Existence and Multiplicity Results for Second-Order Nonlinear Differential Equations with Multivalued Boundary Conditions

In this paper, we consider the following second-order nonlinear differential equations’ problem: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) , , u t f t u t u t u t ′ ′ ′ − Φ = +Ξ a.e on [ ] 0,T Ω = 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, α and β are assumed respectively an ordered pair of lower and upper solutions of the problem, secondly α and β 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.

In this paper, we consider the following second-order nonlinear differential equations' problem: 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, α and β are assumed respectively an ordered pair of lower and upper solutions of the problem, secondly α and β 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.

Introduction
This paper is devoted to the study of the following problem: 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]: , , a.e on 0, where 1 2 , , B B Ξ and f are defined as in problem (1) and for all z ∈  , ( ) 1 a z = . So, in [5] [8] [9], the authors deal with the homogeneous operator differential p-laplacian ] [ : 0, a → +∞  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, α and β 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, α and β 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.

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 ( ) is everywhere defined and single-valued, we say that A is demi-continuous, if for every sequence ( ) 1 n n is monotone and demi-continuous, then it is also maximal monotone. A map ( ) ( ) 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.

Let
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.
is said to be an upper solution of the problem (1), if: is said to be a lower solution of problem (1), if: a.e on 0, Now, let us specify what we mean by strict lower and strict upper solutions of problem (1).
Definition 3. A lower solution α of (1) is said to be strict if all solution u of Proposition 5. Let α be a lower solution of (1) such that: , there exists 0 0 ε > and 0 Ω is an open interval such that 0 0 t ∈ Ω and: then α is a strict lower solution of (1).
Proof. Let u be a solution of problem (1) such that ( ) ( ) Let us assume that u is not strict, then there exists Also we have: and it follows that: . It follows that: which contradicts 4. Then, 0 t does not exist. So, A = ∅ .  Proposition 6. Let β be an upper solution of (1) such that: , there exist 0 0 ε > and 0 Ω is an open interval such that 0 0 t ∈ Ω and: then β is a strict upper solution of (1).
Proof. Let u be a solution of problem (1) such that ( ) ( ) for all t ∈ Ω .
Let us assume that u is not strict, then there exists there exist 0 Ω and 0 0 >  according to (i). We can choose Since Φ is an increasing homeomorphism, we have Also we have: and it follows that: , we have: . It follows that: which contradicts (7). Then, 0 t does not exist. So, E = ∅ .  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): ( ) , 0 d d > such that for a.e t ∈ Ω and for all x ∈  : 3). Our operator Φ 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.
 and all y ∈  , we have: a Borel measurable non-decreasing functions such that: iv) for every 0 r > , there exists ( ) q r L γ ∈ Ω such that for a.e t ∈ Ω and for all , x y ∈  with , x y r ≤ we have: Remark 10. There exist functions proper, convex and lower semi-continuous which are not identically equal to +∞ such that  ( ) Without any loss of generality, we assume that 2 1 t t ≤ .
We obtain: We have: a.e on , , we have:  Now, we introduce the truncation map: and the penalty function : Moreover, for almost all t ∈ Ω and all , x y ∈  , we have: the Nemitsky operators corresponding to 1 f and Λ respectively. We set a.e on 0, We show that problem (10) We consider the function y defined by ( ) ( ) ( ) y t u t t γ = − and rewrite (11) in the terms of this function.
This is a homogeneous Dirichlet problem for (11). To solve (12), let 1 V be the non-linear operator defined by: V is strictly monotone.
• Let us show that 1 V is demicontinuous.
Recall that an operator monotone and demicontinuous is maximal monotone.
So 1 V is maximal monotone.
Let us show that 1 V is coercive.
Hence, using the hypotheses (b) and (c) on Φ , we obtain: whence: ( ) We claim that ∆ is monotone.
Indeed, for ( ) ( ) 2 1 1 2 2 , , , v w v w ∈  , we have: where 2 is the scalar product in Because of monotonicity of the operators Φ and Now, we consider the following sequence of problems: Therefore passing to the limit as n → +∞ , we have: is continuous). So, ∆ is continuous. We claim that ∆ is coercive.
, v w ∈  , we have: for all t ∈ Ω . In particular, we have: We infer that ∆ is maximal monotone (being continuous, monotone) and coercive. Thus ∆ is surjective. Now, we consider ( ) Since ∆ is coercive and B is maximal monotone, we deduce that θ is coercive. Also, θ is maximal monotone (see Brezis [15] Corollary 2.7, p. 36 or Zeidler [14] Theorem 32.I, p. 897). So θ is surjective. We infer that we can find ( ) 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: We denote by .,. p the duality brackets between the pair Let us show that H ϑ + surjective implies ϑ is maximal monotone For this purpose, we suppose that, for some ( ) p y L ∈ Ω and some ( ) Because of (14), we can find 1 u D ∈ such that: We use this in (16) with 1 u u = , we obtain: Because H is strictly monotone, from (16), we conclude that By integration by part, we obtain: From (21) and (18), we infer that: By hypothesis b) on Φ , we have: It follows from (22) and (23) that:  : Moreover, for all a.e on 0, a.e on 0, a.e on 0, Therefore, u is a fixed point of K.
On the other hand, if u D ∈ and u is a fixed point of K, then we have a.e on [ ] 0,T . Hence, u is solution of (1).
Finally, by lemma 1, we have: • Let us show that K is continuous. Also from the monotone convergence theorem, we have: . Using the previous arguments and the dominated convergence theorem, we have: . Therefore, K is continuous.
• Let us show that K is completely continuous. Let Π be a bounded set of ( ) such that:

Existence Results with Ordered Pair of Lower and Upper Solutions
We consider the operator ( ) ( ) K is the operator associated to the problem (1).
We consider the following auxilary boundary problem: a.e on 0, A solution of problem (24)  : ) is a lower solution of the problem (1), we have: Soustraying (25) from (24), we obtain: T The integration by parts of the left-hand side in inequality, yields: We set Also, from the boundary conditions in (24) Also, since Φ is an increasing homeomorphism, we have:  which is solution of problem (24). It follows, by the lemma 2, that u is also solution of problem (1).
We assume that α is a strict lower solution and β is a strict upper solu- , that implies that u is a solution of (1), and (39) and (40)