Journal of Applied Mathematics and Physics
Vol.07 No.06(2019), Article ID:93438,29 pages
10.4236/jamp.2019.76091
Existence and Multiplicity Results for Second-Order Nonlinear Differential Equations with Multivalued Boundary Conditions
Droh Arsene Behi, Assohoun Adje
Faculty of Mathematics and Computer Science, Felix Houphouet Boigny University, Cocody Abidjan, Ivory Coast
Copyright © 2019 by author(s) and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: May 16, 2019; Accepted: June 27, 2019; Published: June 30, 2019
ABSTRACT
In this paper, we consider the following second-order nonlinear differential equations’ problem: a.e on with a discontinuous perturbation and multivalued boundary conditions. By combining lower and upper solutions method, theory of monotone operators and theory of topological degree, we show the existence of solutions of the investigated problem in two cases. At first, and are assumed respectively an ordered pair of lower and upper solutions of the problem, secondly and are assumed respectively non ordered pair of lower and upper solutions of the problem. Moreover, we show multiplicity results when the problem admits a pair of lower and strict lower solutions and a pair of upper and strict upper solutions. We also show that our method of proof stays true for a periodic problem.
Keywords:
-Laplacian, Lower and Upper Solutions, Maximal Monotone Maps, Bernstein-Nagumo-Wintner Growth Condition, Leray-Schauder Topological Degree
1. Introduction
This paper is devoted to the study of the following problem:
(1)
where and are maximal monotone graphs in and are some multifunctions which describe the boundary conditions, is a Caratheodory function and is an increasing homeomorphism such that .
The tool of investigation for this problem is lower and upper solution’s method. This method provides a precious tool to get existence results for first and second order initial and boundary value problems. The method allows to generate monotone iterative techniques which provide constructive methods (amenable numerical treatment), to obtain solutions. The method was first introduced by Perron [1] . Later Nagumo [2] used upper and lower solutions to study second-order differential equations with Dirichlet boundary conditions. Since then many authors have used that method primarily in the context of single-valued differential equations with linear boundary conditions (Dirichlet, Neumann, Sturm-Liouville or periodic). Very recently, in 2017, combining lower and upper solutions method and theory of topological degree, Goli-Adjé [3] established existence and multiplicity results for the considered problem under Neumann-Steklov boundary value conditions. Shortly before, in 2013, Khattabi-Frigon-Ayyadi [4], by lower and upper solutions method and the fixed point index theory, obtained existence and multiplicity results for the problem under various boundary value conditions (Dirichlet, periodic or Neumann). Other authors have investigated the second-order differential equation with multivalued boundary conditions driven by maximal monotone operators, in this direction, see [5] [6] [7] [8] [9] and references therein. In [6] [7], the problems unify classical problems of Dirichlet, periodic and Neumann and in [5] [8] [9] the problems unify classical problems of Dirichlet, Neumann and Sturm-Louiville. To our knowledge, the lower and upper solutions method for differential inclusions formulation of problems of type (1) was initiated by Bader-Papageorgiou [5] in 2002. Soon after, in 2006, Staicub and Papageorgiou [9] extended the study of that problem to gradient systems with a discontinuous nonlinearity. In 2007, Kyritsi and Papageorgiou [8], in their book (see [8] the problem (5.111), p. 390) investigated the following single-valued version of the problem in Staicub-Papageorgiou [9] :
where and f are defined as in problem (1) and for all ,. So, in [5] [8] [9], the authors deal with the homogeneous operator differential p-laplacian for all . But they do not establish multiplicity results.
The goal of this paper is to extend the work of Kyritsi-Papageorgiou [8] to a large class of problems. Indeed, we deal with a non-homogeneous operator -laplacian which contains, for example, some versions of -Laplacian operators like the case when, for all , with is a continuous map. Moreover, to obtain multiplicity results, we combine lower and upper method used in [5] [8] [9] and the one of Goli-Adjé [3] . So, our aim in this paper is to study existence and multiplicity results concerning solutions of problem (1).
After introducing notations, preliminary results and auxiliary results in Section 2 and Section 3, in Section 4, and are assumed respectively an ordered pair of lower and upper solutions of the problem. By combining lower and upper solutions method and theory of topological degree we obtain existence results.
