Boundary value problems for strongly nonlinear equations under a Wintner-Nagumo growth condition
- Cristina Marcelli^{1}Email author and
- Francesca Papalini^{1}
Received: 4 August 2017
Accepted: 30 November 2017
Published: 12 December 2017
Abstract
Keywords
MSC
1 Introduction
The usual technique in this context is generally based on the method of lower and upper solutions, combined with some Nagumo-type growth condition, which is needed to ensure an a priori bound for the derivatives of the solutions in order to apply a suitable fixed point result. In the above quoted papers, the presence of the nonlinear term a inside the differential operator influences the requirements about the differential operator Φ, which was assumed to be homogeneous, or having at most linear growth at infinity.
In this context, by using a different approach with respect to [4], we are able to prove existence results for solutions of (1) subjected to very general boundary value conditions including, as particular cases, Dirichlet, periodic, Sturm-Liouville and Neumann problems. Our results extend those in [4] both for the presence of the function a inside the differential operators and for the great generality of the structure on the boundary conditions. Finally, we also provide some examples of application of our results, in which the operator Φ is not homogeneous and grows exponentially at infinity.
2 Preliminaries and auxiliary results
In this context, a solution for equation (4) is a function \(x \in C^{1}(I)\) with \(x(t)\in J\) for every \(t\in I\) such that the map \(t\mapsto a(t,x(t)) \Phi(x'(t))\) is absolutely continuous in I and \((a(t,x(t))\Phi(x'(t)))' = f(t,x(t),x'(t)) \) for a.e. \(t\in I\).
In what follows, we investigate the existence of solutions for equation (4) satisfying different boundary conditions. Our approach is based on fixed point techniques suitably combined to the method of upper and lower solutions, according to the following definition.
Theorem 1
Then, for every \(\nu_{1}, \nu_{2} \in \mathbb {R}\), there exists a function \(u \in C^{1}(I)\) with \(A_{u} \cdot(\Phi\circ u')\in W^{1,1}(I)\), a solution of problem (5).
3 The Dirichlet problem
Theorem 2
Finally, assume that there exists a well-ordered pair α and β of lower and upper solutions in I for equation (4).
Proof
Claim 1
There exists a solution to problem (19).
Claim 2
The solution u of problem (19) belongs to the functional interval \([\alpha,\beta]\).
Claim 3
\(\vert u'(t) \vert \le N\) for every \(t\in I\).
Summarizing, taking account of (20), we conclude that \(x_{c,d}:=u\) is a solution of problem (D) satisfying (12). □
As we mentioned in Introduction, the Wintner-Nagumo condition (11) is weaker than similar growth conditions assumed in many papers related to boundary value problems of similar type, also with respect to results in which the differential operator does not contain the function a. Moreover, it allows us to widen the range of the differential operators Φ we can consider that can be very general, not necessarily homogeneous, nor having polynomial growth. For instance, in the following example, we apply the existence result to an operator Φ having exponential growth.
Example 1
The following example shows an equation governed by a differential operator of the type considered in [10] and [8], and with a right-hand side which satisfies condition (11), but not other Nagumo-type growth conditions as those assumed in the mentioned papers.
Example 2
4 General nonlinear boundary conditions
We consider now more general boundary conditions in order to deal with periodic, Neumann, Sturm-Liouville boundary conditions for equation (4), using the result obtained in the previous section for Dirichlet problems, following the idea developed in [1]. To this aim, in what follows we adopt the notation \((D_{c,d})\) to denote the Dirichlet problem \((D)\), when we need to emphasize the values of the boundary conditions.
The following lemma provides a compactness-type result for the solutions of equation (4) obtained by means of Theorem 2.
Lemma 1
Proof
By applying Theorem 2 we are able to prove an existence result also for the general problem (22).
Theorem 3
Then problem (22) admits a solution x, belonging to the functional interval \([\alpha, \beta]\), such that \(\Vert x \Vert _{C^{1}}\le\Lambda\), where Λ is the constant given by Theorem 2 (see (12)), with respect to \(M:=\max\{{\max_{t \in I} \vert \alpha(t) \vert }, {\max_{t \in I} \vert \beta(t) \vert }\}\).
Proof
As an immediate consequence of Theorem 3, the following existence result holds.
Theorem 4
Proof
The assertion is an immediate consequence of Theorem 3, taking \(g(u,v,w,z):=w-z\) and \(h(r):=r\). □
Theorem 5
Then problem (28) has a solution belonging to the functional interval \([\alpha, \beta]\).
Proof
For every \(c\in[\alpha(0), \beta(0)]\) and \(d\in[\alpha(T), \beta(T)]\), let \(x_{c,d}\) denote a solution of problem \((D_{c,d})\) belonging to the functional interval \([\alpha,\beta]\), whose existence is ensured by Theorem 2 satisfying (12).
Observe now that, taking account of (29), when \(d=\alpha (T)\), we have \(q(y(T),y'(T))\ge q(\alpha(T),\alpha'(T))\ge0\) for every solution y of problem (\(Q_{d}\)); whereas for \(d=\beta(T)\), we have \(q(y(T),y'(T))\le q(\beta(T),\beta'(T))\le0\) for every solution y of problem (\(Q_{d}\)). Our goal is to show that there exists a value \(d^{*}\in[\alpha (T),\beta(T)]\) and a solution \(y_{*}\) of problem \((Q_{d^{*}})\) such that \(q(y_{d^{*}}(T),y_{d^{*}}'(T))= 0\).
Remark 2
Thus, Theorem 5 gives a condition for the existence of solutions for both these problems.
5 Conclusions
Declarations
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Funding
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Authors’ contributions
All authors read and approved the final manuscript. The contribution of the authors is equal.
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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