- Research
- Open access
- Published:
New existence results for some periodic and Neumann-Steklov boundary value problems with ϕ-Laplacian
Boundary Value Problems volume 2016, Article number: 170 (2016)
Abstract
We study the existence of solutions of the quasilinear equation
with periodic or nonlinear Neumann-Steklov boundary conditions, where \(\phi: \,]{-}a, a[\rightarrow\mathbb{R}\) with \(0 < a< +\infty\) is an increasing homeomorphism such that \(\phi(0)=0\). Combining some sign conditions and the lower and upper solution method, we obtain the existence of solutions when there exists one lower solution or one upper solution.
1 Introduction
This work is devoted to the study of the existence of solutions of the quasilinear equation
with periodic boundary conditions
or nonlinear Neumann-Steklov boundary conditions
where \(\phi: \,]{-}a, a[\,\rightarrow\mathbb{R}\) with \(0 < a< +\infty\) is an increasing homeomorphism such that \(\phi(0)=0\), \(g_{0},g_{T}\): \(\mathbb{R}\longrightarrow\mathbb{R}\) are continuous functions, and \(f:[0,T]\times\mathbb{R}^{2}\rightarrow\mathbb{R}\) is assumed to be an \(L^{1}\)-Carathéodory function.
Generally, in the lower and upper solution method, to show the existence of a solution of a problem, we need the existence of at least one lower solution and at least one upper solution. In the case of the method of sign conditions, we usually need two sign conditions to show the existence of at least one solution of a problem.
In 2007, Bereanu and Mawhin [1] proved, for continuous f, the existence of solutions of problem (1)-(2) under some sign conditions (see [1] Theorem 2) and when there exist a lower solution and an upper solution, ordered or not, of problem (1)-(2) (see [1] Theorem 4).
In 2008, Bereanu and Mawhin [2] proved, for continuous f, the existence of solutions of problem (1)-(3) under some sign conditions (see [2] Theorem 2) and when there exist a lower solution and an upper solution, ordered or not, of problem (1)-(3) (see [2] Theorem 4).
In the following results, we prove the existence of solutions of (1)-(2) and (1)-(3) when we have only one sign condition and only one lower solution or only one upper solution.
After introducing notation and preliminary results in Section 2, in Section 3, combining one sign condition and the existence of only one lower solution or only one upper solution of problem (1)-(2), we prove the existence of at least one solution of problem (1)-(2).
In Section 4, combining one sign condition and the existence of only one lower solution or only one upper solution of problem (1)-(3), we prove the existence of at least one solution of problem (1)-(3).
The results of this section enable us to obtain that, for some forced relativistic pendulum equations with friction and Neumann-Steklov boundary conditions, the existence of a lower solution or the existence of an upper solution is sufficient to obtain the existence of a solution.
2 Notation and preliminaries
We denote:
-
\(C=C([0, T])\), the Banach space of continuous functions on \([0, T]\);
-
\(\|u\|_{C}=\|u\|_{\infty}= \max\{|u(t)|; t \in[0, T]\}\), the norm of C;
-
\(C^{1}=C^{1}([0, T])\), the Banach space of continuous functions on \([0, T]\) having continuous first derivative on \([0, T]\);
-
\(\|u\|_{C^{1}} = \|u\|_{C} + \|u'\|_{C}\), the norm of \(C^{1}\);
-
\(AC=AC([0, T])\), the set of absolutely continuous functions on \([0, T]\);
-
\(L^{1}=L^{1}(0, T)\), the Banach space of Lebesgue-integrable functions on \([0, T]\);
-
\(\|x\|_{L^{1}}=\int_{0}^{T}|x(t)|\,dt\), the norm of \(L^{1}\);
-
\(B_{r}\), the open ball of \(C^{1}\) with center 0 and radius r;
-
\(d_{LS}\), the Leray-Schauder degree, and \(d_{B}\), the Brouwer degree;
-
\(u_{L} = \min_{[0,T]} u\) and \(u_{M} =\max_{[0,T]} u\) for \(u \in C\);
-
\(\operatorname{Range} (u) = \{y\in\mathbb{R}; y=u(t) \mbox{ with } t\in [0,T]\}\) for \(u \in C\).
