ABSTRACT

In this paper, we consider the following second-order nonlinear differential equations’ problem: $-{\left(\Phi \left(u^{\prime }\left(t\right)\right)\right)}^{\prime }=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right)+\Xi \left(u\left(t\right)\right)$ a.e on $\Omega =\left[0,T\right]$ 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, $\alpha$ and $\beta$ are assumed respectively an ordered pair of lower and upper solutions of the problem, secondly $\alpha$ and $\beta$ 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:

$\Phi$ -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:

$\{\begin{array}{l}-{\left(\Phi \left(u^{\prime }\left(t\right)\right)\right)}^{\prime }=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right)+\Xi \left(u\left(t\right)\right)\text{a}\text{.e}\text{on}\Omega =\left[0,T\right]\\ u^{\prime }\left(0\right)\in B_1\left(u\left(0\right)\right),-u^{\prime }\left(T\right)\in B_2\left(u\left(T\right)\right)\end{array}$ (1)

where $B_1$ and $B_2$ are maximal monotone graphs in ${ℝ}^2$ and are some multifunctions which describe the boundary conditions, $f\mathrm{<mo>\; :\; </mo>}\Omega \times {ℝ}^2\to ℝ$ is a Caratheodory function and $\Phi$ is an increasing homeomorphism such that $\Phi \left(0\right)=0$.

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] :

$\{\begin{array}{l}-\left(a\left(u^{\prime }\left(t\right)\right){\Phi }_p\left(u^{\prime }\left(t\right)\right)\right)=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right)+\Xi \left(u\left(t\right)\right)\text{a}\text{.e}\text{on}\Omega =\left[0,T\right]\\ u^{\prime }\left(0\right)\in B_1\left(u\left(0\right)\right),-u^{\prime }\left(T\right)\in B_2\left(u(T)\right)\end{array}$

where $B_1\mathrm{,}{B}_2\mathrm{,}\Xi$ and f are defined as in problem (1) and for all $z\in ℝ$,$a\left(z\right)=1$. So, in [5] [8] [9], the authors deal with the homogeneous operator differential p-laplacian ${\Phi }_p\left(z\right)={\left|z\right|}^{p-2}z$ for all $z\in ℝ$. 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 $\Phi$ -laplacian which contains, for example, some versions of $\Phi$ -Laplacian operators like the case when, for all $z\in ℝ$,$\Phi \left(z\right)=a\left(z\right){\Phi }_p\left(z\right)$ with $a\mathrm{<mo>\; :\; </mo>}ℝ\to \left]\mathrm{0,}+\infty \right[$ 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, $\alpha$ and $\beta$ 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, $\alpha$ and $\beta$ 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 $W^{\mathrm{1,}p}\left(\Omega \right)$ and $L^p\left(\Omega \right)$ are respectively equipped with the norms:

$\Vert y\Vert ={\left({\displaystyle {\int }_0^T}{\left|y\left(t\right)\right|}^p+{\left|y^{\prime }\left(t\right)\right|}^p\right)}^{\frac{1}{p}}$ and ${\Vert u\Vert }_p={\left({\displaystyle {\int }_0^T}{\left|u\left(t\right)\right|}^p\text{d}t\right)}^{\frac{1}{p}}$.

The space of continuous function $C\left(\Omega \right)$ is endowed with the norm:

${\Vert u\Vert }_{\infty }\mathrm{<mo>\; =\; </mo>}\max \left\{\left|u\left(t\right)\right|\mathrm{<mo>\; :\; </mo>}t\in \Omega \right\}$.

We denote: $\rightharpoonup$ : the weak convergence; $\to$ : the strong convergence; $\left|x\right|$ : absolute value of x; $s^{+}=\max \left\{s,0\right\}$ ; $\text{R}\left(A\right)$ : image of operator A; $d_{LS}$ : Leray-Schauder’s degree; ${ℝ}_{+}\backslash \left\{0\right\}={ℝ}_{+}^{\ast }$ ; $P\left(X^{\mathrm{<mo>\; *\; </mo>}}\right)$ : the family of subsets of space $X^{\mathrm{<mo>\; *\; </mo>}}$.