In Section 5, and are assumed respectively non-ordered pair of lower and upper solutions of the problem. Also, by combining lower and upper solutions method and topological degree theory, we obtain existence results.
In Section 6, using the aforementioned method in Section 4 and 5, we show multiplicity results at first when the problem admits a pair of lower and strict lower solutions and a pair of upper and strict upper solutions, secondly when the problem admits two lower solutions and a strict upper solution or when the problem admits a strict lower solution and two upper solutions.
In Section 7, we give an example of application and we show also, as in [8], that our method stays true for the periodic problem.
In Section 8, we give a conclusion.
2. Preliminaries
In this section, we introduce our terminology and notations. We also recall some basic definitions and facts from multivalued analysis that we will need in the sequel. Our main sources are the books of Hu-Papageorgiou [10] and Zeidler [11] .
The Sobolev spaces and are respectively equipped with the norms:
and .
The space of continuous function is endowed with the norm:
.
We denote: : the weak convergence; : the strong convergence; : absolute value of x; ; : image of operator A; : Leray-Schauder’s degree; ; : the family of subsets of space .
Let X be a reflexive Banach space and the topological dual of X. A map is said to be monotone, if for all and for all , we have by we denote the duality brackets for the pair . If additionally, the fact that implies that , then we say that A is strictly monotone. The map A is said to be maximal monotone, if it is monotone and for all ,, the fact that implies that and . It is clear from this definition that A is maximal monotone if and only if its graph is maximal with respect to inclusion among the graphs of monotone maps. If A is maximal monotone, for any , the set is non-empty, closed and convex. Moreover, is demi-closed, i.e., if , either in X and in , or in X and in , then . If is everywhere defined and single-valued, we say that A is demi-continuous, if for every sequence such that in X, we have that in . If map is monotone and demi-continuous, then it is also maximal monotone. A map is said to be coercive, if is bounded or if is unbounded and we have that
A maximal monotone and coercive map is surjective. Let be Banach spaces and . We say:
a) L is “completely continuous”, if in Y implies in Z and
b) L is “compact”, if it is continuous and maps bounded sets into relatively compact sets.
In general, these two notions are distinct. However, if Y is reflexive, then complete continuity implies compactness. Moreover, if Y is reflexive and L is linear, then the two notions are equivalent.
3. Auxiliary Results
Let such that . First, let us define what we mean by solution of problem (1).
Definition 1. A function such that is said to be a solution of the problem (1), if it verifies (1).
Next, we introduce the notions of upper and lower solutions of problem (1).
Definition 2.
a) A function such that is said to be an upper solution of the problem (1), if:
b) A function such that is said to be a lower solution of problem (1), if:
Now, let us specify what we mean by strict lower and strict upper solutions of problem (1).
Definition 3. A lower solution of (1) is said to be strict if all solution u of (1) with is such that .
Definition 4. An upper solution of (1) is said to be strict if all solution u of (1) with is such that .
Proposition 5. Let be a lower solution of (1) such that:
i) For all , there exists and is an open interval such that and:
for all ;
ii) ;
iii) ;
then is a strict lower solution of (1).
Proof. Let u be a solution of problem (1) such that for all .
Let us assume that u is not strict, then there exists such that . Whence
A is closed and bounded. Let . Then
a) If , then and there exist and according to (i). We can choose such that ,, and
Therefore, for almost ,
Since is an increasing homeomorphism, we have
and
(2)
Also we have:
which contradicts (2).
b) We suppose that , then and it follows that:
(3)
Since , because of the monotonicity of , if , we have:
Then,
So
which contradicts (3).
c) We suppose that , then . It follows that:
(4)
Since , because of the monotonicity of , if , we have:
Then, .
So
which contradicts 4. Then, does not exist. So, .
Proposition 6. Let be an upper solution of (1) such that:
i) For all , there exist and is an open interval such that and:
for all ;
ii) ;
iii) ;
then is a strict upper solution of (1).
Proof. Let u be a solution of problem (1) such that for all .
