On Nonlinear Boundary Value Problems for Functional Difference Equations with -Laplacian
Sufficient conditions for the existence of solutions of nonlinear boundary value problems for higher-order functional difference equations with -Laplacian are established by making of continuation theorems. We allow to be at most linear, superlinear, or sublinear in obtained results.
The existence of solutions of boundary value problems for finite difference equations were studied by many authors, one may see the text books [1, 2], the papers [3–5] and the references therein. We present some representative ones, which are the motivations of this paper.
In papers [3, 4], using Krasnoselskii fixed point theorem and Leggett-Williams fixed point theorem, respectively, Karakostas studied the existence of three positive solutions of the problems consisting of the functional differential equation
and one of the following three pairs of conditions:(H1)for each the function is continuous nondecreasing and such that and at least one of the following:(H2),(H3),(H4),(H5).
subject to one of the following boundary conditions:
The question follows: under what conditions above BVP (1.3) has solutions if (H1)–(H5) are not satisfied?
Particular significance lies in the fact that when a BVP is discretized, strange and interesting changes can occur in the solutions. For example, properties such as existence, uniqueness and multiplicity of solutions may not be shared between the continuous differential equation and its related discrete difference equation. Moreover, when investigating difference equations, as opposed to differential equations, basic ideas from calculus are not necessarily available to use, such as the intermediate value theorem, the mean value theorem and Rolles theorem. Thus, new challenges are faced and innovation is required .
In recent paper , Liu studied the solvability of the following problem consisting of the higher-order functional difference equation and boundary conditions
where , with and , , , are sequences,
is continuous in for each . Two cases, that is, or are considered in .
subject to the following boundary conditions
where , for and with , the inverse function is denoted by , an integer, are continuous and satisfy for all , , , , are integer vectors,
is continuous about for each . Boundary condition (1.8) is called nonlinear Sturm-Liouville type conditions.
The purposes of this paper are to establish sufficient conditions for the existence of at least one solutions of BVP (1.7)-(1.8). It is interesting that we allow that to be sublinear, at most linear or superlinear. We do not need the assumptions (H2)–(H5) imposed on .
2. Main Results
To get existence results for solutions of BVP (1.7)-(1.8), we need the following fixed point theorem, which was used to solve multi-point boundary value problems for differential equations in many papers but not used to solve boundary value problems for difference equations.
Let and be real Banach spaces, let be a Fredholm operator of index zero. If is an open bounded subset of , , the map will be called -compact on if is bounded and compact.
Lemma 2.1 (see ). Let and be Banach spaces. Suppose is a Fredholm operator of index zero with , is -compact on any open bounded subset of . If is an open bounded set with and
then there is at least one so that .
Let be endowed with the norm
Let be endowed with the norm for .
It is easy to see that and are real Banach spaces.
Let and and by for all .
Since , are continuous, it is easy to show that(i) is a solution of implies that is a solution of BVP (1.7)-(1.8),(ii),(iii) is a Fredholm operator of index zero and is -compact on each with being an open bounded subset of .
Lemma 2.2. for all and , where is defined by for and for .
Proof. We have the following cases.
Case 1 (). Without loss of generality, suppose . Let then and for , we get We get that for all and so for all . Hence .
Case 2 (). It is easy to see that The proof is complete.
Proof. To apply Lemma 2.1, we divide the proof into two steps.
Let . For , we have , , then
Step 1. We will show that if for some , then is bounded. Indeed, we see that Since we get
It is easy to see from (2.12) and the definition of and that
So we get It follows from the assumptions that For , we have Holder’s inequality It follows from Lemma 2.2 that It follows from (2.11) that there is such that .
Hence for all which proves Step 1.Step 2. We will show that the set is bounded. We first prove that there exists such that . In fact, if for all , then implies that is increasing on , so . Then assumption (A) and imply that and . This contradicts . If for all , the similar contradiction can be deduced. Hence there exists such that .
It follows that there is a constant such that
Then Hence Then for one sees that For , we get It follows that Then On the other hand, since implies that then It follows that Hence Step 2 is proved, namely is bounded. Let be an open bounded subset of centered at zero, it is easy to see that for all and It follows from Lemma 2.1 that equation has at least one solution , then is a solution of BVP (1.7)-(1.8). The proof is complete.
3. An Example
In this section, we present an example to illustrate the main results in Section 2.
Example 3.1. Consider the following BVP: where for and with a constant, is an integer, , are sequences. Corresponding to the assumptions of Theorem 2.3, we set It is easy to see that assumptions (A) and (B) in Theorem 2.3 hold. It follows from Theorem 2.3 that (3.1) has at least one solution if
The second author was partially supported by Natural Science Foundation of Hunan province, China (no. 06JJ5008) and Natural Science Foundation of Guangdong province (no. 7004569).
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