii)

$$QN_{\lambda}u=0 \Leftrightarrow QNu=0$$, $$\lambda\in(0,1)$$,

3. (iii)

$$R(\cdot,u)$$ is the zero operator and $$R(\cdot, \lambda )|_{\varSigma_{\lambda}}=(I-P)|_{\varSigma_{\lambda}}$$,

4. (iv)

$$M[P+R(\cdot, \lambda)]=(I-Q)N_{\lambda}$$.

### Lemma 2.1

()

The following are properties of the function$$\varphi_{p} : \mathbb{R} \to\mathbb{R}$$:

1. (i)

It is continuous, monotonically increasing and invertible. Its inverse$$\varphi_{p} ^{-1} =\varphi_{q}$$, where$$q >1$$and satisfies$$\frac{1}{p}+\frac{1}{q}=1$$.

2. (ii)

For any$$x, y >0$$,

1. (a)

$$\varphi_{p} (x +y) \leq\varphi_{p} (x) + \varphi_{p}(y)$$, if$$1 < p <2$$,

2. (b)

$$\varphi_{p}(x+y) \leq2^{p-2}(\varphi_{p}(x) + \varphi _{p}(y))$$, if$$p \geq2$$.

### Theorem 2.1

()

Let$$(U, \Vert\cdot\Vert_{U})$$and$$(Z, \Vert\cdot\Vert_{Z})$$be two Banach spaces and$$\varOmega\subset U$$an open and bounded set. If the following holds:

($$A_{1}$$):

The operator$$M: U \cap \operatorname {dom}M \to Z$$is a quasi-linear,

($$A_{2}$$):

the operator$$N_{\lambda}:\overline{\varOmega} \to Z$$, $$\lambda\in[0,1]$$isM-compact,

($$A_{3}$$):

$$Mu \neq N _{\lambda}u$$, for$$\lambda\in(0,1)$$, $$u \in\partial\varOmega\cap \operatorname {dom}M$$,

($$A_{4}$$):

$$\deg\{JQN, \varOmega\cap\ker M,0 \} \neq0$$, where the operator$$J:Z_{1} \to U_{1}$$is a homeomorphism with$$J(\theta)=\theta$$and deg is the Brouwer degree,

then the equation$$Mu = Nu$$has at least one solution inΩ̅.

Let

\begin{aligned} U = \Bigl\{ u \in C^{2}[0,+\infty): u, \varphi_{p} \bigl(u'\bigr) \in \mathit{AC}[0,+\infty ), \lim_{t \to +\infty }e^{-t} \bigl\vert u^{(i)}(t) \bigr\vert \text{ exist, } i=0,1 \Bigr\} , \end{aligned}

with the norm $$\Vert u \Vert= \max\{\Vert u \Vert_{\infty}, \Vert u' \Vert_{\infty}\}$$ defined on U where $$\Vert u \Vert_{\infty} =\sup_{t \in[0,+\infty )}e^{-t}|u|$$. The space $$(U, \Vert\cdot\Vert)$$ by a standard argument is a Banach Space.

Let $$Z = L^{1}[0,+\infty )$$ with the norm $$\Vert w \Vert_{L^{1}} = \int_{0} ^{+\infty }|w(v)|\,dv$$. Define M as a continuous operator such that $$M:\operatorname {dom}M \subset U \to Z$$ where

\begin{aligned} \begin{aligned} \operatorname {dom}M &= \Biggl\{ u \in U: \bigl(\varphi_{p} \bigl(u' \bigr)\bigr)' \in L^{1}[0,+\infty ), \varphi _{p} \bigl(u'(0)\bigr)= \int_{0}^{+\infty }v(t)\varphi_{p} \bigl(u'(t)\bigr)\,dt, \\ &\quad \lim_{t \to +\infty } \bigl(\varphi_{p} \bigl(u'(t)\bigr)\bigr)= \sum_{j=1}^{m} \beta_{j} \int _{0} ^{\eta_{j}} \varphi_{p} \bigl(u'(t)\bigr)\,dt \Biggr\} \end{aligned} \end{aligned}

and $$Mu = (\varphi_{p}(u'(t)))'$$. We will define the operator $$N_{\lambda}u : \overline{\varOmega} \to Z$$ by

$$N_{\lambda}u = -\lambda g\bigl(t, u(t),u'(t)\bigr), \quad \lambda\in[0,1], t \in[0,+\infty ),$$

where $$\varOmega\subset U$$ is an open and bounded set. Then the boundary value problem (1.1) in abstract form is $$Mu=Nu$$.