We introduce:
-
the continuous operators \(P,K: C\rightarrow C\) defined by
$$P(u )=Pu = u(0) \quad\mbox{and}\quad K(u)=T^{-1}\bigl[g_{T} \bigl(u(T)\bigr)-g_{0}\bigl(u(0)\bigr)\bigr]; $$ -
the continuous operators \(Q,H : L^{1} \rightarrow C \) defined by
$$Q(u)= Qu =\frac{1}{T} \int_{0}^{T}u(s)\,ds \quad\mbox{and}\quad (Hu) (t)= \int_{0}^{t}u(s)\,ds, \quad\forall t\in[0,T]. $$
Definition 2.1
\(f:[0,T]\times\mathbb{R}^{2}\rightarrow\mathbb{R}\) is \(L^{1}\)-Carathéodory if:
-
(i)
\(f(\cdot, x, y) : [0,T] \rightarrow\mathbb{R}\) is measurable for all \((x, y) \in\mathbb{R}^{2}\);
-
(ii)
\(f(t, \cdot,\cdot) : \mathbb{R}^{2}\rightarrow\mathbb{R}\) is continuous for a.e. \(t \in[0, T]\);
-
(iii)
for each compact set \(A \subset\mathbb{R}^{2}\), there is a function \(\mu_{A} \in L^{1}\) such that \(|f(t, x, y)| \leq\mu_{A}(t)\) for a.e. \(t \in[0, T]\) and all \((x, y) \in A\).
Definition 2.2
A solution of problem (1)-(2) (resp (1)-(3)) is a function \(u\in C^{1}\) satisfies (1)-(2) (resp (1)-(3)) such that \(\phi(u')\in AC\) and \(\|u'\| _{\infty}< a\).
Definition 2.3
A function \(\alpha\in C^{1}\) is a lower solution of problem (1)-(2) if \(\|\alpha'\|_{\infty}< a\), \(\phi(\alpha')\in AC\),
Definition 2.4
A function \(\beta\in C^{1}\) is an upper solution of problem (1)-(2) if \(\|\beta'\|_{\infty}< a\), \(\phi(\beta')\in AC\),
Definition 2.5
A function \(\alpha\in C^{1}\) is a lower solution of problem (1)-(3) if \(\|\alpha'\|_{\infty}< a\), \(\phi(\alpha')\in AC\),
Definition 2.6
A function \(\beta\in C^{1}\) is an upper solution of problem (1)-(3) if \(\|\beta'\|_{\infty}< a\), \(\phi(\beta')\in AC\),
Remark 2.1
It is standard to show that the Nemytskii operator associated to f, \(N_{f}: C^{1}\rightarrow L^{1}\), defined by
is continuous and sends bounded sets into bounded sets.
3 Existence of solutions of periodic problem
3.1 Existence of solutions under two sign conditions
Lemma 3.1
For each \(h \in C\), there exists a unique \(\varrho:= Q_{\phi}(h)\in \operatorname{Range} (h)\) such that
Moreover, the function \(Q_{\phi}:C\rightarrow\mathbb{R}\) is continuous.
Proof
See [1], the proof of Lemma 1. □
Now, consider the family of boundary value problems \((P_{\lambda})\), \(\lambda\in[0, 1]\),
For each \(\lambda\in[0, 1]\), problem \((P_{\lambda})\) can be written equivalently
For each \(\lambda\in[0, 1]\), we associate with \((P_{\lambda})\) the nonlinear operator \(M(\lambda, \cdot)\), where M is defined on \([0, 1] \times C^{1}\) by
Using the Arzelà-Ascoli theorem, we get that M is completely continuous.
