Bernsteintype theorem for ϕLaplacian
Abstract
In this paper we obtain a solution to the secondorder boundary value problem of the form \(\frac{d}{dt}\varPhi'(\dot{u})=f(t,u,\dot{u})\), \(t\in [0,1]\), \(u\colon \mathbb {R}\to \mathbb {R}\) with Sturm–Liouville boundary conditions, where \(\varPhi\colon \mathbb {R}\to \mathbb {R}\) is a strictly convex, differentiable function and \(f\colon[0,1]\times \mathbb {R}\times \mathbb {R}\to \mathbb {R}\) is continuous and satisfies a suitable growth condition. Our result is based on a priori bounds for the solution and homotopical invariance of the Leray–Schauder degree.
Keywords
ΦLaplacian Boundary value problem Fixed point A priori bounds Leray–Schauder degree1 Introduction
The solvability of various secondorder twopoint BVPs with p or ΦLaplacian has been discussed extensively in the literature, see the recent works [1, 2, 3, 4, 5, 6, 7, 8, 9] for results, methods, and references.
The uniqueness of the solution to (1.4), (BC) follows from the assumption that the partial derivatives \(f_{u}\) and \(f_{v}\) exist, are bounded, and \(f_{u}\geq 0\) on \([0,1]\times \mathbb {R}^{2}\).
In 1983, Baxley [12] proved Bernsteintype theorems for boundary value problems for (1) with nonlinear boundary conditions. In 1988, Frigon and O’Regan [13] established existence results of this type for (1), (2) and (1), (BC).
 (\(\varPhi _{1}\))

Φ is strictly convex, differentiable and \(\varPhi (x)/\vert x\vert \to \infty \) as \(\vert x\vert \to \infty \);
 (\(\varPhi _{2}\))

\(\varPhi (0) = \varPhi '(0) = 0\);
 (\(\varPhi _{3}\))

\((\varPhi ')^{1}\) is continuously differentiable;
 (\(\varPhi _{4}\))
 there exists a constant \(K_{\varPhi }>1\) such that$$ K_{\varPhi }\varPhi (x)\leq \varPhi '(x) x\quad \text{for all $x\in \mathbb {R}$.} $$
 (\(f_{1}\))
 There exists a constant \(M>0\) such that$$ x f(t,x,0) > 0 \quad \text{for $ \vert x \vert >M$,} $$
 (\(f_{2}\))
 There exist positive functions S, T bounded on bounded sets such that$$ \bigl\vert f(t,x,v) \bigr\vert \leq S(t,x) \bigl(\varPhi '(v) \cdot v\varPhi (v)\bigr) + T(t,x). $$
Now, we can state our main result.
Main Theorem
Suppose thatΦandfsatisfy (\(\varPhi _{1}\))–(\(\varPhi _{4}\)) and (\(f_{1}\)), (\(f_{2}\)), respectively. Then problem (P), (BC) has at least one solution in\(C^{2}([0,1],\mathbb {R})\).
To establish the validity of the above result, we apply the Leray–Schauder degree theory on a suitable constructed map. To define its domain, we use a priori bounds.
To prove the existence, we use topological methods. This approach has already been used by many authors. In [11] and [13] the authors considered the case of a Laplace operator with various boundary conditions. Generalizations to the pLaplacian and to the operator defined by an arbitrary increasing homeomorphism were developed in [3] and [5], respectively. The main idea in the paper [11] was to use the topological transversality theorem. This is a fixed point type theorem (see [15]). We decided to use an approach via Leray–Schauder degree theory instead, since it is essentially equivalent but the degree theory is familiar to a broader audience.
However, in [3] and [5] authors subject the equation to very specific boundary conditions, namely \(u(0) = A\), \(\dot{u}(1) = B\). In order to show the existence for general Sturm–Liouville conditions, more effort has to be put in as can be seen below.
2 Auxiliary results
Lemma 2.1
 (1)
for every\(v \in X\), function\(g_{v}\colon \mathbb {R}\to \mathbb {R}\), defined by\(g_{v}(c) = G(v,c)\), is an increasing homeomorphism;
 (2)
if\(\{v_{n}\}\)is bounded and\(b_{n} \to \pm \infty \), then\(G(v_{n},b_{n}) \to \pm \infty \).
Proof
Suppose that function c is not continuous, i.e., there exist \(\epsilon > 0\) and a sequence \(v_{n}\) converging to some \(v_{0}\) such that \(\vert c(v_{n})  c(v_{0})\vert > \epsilon \). By the definition of c, \(G(v_{n},c(v_{n})) = C\). In particular, both \(v_{n}\) and \(G(v_{n},c(v _{n}))\) are bounded. This, together with (2), implies that \(c(v_{n})\) is bounded. Take a subsequence \(c(v_{n_{k}})\) which converges to some \(c'\). Note that \(c' \neq c(v_{0})\) because \(\vert c(v_{n})  c(v_{0})\vert > \epsilon \). By the continuity of G, we have \(G(v_{n_{k}}, c(v_{n _{k}})) \to G(v_{0},c')\). But \(G(v_{n_{k}}, c(v_{n_{k}})) = C\) and \(G(v_{0},c') \neq G(v_{0},c(v_{0})) = C\) by (1). A contradiction. □
If \(g_{v}\) is differentiable and \(g_{v}'\) is positive, then the conclusion follows from implicit function theorem. However, in the problem that we consider, \(g'_{v}\) is only nonnegative.
Remark 2.2
Note that this trivializes in [3, 5]. For boundary conditions considered therein \(c_{1}\) and \(c_{2}\) are constants independent of v. We cannot proceed in such a way here.
For every v, we would like to choose \(c_{1}\) and \(c_{2}\) in such a way that \(u = \hat{K}(v,c_{1},c_{2})\) is an element of \(C^{1}_{\mathrm{BC}}\). Moreover, we need that \(c_{1}\) and \(c_{2}\) depend continuously on v.
Lemma 2.3
Let (\(\varPhi _{1}\)) and (\(\varPhi _{3}\)) hold. Then, for every fixed\(v \in C^{0}([0,1])\), there exists a unique pair of constants\(c_{1}(v)\), \(c_{2}(v)\)such that\(\hat{K}(v,c_{1}(v),c_{2}(v)) \in C^{1}_{\mathrm{BC}}([0,1])\). Moreover, the functions\(c_{1},c_{2}\colon C ^{0}([0,1]) \to \mathbb {R}\)are continuous.
Proof
The next lemma is a variant of [13, Theorem 3.3].
Lemma 2.4
Proof

