Second-order initial value problems with singularities
© Kelevedjiev and Popivanov; licensee Springer. 2014
Received: 30 January 2014
Accepted: 13 June 2014
Published: 26 September 2014
Using barrier strip arguments, we investigate the existence of -solutions to the initial value problem , , , which may be singular at and .
MSC: 34B15, 34B16, 34B18.
Here the scalar function is defined on a set of the form , where , , , , , , , and so it may be singular at and .
have been investigated by Rachůnková and Tomeček –. For example in , the authors have discussed the set of all solutions to this problem with a singularity at . Here , with , for and , f is locally Lipschitz on with the properties and for , where is a suitable constant.
where g, h, r, and w are suitable functions.
where k, F, and G are suitable functions.
The solvability of various IVPs has been studied also by Bobisud and O’Regan , Bobisud and Lee , Cabada and Heikkilä , Cabada et al. , , Cid , Maagli and Masmoudi , and Zhao . Existence results for problem (1.1), (1.2) with a singularity at the initial value of have been reported in Kelevedjiev-Popivanov .
where is suitable. Finally, we apply the Arzela-Ascoli theorem on the sequence of -solutions thus constructed to (1.3) to extract a uniformly convergent subsequence and show that its limit is a -solution to singular problem (1.1), (1.2). In the case , we establish -solutions with important properties - monotony and positivity.
which may be singular at . Note that despite the more general equation of this problem, the conditions imposed here as well as the results obtained are not consequences of those in .
2 Topological transversality theorem
In this short section we state our main tools - the topological transversality theorem and a theorem giving an important property of the constant maps.
So, let X be a metric space and Y be a convex subset of a Banach space E. Let be open in Y. The compact map is called admissible if it is fixed point free on ∂U. We denote the set of all such maps by .
A map F in is essential if every map G in such that has a fixed point in U. It is clear, in particular, every essential map has a fixed point in U.
(, Chapter I, Theorem 2.2])
Letbe fixed andbe the constant mapfor. Then F is essential.
We say that the homotopy, , is compact if the mapgiven byforis compact.
(, Chapter I, Theorem 2.6])
are compact maps.
- (iii), , is a compact homotopy joining F and G, i.e.
, , is fixed point free on ∂U.
Then, , has at least one fixed point in U and in particular there is asuch that.
3 Nonsingular problem
where , .
- (R)There exist constants , , , , , and a sufficiently small such that
Our first result ensures bounds for the eventual -solutions to (3.2). We need them to prepare the application of the topological transversality theorem.
Let us mention that some analogous results have been obtained in Kelevedjiev . For completeness of our explanations, we present the full proofs here.
Now we prove an existence result guaranteeing the solvability of IVP (3.1).
Let (R) hold. Then nonsingular problem (3.1) has at least one non-decreasing solution in.
Clearly, is continuous since is continuous.
which is the operator form of family (3.2). So, each fixed point of is a solution to (3.2), which, according to Lemma 3.1, lies in U. Consequently, the homotopy is fixed point free on ∂U.
Finally, is a constant map mapping each function to . Thus, according to Theorem 2.1, is essential.
from which its monotony follows. □
The validity of the following results follows similarly.
Letand let (R) hold for. Then problem (3.1) has at least one strictly increasing solution in.
Let () and let (R) hold for. Then problem (3.1) has at least one positive (nonnegative) non-decreasing solution in.
Let, and let (R) hold for. Then problem (3.1) has at least one strictly increasing solution inwith positive values for.
4 A problem singular at x and
In this section we study the solvability of singular IVP (1.1), (1.2) under the following assumptions.
where T, and are as in (S1).
Notice, for , that we have .
we will consider the proof for an arbitrary fixed , considering two cases. Namely, for is the first case and the second one is for with for some .
from which the assertion follows immediately. □
Having this lemma, we prove the basic result of this section.
which contradicts to the fact that and . This contradiction proves that (4.11) is true.
is a -solution to (1.1), (1.2). This function is strictly increasing because for , and the bounds for and follows immediately from the corresponding bounds for and . □
The following results provide information about the presence of other useful properties of the assured solutions. Their correctness follows directly from Theorem 4.2.
Letand let (S1) and (S2) hold. Then the singular IVP (1.1), (1.2) has at least one strictly increasing solution inwith positive values for.
Let us note that here , and .
and for .
we really can apply Theorem 3.2 to conclude that the considered problem has a strictly increasing solution with on for each .
and on , which means that (S2) also holds. By Theorem 4.3, the considered IVP has at least one positive strictly increasing solution in .
The work is partially supported by the Sofia University Grant 158/2013 and by the Bulgarian NSF under Grant DCVP - 02/1/2009.
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