Lemma 3.2
Assume that there exist \(R>0\) and \(\varepsilon\in\{-1,1\}\) such that
and
Then, for all sufficiently large \(\rho>0\),
and problem (1)-(2) has at least one solution.
Proof
Assume that there exists \((\lambda,u)\in[0,1]\times C^{1}\) such that \(M(\lambda,u)=u\).
We have \(u(0)=u(0)+[QN_{f}(u)]\). It follows that
Since
we get \(\| u'\|_{\infty}< a\). If \(u_{L}\geq R\) or \(u_{M}\leq -R\), by (15) and (16) we have
which contradicts (17); therefore, \(u_{L}< R\) and \(u_{M}>-R\). Since u is continuous on \([0,T]\), there exists \((t_{1},t_{2})\in[0,T]^{2}\) such that \(u_{L}= u(t_{1})\) and \(u_{M}=u(t_{2})\). We have
Using (19), we have
It follows that \(\| u\|_{\infty}< R+aT\).
Since \(\| u'\|_{\infty}< a\) and \(\| u\| _{\infty}< R+aT\), we have
Let M be the operator given by (14), and let \(\rho >R+a(T+1)\). Using (20) and the homotopy invariance of the Leray-Schauder degree, we have
But the range of the mapping \(u\mapsto P(u)+QN_{f}(u)\) is contained in the subspace of constant functions isomorphic to \(\mathbb{R}\), so, using the reduction property of Leray-Schauder degree [3], we have
By the existence property of the Leray-Schauder degree there exists \(u\in B_{\rho}\) such that \(u=M(1,u)\), which is a solution of problem (1)-(2). □
Let us decompose any \(u \in C^{1}\) in the form \(u= \overline{u}+ \widetilde{u}\) (\(\overline{u}=u(0)\), \(\widetilde{u}(0)=0\)), and let \(\widetilde{C}^{1} = \{u \in C^{1}: u(0) = 0\}\).
Lemma 3.3
The set \(\mathfrak{S}\) of solutions \((\overline{u},\widetilde{u})\in \mathbb{R}\times\widetilde{C^{1}}\) of problem
contains a continuum subset \(\mathcal{C}\) whose projection on \(\mathbb {R}\) is \(\mathbb{R}\) and whose projection on \(\widetilde{C^{1}}\) is contained in the ball \(B_{a(T+1)}\).
Proof
The proof is similar to the proof of Lemma 4 in [1]. □
Theorem 3.1
Assume that there exist \(R>0\) and \(\varepsilon\in\{-1,1\}\) such that
and
Then problem (1)-(2) admits at least one solution.
Proof
The proof is similar to the proof of Theorem 2 in [1].
Let us consider the continuum \(\mathcal{C}\) given by Lemma 3.3. We have \(\mathcal{C}\neq\emptyset\) (see the proof of Lemma 4 in [1]). Let \((\overline{u},\widetilde{u})\in\mathcal{C}\). Using Lemma 3.3, it follows that
Let \(v_{1}=R+aT+\widetilde{u}\) and \(v_{2}=-R-aT+\widetilde{u}\).
Since
for all \(t\in[0,T]\), we have
Applying (22), we have \(\varepsilon\{QN_{f}(v_{1})\}\geq0\), and applying (23), we have \(\varepsilon\{QN_{f}(v_{2})\}\leq0\).
We deduce, using the intermediate value theorem for a continuous functions on a connected set, that there exists \((\overline {w},\widetilde{w})\in\mathcal{C}\) such that
Therefore, \(w=\overline{w}+\widetilde{w}\) is a solution of problem (1)-(2). □
3.2 Existence of solutions under one sign condition and only one lower solution or only one upper solution
For \(\alpha\in C^{1}\), let us define two functions \(\gamma_{1} : [0,T]\times\mathbb{R} \rightarrow\mathbb{R}\) and \(\gamma _{2} : [0,T]\times\mathbb{R} \rightarrow\mathbb{R}\) by
We introduce the following lemma (see [4], Lemma 6.3 and Corollary 6.4).