Throughout the paper we will assume the hypotheses:

($$\phi_{1}$$):

$$\sum_{j=1}^{m} \beta_{j} \eta_{j} = \int_{0}^{+\infty }v(t)\, dt=1$$;

($$\phi_{2}$$):
$$C = \left| \textstyle\begin{array}{c@{\quad}c} Q_{1}e^{-t} & Q_{2}e^{-t} \\ Q_{1}te^{-t} & Q_{2}te^{-t} \end{array}\displaystyle \right| := \left| \textstyle\begin{array}{c@{\quad}c}c_{11} & c_{12} \\ c_{21} & c_{22} \end{array}\displaystyle \right| =c_{11}\cdot c_{22} - c_{12} \cdot c_{21} \neq0,$$

where

$$Q_{1}w =\int_{0}^{+\infty }v(t) \int_{0}^{t} w(s)\,ds\,dt,$$

and

$$Q_{2}w=\sum_{j=1}^{m} \beta_{j}\int_{0}^{\eta_{j}}\int_{t}^{+\infty }w(s)\,ds\,dt.$$

It is obvious that $$\ker M = \{u \in \operatorname {dom}M:u=a +bt: a, b \in\mathbb {R}, t \in[0,+\infty )\}$$ and $$\operatorname {Im}M = \{w:w \in Z, Q_{1}w = Q_{2}w=0\}$$.

Clearly, $$\ker M=2$$ is linearly homeomorphic to $$\mathbb{R}^{2}$$ and $$\operatorname {Im}M \subset Z$$ is closed, hence, the operator $$M:\operatorname {dom}M \subset U \to Z$$ is quasi-linear.

We next define the projector $$P:U \to U_{1}$$ as

$$Pu(t)=u(0) + u'(0)t, \quad u \in U,$$
(2.1)

and the operators $$\Delta_{1}, \Delta_{2} : Z \to Z_{1}$$ as

$$\Delta_{1}w=\frac{1}{C}(\delta_{11}Q_{1}w + \delta_{12}Q_{2}w)e^{-t},$$

and

$$\Delta_{2}w=\frac{1}{C}(\delta_{21}Q_{1}w + \delta _{22}Q_{2}w)e^{-t},$$

where $$\delta_{ij}$$ is the co-factor of $$c_{ij}$$, $$i,j=1,2$$. Then the operator $$Q: Z \to Z_{1}$$ will be defined as

$$Qw = (\Delta_{1}w) + (\Delta_{2}w) \cdot t$$
(2.2)

where $$Z_{1}$$ is the complement space of ImM in Z. Then the operator $$Q: Z \to Z_{1}$$ can easily be shown to be a semi-projector.

Let the operator $$R:U \times[0,1] \to U_{2}$$ be defined by

\begin{aligned} R(u,\lambda) (t)&= \int_{0}^{t} \varphi_{q} \biggl( \varphi_{p}\bigl(u'(0)\bigr) - \int _{0}^{\tau}\lambda\bigl(g \bigl(s,u(s),u'(s)\bigr) - QNu(s)\bigr)\,ds \biggr)\,d\tau- u'(0)t, \end{aligned}

where $$U_{2}$$ is the complement space of kerM in U.

### Lemma 2.2

Ifgis a$$L^{1}[0,+\infty )$$-Carathéodory function, then$$R:U \times[0,1] \to U_{2}$$isM-compact.