Let us assume that u is not strict, then there exists such that . Whence
E is closed and bounded. Let . Then
a) If there exist and according to (i). We can choose such that , and
Therefore, for almost ,
Since is an increasing homeomorphism, we have
and
(5)
Also we have:
which contradicts (5).
b) We suppose that , then and it follows that:
(6)
Since , because of the monotonicity of , if , we have:
Then
So
which contradicts (6).
c) We suppose that , then . It follows that:
(7)
Since , because of the monotonicity of , if , we have:
. Then .
So
which contradicts (7). Then, does not exist. So, .
Remark 7. In general, for a given problem, there is not a methodology (single valued and multivalued alike) which allows generating a lower and upper solutions. But, one should try simple functions such as constants, linear, quadratic, exponentials, eigenfunctions of simple operator, etc.
We make the following hypotheses on the data of (1):
: There exists a lower solution and an upper solution .
is an increasing continuous map such that:
a) ;
b) there exists such that: for all ;
c) there exist such that for a.e and for all :
Remark 8. Suppose that . Then this function satisfies hypothesis . This function corresponds to the one-dimensional operator p-Laplacian. Another interesting case which satisfies hypothesis is when is defined by with continuous, for all and is strictly increasing on and . For example, we can have
It is well-known that under the monotonicity condition and the hypotheses (a) and (b), is an increasing homeomorphism from onto . And is strictly monotone and as (See Deimling [12] chap. 3). Our operator is a slightly more restrictive version of the scalar case of the operator used by Sophia Kirytsi-N. Matzakos [13] and Manasevich-Mawhin [14] where growth condition is not assumed. Nevertheless, it incorporates the operator p-laplacian and many other classes of operators.
is a fonction such that:
i) for all , is measurable;
ii) for a.e , is continuous;
iii) for a.e , and all , we have:
where and a Borel measurable non-decreasing functions such that:
with ;
iv) for every , there exists such that for a.e and for all with we have: .
Remark 9. Hypothesis (iii) is known as a Bernstein-Nagumo-Wintner growth condition and produces a uniform a priori bound of the derivatives of the solutions of problem (1). And the hypotheses (i), (ii) and (iv) are well known as -Carathéodory conditions.
: and : are maximal monotone maps such that .
Remark 10. There exist functions proper, convex and lower semi-continuous which are not identically equal to such that . More exactly, there exists some increasing positives functions and such that (the minimum absolute value element in the closed, convex set ). Then ,. We have for all , where
and ,.
: is a function that maps bounded sets to bounded sets and there exists such that is increasing.
Remark 11. We emphasize that need not be continuous.
Lemma 1. If and hypotheses and (iii) hold,
and if
for all
then, there exists (depending only on ) such that: for all .
Proof. Set (See hypothesis ). By hypothesis (iii), we can find such that
We claim that for all . Let’s suppose that it is not the case. Then, we can find such that
By the mean value theorem, there exists such that . Without any loss of generality, we assume that . We obtain:
.
Since , by the intermediate value theorem, there exists and with such that and . We have:
Thus:
and then
Setting , we have:
which contradicts the choice of .
Now, we introduce the truncation map: defined by
(8)
where and the penalty function defined by:
(9)
We set . Note that for ae and all , we have . Moreover, for almost all and all , we have: with . For every , set
and
the Nemitsky operators corresponding to and respectively. We set for every .
Proposition 12. If hypothesis (ii) holds, then: is continuous.
Proof. Since and are Nemitsky operators, it is standard to show that they are continuous. It follows that G is continuous.
We introduce the set
and then we define the non-linear operator: by
Proposition 13. If the hypotheses and hold, then is maximal monotone.
Proof. Given , we consider the following nonlinear boundary value problem:
(10)
We show that problem (10) has a unique solution . To this end, given , we consider the following two-point boundary value problem:
(11)
We set . Then and . We consider the function y defined by and rewrite (11) in the terms of this function.
(12)
This is a homogeneous Dirichlet problem for (11). To solve (12), let be the non-linear operator defined by:
where denote the duality brackets for the pair .
Let us show that is strictly monotone.
Let . We have
Then
We know that is monotone. Moreover, it is easy to show that is strictly monotone. Whence
Therefore, is strictly monotone.
• Let us show that is demicontinuous.
Using the extended dominated convergence theorem (see for example Hu-Papageorgiou 10, Theorem A.2.54, p. 907), it follows easily that is demicontinuous.