Lemma 3.4
For \(u \in C^{1}\), the following three properties are true.
-
(a)
For \(i\in\{1,2\}\), \(\frac{d}{dt}\gamma_{i}(t,u(t))\) exists for a.e. \(t \in[0, T ]\).
-
(b)
$$\frac{d}{dt}\gamma_{1}\bigl(t,u(t)\bigr)= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \alpha'(t)& \textit{if } u(t)< \alpha(t),\\ u'(t)&\textit{if }u(t)\geq\alpha(t), \end{array}\displaystyle \right . $$
and
$$\frac{d}{dt}\gamma_{2}\bigl(t,u(t)\bigr)= \left \{ \textstyle\begin{array}{@{}l@{\quad}l} \alpha'(t)& \textit{if } u(t)>\alpha(t),\\ u'(t)&\textit{if }u(t)\leq\alpha(t). \end{array}\displaystyle \right . $$ -
(c)
For \(i\in\{1,2\}\), if \((u_{n})_{n}\subset C^{1}\) is such that \(u_{n}\rightarrow u\) in \(C^{1}\), then \(\gamma_{i}(\cdot,u_{n})\rightarrow \gamma_{i}(\cdot,u)\) in C, and for almost every \(t\in[0,T]\), \(\lim_{n\rightarrow\infty}\frac{d}{dt}\gamma _{i}(t,u_{n}(t))=\frac{d}{dt}\gamma_{i}(t,u(t))\).
Theorem 3.2
Assume that:
- (i)
-
(ii)
there exists \(R>0 \) such that
$$ u_{L}\geq R \quad\textit{and}\quad \bigl\| u' \bigr\| _{\infty}< a \quad\Rightarrow\quad \int _{0}^{T}f\bigl(t,u(t),u'(t) \bigr)\,dt> 0. $$(24)
Then problem (1)-(2) admits at least one solution.
Proof
Step 1: The modified problem.
Consider the function \(\delta: \mathbb{R}\rightarrow\mathbb{R}\) given by \(\delta(x)=\max\{-a, \min\{x, a\}\}\). Consider the function \(f^{\ast} : [0,T] \times\mathbb {R}^{2}\longrightarrow\mathbb{R}\) given by
which is an \(L^{1}\)-Carathéodory function. Consider the modified problem
Step 2: Any solution of problem ( 26 ) is a solution of problem ( 1 )-( 2 ).
Let u be a solution of problem (26). We prove that \(\alpha(t) \leq u(t)\) for all \(t\in[0,T]\).
Let us assume on the contrary that, for some \(t_{0} \in[0,T]\),
If \(t_{0}\in]0,T[\), then \(\alpha'(t_{0}) = u'(t_{0})\); hence, \(\phi(\alpha '(t_{0}))=\phi(u'(t_{0}))\). We can find \(\omega>0\) such that for all \(t\in \,]t_{0},t_{0}+\omega[\), \(\alpha(t)>u(t)\). We have
It follows that, for all \(t\in\,]t_{0},t_{0}+\omega[\),
Since ϕ is an increasing homeomorphism, \(\phi(\alpha'(t))-\phi(u'(t))>0\Rightarrow\alpha'(t)-u'(t)>0\), a contradiction.
If \(t_{0}\in\{0,T\}\), then \(\alpha'(0) - u'(0)=0=\alpha'(T) - u'(T)\). We can find \(\omega>0\) such that for all \(t\in\,]0,\omega[\), \(\alpha(t)>u(t)\). We have
It follows that, for all \(t\in\,]0,\omega[\),
Since ϕ is an increasing homeomorphism, \(\phi(\alpha'(t))-\phi(u'(t))>0\Rightarrow\alpha'(t)-u'(t)>0\), a contradiction.
