The solvability of nonhomogeneous boundary value problems with ϕ-Laplacian operator
© Chen and Ma; licensee Springer. 2014
Received: 13 October 2013
Accepted: 1 April 2014
Published: 10 April 2014
We treat the nonhomogeneous boundary value problems with ϕ-Laplacian operator , , , , where () is an increasing homeomorphism such that , , , , , and is continuous. We will show that even if some of the τ and are negative, the boundary value problem with singular ϕ-Laplacian operator is always solvable, and the problem with a bounded ϕ-Laplacian operator has at least one positive solution.
where () is an increasing homeomorphism such that , , , , , and is continuous.
- (1)(regular unbounded): The p-Laplacian operator
- (2), (singular unbounded): The relativistic operator
- (3), (regular bounded): The one dimensional mean curvature operator
The study of the ϕ-Laplacian equations is a classical topic that has attracted the attention of many experts because of its interest in applications. Since 2004, in a number of papers, Bereanu and Mawhin have considered such problems with Dirichlet, Neumann or periodic boundary conditions (see, for example, [1–3] and the references therein). In these papers, the various boundary value problems are reduced to the search for fixed point of some nonlinear operators defined on Banach spaces. In particular, they have also considered some boundary value problems with nonhomogeneous boundary conditions, and they obtained the existence of solutions by the use of the Schauder fixed point theorem (see [2, 3]). Recently, Torres  proved the existence of a solution of a forced Liénard differential equation with ϕ-Laplacian by means of Schauder fixed point theorem.
We note here that many nonlinear differential problems require the search of positive, meaningful, solutions. The existence of positive solutions for ordinary differential equations and p-Laplacian equations have been studied by several authors and many interesting results has been obtained (only to mention some of them; see [5–7], and the references therein). If the coefficient occurring in the boundary conditions takes a negative value, then the existence of a positive solution for a BVP with ϕ-Laplacian operator is less considered because it is sometimes difficult to construct a corresponding cone for applying the fixed point theorem.
Some of the τ and coefficients in (1.1) are allowed to take a negative value.
In order to obtain a positive solution of the problem (1.1), we make a change of variable and generate two first-order differential equations and the corresponding nonlinear operator B. The new method can be used for the differential domains and ranges of and give an a priori estimate of the solution.
This paper is organized as follows. In Section 2, we give some preliminaries and lemmas. In Section 3, we show some theorems on the existence of (positive) solutions of the differential equations (1.1). Moreover, an example is given to illustrate our results.
2 Preliminaries and lemmas
Remark 2.1 From the deduction of (2.2)-(2.7), we find that if is well defined and x is a fixed point of the nonlinear operator B, then in (2.6) is a solution of the problem (1.1).
It is easy to verify that is continuous and takes bounded sets into bounded sets.
Lemma 2.1 (See )
for and for , or
for and for ,
then S has at least one fixed point in .
3 The main result
- (i)The nonlinearity satisfies
- (iii)There exists a positive constant r such that(3.1)
If , for all , then the condition (iv) clearly holds.
- (2)If , instead of conditions (i) and (ii), we assume that for any , the following inequalities hold:
Case I. Singular ϕ-Laplacian operator: ().
Theorem 3.1 If f is continuous, then the problem (1.1) has at least one solution.
Thus . Utilizing (2.9) and Arzela-Ascoli theorem, it is easy to verify that the nonlinear operator S is a completely continuous operator. Therefore, the nonlinear operator S has at least one fixed point by Schauder fixed point theorem.
Consequently, we conclude that the fixed point of S is a solution of the problem (1.1). □
Remark 3.2 Theorem 3.1 shows that if ϕ is singular () and f is continuous on , then the problem (1.1) is always solvable.
Case II. Bounded ϕ-Laplacian operator: , .
Theorem 3.2 If the conditions (i)-(iv) hold, then the nonlinear operator B defined by (2.7) has at least one fixed point. Further, the problem (1.1) has at least one positive solution.
Clearly, the nonlinear operator is well defined.
Consequently, we get from Remark 2.1 that the problem (1.1) has a positive solution . □
Remark 3.3 In order to prove the existence of a positive solution of the problem (1.1), we make a change of variable and introduce a first-order differential equation, and investigate the existence of a fixed point of the corresponding nonlinear operator B. The technique can be used for the different domains and ranges of and give an a priori estimate of the solution.
for any , then the problem (1.1) has at least one solution.
Then the operator has at least one fixed point by the use of Schauder fixed point theorem. Applying expression (2.6), we conclude that the problem (1.1) has at least one solution. □
Remark 3.4 Observe that the solution provided by Theorem 3.3 could be trivial or negative.
for any , then problem (1.1) has no solution.
Thus, it implies that there exists a neighborhood such that for any . This implies that the nonlinear operator is not well defined, since the domain of is the interval and thus a solution of (1.1) cannot exist. □
Then conditions (ii)-(iv) also hold. Therefore, we find from Theorem 3.2 that the differential equation (3.3) has at least one positive solution.
The authors would wish to express their appreciations to the anonymous referee for his/her valuable suggestions, which have greatly improved this paper. The work was partially supported by NSFC of China (No. 11201248), K.C. Wong Fund of Ningbo University.
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