Recall that an operator monotone and demicontinuous is maximal monotone. So is maximal monotone.
Let us show that is coercive.
For we have:
Hence, using the hypotheses (b) and (c) on , we obtain:
with .
whence:
, for some .
Therefore, is coercive.
Recall that an operator maximal monotone which is coercive is surjective.
Moreover, since is strictly monotone, we infer that there exists a unique such that . It follows easily that and it solves the problem (12). Then and it is the solution of the problem (11). We can define the solution map which to each pair assigns the unique solution of the problem (11). Let be defined by:
We claim that is monotone.
Indeed, for , we have:
where is the scalar product in .
From (11), we have . Because of monotonicity of the operators and , we have the monotonicity of .
We claim that is continuous.
Indeed, let and be real sequences converging respectively to b and e.
Let us set that
Now, we consider the following sequence of problems:
(13)
We claim that is bounded.
Let us multiply (13) by and integrate on , we obtain:
By using green’s formula, we obtain:
with .
whence:
for some .
Furthermore, using the Cauchy-Schwartz inequality and then the triangular inequality, we obtain the following inequalities:
Then:
So
for some .
Therefore, the sequence is bounded. It follows immediately that the sequences is bounded. So directly from the problem (11), we get that the sequence is bounded. By integration, we obtain . So the sequence is bounded. Then we have respectively
Due to the compact embedding of in , we have:
Since is an increasing homeomorphism, exists and is continuous. So, we have: in . Whence (i.e. ). Therefore passing to the limit as , we have:
(i.e. is continuous). So, is continuous.
We claim that is coercive.
For , we have:
Then
where denote the euclidean norm in .
Since , by mean value theorem, there exists such that .
As , we have:
for all . In particular, we have:
Hence
Therefore is coercive.
We infer that is maximal monotone (being continuous, monotone) and coercive. Thus is surjective. Now, we consider with operator B defined by for all . Since is coercive and B is maximal monotone, we deduce that is coercive. Also, is maximal monotone (see Brezis [15] Corollary 2.7, p. 36 or Zeidler [14] Theorem 32.I, p. 897). So is surjective. We infer that we can find such that . Since , we can find such that and and and . So and . It follows that . Therefore is the solution of the problem (10).
Let be the operator defined by:
Since is continuous and monotone, then H is continuous and monotone. Therefore H is maximal monotone. Moreover it is evident to see that H is strictly monotone.
Since in (10) the choice of h is arbitrary, then by the previous arguments, we have:
(14)
We denote by the duality brackets between the pair .
Let us show that surjective implies is maximal monotone
For this purpose, we suppose that, for some and some :
(15)
Because of (14), we can find such that:
We use this in (16) with , we obtain:
(16)
Because H is strictly monotone, from (16), we conclude that and . So is maximal monotone. In addition, since is monotone, we have . Whence the operator is maximal monotone, strictly monotone and coercive. Therefore is well defined, single valued, and maximal monotone (From into ).
Proposition 14. If hypothesis holds, then is completly continuous.
Proof. Suppose that in . We have to show that in . let us set for all . We have
(17)
By integration by part, we obtain:
(18)
Since , we have and for all . We recall that , then:
(19)
Moreover, the map being monotone, we have:
(20)
From (19) and (20), we obtain:
(21)
From (21) and (18), we infer that:
(22)
By hypothesis b) on , we have:
(23)
It follows from (22) and (23) that:
whence:
for some .
Therefore the sequence is bounded. Then we can find a convergente subsequence of . So in . Due to the compact embedding of in , we have in . Since is bounded, we have and are bounded. It follows immediatly that: is bounded. Then imply that is bounded. Whence, by integration, is bounded. So we can suppose that in . Due to the compact embedding of in , we obtain in . Since is an increasing homeomorphism, exists and is continuous. So, we have for all . Then for all . It follows that in (By Lebesgue dominated convergence theorem). We infer that: . It follows
But recall that in . Thus in . This prove that the operator L is completly continuous.
Proposition 15. If the conditions in lemma 1 hold, then a function is solution of (1) if and only if and u is a fixed point of defined by:
with the operator Q defined by: .
Moreover, for all , and K is continuous and completly continuous.