Let X be a reflexive Banach space and $X^{\mathrm{<mo>\; *\; </mo>}}$ the topological dual of X. A map $A\mathrm{<mo>\; :\; </mo>}D\left(A\right)\subseteq X\to P\left(X^{\mathrm{<mo>\; *\; </mo>}}\right)$ is said to be monotone, if for all $x\mathrm{,}y\in D\left(A\right)$ and for all $x^{\mathrm{<mo>\; *\; </mo>}}\in A\left(x\right)\mathrm{,}{y}^{\mathrm{<mo>\; *\; </mo>}}\in A\left(y\right)$, we have $\langle x^{\mathrm{<mo>\; *\; </mo>}}-y^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}x-y\rangle \ge 0$ by $\langle \mathrm{.}\rangle$ we denote the duality brackets for the pair $\left(X\mathrm{,}{X}^{\mathrm{<mo>\; *\; </mo>}}\right)$. If additionally, the fact that $\langle x^{*}-y^{*},x-y\rangle =0$ implies that $x=y$, then we say that A is strictly monotone. The map A is said to be maximal monotone, if it is monotone and for all $x\in D\left(A\right)$,$x^{\mathrm{<mo>\; *\; </mo>}}\in A\left(x\right)$, the fact that $\langle x^{\mathrm{<mo>\; *\; </mo>}}-y^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}x-y\rangle \ge 0$ implies that $y\in D\left(A\right)$ and $y^{\mathrm{<mo>\; *\; </mo>}}\in A\left(y\right)$. It is clear from this definition that A is maximal monotone if and only if its graph $\text{Gr}A=\left\{\left(x\mathrm{,}{x}^{\mathrm{<mo>\; *\; </mo>}}\right)\in X\times X^{\mathrm{<mo>\; *\; </mo>}}\mathrm{<mo>\; :\; </mo>}{x}^{\mathrm{<mo>\; *\; </mo>}}\in A\left(x\right)\right\}$ is maximal with respect to inclusion among the graphs of monotone maps. If A is maximal monotone, for any $x\in D\left(A\right)$, the set $A\left(x\right)$ is non-empty, closed and convex. Moreover, $\text{Gr}A$ is demi-closed, i.e., if $\left(x_n\mathrm{,}{x}_n^{\mathrm{<mo>\; *\; </mo>}}\right)\in \text{Gr}A\mathrm{,}n\ge 1$, either $x_n\to x$ in X and $x_n^{\mathrm{<mo>\; *\; </mo>}}\rightharpoonup x^{\mathrm{<mo>\; *\; </mo>}}$ in $X^{\mathrm{<mo>\; *\; </mo>}}$, or $x_n\rightharpoonup x$ in X and $x_n^{\mathrm{<mo>\; *\; </mo>}}\to x^{\mathrm{<mo>\; *\; </mo>}}$ in $X^{\mathrm{<mo>\; *\; </mo>}}$, then $\left(x\mathrm{,}{x}^{\mathrm{<mo>\; *\; </mo>}}\right)\in \text{Gr}A$. If $A\mathrm{<mo>\; :\; </mo>}X\to X^{\mathrm{<mo>\; *\; </mo>}}$ is everywhere defined and single-valued, we say that A is demi-continuous, if for every sequence ${\left(x_n\right)}_{n\ge 1}$ such that $x_n\to x$ in X, we have that $A\left(x_n\right)\rightharpoonup A\left(x\right)$ in $X^{\mathrm{<mo>\; *\; </mo>}}$. If map $A\mathrm{<mo>\; :\; </mo>}X\supseteq D\left(A\right)\to X^{\mathrm{<mo>\; *\; </mo>}}$ is monotone and demi-continuous, then it is also maximal monotone. A map $A\mathrm{<mo>\; :\; </mo>}X\supseteq D\left(A\right)\to P\left(X^{\mathrm{<mo>\; *\; </mo>}}\right)$ is said to be coercive, if $D\left(A\right)\subseteq X$ is bounded or if $D\left(A\right)$ is unbounded and we have that

$\frac{\inf \left\{{\langle x^{\mathrm{<mo>\; *\; </mo>}}\mathrm{,}x\rangle }_X\mathrm{<mo>\; :\; </mo>}{x}^{\mathrm{<mo>\; *\; </mo>}}\in A\left(x\right)\right\}}{{\Vert x\Vert }_X}\to +\infty \text{as}{\Vert x\Vert }_X\to +\infty \mathrm{,}x\in D\left(A\right)\mathrm{.}$

A maximal monotone and coercive map is surjective. Let $Y\mathrm{,}Z$ be Banach spaces and $L\mathrm{<mo>\; :\; </mo>}Y\to Z$. We say:

a) L is “completely continuous”, if $y_n\rightharpoonup y$ in Y implies $L\left(y_n\right)\to L\left(y\right)$ 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 $p\mathrm{,}q\in {\mathbb{N}}^{\mathrm{<mo>\; *\; </mo>}}$ such that $\frac{1}{p}+\frac{1}{q}=1,p\ge 2$. First, let us define what we mean by solution of problem (1).

Definition 1. A function $u\in C^1\left(\Omega \right)$ such that $\Phi \left(u^{\prime }\left(\mathrm{.}\right)\right)\in W^{\mathrm{1,}q}\left(\left(\mathrm{0,}T\right)\right)$ 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 $\beta \in C^1\left(\Omega \right)$ such that $\Phi \left({\beta }^{\prime }\left(\mathrm{.}\right)\right)\in W^{\mathrm{1,}q}\left(\left(\mathrm{0,}T\right)\right)$ is said to be an upper solution of the problem (1), if:

$\{\begin{array}{l}-\left(\Phi \left({\beta }^{\prime }\left(t\right)\right)\right)\ge f\left(t\mathrm{,}\beta \left(t\right)\mathrm{,}{\beta }^{\prime }\left(t\right)\right)+\Xi \left(\beta \left(t\right)\right)\text{a}\text{.eon}\Omega =\left[\mathrm{0,}T\right]\\ {\beta }^{\prime }\left(0\right)\in B_1\left(\beta \left(0\right)\right)-{ℝ}_{+}\mathrm{,}-{\beta }^{\prime }\left(T\right)\in B_2\left(\beta \left(T\right)\right)-{ℝ}_{+}\mathrm{.}\end{array}$

b) A function $\alpha \in C^1\left(\Omega \right)$ such that $\Phi \left({\alpha }^{\prime }\left(\mathrm{.}\right)\right)\in W^{\mathrm{1,}q}\left(\left(\mathrm{0,}T\right)\right)$ is said to be a lower solution of problem (1), if:

$\{\begin{array}{l}-\left(\Phi \left({\alpha }^{\prime }\left(t\right)\right)\right)\le f\left(t\mathrm{,}\alpha \left(t\right)\mathrm{,}{\alpha }^{\prime }\left(t\right)\right)+\Xi \left(\alpha \left(t\right)\right)\text{a}\text{.eon}\Omega =\left[\mathrm{0,}T\right]\\ {\alpha }^{\prime }\left(0\right)\in B_1\left(\alpha \left(0\right)\right)+{ℝ}_{+}\mathrm{,}-{\alpha }^{\prime }\left(T\right)\in B_2\left(\alpha \left(T\right)\right)+{ℝ}_{+}\mathrm{.}\end{array}$

Now, let us specify what we mean by strict lower and strict upper solutions of problem (1).

