On the solvability of a Neumann boundary value problem for the differential equation
© Palamides et al.; licensee Springer 2012
Received: 5 July 2012
Accepted: 6 July 2012
Published: 23 July 2012
Using barrier strip arguments, we investigate the existence of -solutions to the Neumann boundary value problem , , .
Keywordsboundary value problem equation unsolved with respect to the second derivative Neumann boundary conditions existence
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 [1–5] for results and references.
The solvability of the homogeneous Neumann problem for the equation , under appropriate conditions on f, has been studied in [6–8]. Results, concerning the existence of solutions to the homogeneous and nonhomogeneous Neumann problem for the equation can be found in  and  respectively. BVPs for the same equation with various linear boundary conditions have been studied in [9, 11–13]. The results of  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 . For results, which guarantee the solvability of the Dirichlet BVP for the same equation, in the scalar and in the vector cases, see  and  respectively.
Concerning the kind of the nonlinearity of the function , we note that it is assumed sublinear in , semilinear in  and linear with respect to x, p and q in [8, 12]. Finally, in  and 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 , we prove our main existence result.
2 Basic hypotheses
where is as in H1.
H3. The functions and are continuous for , where , and is as in H1.
3 Auxiliary lemmas
where and prove the following three auxiliary results.
where , , , and , , , .
and the proof of the lemma is completed. □
, , .
- (b)By the mean value theorem, for each there is a such that
which proves (b) and completes the proof of the lemma. □
- (a)the BVP
It follows immediately from for . □
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 .
Theorem 4.1 Let H 1, H 2 and H 3 hold. Then problem (N) has at least one solution in .
From the fact that , it follows that is an essential map (see, ). By the Topological transversality theorem (see, ), 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.
is continuous and and . Thus H1 holds for 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 .
In memory of Professor Myron K. Grammatikopoulos, 1938-2007.
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 02-75/2008 and DO 02-115/2008.
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