### Proof

Let the set $$\varOmega\subset U$$ be nonempty, open and bounded, then, for $$u \in\overline{\varOmega}$$, there exists a constant $$k >0$$ such that $$\Vert u \Vert< k$$. Since g is an $$L^{1}[0,+\infty )$$-Carathéodory function, there exists $$\psi_{k} \in L^{1}[0,+\infty )$$ such that, for a.e. $$t \in[0,+\infty )$$ and $$\lambda\in[0,1]$$, we have

\begin{aligned} \Vert N_{\lambda}u \Vert _{L^{1}}+ \Vert QN _{\lambda }u \Vert _{L^{1}}&= \int_{0}^{+\infty } \bigl\vert N_{\lambda}u(v) \bigr\vert \,dv + \int_{0}^{+\infty } \bigl\vert QN _{\lambda}u(v) \bigr\vert \,dv \\ & \leq \Vert \psi_{k} \Vert _{L^{1}}+ \Vert QNu \Vert _{L^{1}}. \end{aligned}

Now for any $$u \in\overline{\varOmega}$$, $$\lambda\in[0,1]$$, we have

\begin{aligned} \begin{aligned} [b]\bigl\Vert R(u,\lambda) \bigr\Vert _{\infty} &= \sup_{t \in[0,+\infty )}e^{-t} \bigl\vert R(u,\lambda) (t) \bigr\vert \leq\frac{1}{e} \varphi_{q} \bigl(\varphi_{p}(k) + \Vert Nu_{\lambda} \Vert _{L^{1}} + \Vert QN_{\lambda}u \Vert _{L^{1}}\bigr)+k \\ &\leq\varphi_{q} \bigl( \varphi_{p}(k)+ \Vert \psi_{k} \Vert _{L^{1}} + \Vert QNu \Vert _{L^{1}}\bigr) +k< +\infty \end{aligned} \end{aligned}
(2.3)

and

\begin{aligned} \begin{aligned}[b] \bigl\Vert R'(u, \lambda) \bigr\Vert _{\infty} &= \sup_{t \in[0,+\infty )}e^{-t} \bigl\vert R'(u,\lambda) (t) \bigr\vert \\ &\leq\varphi_{q} \bigl(\varphi_{p}(k)+ \Vert \psi_{k} \Vert _{L^{1}} + \Vert QNu \Vert _{L^{1}}\bigr)+k < +\infty . \end{aligned} \end{aligned}
(2.4)

Therefore it follows from (2.3) and (2.4) that $$R(u, \lambda)\overline{\varOmega}$$ is uniformly bounded.

Next we show that $$R(u, \lambda)\overline{\varOmega}$$ is equicontinuous in a compact set. Let $$u \in\overline{\varOmega}$$, $$\lambda\in[0,1]$$. For any $$T \in[0,+\infty )$$, with $$t_{1}, t_{2} \in [0,T]$$ where $$t_{1} < t_{2}$$, we have

\begin{aligned} &\bigl\vert e^{t_{2}}R(u, \lambda) (t_{2})-e^{t_{1}}R(u,\lambda) (t_{1}) \bigr\vert \\ &\quad= \biggl\vert e^{t_{2}} \int_{0}^{t_{2}} \varphi_{q} \biggl( \varphi _{p}\bigl(u'(0)\bigr)- \int_{0}^{\tau} \lambda\bigl(g \bigl(s,u(s),u'(s)\bigr)-QNu(s)\bigr)\,ds \biggr)\,d \tau-u'(0)t_{2}e^{-t_{2}} \\ &\qquad - e^{-t_{1}} \int_{0}^{-t_{1}} \varphi_{q} \biggl( \varphi_{p}\bigl(u'(0)\bigr)- \int _{0}^{\tau} \lambda\bigl(g \bigl(s,u(s),u'(s)\bigr) -QNu(s)\bigr)\,ds \biggr)\,d\tau+ u'(0)t_{1}e^{t_{1}} \biggr\vert \\ &\quad\leq \bigl\vert e^{t_{2}}-e^{-t_{1}} \bigr\vert \int_{0}^{t_{1}} \varphi_{q} \biggl( \varphi _{p}\bigl( \bigl\vert u'(0) \bigr\vert \bigr)+ \int_{0}^{\tau} \lambda \bigl\vert g \bigl(s,u(s),u'(s)\bigr)-QNu(s) \bigr\vert \,ds \biggr)\,d\tau \\ &\qquad + e^{-t_{2}} \int_{t_{1}}^{t_{2}} \varphi_{q} \biggl( \varphi _{p}\bigl( \bigl\vert u'(0) \bigr\vert \bigr)+ \int_{0}^{\tau} \lambda \bigl\vert g \bigl(s,u(s),u'(s)\bigr)-QNu(s) \bigr\vert \,ds \biggr)\,d\tau \\ &\qquad + \bigl\vert t_{1}e^{-t_{1}}-t_{2}e^{-t_{2}} \bigr\vert \bigl\vert u'(0) \bigr\vert \\ &\quad\leq\bigl(e^{t_{2}}-e^{-t_{1}}\bigr)\varphi_{q} \bigl( \varphi_{p}(k) + \Vert \psi_{k} \Vert _{L^{1}} + \Vert QNu \Vert _{L^{1}}\bigr)t_{1} \\ &\qquad+ e^{-t_{2}}\varphi_{q} \bigl( \varphi _{p}(k) + \Vert \psi_{k} \Vert _{L^{1}} + \Vert QNu \Vert _{L^{1}}\bigr) (t_{2} -t_{1}) + \bigl\vert t_{1}e^{-t_{1}}-t_{2}e^{-t_{2}} \bigr\vert r \\&\quad\to 0, \quad\text{as } t_{1} \to t_{2}, \end{aligned}
(2.5)