Definition 3. A lower solution $\alpha$ of (1) is said to be strict if all solution u of (1) with $u\left(t\right)\ge \alpha \left(t\right)\mathrm{,}\forall t\in \left[\mathrm{0,}T\right]$ is such that $u\left(t\right)>\alpha \left(t\right),\forall t\in \left[0,T\right]$.

Definition 4. An upper solution $\beta$ of (1) is said to be strict if all solution u of (1) with $u\left(t\right)\le \beta \left(t\right)\mathrm{,}\forall t\in \left[\mathrm{0,}T\right]$ is such that $u\left(t\right)<\beta \left(t\right),\forall t\in \left[0,T\right]$.

Proposition 5. Let $\alpha$ be a lower solution of (1) such that:

i) For all $t_0\in \left]\mathrm{0,}T\right[$, there exists ${\epsilon }_0>0$ and ${\Omega }_0$ is an open interval such that $t_0\in {\Omega }_0$ and:

$-\left(\Phi \left({\alpha }^{\prime }\left(t\right)\right)\right)\le f\left(t\mathrm{,}x\mathrm{,}y\right)+\Xi \left(x\right)\text{a}\text{.e}t\in {\Omega }_0\mathrm{,}$

for all $\left(x\mathrm{,}y\right)\in \left[\alpha \left(t\right)\mathrm{,}\alpha \left(t\right)+{\epsilon }_0\right]\times \left[{\alpha }^{\prime }\left(t\right)-{\epsilon }_0\mathrm{,}{\alpha }^{\prime }\left(t\right)+{\epsilon }_0\right]$ ;

ii) ${\alpha }^{\prime }\left(0\right)\in B_1\left(\alpha \left(0\right)\right)+{ℝ}_{+}^{\ast }$ ;

iii) $-{\alpha }^{\prime }\left(T\right)\in B_2\left(\alpha \left(T\right)\right)+{ℝ}_{+}^{\ast }$ ;

then $\alpha$ is a strict lower solution of (1).

Proof. Let u be a solution of problem (1) such that $\alpha \left(t\right)\le u\left(t\right)$ for all $t\in \Omega$.

Let us assume that u is not strict, then there exists $\bar{t}\in \left[\mathrm{0,}T\right]$ such that $\alpha \left(\bar{t}\right)=u\left(\bar{t}\right)$. Whence

$A=\left\{t\in \left[0,T\right]:\alpha \left(t\right)=u\left(t\right)\right\}\ne \varnothing$

A is closed and bounded. Let $t_0=\min A$. Then

$\underset{\left[0,T\right]}{\min }\left[u\left(t\right)-\alpha \left(t\right)\right]=\alpha \left(t_0\right)-u(t0)$

a) If $t_0\in \left]\mathrm{0,}T\right[$, then $u^{\prime }\left(t_0\right)-{\alpha }^{\prime }\left(t_0\right)=0$ and there exist ${\Omega }_0$ and ${ϵ}_0>0$ according to (i). We can choose $t_1\in {\Omega }_0$ such that $t_1<t_0$,$u^{\prime }\left(t_1\right)<{\alpha }^{\prime }\left(t_1\right)$, and

$\forall t\in \left[t_1,t_0\right],\left(u\left(t\right),u^{\prime }\left(t\right)\right)\in \left]\alpha \left(t\right),\alpha \left(t\right)+{ϵ}_0\right[\times \left]{\alpha }^{\prime }\left(t\right)-{ϵ}_0,{\alpha }^{\prime }\left(t\right)+{ϵ}_0\right[.$

Therefore, for almost $t\in \left[t_1\mathrm{,}{t}_0\right]$,

$-{\left(\Phi \left({\alpha }^{\prime }\left(t\right)\right)\right)}^{\prime }-f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right)-\Xi \left(u\left(t\right)\right)\le 0.$

Since $\Phi$ is an increasing homeomorphism, we have

$\Phi \left(u^{\prime }\left(t_0\right)\right)-\Phi \left({\alpha }^{\prime }\left(t_0\right)\right)=0$

and

$\Phi \left(u^{\prime }\left(t_1\right)\right)<\Phi \left({\alpha }^{\prime }\left(t_1\right)\right)$ (2)

Also we have:

$\begin{array}{l}\Phi \left(u^{\prime }\left(t_1\right)\right)-\Phi \left({\alpha }^{\prime }\left(t_1\right)\right)\\ =-{\displaystyle {\int }_{t_1}^{t_0}}{\left(\Phi \left(u^{\prime }\left(s\right)\right)\right)}^{\prime }-{\left(\Phi \left({\alpha }^{\prime }\left(s\right)\right)\right)}^{\prime }\text{d}s\\ =-{\displaystyle {\int }_{t_1}^{t_0}}-f\left(s,u\left(s\right),u^{\prime }\left(s\right)\right)-\Xi \left(u\left(s\right)\right)-{\left(\Phi \left({\alpha }^{\prime }\left(s\right)\right)\right)}^{\prime }\text{d}s\ge 0\end{array}$

which contradicts (2).

b) We suppose that $t_0=0$, then ${\alpha }^{\prime }\left(0\right)=u^{\prime }\left(0\right)$ and it follows that:

$\Phi \left(u^{\prime }\left(0\right)\right)-\Phi \left({\alpha }^{\prime }\left(0\right)\right)\ge 0.$ (3)

Since ${\alpha }^{\prime }\left(0\right)\in B_1\left(\alpha \left(0\right)\right)+e_0,e_0>0$, because of the monotonicity of $B_1$, if $\alpha \left(0\right)\ge u\left(0\right)$, we have:

${\alpha }^{\prime }\left(0\right)>u^{\prime }\left(0\right).$ Then, $\Phi \left({\alpha }^{\prime }\left(0\right)\right)>\Phi \left(u^{\prime }(0)\right)$

So $\Phi \left(u^{\prime }\left(0\right)\right)-\Phi \left({\alpha }^{\prime }\left(0\right)\right)<0$

which contradicts (3).

c) We suppose that $t_0=T$, then $u^{\prime }\left(T\right)={\alpha }^{\prime }\left(T\right)$. It follows that:

$\Phi \left(u^{\prime }\left(T\right)\right)-\Phi \left({\alpha }^{\prime }\left(T\right)\right)\le 0.$ (4)

Since $-{\alpha }^{\prime }\left(T\right)\in B_2\left(\alpha \left(T\right)\right)+e_0,e_0>0$, because of the monotonicity of $B_1$, if $\alpha \left(T\right)\ge u\left(T\right)$, we have:

${\alpha }^{\prime }\left(T\right)<u^{\prime }\left(T\right).$ Then, $\Phi \left({\alpha }^{\prime }\left(T\right)\right)<\Phi \left(u^{\prime }\left(T\right)\right)$.

