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.

1 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 [1–3].

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 [4] obtained existence results for a resonant multi-point second-order boundary value problem on the half-line, Capitanelli, Fragapane and vivaldi [5] addressed regularity results for p-Laplacians in pre-fractal domains, while Jiang and Kosmatov [6] considered resonant p-Laplacian problems with functional boundary conditions. For other work on resonant problems without p-Laplacian operator, see [7–10], while for problems with the p-Laplacian operator, see [11–16]. In [17], 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.

2 Preliminaries

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

Definition 2.1

([11])

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

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

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

• (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

([18])

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:

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

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

Definition 2.3

([19])

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

• (i) $P^{2} =P$;

• (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

• (i) $Q^{2} = Q$;

• (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

([20])

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:

• (i) X is bounded in U;

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

• (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

([18])

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:

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

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

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

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

Lemma 2.1

([19])

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

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

• (ii) For any$x, y >0$,

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

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

Theorem 2.1

([18])

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. □

3 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

$$\{\begin{array}{c}{\delta }_{11}(-\lambda |a|+(1-\lambda )Q_{1}N(a+bt))+{\delta }_{12}(-\lambda |b|+(1-\lambda )Q_{2}N(a+bt))=0,\\ {\delta }_{21}(-\lambda |a|+(1-\lambda )Q_{1}N(a+bt))+{\delta }_{22}(-\lambda |b|+(1-\lambda )Q_{2}N(a+bt))=0.\end{array}$$

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.

4 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.