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.
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1 Introduction
The following second-order p-Laplacian boundary value problem will be considered in this work:
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
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:
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\(p \geq2\).
- (a)
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
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
and \(Mu = (\varphi_{p}(u'(t)))'\). We will define the operator \(N_{\lambda}u : \overline{\varOmega} \to Z\) by
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
and the operators \(\Delta_{1}, \Delta_{2} : Z \to Z_{1}\) as
and
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
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
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
Now for any \(u \in\overline{\varOmega}\), \(\lambda\in[0,1]\), we have
and
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
and
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 \(\lim_{t \to +\infty }e^{-t}=0\),
Hence,
and
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
Therefore, condition (iii) of definition (2.5) holds.
Let \(u \in\overline{\varOmega}\). Since \(Mu = (\varphi_{p}(u'(t)))'\) we have
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
Hence,
Also, from \(Mu = N_{\lambda}u\), we get
In view of (3.1), we have
If \(1 < p \leq2\), it follows from Lemma 2.1 that
If \(p >2\) then, by Lemma 2.1, we get
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
If (3.3) holds, for any \(u(t) = a + bt \in\varOmega_{3}\), from \(-\lambda u + (1-\lambda)JQNu =0\), we obtain
Since \(C \neq0\),
From (3.10), when \(\lambda=1\), \(a = b =0\). When \(\lambda=0\),
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
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
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:
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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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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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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DOI: https://doi.org/10.1186/s13661-020-01415-3