In consequence, we have that \(\alpha(t) \leq u(t)\) for all \(t\in [0,T]\). Therefore, u is a solution of problem (1)-(2).
Step 3: Existence of solutions of problem ( 26 ).
Let \(\Delta=[\alpha_{L},\alpha_{M}]\times[-a,a]\). Since f is an \(L^{1}\)-Carathéodory function, there exists \(\varphi\in L^{1}\) such that, for a.e. \(t \in[0, T]\) and all \((x, y) \in\Delta\), \(|f(t,x;y)|\leq \varphi(t)\). Let \(R_{1}=\max\{|\alpha_{L}|,|\alpha_{L}-\frac{1}{T}\|\varphi \|_{L^{1}}|,R+aT\}\). By (24), if \(u\in C^{1}\) is such that \(\|u'\|_{\infty}< a\) and \(u_{L}\geq R_{1}\), then
Moreover, if \(u\in C^{1}\) is such that \(\|u'\|_{\infty}< a\) and \(u_{M}\leq -R_{1}\), then
It follows that
Using (27), (28), and Theorem 3.1, we deduce that problem (26) has at least one solution, which is also a solution of problem (1)-(2) by step 2. □
Theorem 3.3
Assume that:
- (i)
-
(ii)
there exists \(R>0\) such that
$$ u_{M}\leq-R \quad\textit{and}\quad \bigl\| u' \bigr\| _{\infty}< a\quad \Rightarrow\quad \int _{0}^{T}f\bigl(t,u(t),u'(t) \bigr)\,dt< 0. $$(29)
Then problem (1)-(2) admits at least one solution.
Proof
The proof is similar to the proof of Theorem 3.2. □
4 Existence of solutions of Neumann-Steklov problem
4.1 Existence of solutions under two sign conditions
Consider the family of boundary value problems \((P_{\lambda})\), \(\lambda\in[0, 1]\):
For each \(\lambda\in[0, 1]\), problem \((P_{\lambda})\) can be written equivalently
For each \(\lambda\in[0, 1]\), we associate with \((P_{\lambda})\) the nonlinear operator \(M(\lambda, \cdot)\), where M is defined on \([0, 1] \times C^{1}\) by
with
Using the Arzelà-Ascoli theorem, we get that M is completely continuous.
Lemma 4.1
Assume that there exist \(R>0\) and \(\varepsilon\in\{-1,1\}\) such that
and
Then, for all sufficiently large \(\rho>0\),
and problem (1)-(3) has at least one solution.
Proof
Assume that there exists \((\lambda,u)\in[0,1]\times C^{1}\) such that \(M(\lambda,u)=u\).
We have
It follows that
Since
we have \(\| u'\|_{\infty}< a\). If \(u_{L}\geq R\) or \(u_{M}\leq-R\), then by (33) and (34) we have
which contradicts (35); therefore, \(u_{L}< R\) and \(u_{M}>-R\). Since u is continuous on \([0,T]\), there exists \((t_{1},t_{2})\in[0,T]^{2}\) such that \(u_{L}= u(t_{1})\) and \(u_{M}=u(t_{2})\). We have
Using (37), we have
It follows that \(\| u\|_{\infty}< R+aT\). Since \(\| u'\|_{\infty}< a\) and \(\| u\| _{\infty}< R+aT\), we have
Let M be the operator given by (31) and let \(\rho>R+a(T+1)\). Using (38) and the homotopy invariance of the Leray-Schauder degree, we have
But the range of the mapping \(u\mapsto P(u)+QN_{f}(u)-K(u)\) is contained in the subspace of constant functions isomorphic to \(\mathbb {R}\), so, using the reduction property of Leray-Schauder degree [3], it follows that
Then, by the existence property of the Leray-Schauder degree there exists \(u\in B_{\rho}\) such that \(u=M(1,u)\), which is a solution of problem (1)-(3). □
Let us decompose any \(u \in C^{1}\) in the form \(u= \overline{u}+ \widetilde{u}\) (\(\overline{u}=u(0)\), \(\widetilde{u}(0)=0\)), and let \(\widetilde{C}^{1} = \{u \in C^{1}: u(0) = 0\}\).