So $\Phi \left(u^{\prime }\left(T\right)\right)-\Phi \left({\alpha }^{\prime }\left(T\right)\right)>0$

which contradicts 4. Then, $t_0$ does not exist. So, $A=\varnothing$.$\square$

Proposition 6. Let $\beta$ be an upper solution of (1) such that:

i) For all $t_0\in \left]\mathrm{0,}T\right[$, there exist ${\epsilon }_0>0$ and ${\Omega }_0$ is an open interval such that $t_0\in {\Omega }_0$ and:

$-\left(\Phi \left({\beta }^{\prime }\left(t\right)\right)\right)\ge f\left(t\mathrm{,}x\mathrm{,}y\right)+\Xi \left(x\right)\text{a}\text{.e}t\in {\Omega }_0\mathrm{,}$

for all $\left(x\mathrm{,}y\right)\in \left[\beta \left(t\right)-{\epsilon }_0\mathrm{,}\beta \left(t\right)\right]\times \left[{\beta }^{\prime }\left(t\right)-{\epsilon }_0\mathrm{,}{\beta }^{\prime }\left(t\right)+{\epsilon }_0\right]$ ;

ii) ${\beta }^{\prime }\left(0\right)\in B_1\left(\beta \left(0\right)\right)-{ℝ}_{+}^{\ast }$ ;

iii) $-{\beta }^{\prime }\left(T\right)\in B_2\left(\beta \left(T\right)\right)-{ℝ}_{+}^{\ast }$ ;

then $\beta$ is a strict upper solution of (1).

Proof. Let u be a solution of problem (1) such that $u\left(t\right)\le \beta \left(t\right)$ for all $t\in \Omega$.

Let us assume that u is not strict, then there exists $\bar{t}\in \left[\mathrm{0,}T\right]$ such that $\beta \left(\bar{t}\right)=u\left(\bar{t}\right)$. Whence

$E=\left\{t\in \left[0,T\right]:u\left(t\right)=\beta \left(t\right)\right\}\ne \varnothing$

E is closed and bounded. Let $t_0=\min E$. Then

$u^{\prime }\left(t_0\right)-{\beta }^{\prime }\left(t_0\right)=0$

a) If $t_0\in \left]\mathrm{0,}T\right[$ there exist ${\Omega }_0$ and ${ϵ}_0>0$ according to (i). We can choose $t_1\in {\Omega }_0$ such that $t_1<t_0$,$u^{\prime }\left(t_1\right)>{\beta }^{\prime }\left(t_1\right)$ and

$\forall t\in \left[t_1,t_0\right],\left(u\left(t\right),u^{\prime }\left(t\right)\right)\in \left]\beta \left(t\right)-{ϵ}_0,\beta \left(t\right)\right[\times \left]{\beta }^{\prime }\left(t\right)-{ϵ}_0,{\beta }^{\prime }\left(t\right)+{ϵ}_0\right[.$

Therefore, for almost $t\in \left[t_1\mathrm{,}{t}_0\right]$,

$-{\left(\Phi \left({\beta }^{\prime }\left(t\right)\right)\right)}^{\prime }-f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right)-\Xi \left(u\left(t\right)\right)\ge 0.$

Since $\Phi$ is an increasing homeomorphism, we have

$\Phi \left({\beta }^{\prime }\left(t_0\right)\right)-\Phi \left(u^{\prime }\left(t_0\right)\right)=0$

and

$\Phi \left(u^{\prime }\left(t_1\right)\right)>\Phi \left({\beta }^{\prime }\left(t_1\right)\right)$ (5)

Also we have:

$\begin{array}{l}\Phi \left(u^{\prime }\left(t_1\right)\right)-\Phi \left({\beta }^{\prime }\left(t_1\right)\right)\\ =-{\displaystyle {\int }_{t_1}^{t_0}}{\left(\Phi \left(u^{\prime }\left(s\right)\right)\right)}^{\prime }-{\left(\Phi \left({\beta }^{\prime }\left(s\right)\right)\right)}^{\prime }\text{d}s\\ =-{\displaystyle {\int }_{t_1}^{t_0}}{\left(-\Phi \left({\beta }^{\prime }\left(s\right)\right)\right)}^{\prime }-f\left(s,u\left(s\right),u^{\prime }\left(s\right)\right)-\Xi \left(u\left(s\right)\right)\text{d}s\le 0\end{array}$

which contradicts (5).

b) We suppose that $t_0=0$, then ${\beta }^{\prime }\left(0\right)=u^{\prime }\left(0\right)$ and it follows that:

$\Phi \left(u^{\prime }\left(0\right)\right)-\Phi \left({\beta }^{\prime }\left(0\right)\right)\le 0.$ (6)

Since ${\beta }^{\prime }\left(0\right)\in B_1\left(\beta \left(0\right)\right)-e_0,e_0>0$, because of the monotonicity of $B_1$, if $\beta \left(0\right)\le u\left(0\right)$, we have:

${\beta }^{\prime }\left(0\right)<u^{\prime }\left(0\right).$ Then $\Phi \left({\beta }^{\prime }\left(0\right)\right)<\Phi \left(u^{\prime }(0)\right)$

