Research

# On a certain way of proving the solvability for boundary value problems

Arnold Y Lepin

Author Affiliations

Institute of Mathematics and Computer Science, University of Latvia, Riga, Latvia

Boundary Value Problems 2014, 2014:111  doi:10.1186/1687-2770-2014-111

 Received: 13 December 2013 Accepted: 28 April 2014 Published: 13 May 2014

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

### Abstract

A certain way of replacing a given boundary value problem by another one, a solution of which solves also the original problem, is considered.

MSC: 34B15.

##### Keywords:
boundary value problems; upper and lower functions

### Research

Consider the solvability of the boundary value problem (BVP)

(1)

(2)

where is strictly increasing in for fixed t and x, satisfies the Caratheodory conditions, that is, is measurable in I for fixed , is continuous on for fixed , and for any compact set there exists function such, that for any , the estimate holds, , , α is the lower function, β the upper function.

This boundary value problem is replaced by another one, which is dependent on the parameter , ,

(3)

where is strictly increasing in for fixed t and x, and satisfies the Caratheodory conditions.

Definition 1 A function is a solution of (1), if is absolutely continuous on I and (1) is satisfied almost everywhere on I.

We provide below definitions of generalized upper and lower functions and the generalized solution along with Theorem 1 from [1-3]. This is needed to prove the main result.

Definition 2 The class consists of functions , which possess the property: for any there exist the left derivative and the limit , and ; for any there exist the right derivative and the limit , and , and, for any , .

The class consists of functions , which possess the following property: for any there exist the left derivative and the limit , and ; for any there exist the right derivative and the limit , and , and, for any , .

Definition 3 We call a bounded function a generalized lower function and write , if in any interval , where this function satisfies the Lipschitz condition, for any and where the derivative exists, the inequality

holds. We will call a bounded function a generalized upper function and write , if in any interval , where this function satisfies the Lipschitz condition, for any and where the derivative exists, the inequality

holds.

A function will be called a generalized solution, if .

A generalized solution has a derivative at any point, possibly infinite, either −∞ or +∞, and is continuous on ; if in some interval the derivative does not attain the values −∞ or +∞, then x is a solution of (1) in this interval.

Theorem 1Let, and. Then for anyandthere exists a generalized solution of the Dirichlet problem

(4)

In addition to conditions on α and β the compactness conditions are needed for solvability of the boundary value problem (1)-(2). The Nagumo condition [4] for φ-Laplacian and the Schrader condition [5] are sufficient conditions for compactness of a set of solutions. We accept the following compactness conditions.

Definition 4 We say that the compactness condition is fulfilled, if for all and any generalized solution of the Dirichlet problem (4) is a solution.

It is clear that this condition is weaker than the Schrader condition.

A set of solutions of the Dirichlet problem (4) will be denoted by S.

Remark 1 If , , and the compactness condition is fulfilled, then the Dirichlet problem (4) has a solution.

Theorem 2Let, and the compactness condition be fulfilled. If the boundary value problem (3) has a solutionfor alland for

then there existssuch thatsolves the boundary value problem (1)-(2).

Proof Notice that the results in [6] imply that . Suppose the contrary. Let the sequence , where , tend to infinity. Consider the sequence , where ,  . We can assume, without loss of generality, that it converges in any rational points of the interval to the function u, located between α and β. Notice that without loss of generality for any interval it follows from the boundedness of u and the Mean Value Formula that there exists an interval such that

It is clear that , , and u satisfy the Lipschitz condition with constant L in . The u can be extended by continuity to the entire interval , and thus we obtain a function u that satisfies the Lipschitz condition. It follows from the Lipschitz condition that converges to for any . It is clear that the derivatives converge to the derivative for any . Therefore, is a solution of (1) in the interval . Continuing the construction of on both sides, one gets a solution of (1) on the maximal interval . If , then is either −∞ or +∞. Similarly, if , then is either −∞ or +∞. If and is not −∞ or +∞, then can be continued to a. Similarly, if and is not −∞ nor +∞, then can be continued to b. By repeating this construction, find an open set in I, where the function is defined and is a solution of (1) on intervals from . A set is closed and nowhere dense. For the limit is equal to −∞ or +∞. Indeed, assuming the contrary and acting as above, we get . Extend to irrational points of . If , then , and in the remaining cases . The above limits exist since is monotone in neighborhood of any point from . Similarly we get for , and . Therefore is a generalized solution of (1). It follows from the compactness condition that is a solution of (1). Let us show that the sequence uniformly converges to . Suppose the contrary is true. We assume, without loss of generality, that there exist and a sequence , where , such that , and . Consider the case ,  . We can assume, without loss of generality, that ,  , and this contradicts the equality . The uniform convergence is proved. We can conclude now that all are the solutions of the boundary value problem (1)-(2). □

Remark 2 Theorem 2 gives the possibility to prove the solvability of boundary value problems if the solvability of more simple boundary value problems is known.

Remark 3 If and the inequalities hold for a solution x of the boundary value problem (1)-(2), then the compactness condition (Definition 4) can be weakened.

Definition 5 We will say that the compactness condition holds if for any and all generalized solutions of the problem

are classical solutions.

Example One way to use Theorem 2 is to verify that for all , and , , the following conditions are satisfied:

where if , if , if .

### Competing interests

The author declares that he has no competing interests.

### Authors’ contributions

The author participated in drafting, revising and commenting on the manuscript. The author read and approved the final manuscript.

### Acknowledgements

The author sincerely thanks the reviewers for their valuable suggestions and useful comments. This research was supported by the Institute of Mathematics and Computer Science, University of Latvia.

### References

1. Lepin, LA: Generalized solutions and solvability of boundary value problems for the second order differential equations. Differ. Equ.. 18, 1323–1330 (1982)

2. Lepin, AY, Lepin, LA: Boundary Value Problems for Ordinary Differential Equations, Zinatne, Riga (1988) (Russian)

3. Lepin, AY, Lepin, LA: Generalized lower and upper functions for φ-Laplacian equations. Differ. Equ. (2014, in press)

4. Nagumo, M: Über die Differentialgleichung . Proc. Phys. Math. Soc. Jpn.. 19(3), 861–866 (1937)

5. Schrader, KW: Existence theorems for second order boundary value problems. J. Differ. Equ.. 5(3), 572–584 (1969). Publisher Full Text

6. Lepin, AY: Compactness of generalized solutions between the generalized lower and upper functions. Proc. LUMII Math. Differ. Equ.. 11, 22–24 (2011)