and

\begin{aligned} \begin{aligned}[b] &\bigl\vert e^{-t_{2}}R'(u, \lambda) (t_{2})-e^{-t_{1}}R'(u,\lambda) (t_{1}) \bigr\vert \\ &\quad= \biggl\vert e^{t_{2}}\varphi_{q} \biggl( \varphi_{p}\bigl(u'(0)\bigr)- \int_{0}^{t_{2}} \lambda\bigl(g \bigl(s,u(s),u'(s)\bigr)-QNu(s)\bigr)\,ds \biggr) -u'(0)e^{-t_{2}} \\ & \qquad- e^{-t_{1}}\varphi_{q} \biggl( \varphi_{p} \bigl(u'(0)\bigr)- \int_{0}^{t_{1}} \lambda \bigl(g \bigl(s,u(s),u'(s)\bigr) -QNu(s)\bigr)\,ds \biggr) + u'(0)e^{-t_{1}} \biggr\vert \\ &\quad\leq\bigl(e^{t_{2}}-e^{-t_{1}}\bigr)\varphi_{q} \bigl( \varphi_{p}(k) + \Vert \psi_{k} \Vert _{L^{1}} + \Vert QNu \Vert _{L^{1}}\bigr) + \bigl(e^{-t_{1}}-e^{-t_{2}}\bigr)k \\ &\quad\to0, \quad\text{as } t_{1} \to t_{2}. \end{aligned} \end{aligned}
(2.6)

Thus, (2.5) and (2.6) show that $$R(u,\lambda )\overline{\varOmega}$$ is equicontinuous on $$[0,T]$$.

We will now prove that $$R(u,\lambda)\overline{\varOmega}$$ is equiconvergent at ∞. Since $$\lim_{t \to +\infty }e^{-t}=0$$,

\begin{aligned} \lim_{t \to +\infty } e^{-t}R(u,\lambda) (t)= \lim _{t \to +\infty } e^{-t}R'(u,\lambda) (t)=0. \end{aligned}

Hence,

\begin{aligned} \begin{aligned}[b] &\Bigl\vert e^{-t}R(u, \lambda) (t)-\lim_{t \to +\infty }e^{-t}R(u,\lambda) (t) \Bigr\vert \\ &\quad= \biggl\vert e^{-t} \int_{0}^{t} \varphi_{q} \biggl( \varphi_{p}\bigl(u'(0)\bigr) - \int _{0}^{\tau} \lambda\bigl(g \bigl(s,u(s),u'(s)\bigr)-QNu(s)\bigr)\,ds \biggr)\,d\tau -te^{-t}u'(0) -0 \biggr\vert \hspace{-24pt} \\ &\quad\leq te^{-t} \varphi_{q} \bigl( \varphi_{p}(k) + \Vert \psi_{k} \Vert _{L^{1}} + \Vert QNu \Vert _{L^{1}}\bigr) + kte^{-t} \\&\quad\to0, \quad\text{uniformly as } t \to +\infty , \end{aligned} \end{aligned}
(2.7)