So $\Phi \left(u^{\prime }\left(0\right)\right)-\Phi \left({\beta }^{\prime }\left(0\right)\right)>0$

which contradicts (6).

c) We suppose that $t_0=T$, then $u^{\prime }\left(T\right)={\beta }^{\prime }\left(T\right)$. It follows that:

$\Phi \left(u^{\prime }\left(T\right)\right)-\Phi \left({\beta }^{\prime }\left(T\right)\right)\ge 0.$ (7)

Since $-{\beta }^{\prime }\left(T\right)\in B_2\left(\beta \left(T\right)\right)-e_0,e_0>0$, because of the monotonicity of $B_1$, if $\beta \left(T\right)\le u\left(T\right)$, we have:

${\beta }^{\prime }\left(T\right)>u^{\prime }\left(T\right)$. Then $\Phi \left({\beta }^{\prime }\left(T\right)\right)>\Phi \left(u^{\prime }\left(T\right)\right)$.

So $\Phi \left(u^{\prime }\left(T\right)\right)-\Phi \left({\beta }^{\prime }\left(T\right)\right)<0$

which contradicts (7). Then, $t_0$ does not exist. So, $E=\varnothing$.$\square$

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):

$\left(H_0\right)$ : There exists a lower solution $\alpha \in C^1\left(\Omega \right)$ and an upper solution $\beta \in C^1\left(\Omega \right)$.

$\left(H_{\Phi }\right)$ $\Phi \mathrm{<mo>\; :\; </mo>}ℝ\to ℝ$ is an increasing continuous map such that:

a) $\Phi \left(0\right)=0$ ;

b) there exists $d_1>0$ such that: $\Phi \left(x\right)x\ge d_1{\left|x\right|}^p$ for all $x\in ℝ$ ;

c) there exist $d_2,d_3>0$ such that for a.e $t\in \Omega$ and for all $x\in ℝ$ :

$\left|\Phi \left(x\right)\right|\le d_2+d_3{\left|x\right|}^{p-1}\mathrm{.}$

Remark 8. Suppose that $\Phi \left(z\right)={\Phi }_p\left(z\right)={\left|z\right|}^{p-2}z,p\ge 2$. Then this function satisfies hypothesis $\left(H_{\Phi }\right)$. This function corresponds to the one-dimensional operator p-Laplacian. Another interesting case which satisfies hypothesis $\left(H_{\Phi }\right)$ is when $\Phi$ is defined by $\Phi \left(z\right)=\phi \left(\left|z\right|\right){\left|z\right|}^{p-2}z$ with $\phi \mathrm{<mo>\; :\; </mo>}{ℝ}_{+}\to {ℝ}_{+}$ continuous, $\phi \left(y\right)\ge k>0$ for all $y\ge 0$ and $y\mapsto \phi \left(y\right)y^{p-1}$ is strictly increasing on ${ℝ}_{+}$ and $\phi \left(y\right)y^{p-1}\le d_2+d_3{y}^{p-1}$. For example, we can have

$\phi \left(y\right)=\frac{\sqrt{1+{\left(1+y^{p-1}\right)}^2}}{1+y^{p-1}}\text{or}\phi \left(y\right)=1+\frac{1}{1+y^{p-1}}\mathrm{.}$

It is well-known that under the monotonicity condition and the hypotheses (a) and (b), $\Phi$ is an increasing homeomorphism from $ℝ$ onto $ℝ$. And ${\Phi }^{-1}$ is strictly monotone and $\left|{\Phi }^{-1}\left(y\right)\right|\to +\infty$ as $\left|y\right|\to +\infty$ (See Deimling [12] chap. 3). Our operator $\Phi$ 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 $\left|\Phi \left(x\right)\right|\le d_2+d_3{\left|x\right|}^{p-1}$ is not assumed. Nevertheless, it incorporates the operator p-laplacian and many other classes of operators.

$\left(H_f\right)$ $f\mathrm{<mo>\; :\; </mo>}\Omega \times ℝ\times ℝ\to ℝ$ is a fonction such that:

i) for all $x\mathrm{,}y\in ℝ$,$t\mapsto f\left(t\mathrm{,}x\mathrm{,}y\right)$ is measurable;

ii) for a.e $t\in \Omega$,$\left(x\mathrm{,}y\right)\mapsto f\left(t\mathrm{,}x\mathrm{,}y\right)$ is continuous;

iii) for a.e $t\in \Omega$,$x\in \left[\alpha \left(t\right)\mathrm{,}\beta \left(t\right)\right]$ and all $y\in ℝ$, we have:

$\left|f\left(t,x,y\right)\right|<\eta \left(\left|\Phi \left(y\right)\right|\right)\left(\psi \left(t\right)+c\left|y\right|\right)$

where $\psi \in L^1{\left(\Omega \right)}_{+},c>0$ and $\eta \mathrm{<mo>\; :\; </mo>}{ℝ}_{+}\to {ℝ}_{+}\backslash \left\{0\right\}$ a Borel measurable non-decreasing functions such that:

$\begin{array}{c}{\displaystyle {\int }_{\Phi \left(\zeta \right)}^{+\infty }}\frac{\text{d}s}{\eta \left(s\right)}>{\Vert \psi \Vert }_1+c\left(\underset{\Omega }{\max }\beta -\underset{\Omega }{\min }\alpha \right)\\ +\frac{T}{\eta \left(\zeta \right)}\sup \left\{\left|\Xi \left(z\right)\right|:\left|z\right|\le \max \left\{{\Vert \alpha \Vert }_{\infty },{\Vert \beta \Vert }_{\infty }\right\}\right\}\end{array}$

with $\zeta =\frac{\max \left\{\left|\alpha \left(T\right)-\beta \left(0\right)\right|\mathrm{,}\left|\alpha \left(0\right)-\beta \left(T\right)\right|\right\}}{T}$ ;

iv) for every $r>0$, there exists ${\gamma }_r\in L^q\left(\Omega \right)$ such that for a.e $t\in \Omega$ and for all $x,y\in ℝ$ with $\left|x\right|,\left|y\right|\le r$ we have: $\left|f\left(t,x,y\right)\right|\le {\gamma }_r\left(t\right)$.

