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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 [13].

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 [710], while for problems with the p-Laplacian operator, see [1116]. 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.

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:

  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

([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:

  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

([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

  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

([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:

  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

([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:

  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

([19])

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

([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. □

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|+(1λ)Q1N(a+bt))+δ12(λ|b|+(1λ)Q2N(a+bt))=0,δ21(λ|a|+(1λ)Q1N(a+bt))+δ22(λ|b|+(1λ)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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OF conceived the idea. SA supervised the work. All authors discussed and contributed to the final manuscript.

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Imaga, O.F., Iyase, S.A. Existence of solution for a resonant p-Laplacian second-order m-point boundary value problem on the half-line with two dimensional kernel. Bound Value Probl 2020, 114 (2020). https://doi.org/10.1186/s13661-020-01415-3

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MSC

  • 70K30
  • 34B10
  • 34B15
  • 35J92

Keywords

  • Coincidence degree
  • Half-line
  • Integral boundary value problem
  • m-Point
  • p-Laplacian
  • Resonance