and

\begin{aligned} \begin{aligned}[b] &\Bigl\vert e^{-t}R'(u, \lambda) (t)-\lim_{t \to +\infty }e^{-t}R'(u, \lambda) (t) \Bigr\vert \\ &\quad= \biggl\vert e^{-t}\varphi_{q} \biggl( \varphi_{p}\bigl(u'(0)\bigr) - \int_{0}^{t} \lambda\bigl(g \bigl(s,u(s),u'(s)\bigr)-QNu(s)\bigr)\,ds \biggr) -e^{-t}u'(0) - 0 \biggr\vert \\ &\quad\leq e^{-t} \varphi_{q} \bigl( \varphi_{p}(k) + \Vert \psi_{k} \Vert _{L^{1}} + \Vert QNu \Vert _{L^{1}}\bigr) + ke^{-t} \\&\quad\to0, \quad\text{uniformly as } t \to +\infty . \end{aligned} \end{aligned}
(2.8)

Therefore $$R(u,\lambda)\overline{\varOmega}$$ is equiconvergent at +∞. It then follows from Definition 2.4 that $$R(u,\lambda)$$ is compact. □

### Lemma 2.3

The operator$$N_{\lambda}$$isM-compact.

### Proof

Since Q is a semi-projector, $$Q(I-Q)N_{\lambda}(\overline{\varOmega })=0$$. Hence, $$(I-Q)N_{\lambda}(\overline{\varOmega})\subset\ker Q = \operatorname {Im}M$$. Conversely, let $$w \in \operatorname {Im}M$$, then $$w=w -Qw = (I-Q)w \in (I-Q)Z$$. Hence, condition (i) of definition (2.5) is satisfied. It can easily be shown that condition (ii) of Definition 2.5 holds.

Let $$u \in\varSigma_{\lambda}=\{u \in\overline{\varOmega}:Mu = N_{\lambda}u\}$$, then $$N_{\lambda}u \in \operatorname {Im}M$$. Hence, $$QN_{\lambda }u=0$$ and $$R(u,0)(t)=0$$. From $$(\varphi_{p}(u'(t)))' + g(t, u(t),u'(t))=0$$, $$t \in(0,+\infty)$$, we have

\begin{aligned} R(u,\lambda) (t)&= \int_{0}^{t} \varphi_{q} \biggl( \varphi_{p}\bigl(u'(0)\bigr)- \int _{0}^{\tau} \lambda g \bigl(s,u(s),u'(s) \bigr)\,ds \biggr)\,d\tau- u'(0)t \\ &= \int_{0}^{t} \varphi_{q} \bigl( \varphi_{p}\bigl(u'(0)\bigr)+ \varphi_{p} \bigl(u'(\tau )\bigr)-\varphi_{p}\bigl(u'(0) \bigr) \bigr)\,d\tau- u'(0)t \\ &= u(t) - u(0)-u'(0)t=u(t)-Pu(t)=\bigl[(I-P)u\bigr](t). \end{aligned}

Therefore, condition (iii) of definition (2.5) holds.

Let $$u \in\overline{\varOmega}$$. Since $$Mu = (\varphi_{p}(u'(t)))'$$ we have

\begin{aligned} M\bigl[Pu +R(u,\lambda)\bigr](t)&= \big(\varphi_{p}\bigl(\bigl[Pu + R(u,\lambda)\bigr]\big)'(t)\bigr)' \\ &= \biggl(\varphi_{p} \biggl[u(0)+u'(0)t + \int_{0}^{t} \varphi_{q} \biggl( \varphi_{p}\bigl(u'(0)\bigr)- \int_{0}^{\tau} \lambda\bigl(g \bigl(s,u(s),u'(s)\bigr) \\ &\quad - QN(s)\bigr)\,ds \biggr) \,d\tau-u'(0)t \biggr]' \biggr)' \\ &= \biggl(\varphi_{p}\bigl(u'(0)\bigr)- \int_{0}^{\tau} \lambda\bigl(g \bigl(s,u(s),u'(s)\bigr) - QN(s)\bigr)\,ds \biggr)'=(I-Q)N_{\lambda}(t), \end{aligned}

that is, condition (iv) of definition (2.5) holds. Hence, $$N_{\lambda}$$ is M-compact in Ω̅. □

## Existence result

In this section, the conditions for existence of solutions for boundary value problem (1.1) will be stated and proved.