Remark 9. Hypothesis $\left(H_f\right)$ (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 $\left(H_f\right)$ (i), (ii) and (iv) are well known as $L^p$ -Carathéodory conditions.

$\left(H_B\right)$ : $B_1$ and $B_2$ : $ℝ\to P\left(ℝ\right)$ are maximal monotone maps such that $0\in B_1\left(0\right)\cap B_2\left(0\right)$.

Remark 10. There exist functions $E_1\mathrm{,}{E}_2\mathrm{<mo>\; :\; </mo>}ℝ\to ℝ\cup \left\{+\infty \right\}$ proper, convex and lower semi-continuous which are not identically equal to $+\infty$ such that $B_1=\partial E_1,B_2=\partial E_2$. More exactly, there exists some increasing positives functions $p_1$ and $p_2$ such that $p_i\left(s\right)=\text{Proj}\left(\mathrm{0\; <mo>\; ;\; </mo>}{B}_i\left(s\right)\right)$ (the minimum absolute value element in the closed, convex set $B_i\left(s\right)$ ). Then $E_i\left(s\right)={\displaystyle {\int }_0^s}{p}_i\left(t\right)\text{d}t$,$i=1,2$. We have $B_i\left(s\right)=\left[p_i\left(s-\right)\mathrm{<mo>\; ;\; </mo>}{p}_i\left(s+\right)\right]$ for all $s\in ℝ$, where

$p_i\left(s-\right)={\lim }_{\epsilon \to 0^{+}}{p}_i\left(s-\epsilon \right)$ and $p_i\left(s+\right)={\lim }_{\epsilon \to 0^{+}}{p}_i\left(s+\epsilon \right)$,$i=1,2$.

$\left(H_{\Xi }\right)$ : $\Xi \mathrm{<mo>\; :\; </mo>}ℝ\to ℝ$ is a function that maps bounded sets to bounded sets and there exists $M>0$ such that $x\to \Xi \left(x\right)+Mx$ is increasing.

Remark 11. We emphasize that $\Xi$ need not be continuous.

Lemma 1. If $u\in C^1\left(\Omega \right)$ and hypotheses $\left(H_{\Phi }\right)$ and $\left(H_f\right)$ (iii) hold,

$-{\left(\Phi \left(u^{\prime }\left(t\right)\right)\right)}^{\prime }=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right)+\Xi \left(u\left(t\right)\right)\text{a}\text{.eon}\Omega =\left[\mathrm{0,}T\right]$

and if

$\alpha \left(t\right)\le u\left(t\right)\le \beta \left(t\right)$ for all $t\in \Omega$

then, there exists $M_1>0$ (depending only on $\alpha \mathrm{,}\beta \mathrm{,}\eta \mathrm{,}\psi \mathrm{,}\Xi \mathrm{,}c$ ) such that: $\left|u^{\prime }\left(t\right)\right|\le M_1$ for all $t\in \Omega$.

Proof. Set $\mu =\left(\frac{1}{\eta \left(\zeta \right)}\right)\sup \left\{\left|\Xi \left(z\right)\right|:\left|z\right|\le \max \left\{{\Vert \alpha \Vert }_{\infty },{\Vert \beta \Vert }_{\infty }\right\}\right\}$ (See hypothesis $H_{\Xi }$ ). By hypothesis $\left(H_f\right)$ (iii), we can find $M_1>\zeta$ such that

${\displaystyle {\int }_{\Phi \left(\zeta \right)}^{\Phi \left(M_1\right)}}\frac{\text{d}s}{\eta \left(s\right)}>{\Vert \psi \Vert }_1+c\left(\underset{\Omega }{\max }\beta -\underset{\Omega }{\min }\alpha \right)+\mu T\mathrm{.}$

We claim that $\left|u^{\prime }\left(t\right)\right|\le M_1$ for all $t\in \Omega$. Let’s suppose that it is not the case. Then, we can find $t_1\in \Omega$ such that

$\left|u^{\prime }\left(t_1\right)\right|>M_1\mathrm{.}$

By the mean value theorem, there exists $t_2\in \left(\mathrm{0,}T\right)$ such that $u\left(T\right)-u\left(0\right)=u^{\prime }\left(t_2\right)T$. Without any loss of generality, we assume that $t_2\le t_1$. We obtain:

$\left|u^{\prime }\left(t_2\right)\right|=\frac{1}{T}\left|u\left(T\right)-u\left(0\right)\right|\le \frac{1}{T}\max \left\{\left|\alpha \left(T\right)-\beta \left(0\right)\right|,\left|\beta \left(T\right)-\alpha \left(0\right)\right|\right\}$ $\Rightarrow \left|u^{\prime }\left(t_2\right)\right|\le \zeta <M_1$.

