Periodic solutions for implicit evolution inclusions

TL;DR

This paper proves the existence of periodic solutions for a class of nonlinear implicit evolution inclusions in Hilbert spaces, using approximation methods and surjectivity results for parabolic operators.

Contribution

It introduces a novel approach combining approximation techniques and surjectivity results to establish periodic solutions for nonlinear implicit evolution inclusions.

Findings

Existence of periodic solutions established.

Applicable to nonlinear, nonmonotone, time-varying set-valued maps.

Method extends to evolution equations in Hilbert spaces.

Abstract

We consider a nonlinear implicit evolution inclusion driven by a nonlinear, nonmonotone, time-varying set-valued map and defined in the framework of an evolution triple of Hilbert spaces. Using an approximation technique and a surjectivity result for parabolic operators of monotone type, we show the existence of a periodic solution.

Equations

$$\left\{\begin{array}[]{ll}\frac{d}{dt}(Bu(t))+A(t,u(t))\ni 0\ \mbox{for almost all}\ t\in T=[0,b]\\ B(u(0))=B(u(b)).\end{array}\right\}$$

$$X\hookrightarrow H\hookrightarrow X^{\ast }$$

$$|\cdot |⩽c_1||\cdot ||\text{mbox}and||\cdot |{|}_{\ast }⩽c_2|\cdot |\text{mbox}forsome{c}_1,c_2>0.$$

$$⟨\cdot ,\cdot ⟩{|}_{H\times X}=(\cdot ,\cdot ).$$

$$W_p(0,b)=\{u\in L^p(T,X):u^{\prime }\in L^{p^{\prime }}(T,X^{\ast })\}.$$

$$W_p(0,b)\subseteq A{C}^{1,p^{\prime }}(T,X^{\ast })=W^{1,p^{\prime }}((0,b),X^{\ast }).$$

$$\frac{d}{dt}(u(t),v(t))=⟨u^{\prime }(t),v(t)⟩+⟨u(t),v^{\prime }(t)⟩\text{mbox}foralmostallt\in T.$$

$$\text{mbox}GrG=\{(t,y)\in T\times Y:y\in G(t)\}\in {\mathcal{L}}_T\otimes B(Y),$$

$$P_{f(c)}(X)=\{C\subseteq Y:C\text{mbox}isnonempty,closed(andconvex)\}.$$

$$|C|=\sup \left\{||c|{|}_Y:c\in C\right\}.$$

$$\underset{n\to \infty }{\mathrm{limsup}}⟨y_n^{\ast },y_n-y{⟩}_{Y^{\ast }Y}⩽0,$$

$$|A(t,x)|⩽c_1(t)+c_2||x|{|}^{p-1}$$

$$\inf \left\{⟨u^{\ast },x⟩:u^{\ast }\in A(t,x)\right\}⩾c_3||x|{|}^p-c_4(t),$$

$$(u,v{)}_{\ast }=⟨(ϵJ+B{)}^{-1}u,v⟩\text{mbox}forallu,v\in X^{\ast }.$$

$$A_{ϵ}(t,v)=A(t,(ϵJ+B{)}^{-1}v).$$

$${\hat{A}}_{ϵ}(v)=\{u\in L^{p^{\prime }}(T,V^{\ast }):u(t)\in A_{ϵ}(t,v(t))\text{mbox}foralmostallt\in T\}.$$

$$W_p^{per}((0,b),V^{\ast })=\{u\in L^p(T,V^{\ast }):u^{\prime }\in L^{p^{\prime }}(T,V^{\ast }),u(0)=u(b)\}.$$

$$L(u)=u^{\prime }.$$

$$s_n\to u\text{mbox}in{L}^p(T,V^{\ast }),s_n(t)\to u(t)\text{mbox}foralmostallt\in T,$$

$$|s_n(t){|}_{\ast }⩽|u(t){|}_{\ast }\text{mbox}foralmostallt\in T,\text{mbox}andforalln\in \mathbb{N}.$$

$$t\mapsto A_{ϵ}(t,s_n(t))=A(t,(ϵJ+B{)}^{-1}{s}_n(t))$$

$$v_n\stackrel{w}{\to }v\text{mbox}in{L}^{p^{\prime }}(T,V^{\ast })\text{mbox}asn\to \infty .$$

$$v(t)\in \overline{\mathrm{conv}}w-\underset{n\to \infty }{\mathrm{limsup}}{A}_{ϵ}(t,s_n(t))\subseteq A_{ϵ}(t,u(t))\text{mbox}foralmostallt\in T,$$

$$\Rightarrow v\in {\hat{A}}_{ϵ}(u)\text{mbox}andso{\hat{A}}_{ϵ}(\cdot )\text{mbox}hasnonemptyvalues.$$

$$((v,u){)}_{\ast }={\int }_0^b(v(t),u(t){)}_{\ast }dt\text{mbox}forallu\in L^p(T,V^{\ast }),v\in L^{p^{\prime }}(T,V^{\ast }).$$