Proof. If u is a solution of (1), then because of hypothesis (iii). It follows that . So, . We have also and . Hence, .
Furthermore, we have:
Therefore, u is a fixed point of K.
On the other hand, if and u is a fixed point of K, then we have ,, and a.e on a.e on . Hence, u is solution of (1).
Finally, by lemma 1, we have: ,.
• Let us show that K is continuous.
Let in . Then, there exists such that and .
We will show that in . That’s mean in and in . For , we have:
Since and N are continuous respectively in and , we have:
Also from the monotone convergence theorem, we have:
Since in , it follows that in . Using the previous arguments and the dominated convergence theorem, we have:
It follows that
Since is an increasing homeomorphism, exists and is continuous. Finally, we have:
By integration, we obtain: in . Therefore, K is continuous.
• Let us show that K is completely continuous.
Let be a bounded set of . We set . Since is bounded, there exist such that:
It follows that:
Therefore, there exist such that .
For and .
We infer that for all , there exists such that
It suffices to take . Therefore, being an increasing homeomorphism, for all , it exists , such that for all ,, if , then
is uniformly equicontinuous and is bounded on . By Ascoli-Arzela’s theorem, is relatively compact in . Since K is continuous and is relatively compact in for every bounded subset of , K is completely continuous.
4. Existence Results with Ordered Pair of Lower and Upper Solutions
We consider the operator defined by:
Evidently, is bounded (i.e., maps bounded sets to bounded ones) and is continuous.
Theorem 16. Suppose that there exists a lower solution and an upper solution such that ,.
Then the problem (1) admits at least one solution u, such that:
Moreover, if are strict, then
where
K is the operator associated to the problem (1).
We consider the following auxilary boundary problem:
(24)
A solution of problem (24) is a function such that and satisfied (24).
The problem (24) is equivalent to the fixed point problem and with defined by:
We have:
Lemma 2. All solution u of (24) is such that ,.
Proof. Since is a lower solution of the problem (1), we have:
(25)
Soustraying (25) from (24), we obtain:
(26)
We multiply (26) by and then integrate on . We obtain:
(27)
The integration by parts of the left-hand side in inequality, yields:
(28)
We set
(29)
Also, from the boundary conditions in (24) and (25), we have:
If , then from the monotony of (See hypothesis ), we have:
. Whence .
So, it follows that
(30)
In a similar way, using the boundary conditions and with , if , we have:
. We infer that .
It follows that
(31)
Also, since is an increasing homeomorphism, we have:
(32)
where .
Using the inequalities (30), (31) and (32) in the first member of (27), we obtain:
(33)
Furthermore:
(34)
Also from the definiton of the penalty map , if (By , we denote the Lebesgue mesure in ), then:
(35)
Finally, by virtue of hypothesis and since , we see that:
(36)
Using the inequalities, (34), (35) and (36) in the second member of (27), we infer that:
(37)
We consider (27) and using (33) and (34), we have a contradiction when . Therefore, for all ,. In a similar fashion we show that for all . Thus .
Proof. theorem 16: As in the proof of the proposition 15, we can show the complete continuity of the operator . Moreover, for all . Therefore, by Leray-Schauder’s theorem, we can say that the operator has a fixed point u in the open ball which is solution of problem (24). It follows, by the lemma 2, that u is also solution of problem (1).
We assume that is a strict lower solution and is a strict upper solution of (1). Let
be quite a few such that
for some .
Because is completely continuous, we can compute the degree of . The function H defined by is compact on . We assume that there exist and such that , then . But , so which contradict the fact that . We can apply the homotopic invariance degre property of Leray-Schauder to obtain:
Indeed, let us recall the following set:
By definitions of strict lower and strict upper solutions, and cannot be solution of problem (24). Therefore, (24) has not solution on the boundary of . By using the additivity and excision properties of Leray-Schauder degree, we obtain:
Furthermore, since K is completely continuous operator associate to (1). That equates to on , we have
5. Existence Results with Non Ordered Lower and Upper Solutions
Theorem 17. We assume that there exists a lower solution and an upper solution of (1) such that
such that . (38)
Then the problem (1) admits at least one solution u, such that:
for some (39)
and
(40)
Proof. We set .
