Keywords:boundary value problem; equation unsolved with respect to the second derivative; Neumann boundary conditions; existence
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
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
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
In view of H1, from (3.4) we have
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
and the proof of the lemma is completed. □
By Lemma 3.1, we have
respectively. The obtained contradictions show that
which proves (a).
which proves (b) and completes the proof of the lemma. □
(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
Finally, from H3 we have that
are equivalent. Now from Lemma 3.1 we have
and Lemma 3.2 yields
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 .
Proof First, we observe that according to Lemma 3.3, the family of boundary value problems
Since , , is a continuous, linear, one-to-one 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, ). 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.
Example 4.2 Consider the boundary value problem
Thus, H2 also holds.
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same engagement.
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.
Cabada, A, Pouso, R: Existence result for the problem with periodic and Neumann boundary conditions. Nonlinear Anal.. 30, 1733–1742 (1997). Publisher Full Text
Chu, J, Lin, X, Jiang, D, O’Regan, D, Agarwal, RP: Positive solutions for second-order superlinear repulsive singular Neumann boundary value problems. Positivity. 12, 555–569 (2008). Publisher Full Text
Rachu̇nková, I, Staněk, S, Weinmüller, E, Zenz, M: Neumann problems with time singularities. Comput. Math. Appl.. 60, 722–733 (2010). Publisher Full Text
Hetzer, G, Stallbohm, V: Eine Existenzaussage für asymptotisch lineare Störungen eines Fredholmoperators mit Index 0. Manuscr. Math.. 21, 81–100 (1977). Publisher Full Text
Petryshyn, WV: Fredholm theory for abstract and differential equations with noncompact nonlinear perturbations of Fredholm maps. J. Math. Anal. Appl.. 72, 472–499 (1979). Publisher Full Text
Petryshyn, WV, Yu, ZS: Solvability of Neumann BV problems for nonlinear second order ODE’s which need not be solvable for the highest order derivative. J. Math. Anal. Appl.. 91, 244–253 (1983). Publisher Full Text
Petryshyn, WV: Solvability of various boundary value problems for the equation . Pac. J. Math.. 122, 169–195 (1986). Publisher Full Text
Kelevedjiev, P, O’Regan, D, Popivanov, N: On a singular boundary value problem for the differential equation using barrier strips. Nonlinear Anal.. 65, 1348–1361 (2006). Publisher Full Text
Fitzpatrick, PM: Existence results for equations involving noncompact perturbation of Fredholm mappings with applications to differential equations. J. Math. Anal. Appl.. 66, 151–177 (1978). Publisher Full Text
Fitzpatrick, PM, Petryshyn, WV: Galerkin method in the constructive solvability of nonlinear Hammerstein equations with applications to differential equations. Trans. Am. Math. Soc.. 238, 321–340 (1978)
Mao, Y, Lee, J: Two point boundary value problems for nonlinear differential equations. Rocky Mt. J. Math.. 26, 1499–1515 (1996). Publisher Full Text
Marano, SA: On a boundary value problem for the differential equation . J. Math. Anal. Appl.. 182, 309–319 (1994). Publisher Full Text
Petryshyn, WV, Yu, ZS: Periodic solutions of nonlinear second-order differential equations which are not solvable for the highest-order derivative. J. Math. Anal. Appl.. 89, 462–488 (1982). Publisher Full Text
Grammatikopoulos, MK, Kelevedjiev, PS, Popivanov, N: On the solvability of a singular boundary-value problem for the equation . J. Math. Sci.. 149, 1504–1516 (2008). Publisher Full Text