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On a certain way of proving the solvability for boundary value problems
Boundary Value Problems volume 2014, Article number: 111 (2014)
A certain way of replacing a given boundary value problem by another one, a solution of which solves also the original problem, is considered.
Consider the solvability of the boundary value problem (BVP)
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 , ,
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.
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
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 1 Let , and . Then for any and there exists a generalized solution of the Dirichlet problem
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  for φ-Laplacian and the Schrader condition  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 2 Let , and the compactness condition be fulfilled. If the boundary value problem (3) has a solution for all and for
then there exists such that solves the boundary value problem (1)-(2).
Proof Notice that the results in  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 .
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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.
The author declares that he has no competing interests.
The author participated in drafting, revising and commenting on the manuscript. The author read and approved the final manuscript.