### Theorem 3.1

Assumegis a$$L^{[}0,+\infty )$$-Carathéodory function and the following hypotheses hold:

($$H_{1}$$):

there exist functions$$x_{1}(t), x_{2}(t), x_{3}(t) \in L^{1}[0,+\infty )$$such that, for a.e. $$t \in[0,+\infty )$$,

$$\bigl\vert g\bigl(t,u,u'\bigr) \bigr\vert \leq e^{-t}\bigl(x_{1}(t) \vert u \vert ^{p-1} + x_{2}(t) \bigl\vert u' \bigr\vert ^{p-1}\bigr) + x_{3}(t),$$
(3.1)
($$H_{2}$$):

for$$u \in \operatorname {dom}M$$there exists a constant$$A_{0} >0$$, such that, if$$|u(t)|>A_{0}$$for$$t \in[0,+\infty )$$or$$|u'(t)|>A_{0}$$for$$t \in[0,+\infty ]$$, then either

$$Q_{1}Nu(t) \neq0 \quad\textit{or} \quad Q_{2}Nu(t) \neq0, \quad t \in [0,+\infty ),$$
(3.2)
($$H_{3}$$):

there exists a constant$$l>0$$such that, for$$|a| >l$$or$$|b|>l$$either

$$Q_{1}N(a +bt) + Q_{2}N(a +bt) < 0, \quad t \in[0,+\infty ),$$
(3.3)

or

$$Q_{1}N(a +bt) + Q_{2}N(a +bt) >0, \quad t \in[0,+\infty ),$$
(3.4)

where$$a, b \in\mathbb{R}$$, $$|a| + |b| > l$$and$$t \in[0,+\infty )$$.

Then the boundary value problem (1.1) has at least one solution, provided

$$2^{2q-4}\bigl( \Vert x_{2} \Vert _{L^{1}} + 2^{q-2} \Vert x_{1} \Vert _{L^{1}}\bigr) < 1, \quad\textit{for } 1 < p \leq2,$$

or

$$\varphi_{q}\bigl( \Vert x_{1} \Vert _{L^{1}} + \Vert x_{2} \Vert _{L^{1}}\bigr) < 1, \quad \textit{for } p>2.$$

The following lemmas are also needed to prove our main result.

### Lemma 3.1

The set$$\varOmega_{1} = \{ u \in \operatorname {dom}M :Mu = N_{\lambda}u \textit{ for some } \lambda\in(0,1)\}$$is bounded.

### Proof

Let $$u \in\varOmega_{1}$$ then $$N_{\lambda}u \in \operatorname {Im}M= \ker Q$$. Hence, $$QN_{\lambda}u = 0$$ and $$QNu=0$$. It follows from $$H_{2}$$ that there exist $$t_{0}, t_{1} \in[0,+\infty )$$, such that $$|u(t_{0})| \leq A_{0}$$ and $$|u'(t_{1})| \leq A_{0}$$. From $$u(t)=u(t_{0}) + \int_{t_{0}}^{t}u'(v)\,dv$$, we have

\begin{aligned} \bigl\vert u(t) \bigr\vert = \biggl\vert u(t_{0}) - \int_{t_{0}}^{t}u'(s)\,ds \biggr\vert \leq A_{0} + \vert t-t_{0} \vert \bigl\Vert u' \bigr\Vert _{\infty}. \end{aligned}

Hence,

$$\Vert u \Vert _{\infty} = \sup_{t \to\infty}e^{-t} \bigl\vert u(t) \bigr\vert \leq A_{0} + \bigl\Vert u' \bigr\Vert _{\infty}.$$
(3.5)

Also, from $$Mu = N_{\lambda}u$$, we get

$$\varphi_{p}\bigl(u'(t)\bigr)=- \int_{t_{1}}^{t} \lambda g\bigl(s,u(s),u'(s) \bigr)\,ds + \varphi _{p}\bigl(u(t_{1})\bigr).$$

