Nonlinear and Variational Analysis and their Applications 2021View this Special Issue
Iterative Solutions for the Differential Equation with -Laplacian on Infinite Interval
This paper systematically investigates a class of fourth-order differential equation with -Laplacian on infinite interval in Banach space. By means of the monotone iterative technique, we establish not only the existence of positive solutions but also iterative schemes under the suitable conditions. At last, we give an example to demonstrate the application of the main result.
The partial differential equation with the -Laplacian operator which is often used to describe, for example, diffusion process , with a spatial symmetric potential , can be reduced to , where . This fact leads us to study the following -Laplacian boundary value problem (BVP): where , , , , are real constants, is a Lebesgue integrable function with , , is continuous and may be singular at , and is a continuous function.
The -Laplacian equation arises quite naturally in the modeling of different physical and natural phenomena. For instance, in fluid mechanics, the shear stress and the velocity gradient of certain fluids obey a relation of the form , where . Here, the real number and (respectively, ) designate a Newtonian (respectively, pseudoplastic, dilatant) fluid. Given is a constant, the resulting equations of motion then involve , which reduces to . Over the last couple of decades, many important results including integral and fractional equations with -Laplacian on certain boundary value conditions had been obtained. We refer the reader to [2–17] and the references cited therein. Liang and Zhang  considered the -point BVP with a -Laplacian operator where , with , and satisfies , , and has countably many singularities in , is a continuous function. The existence of positive solutions is obtained by applying the fixed-point theorem of Leggett-Williams.
Using the fixed point index theory, Xu and Yang in  researched the existence of positive solutions for the fourth order -Laplacian BVP where , is a continuous function.
The motivation for the present work stems from both practical and theoretical aspects. In fact, many mathematical problems in science and engineering are set in unbounded domains, such as unsteady flow of gas through a semi-infinite porous media, the theory of drain flows, plasma physics, in determining the electrical potential in an isolated neutral atom. In all these applications, it is frequent that only positive solutions are useful. In this paper, we study the differential equation with -Laplacian operator as BVP (2); when , BVP (2) becomes the ordinary fourth-order differential equation. The results for the existence of the maximal and minimal solutions to the BVP (2) are established. In addition, we establish iterative schemes for approximating the solutions, which start from the known simple linear functions. However, to the best knowledge of the authors, there are few works in the literature dealing with the existence of positive solutions to boundary value problems of differential equation on infinite intervals with -Laplacian operator by using iterative technique up to now. The goal of the present paper is to fill the gap in this area, so it is interesting and important to study the existence of positive solutions for BVP (2).
2. Preliminaries and Lemmas
The basic space used in this paper is , where is denoted by
Then, is a Banach space equipped with the norm . Define a cone in the Banach space by
Lemma 1 (see ). Let , then is a solution of if and only if is a solution of where
Lemma 2. Green’s function satisfies
(1), for any (2), for any In what follows, we list some conditions for convenience.
is a Lebesgue integrable function with , .
is a continuous function, on and is bounded, for , , is a closed subinterval.
is continuous, on and
By routine discussion, Lemma 3 is valid.
Lemma 3. Assume that holds and ; then, is a solution of if and only if is a solution of
Let , then BVP (2) is divided into the following two parts:
Now, we claim that is well defined for . In fact, for any , there exists , such that . From and the definition of , we have
Thus, by , we know
Together with (15), for any , we can see that
On the other hand, for any , by (13), we have
Therefore, , for any . Hence, is well defined. Obviously, is a positive solution of BVP (2) if and only if is a fixed point of in .
The Arzela-Ascoli theorem fails to work in the Banach space due to the fact that the infinite interval is noncompact. The following compactness criterion will help us to resolve this problem.
Lemma 4 (see [21, 22]). Let be defined as (5) and be any bounded subset of . Then, is relatively compact in if is equicontinuous on any finite subinterval of , and for any given , there exists , such that uniformly with respect to , as .
3. Main Results
Lemma 5. Assume that hold. Then, is completely continuous.
Proof. It is clear that for any , . Thus, . Now, we prove that is continuous and compact, respectively. Let be a bounded subset. Then, there exists , such that , for any . So, for any , we have
So, is bounded in . Moreover, given , for any and , without loss of generality, we may assume that . In fact,
So, for any , there exists , such that for any with , and for any , we have
Hence, is equicontinuous on . Since is arbitrary, is locally equicontinuous on .
Next, we prove that we show that is equiconvergent at . For any , we have So, for any , we have Thus, for any , there exists , for any and for any , such that Consequently, for any and for any , we have Therefore, for any and for any , we have This implies that is equiconvergent at .
Let as ; then, there exists such that , is a natural number set. By , we have where So, for any , we can find a sufficiently large , such that It follows from the Lebesgue dominated convergence theorem and continuity of , we can get So, for the above , there exists , when , we have Therefore, when , which implied that as . Since Then, by the Lebesgue dominated convergence theorem, we get Therefore, is continuous. In conclusion, by Lemma 4, we know that is completely continuous. The proof is completed.☐
Theorem 6. Assume that hold and there exists which satisfies the following condition:
, , .
where Then, BVP (2) has the maximal and minimal positive solutions and on , such that Moreover, for initial values define the iterative sequences and by Then,
Proof. From Lemma 5, we know that is completely continuous. For any with , from the definition of and , we know that . Let . In what follows, we firstly prove . In fact, for any , we have . By , we know that . Also, by , we have
Thus, we have proved that . Let , and then, . Let , ; by Theorem 6, we have . Denote Since , we have . It follows from the complete continuity of that is a sequentially compact set. By , we have
Together with , we also have . By induction, we obtain
Therefore, there exists such that as . Applying the continuity of and , we get that .
On the other hand, let , , and then . Let , ; then, by Theorem 6, we have . Denote . Since , we have . It follows from the complete continuity of that is a sequentially compact set. Since , we have By induction, we get Thus, there exists such that