Lemma 4.2
The set \(\mathfrak{S}\) of solutions \((\overline{u},\widetilde{u})\in \mathbb{R}\times\widetilde{C^{1}}\) of problem
contains a continuum subset \(\mathcal{C}\) whose projection on \(\mathbb {R}\) is \(\mathbb{R}\) and whose projection on \(\widetilde{C^{1}}\) is contained in the ball \(B_{a(T+1)}\).
Proof
The proof is similar to the proof of Lemma 4 in [2]. □
Theorem 4.1
Assume that there exist \(R>0\) and \(\varepsilon\in\{-1,1\}\) such that
and
Then problem (1)-(3) admits at least one solution.
Proof
The proof is similar to that of Theorem 2 in [2] and that of Theorem 3.1. □
4.2 Existence of solutions under one sign condition and only one lower solution or only one upper solution
Theorem 4.2
Assume that:
- (i)
-
(ii)
there exists \(R>0\) such that
$$ \begin{aligned} &u_{L}\geq R \quad\textit{and} \quad\bigl\| u' \bigr\| _{\infty}< a \\ &\quad\Rightarrow\quad \int _{0}^{T}f\bigl(t,u(t),u'(t) \bigr)\,dt-g_{T}\bigl(u(T)\bigr)+g_{0}\bigl(u(0)\bigr)> 0. \end{aligned} $$(42)
Then problem (1)-(3) admits at least one solution.
Proof
Step 1: The modified problem.
Consider the functions \(f^{\ast} : [0,T] \times\mathbb {R}^{2}\longrightarrow\mathbb{R}\), \(g_{0}^{\ast} : \mathbb {R}\longrightarrow\mathbb{R}\), and \(g_{T}^{\ast} : \mathbb {R}\longrightarrow\mathbb{R}\) given by
and
The function \(f^{\ast}\) is an \(L^{1}\)-Carathéodory function, and \(g_{0}^{\ast}\) and \(g_{T}^{\ast}\) are continuous. Consider the modified problem
Step 2: Any solution of problem ( 46 ) is a solution of problem ( 1 )-( 3 ).
Let u be a solution of problem (46). We prove that \(\alpha(t) \leq u(t)\) for all \(t\in[0,T]\).
Let us assume on the contrary that, for some \(t_{0} \in[0,T]\),
If \(t_{0}\in\,]0,T[\), then \(\alpha'(t_{0}) = u'(t_{0})\); hence, \(\phi(\alpha '(t_{0}))=\phi(u'(t_{0}))\). We can find \(\omega>0\) such that for all \(t\in \,]t_{0},t_{0}+\omega[\), \(\alpha(t)>u(t)\). We have
It follows that, for all \(t\in\,]t_{0},t_{0}+\omega[\),
Since ϕ is an increasing homeomorphism, \(\phi(\alpha'(t))-\phi(u'(t))>0\Rightarrow\alpha'(t)-u'(t)>0\), a contradiction.
If \(t_{0}=0\), then
a contradiction with the definition of a lower solution.
If \(t_{0}=T\), then
a contradiction with the definition of a lower solution.
In consequence, we have that \(\alpha(t) \leq u(t)\) for all \(t\in [0,T]\). Therefore, u is a solution of problem (1)-(3).
Step 3: Existence of solutions of problem ( 1 )-( 3 ).