In view of (3.1), we have

\begin{aligned} \begin{aligned}[b] \bigl\vert \bigl(u'(t) \bigr) \bigr\vert &\leq\varphi_{q} \biggl(\varphi_{p}(A_{0})+ \int_{0}^{+\infty } \bigl(x_{1}(t) \bigl\vert \varphi_{p}\bigl(u(t)\bigr) \bigr\vert + x_{2}(t) \bigl\vert \varphi_{p}\bigl(u' \bigr) \bigr\vert + x_{3}(t)\bigr)\,dt \biggr) \\ &\leq\varphi_{q} \bigl(\varphi_{p}(A_{0})+ \Vert x_{1} \Vert _{L^{1}}\varphi _{p} \bigl( \Vert u \Vert _{\infty}\bigr) + \Vert x_{2} \Vert _{L^{1}}\varphi_{p}\bigl( \bigl\Vert u' \bigr\Vert _{\infty}\bigr) + \Vert x_{3} \Vert _{L^{1}} \bigr) \\ &\leq\varphi_{q} \bigl(\varphi_{p}(A_{0})+ \Vert x_{1} \Vert _{L^{1}}\varphi _{p} \bigl(A_{0}+ \bigl\Vert u' \bigr\Vert _{\infty}\bigr) + \Vert x_{2} \Vert _{L^{1}} \varphi _{p}\bigl( \bigl\Vert u' \bigr\Vert _{\infty}\bigr) + \Vert x_{3} \Vert _{L^{1}} \bigr). \end{aligned} \end{aligned}
(3.6)

If $$1 < p \leq2$$, it follows from Lemma 2.1 that

$$\bigl\Vert u' \bigr\Vert _{\infty} \leq\frac{2^{2q-4}[\varphi_{q}( \Vert x_{3} \Vert _{L^{1}}) + A_{0}(1+2^{q-2} \Vert x_{1} \Vert _{L^{1}}}{1-2^{2q-4}( \Vert x_{2} \Vert _{L^{1}} + 2^{q-2} \Vert x_{1} \Vert _{L^{1}})}.$$
(3.7)

If $$p >2$$ then, by Lemma 2.1, we get

$$\bigl\Vert u' \bigr\Vert _{\infty} \leq\frac{A_{0}(1+ \varphi_{q}( \Vert x_{1} \Vert _{L^{1}}) + \varphi_{q}( \Vert x_{3} \Vert _{L^{1}})}{1-\varphi_{q}( \Vert x_{1} \Vert _{L^{1}} + \Vert x_{2} \Vert _{L^{1}})}.$$
(3.8)

Since $$\Vert u \Vert= \max\{\Vert u \Vert_{\infty}, \Vert u' \Vert _{\infty}\} \leq A_{0} + \Vert u' \Vert_{\infty}$$, in view of (3.7) and (3.8), $$\varOmega_{1}$$ is bounded. □

### Lemma 3.2

If$$\varOmega_{2} =\{u \in\ker M:-\lambda u +(1-\lambda)JQNu=0, \lambda\in[0,1]\}$$, $$J: \operatorname {Im}Q \to\ker M$$is a homomorphism, then$$\varOmega_{2}$$is bounded.

### Proof

For $$a, b \in R$$, let $$J: \operatorname {Im}Q \to\ker M$$ be defined by

$$J(a+bt)= \frac{1}{C}\bigl[\delta_{11} \vert a \vert +\delta_{12} \vert b \vert + \bigl(\delta _{21} \vert a \vert + \delta_{22} \vert b \vert \bigr)t)\bigr]e^{-t}.$$
(3.9)

If (3.3) holds, for any $$u(t) = a + bt \in\varOmega_{3}$$, from $$-\lambda u + (1-\lambda)JQNu =0$$, we obtain

${δ11(−λ|a|+(1−λ)Q1N(a+bt))+δ12(−λ|b|+(1−λ)Q2N(a+bt))=0,δ21(−λ|a|+(1−λ)Q1N(a+bt))+δ22(−λ|b|+(1−λ)Q2N(a+bt))=0.$

