On a certain way of proving the solvability for boundary value problems
© Lepin; licensee Springer. 2014
Received: 13 December 2013
Accepted: 28 April 2014
Published: 13 May 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.
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.
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 , .
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.
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.
then there exists such that solves the boundary value problem (1)-(2).
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.
are classical solutions.
where if , if , if .
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.
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