Let \(\Delta=[\alpha_{L},\alpha_{M}]\times[-a,a]\). Since f is an \(L^{1}\)-Carathéodory function, there exists \(\varphi\in L^{1}\) such that, for a.e. \(t \in[0, T]\) and all \((x, y) \in\Delta\), \(|f(t,x;y)|\leq \varphi(t)\).
Let
By (42), if \(u\in C^{1}\) is such that \(\|u'\|_{\infty}< a\) and \(u_{L}\geq R_{1}\), then
Moreover, if \(u\in C^{1}\) is such that \(\|u'\|_{\infty}< a\) and \(u_{M}\leq -R_{1}\), then
It follows that
Using (47), (48), and Theorem 4.1, we deduce that problem (46) has at least one solution, which is also a solution of problem (1)-(3) by step 2. □
Theorem 4.3
Assume that:
- (i)
-
(ii)
there \(R>0\) such that
$$ \begin{aligned} &u_{M}\leq-R\quad \textit{and} \quad\bigl\| u' \bigr\| _{\infty}< a \\ &\quad\Rightarrow\quad \int _{0}^{T}f\bigl(t,u(t),u'(t) \bigr)\,dt-g_{T}\bigl(u(T)\bigr)+g_{0}\bigl(u(0)\bigr)< 0. \end{aligned} $$(49)
Then problem (1)-(3) admits at least one solution.
Proof
The proof is similar to that of Theorem 4.2. □
Corollary 4.1
Assume that:
-
(a)
there exists \(A\in\mathbb{R}\) such that \(f(t,u,v)\geq A\) for a.e. \(t\in[0,T]\) and all \((u,v)\in\mathbb{R}\times[-a,a]\);
-
(b)
\({\lim_{x\rightarrow+\infty }}(g_{0}(x)-g_{T}(x))=+\infty\);
- (c)
Then problem (1)-(3) admits at least one solution.
Proof
By (a) we have
By (b) there exists \(R>0\) such that (42) is true. By Theorem 4.2 problem (1)-(3) admits at least one solution. □
Corollary 4.2
Assume that:
-
(a)
there exists \(A\in\mathbb{R}\) such that \(f(t,u,v)\leq A\) for a.e. \(t\in[0,T]\) and all \((u,v)\in\mathbb{R}\times[-a,a]\);
-
(b)
\({\lim_{x\rightarrow-\infty }}(g_{0}(x)-g_{T}(x))=-\infty\);
- (c)
Then problem (1)-(3) admits at least one solution.
Proof
The proof is similar to that of Corollary 4.1. □
Example 4.1
Consider the problem
We can see that \(|f(t,u,v)|\leq6\) for all \((t,u,v)\in[0,T]\times\mathbb {R}\times[-1,1]\), \(\alpha(t)=0\) is a lower solution, and \({\lim_{x\rightarrow +\infty}} (x^{2}+e^{x}-4)=+\infty\). Using Corollary 4.1, we deduce the existence of at least one solution.
References
Bereanu, C, Mawhin, J: Existence and multiplicity results for some nonlinear problems with singular ϕ-Laplacian. J. Differ. Equ. 243, 536-557 (2007)
Bereanu, C, Mawhin, J: Nonhomogeneous boundary value problems for some nonlinear equations with singular ϕ-Laplacian. J. Math. Anal. Appl. 352, 218-233 (2009)
Mawhin, J: Topological Degree Methods in Nonlinear Boundary Value Problems. CBMS Series, vol. 40. Am. Math. Soc., Providence (1979)
De Coster, C, Habets, P: Two-Point Boundary Value Problems: Lower and Upper Solutions. Elsevier, Amsterdam (2006)
Acknowledgements
The authors would like to thank the reviewers and the editors for their valuable suggestions and comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Goli, K.C.E., Adjé, A. New existence results for some periodic and Neumann-Steklov boundary value problems with ϕ-Laplacian. Bound Value Probl 2016, 170 (2016). https://doi.org/10.1186/s13661-016-0676-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-016-0676-6