Since $u\in C^1\left(\Omega \right)$, by the intermediate value theorem, there exists $t_3$ and $t_4\in \left[t_2\mathrm{,}{t}_1\right)$ with $t_3<t_4$ such that $\left|u^{\prime }\left(t_3\right)\right|=\zeta$ and $\left|u^{\prime }\left(t_4\right)\right|=M_1$. We have:

$\begin{array}{l}-{\left(\Phi \left(u^{\prime }\left(t\right)\right)\right)}^{\prime }=f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right)+\Xi \left(u\left(t\right)\right)\text{a}\text{.e}\text{on}\Omega =\left[\mathrm{0,}T\right]\\ \Rightarrow {\left|\Phi \left(u^{\prime }\left(t\right)\right)\right|}^{\prime }\le \left|{\left(\Phi \left(u^{\prime }\left(t\right)\right)\right)}^{\prime }\right|\le \left|f\left(t,u\left(t\right),u^{\prime }\left(t\right)\right)\right|+\left|\Xi \left(u\left(t\right)\right)\right|\\ \le \eta \left(\left|\Phi \left(u^{\prime }\left(t\right)\right)\right|\right)\left(\psi \left(t\right)+c\left|u^{\prime }\left(t\right)\right|\right)+\left|\Xi \left(u\left(t\right)\right)\right|\text{a}\text{.e}\text{on}\Omega \mathrm{.}\end{array}$

Thus:

$\frac{{\left|\Phi \left(u^{\prime }\left(t\right)\right)\right|}^{\prime }}{\eta \left(\left|\Phi \left(u^{\prime }\left(t\right)\right)\right|\right)}\le \psi \left(t\right)+c\left|u^{\prime }\left(t\right)\right|+\frac{\left|\Xi \left(u\left(t\right)\right)\right|}{\eta \left(\left|\Phi \left(u^{\prime }\left(t\right)\right)\right|\right)}\text{a}\text{.e}\text{on}\left[t_3\mathrm{,}{t}_4\right]$

and then

${\displaystyle {\int }_{t_3}^{t_4}}\frac{{\left|\Phi \left(u^{\prime }\left(t\right)\right)\right|}^{\prime }}{\eta \left(\left|\Phi \left(u^{\prime }\left(t\right)\right)\right|\right)}\text{d}t\le {\Vert \psi \Vert }_1+c\left(\underset{\Omega }{\max }\beta -\underset{\Omega }{\min }\alpha \right)+\mu T\mathrm{.}$

Setting $s=\left|\Phi \left(u^{\prime }\left(t\right)\right)\right|$, we have:

${\displaystyle {\int }_{\Phi \left(\zeta \right)}^{\Phi \left(M_1\right)}}\frac{\text{d}s}{\eta \left(s\right)}\le {\Vert \psi \Vert }_1+c\left(\underset{\Omega }{\max }\beta -\underset{\Omega }{\min }\alpha \right)+\mu T$

which contradicts the choice of $M_1>0$.$\square$

Now, we introduce the truncation map: $\varrho \mathrm{<mo>\; :\; </mo>}\Omega \times ℝ\times ℝ\to {ℝ}^2$ defined by

$\rho \left(t,x,y\right)=\{\begin{array}{ll}\left(\alpha \left(t\right),{\alpha }^{\prime }\left(t\right)\right)\hfill & \text{if}x<\alpha \left(t\right)\hfill \\ \left(\beta \left(t\right),{\beta }^{\prime }\left(t\right)\right)\hfill & \text{if}x>\beta \left(t\right)\hfill \\ \left(x,M_0\right)\hfill & \text{if}\alpha \left(t\right)\le x\le \beta \left(t\right),y>M_0\hfill \\ \left(x,-M_0\right)\hfill & \text{if}\alpha \left(t\right)\le x\le \beta \left(t\right),y<-M_0\hfill \\ \left(x,y\right)\hfill & \text{if}\alpha \left(t\right)\le x\le \beta \left(t\right),\left|y\right|\le M_0\hfill \end{array}$ (8)

where $M_0>\max \left\{M_1,{\Vert {\alpha }^{\prime }\Vert }_{\infty },{\Vert {\beta }^{\prime }\Vert }_{\infty }\right\}$ and the penalty function $\Lambda \mathrm{<mo>\; :\; </mo>}\Omega \times ℝ\to ℝ$ defined by:

$\Lambda \left(t,x\right)=\{\begin{array}{ll}{\Phi }_p\left(\alpha \left(t\right)\right)-{\Phi }_p\left(x\right)\hfill & \text{if}x<\alpha \left(t\right)\hfill \\ 0\hfill & \text{if}\alpha \left(t\right)\le x\le \beta \left(t\right)\hfill \\ {\Phi }_p\left(\beta \left(t\right)\right)-{\Phi }_p\left(x\right)\hfill & \text{if}x>\beta \left(t\right)\hfill \end{array}$ (9)

We set $f_1\left(t\mathrm{,}x\mathrm{,}y\right)=f\left(t\mathrm{,}\varrho \left(t\mathrm{,}x\mathrm{,}y\right)\right)$. Note that for ae $x\in \left[\alpha \left(t\right)\mathrm{,}\beta \left(t\right)\right]$ and all $\left|y\right|<M_0$, we have $f_1\left(t,x,y\right)=f\left(t,x,y\right)$. Moreover, for almost all $t\in \Omega$ and all $x\mathrm{,}y\in ℝ$, we have: $\left|f_1\left(t\mathrm{,}x\mathrm{,}y\right)\right|\le {\gamma }_r\left(t\right)$ with $r=\max \left\{M_0,{\Vert \alpha \Vert }_{\infty },{\Vert \beta \Vert }_{\infty }\right\}$. For every $u\in W^{\mathrm{1,}p}\left(\left(\mathrm{0,}T\right)\right)$, set

$N_1\left(u\right)\left(\mathrm{.}\right)=f_1(\mathrm{.,}u\left(\mathrm{.}\right)\mathrm{,}{u}^{\prime }(.))$

and

$\hat{\Lambda }\left(u\right)(.)=\Lambda (.,u(.))$

the Nemitsky operators corresponding to $f_1$ and $\Lambda$ respectively. We set $G\left(u\right)=N_1\left(u\right)+\hat{\Lambda }\left(u\right)$ for every $u\in W^{\mathrm{1,}p}\left(\left(\mathrm{0,}T\right)\right)$.

Proposition 12. If hypothesis $H\left(f\right)$ (ii) holds, then: $G\mathrm{<mo>\; :\; </mo>}{W}^{\mathrm{1,}p}\left(\left(\mathrm{0,}T\right)\right)\to L^q\left(\Omega \right)$ is continuous.