Since $$C \neq0$$,

\begin{aligned} \begin{gathered} \lambda \vert a \vert =(1 - \lambda)Q_{1}N(a +bt), \\ \lambda \vert b \vert =(1 - \lambda)Q_{2}N(a +bt). \end{gathered} \end{aligned}
(3.10)

From (3.10), when $$\lambda=1$$, $$a = b =0$$. When $$\lambda=0$$,

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

which contradicts (3.3) and (3.4), hence from ($$H_{3}$$), $$|a| \leq l$$ and $$|b| \leq l$$. For $$\lambda\in(0,1)$$, in view of (3.3) and (3.10), we have

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

which contradicts $$\lambda(|a|+|b|) \geq0$$. Hence, ($$H_{3}$$), $$|a| \leq l$$ and $$|b| \leq l$$, thus $$\Vert u \Vert\leq2l$$. Therefore $$\varOmega _{2}$$ is bounded. □

### Proof of Theorem 3.1

Since M is quasi-linear, condition ($$A_{1}$$) of Theorem 2.1 holds, Lemma 2.2 proved ($$A_{2}$$), while Lemma 3.1 shows that ($$A_{3}$$) holds.

Let $$\varOmega\supset\varOmega_{1} \cup\varOmega_{2}$$ be a nonempty, open and bounded set, $$u \in \operatorname {dom}M \cap\partial\varOmega$$, $$H(u,\lambda)=-\lambda u +(1-\lambda)JQNu$$, and J be as defined in Lemma 3.2 then $$H(u,\lambda) \neq0$$. Therefore by the homotopy property of the Brouwer degree

\begin{aligned} \deg\{JQN|_{\overline{\varOmega} \cap\ker M},\varOmega\cap\ker M,0\}&=\deg\bigl\{ H(\cdot, 0), \varOmega\cap\ker M,0\bigr\} \\ &=\deg\bigl\{ H(\cdot,1),\varOmega\cap\ker M,0\bigr\} \\ &=\deg\{-I,\varOmega\cap\ker M,0\} \neq0. \end{aligned}

Hence, condition ($$A_{4}$$) of Theorem 2.1 also holds. □

Since all the conditions of Theorem 2.1 are satisfied, the abstract equation $$Mu=Nu$$ has at least one solution in $$\overline {\varOmega} \cap \operatorname {dom}M$$. Hence, (1.1) has at least one solution.

## Example

Consider the following boundary value problem:

$$\left \{ \textstyle\begin{array}{l} (\varphi_{4}(u'(t)))' + e^{-t-2} \sin t \cdot u^{3}+e^{-t-3}\cos t\cdot u^{\prime3} + \frac{1}{6}e^{-6t}=0, \quad t \in (0,+\infty ), \\ \varphi_{4}(u'(0))=\int_{0}^{+\infty }2e^{-2t}\varphi_{4}(u'(t))\,dt, \qquad \varphi_{4}(u'(+\infty ))= 9\int_{0}^{1/9}\varphi_{4}(u'(t))\,dt. \end{array}\displaystyle \right .$$
(4.1)

Here $$v(t) =2e^{-2t}$$, $$p=4$$, $$q=\frac{4}{3}$$, $$\beta_{1} = 9$$, $$\eta _{1} = \frac{1}{9}$$, $$x_{1}= e^{-t-2}\sin t$$ and $$x_{2}=e^{-t-3}\cos t$$. Therefore, $$\sum_{j=1}^{1}\beta_{j} \eta_{j}=1$$, $$\int_{0}^{+\infty }v(t)\, dt=1$$, $$C \neq0$$ and $$\varphi_{q}(\Vert x_{1} \Vert_{L^{1}} + \Vert x_{2} \Vert_{L^{2}})<1$$. It can easily be seen that conditions ($$H_{1}$$)–($$H_{3}$$) hold. Hence, (4.1) has at least one solution.

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### Acknowledgements

The authors acknowledges Covenant University for the support received from them. The authors are also grateful to the referees for their valuable suggestions.

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### Contributions

OF conceived the idea. SA supervised the work. All authors discussed and contributed to the final manuscript.

### Corresponding author

Correspondence to O. F. Imaga.

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