Abstract
Keywords:
boundary value problem; equation unsolved with respect to the second derivative; Neumann boundary conditions; existence1 Introduction
The purpose of this paper is to establish the existence of solutions to the scalar Neumann boundary value problem (BVP)
where the function and its first derivatives are continuous only on suitable subsets of the set .
The literature devoted to the solvability of singular and nonsingular Neumann BVPs for second order ordinary differential equations whose main nonlinearities do not depend on the second derivative is vast. We quote here only [15] for results and references.
The solvability of the homogeneous Neumann problem for the equation , under appropriate conditions on f, has been studied in [68]. Results, concerning the existence of solutions to the homogeneous and nonhomogeneous Neumann problem for the equation can be found in [9] and [10] respectively. BVPs for the same equation with various linear boundary conditions have been studied in [9,1113]. The results of [14] guarantee the solvability of BVPs for the equation with fully linear boundary conditions. BVPs for the equation with fully nonlinear boundary conditions have been studied in [15]. For results, which guarantee the solvability of the Dirichlet BVP for the same equation, in the scalar and in the vector cases, see [12] and [16] respectively.
Concerning the kind of the nonlinearity of the function , we note that it is assumed sublinear in [6], semilinear in [11] and linear with respect to x, p and q in [8,12]. Finally, in [9] and [17]f is a linear function with respect to q, while with respect to p, it is a quadratic function or satisfies Nagumo type growth conditions respectively.
As in [10,15,18,19], we use sign conditions to establish a priori bounds for x, and , where is a solution to a suitable family of BVPs similar to that in [10,19]. Using these a priori bounds and applying the topological transversality theorem from [20], we prove our main existence result.
2 Basic hypotheses
3 Auxiliary lemmas
In order to obtain our main existence results, we use the constant K from the hypotheses to construct the family of BVPs
where and prove the following three auxiliary results.
Lemma 3.1Let H1 hold andbe a solution to (3.1)_{λ}, . Then
Proof For , problem (3.1)_{0} is of the form
The unique solution to this BVP satisfies the bound
is a solution to the homogeneous boundary value problem
The equation is equivalent to the following one
Hence, by the intermediate value theorem, we obtain consecutively
and
Next, suppose that achieves its maximum at . Then the function has also a maximum at . Consequently, we have
Using the fact that , from (3.2) we obtain
In view of H1, from (3.4) we have
where
Suppose now that . Then, from (3.4) and (3.5) it follows that
if . Multiplying (3.6) and (3.7) by , we obtain
respectively. Finally, since we have . So
which contradicts (3.3). Thus, we infer that if achieves its maximum on , then
Let be the maximum of and suppose that . Following the above reasoning and using the fact that , we obtain
If , then and so is a strictly increasing function for , where is a sufficiently small neighbourhood of . So, we see that
i.e., is a strictly decreasing function for . Therefore, can not be the maximum of on , which is a contradiction. Assume next that . Then similar to the above arguments lead again to a contradiction. Thus, we see that
The inequality
can be obtained in the same manner. Consequently, the eventual solutions of (3.1)_{λ}, satisfy the bound
and the proof of the lemma is completed. □
Lemma 3.2Let H1 and H2 hold andbe a solution to (3.1)_{λ}, . Then:
Proof (a) Suppose there exists a or a such that
By Lemma 3.1, we have
In particular, (3.8) holds for and . Thus, in view of H2, we have
or
respectively. The obtained contradictions show that
and therefore
which proves (a).
(b) By the mean value theorem, for each there is a such that
Since, in view of (a), we have , from the last formula we find that
which proves (b) and completes the proof of the lemma. □
Lemma 3.3Let H1, H2 and H3 hold. Then there exists a functioncontinuous forand such that
(a) the BVP
is equivalent to BVP (3.1)_{λ}.
Proof (a) We write the differential equation from (3.1)_{λ} as
and consider the function
where . Since and , we can use H2 to conclude that
On the other hand, for we have
Finally, from H3 we have that
So, (3.10), (3.11) and (3.12) allow us to apply a wellknown theorem to conclude that there is a unique function which is continuous for and such that the equations
and
are equivalent. Now from Lemma 3.1 we have
and Lemma 3.2 yields
for and . Consequently, equation (3.9) is equivalent to the equation
which yields the first assertion.
4 The main result
Our main result is the following existence theorem, the proof of which is based on the lemmas of the previous sections and the Topological transversality theorem [20].
Theorem 4.1Let H1, H2 and H3 hold. Then problem (N) has at least one solution in.
Proof First, we observe that according to Lemma 3.3, the family of boundary value problems
is equivalent to the family (3.1)_{λ} for . Next define the set
Since , , is a continuous, linear, onetoone map of onto , the map , exists and is continuous. In addition, , , is a continuous and j is a completely continuous embedding. Since is a compact subset of , and , , and , , are continuous on and respectively, the homotopy
is compact. Besides, the equation
which coincides with BVP (3.13)_{λ}. Thus, the fixed points of are solutions to (3.13)_{λ}. But, from Lemma 3.1 and Lemma 3.2 it follows that the solutions to (3.13)_{λ} are elements of U. Consequently, , , is a fixed point free on ∂U, i.e., is an admissible map for all . Finally, we see that the map is a constant map, i.e., , where l is the unique solution to the BVP
From the fact that , it follows that is an essential map (see, [20]). By the Topological transversality theorem (see, [20]), is also essential, i.e., problem (3.13)_{1} has a solution. It is also a solution to (3.1)_{1}, by Lemma 3.3. To complete the proof, remark that problem (3.1)_{1} coincides with the problem (N). □
We conclude with the following example, which illustrates our main result.
Example 4.2 Consider the boundary value problem
It is clear that for the function
is continuous and and . Thus H1 holds for and .
To verify H2 we choose, for example, , and . Next we need the constants L and . Having in mind that and , from
it follows that . On the other hand, from
we have
and
Thus, H2 also holds.
Finally, H3 holds since and are continuous for .
Thus, we can apply Theorem 4.1 to conclude that the considered problem has a solution in .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same engagement.
Acknowledgements
In memory of Professor Myron K. Grammatikopoulos, 19382007.
This research was partially supported by Sofia University Grant N350/2012. The research of N. Popivanov was partially supported by the Bulgarian NSF under Grants DO 0275/2008 and DO 02115/2008.
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