\(u(t)>M\) for \(t\in (t_{0}^{},t_{0}^{+})\),

\(\dot{u}(t)\geq 0\) on \((t_{0}^{},t_{0}]\),

\(\dot{u}(t)\leq 0\) on \([t_{0},t_{0}^{+})\).
Lemma 2.5
Proof
Now we provide bounds for u̇. The proof of the following theorem is based on [13].
Lemma 2.6
Proof
Assume that \(\dot{u}(\mu )=\dot{u}(t_{0})\) and \(\dot{u}(t)\geq 0\) for every \(t\in [\mu ,\nu ]\). The other cases are treated similarly and the same bound is obtained.
3 Proof of the main theorem
4 Examples
Example 4.1
Let \(\varPhi \colon \mathbb {R}\to \mathbb {R}\), \(\varPhi (x)=\frac{1}{p}\vert x\vert ^{p}\), \(1< p\leq 2\). It is easy to see that the function Φ satisfies assumptions (\(\varPhi _{1}\))–(\(\varPhi _{3}\)). Moreover, since \(\frac{1}{p}\vert x\vert ^{p}\leq x^{2}\vert x\vert ^{p2}\), one can take \(K_{\varPhi }=p\) in (\(\varPhi _{4}\)).
Example 4.2
Let \(\varPhi (x)=\sum_{i=1}^{n}\frac{1}{p_{i}}\vert x\vert ^{p_{i}}\), \(1< p_{i} \leq 2\), for \(i=1,2,\ldots,n\). Then Φ satisfies assumptions (\(\varPhi _{1}\))–(\(\varPhi _{3}\)). Assumption (\(\varPhi _{4}\)) is satisfied with \(K_{\varPhi }=\min \{p_{1},\ldots,p _{n}\}\).
Example 4.3
Notes
Acknowledgements
The authors would like to thank Professor Andrzej Granas for suggesting this problem during the seminar in Gdańsk in 2016.
Availability of data and materials
Not applicable.
Authors’ contributions
The three authors contributed equally in this paper. All authors read and approved the final manuscript.
Funding
M. Starostka was partially supported by Grants Beethoven2 and Preludium9 of the National Science Centre, Poland, no. 2016/23/G/ST1/04081 and no. 2015/17/N/ST1/02527.
Competing interests
The authors declare that they have no competing interests.
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