A Fourth-Order Boundary Value Problem with One-Sided Nagumo Condition
© W. Song and W. Gao. 2011
Received: 10 January 2011
Accepted: 9 March 2011
Published: 14 March 2011
The aim of this paper is to study a fourth-order separated boundary value problem with the right-hand side function satisfying one-sided Nagumo-type condition. By making a series of a priori estimates and applying lower and upper functions techniques and Leray-Schauder degree theory, the authors obtain the existence and location result of solutions to the problem.
with being a continuous function.
with , .
the authors in  obtained the existence of solutions with the assumption that satisfies the two-sided Nagumo-type conditions. For more related works, interested readers may refer to [1–14]. The one-sided Nagumo-type condition brings some difficulties in studying this kind of problem, as it can be seen in [15–18].
Motivated by the above works, we consider the existence of solutions when satisfies one-sided Nagumo-type conditions. This is a generalization of the above cases. We apply lower and upper functions technique and topological degree method to prove the existence of solutions by making a priori estimates for the third derivative of all solutions of problems (1.1) and (1.2). The estimates are essential for proving the existence of solutions.
The outline of this paper is as follows. In Section 2, we give the definition of lower and upper functions to problems (1.1) and (1.2) and obtain some a priori estimates. Section 3 will be devoted to the study of the existence of solutions. In Section 4, we give an example to illustrate the conclusions.
2. Definitions and A Priori Estimates
Upper and lower functions will be an important tool to obtain a priori bounds on , , and . For this problem we define them as follows.
define a pair of lower and upper functions of problems (1.1) and (1.2) if the following conditions are satisfied:
(i) , ,
(ii) , ,
that is, lower and upper functions, and their first derivatives are also well ordered.
To have an a priori estimate on , we need a one-sided Nagumo-type growth condition, which is defined as follows.
Let be a continuous function satisfying one-sided Nagumo-type condition in .
for and every , one has .
Hence, . Since can be taken arbitrarily as long as , we conclude that for every provided that .
Taking , we have .
Observe that the estimation depends only on the functions , , , and and it does not depend on the boundary conditions.
3. Existence and Location Result
In the presence of an ordered pair of lower and upper functions, the existence and location results for problems (1.1) and (1.2) can be obtained.
for , for some independent of .
The case is analogous. Thus, for every . In a similar way, we may prove that for every .
with independent of .
Therefore, satisfies the one-sided Nagumo-type condition in with replaced by , with independent of .
The hypotheses of Lemma 2.4 are satisfied with replaced by . So there exists an , depending on and , such that for every . As and do not depend on , we see that is maybe independent of .
For , the problems (3.6) and (3.7) has at least one solution .
has at least one solution in .
The function is a solution of the problems (1.1) and (1.2).
and so for every .
The inequalities and for every can be proved in the same way. Then is a solution of problems (1.1) and (1.2).
4. An Example
The following example shows the applicability of Theorem 3.1 when satisfies only the one-sided Nagumo-type condition.
Then satisfies condition (3.2) and the one-sided Nagumo-type condition with , in .
does not satisfy the two-sided Nagumo condition.
The authors would like to thank the referees for their valuable comments on and suggestions regarding the original manuscript. This work was supported by NSFC (10771085), by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education, and by the 985 Program of Jilin University.
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