We consider the functions ,, and for , the multifunctions are defined respectively by:
where, for , and are defined as in remark 10, is -Caratheodory and and are maximal monotone operators. We consider the following modified problem:
(41)
We can verify that is a lower solution and is upper solution of the problem (41). Let be defined by . We have:
So we can find such that
and
Therefore, is an upper solution of (41).
The function defined by , verifies:
So we can find such that
and
Therefore, is a lower solution of (41). Furthermore,
Let us introduce the sets
and
By using the definition (39), we obtain:
Also we have:
Let us consider:
Then
and
Since all constant function between and is into , is nonempty.
Let be the fixed point operator associated to problem (41) given in the proposition 15. We consider such that and . There exists such that
or .
Let us consider the case .
If , then and there exists such that for all . Moreover . Whence,
,
for all . It follows that u is increasing on . That contradicts the existence of .
If and we obtain the contradiction .
If and we obtain the contradiction .
In the similar fashion, we obtain contradiction with the case . Therefore
(42)
Let such that . It becomes from (42) that ,, and . It follows, there exists such that and , that implies
Then,
So,
(43)
We have two cases:
• 1rst case: We assume that there exists such that . From (43), we infer that , that implies that u is a solution of (1), and (39) and (40) are satisfied. Then, there exists such that or ,
• 2nd case: We assume that for all . Then, as in the proof of theorem 16, we have:
Theorem 18. If there exists a lower solution and an upper solution of the problem (1), then the problem (1) admits at least one solution u such that:
(44)
Proof. If , by theorem 16, the problem (1) admits at least one solution such that:
Moreover, (44) holds.
If
such that (45)
By the theorem 17, the problem (1) admits at least one solution, such that:
6. Multiplicity Results
Theorem 19. We assume that there exists a lower solution and a strict lower solution of problem (1), an upper solution and a strict upper solution of problem (1) such that:
Then the problem (1) admits at least three solutions u, v and w such that:
for some .
Proof. By using the theorem 16 and the fact that and are strict, we can say that the problem (1) admits at least one solution u, such that:
(46)
By using the theorem 16, the problem (1) admits at least one solution w such that:
(47)
(46) and (47) yield and .
In the following theorem, we show existence of at least two solutions of the problem (1).
Theorem 20. We assume that there exists and are two lower solutions of the problem (1), is a strict upper solution of problem (1) such that:
Then the problem (1) admits at least two solutions u and w such that:
and
such that . (48)
Proof. By using the theorem 16 and the fact that is strict, we can say that the problem (1) admits at least one solution u, such that:
(49)
By using the theorem 17, the problem (1) admits at least one solution w such that:
(50)
(49) and (50) yield .
Theorem 21. We assume that there exists a strict lower solutions of the problem (1) and two upper solutions and of the problem (1) such that:
Then the problem (1) admits at least two solutions u and w such that:
and
such that . (51)
Proof. The proof is similar to those of theorem 18.
7. Example and Periodic Problem
7.1. Example
Let us consider the following problem:
(52)
where and are defined as in problem (1). Here,
,
for all and by the remark 8, it satisfies hypothesis . Therefore, theorem 16 and theorem 17 are true for the problem (52). Moreover, by [8] (see example 5.2.25 page 404), this problem unifies classical problems of Dirichlet, Neumann and Sturm-Liouville and go beyong them.
7.2. Periodic Problem
Let us consider the following periodic problem:
(53)
Remark 22. The theorems 16 and 17 stay true for this problem (see 8 remark 5.2.26 page 404 and also [5] section 6 page 23).
8. Conclusion
In this article, by combining lower and upper solutions method, theory of maximal monotone operators and theory of topological degree, we establish existence and multiplicity results for second-order problems with multivalued boundary conditions. We give an example but more examples and applications can be given. In perspective, we will study the same problems under general multivalued boundary conditions. Also, the same problem can be considered for a singular -Laplacian operator.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Behi, D.A. and Adje, A. (2019) Existence and Multiplicity Results for Second-Order Nonlinear Differential Equations with Multivalued Boundary Conditions. Journal of Applied Mathematics and Physics, 7, 1340-1368. https://doi.org/10.4236/jamp.2019.76091
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