Proof. Since $N_1$ and $\hat{\Lambda }$ are Nemitsky operators, it is standard to show that they are continuous. It follows that G is continuous. $\square$

We introduce the set

$D=\left\{u\in C^1\left(\Omega \right):\Phi \left(u^{\prime }\right)\in W^{1,q}\left(0,T\right),u^{\prime }\left(0\right)\in B_1\left(u\left(0\right)\right)\text{and}-u^{\prime }\left(T\right)\in B_2\left(u\left(T\right)\right)\right\}$

and then we define the non-linear operator: $\vartheta \mathrm{<mo>\; :\; </mo>}D\subseteq L^p\left(\Omega \right)\to L^q\left(\Omega \right)$ by

$\vartheta \left(u\right)\left(\mathrm{.}\right)=-{\left(\Phi \left(u^{\prime }\left(\mathrm{.}\right)\right)\right)}^{\prime }\text{for}\text{all}u\in D\mathrm{.}$

Proposition 13. If the hypotheses $\left(H_B\right)$ and $\left(H_{\Phi }\right)$ hold, then $\vartheta$ is maximal monotone.

Proof. Given $h\in L^q\left(\Omega \right)$, we consider the following nonlinear boundary value problem:

$\{\begin{array}{l}-{\left(\Phi \left(u^{\prime }\left(t\right)\right)\right)}^{\prime }+{\Phi }_p\left(u\left(t\right)\right)=h\left(t\right)\text{a}\text{.e}\text{on}\Omega =\left[\mathrm{0,}T\right]\\ u^{\prime }\left(0\right)\in B_1\left(u\left(0\right)\right)\mathrm{,}-u^{\prime }\left(T\right)\in B_2\left(u\left(T\right)\right)\mathrm{.}\end{array}$ (10)

We show that problem (10) has a unique solution $u\in C^1\left(\Omega \right)$. To this end, given $v\mathrm{,}w\in ℝ$, we consider the following two-point boundary value problem:

$\{\begin{array}{l}-{\left(\Phi \left(u^{\prime }\left(t\right)\right)\right)}^{\prime }+{\Phi }_p\left(u\left(t\right)\right)=h\left(t\right)\text{a}\text{.e}\text{on}\Omega =\left[\mathrm{0,}T\right]\\ u\left(0\right)=v\mathrm{,}u\left(T\right)=w\end{array}$ (11)

We set $\gamma \left(t\right)=\left(1-\frac{t}{T}\right)v+\frac{t}{T}w$. Then $\gamma \left(0\right)=v$ and $\gamma \left(T\right)=w$. We consider the function y defined by $y\left(t\right)=u\left(t\right)-\gamma \left(t\right)$ and rewrite (11) in the terms of this function.

$\{\begin{array}{l}-{\left(\Phi \left(y^{\prime }\left(t\right)+{\gamma }^{\prime }\left(t\right)\right)\right)}^{\prime }+{\Phi }_p\left(y\left(t\right)+\gamma \left(t\right)\right)=h\left(t\right)\text{a}\text{.e}\text{on}\Omega =\left[\mathrm{0,}T\right]\\ y\left(0\right)=y\left(T\right)=0.\end{array}$ (12)

This is a homogeneous Dirichlet problem for (11). To solve (12), let $V_1$ be the non-linear operator defined by: $V_1\mathrm{<mo>\; :\; </mo>}{W}_0^{\mathrm{1,}p}\left(\Omega \right)\to W^{-\mathrm{1,}q}(\Omega )$

$\begin{array}{l}{\langle V_1\left(y\right)\mathrm{,}z\rangle }_0={\displaystyle {\int }_0^T}\Phi \left(y^{\prime }\left(t\right)+{\gamma }^{\prime }\left(t\right)\right)z^{\prime }\left(t\right)\text{d}t\\ +{\displaystyle {\int }_0^T}{\Phi }_p\left(y\left(t\right)+\gamma \left(t\right)\right)z\left(t\right)\text{d}t\mathrm{,}\forall y\mathrm{,}z\in W_0^{\mathrm{1,}p}(\Omega )\end{array}$

where ${\langle \rangle }_0$ denote the duality brackets for the pair $\left(W^{-\mathrm{1,}q}\left(\Omega \right)\mathrm{,}{W}_0^{\mathrm{1,}p}\left(\Omega \right)\right)$.

Let us show that $V_1$ is strictly monotone.

Let $y\mathrm{,}z\in W_0^{\mathrm{1,}p}\left(\Omega \right)$. We have

$\begin{array}{l}{\langle V_1\left(y\right)-V_1\left(z\right)\mathrm{,}y-z\rangle }_0\\ ={\langle V_1\left(y\right)\mathrm{,}y-z\rangle }_0-{\langle V_1\left(z\right)\mathrm{,}y-z\rangle }_0\\ ={\displaystyle {\int }_0^T}\Phi \left(y^{\prime }\left(t\right)+{\gamma }^{\prime }\left(t\right)\right)\left(y^{\prime }\left(t\right)-z^{\prime }\left(t\right)\right)\text{d}t+{\displaystyle {\int }_0^T}{\Phi }_p\left(y\left(t\right)+\gamma \left(t\right)\right)\left(y\left(t\right)-z\left(t\right)\right)\text{d}t\\ -{\displaystyle {\int }_0^T}\Phi \left(z^{\prime }\left(t\right)+{\gamma }^{\prime }\left(t\right)\right)\left(y^{\prime }\left(t\right)-z^{\prime }\left(t\right)\right)\text{d}t-{\displaystyle {\int }_0^T}{\Phi }_p\left(z\left(t\right)+\gamma \left(t\right)\right)\left(y\left(t\right)-z\left(t\right)\right)\text{d}t\mathrm{.}\end{array}$