$$\begin{array}[]{ll}``u_{n}\xrightarrow{w}u\ \mbox{in}\ L^{p}(T,V^{*}),u^{\prime}_{n}\xrightarrow{w}u^{\prime}\ \mbox{in}\ L^{p^{\prime}}(T,V^{*})\ \mbox{and}\ v_{n}\in\hat{A}_{\epsilon}(u_{n})\ (\mbox{for all}\ n\in\mathbb{N}),\\ \mbox{such that}\ v_{n}\xrightarrow{w}v\ \mbox{in}\ L^{p^{\prime}}(T,V^{*})\ \mbox{and}\ \limsup_{n\rightarrow\infty}((v_{n},u_{n}-u))_{*}\leqslant 0".\end{array}$$

$$\displaystyle\displaystyle\begin{array}[]{ll}((v_{n},u_{n}-u))_{*}&=\int^{b}_{0}(v_{n}(t),u_{n}(t)-u(t))_{*}dt\ \mbox{(see (\ref{eq4}))}\\ &=\int^{b}_{0}\langle v_{n}(t),(\epsilon J+B)^{-1}(u_{n}-u)(t)\rangle dt\ \mbox{(see (\ref{eq2}))}.\end{array}$$

$$⟨v_n(t),(ϵJ+B{)}^{-1}(u_n-u)(t)⟩=⟨v_n(t),y_n(t)-y(t)⟩$$

$$\{y_n{\}}_{n⩾1}\subseteq L^p(T,X)\text{mbox}isbounded(see(\text{ref}eq5)).$$

$$\begin{array}[]{ll}&y^{\prime}_{n}=((\epsilon J+B)^{-1}u_{n})^{\prime}\\ \Rightarrow&\{y^{\prime}_{n}\}_{n\geqslant 1}\subseteq L^{p^{\prime}}(T,X^{*})\ \mbox{is bounded (see (\ref{eq5}))}.\end{array}$$

$$\{y_n{\}}_{n⩾1}\subseteq W_p(0,b)\text{mbox}isbounded.$$

Full text

aa 2010 Mathematics Subject Classification. Primary: 34A20. Secondary: 35A05, 35R70.

1. Introduction

In this paper we study the following periodic implicit evolution inclusion

[TABLE]

Problem (1) is defined in the framework of an evolution triple $\displaystyle(X,H,X^{*})$ of Hilbert spaces (see Section 2), where $\displaystyle B\in\mathcal{L}(X,X^{*})$ and $\displaystyle A:T\times X\rightarrow 2^{X^{*}}$ is a map measurable in $\displaystyle t\in T$ and such that for almost all $\displaystyle t\in T$, $\displaystyle A(t,\cdot)$ is bounded and pseudo-monotone.

Implicit evolution equations were studied by Andrews, Kuttler & Schillor [1], Barbu [2], Barbu & Favini [4], Favini & Yagi [6], Liu [11], and Showalter [15]. However, in all these works, the operator $\displaystyle A$ was time-invariant and maximal monotone. Moreover, the aforementioned works treat the Cauchy problem. We are not aware of any work on implicit evolution equations treating the periodic problem. We mention also the works of Barbu & Favini [3] and DiBenedetto & Showalter [5], treating the case where $\displaystyle B$ is nonlinear monotone. For this case the hypotheses and the techniques are different.

This paper is strongly influenced by Lions [10]. In fact, our existence result (Theorem 7) is based on a multivalued version of a surjectivity result, which was proved for the first time for single-valued maps by Lions [10, Theorem 1.2, p. 319], see Theorem 4 below. This way we can accommodate the multivalued nature of the map $\displaystyle A(t,x)$ in problem (1). The fact that we allow $\displaystyle A(t,x)$ to be set-valued broadens significantly the applicability of our work. Now we can also treat the subdifferential of continuous but not $\displaystyle C^{1}$-convex functionals, a situation that the single-valued formulation cannot handle. In addition, the presence of the operator $\displaystyle B$ in the time derivative complicates the abstract setting. Since $\displaystyle B$ can be degenerate, this adds an additional level of difficulty in the analysis of problem (1) compared to the applications studied by Lions [10, pp. 321-328]. We overcome the difficulty, using the elliptic regularization technique, also first introduced by Lions.

2. Mathematical background

Suppose that $\displaystyle X$ and $\displaystyle Y$ are Banach spaces and $\displaystyle X$ is continuously and densely embedded into $\displaystyle Y$. Then we know that $\displaystyle Y^{*}$ is continuously embedded into $\displaystyle X^{*}$ and if $\displaystyle X$ is reflexive, then the embedding of $\displaystyle Y^{*}$ into $\displaystyle X^{*}$ is also dense.

Definition 1**.**

By an “evolutions triple”, we mean a triple of spaces

[TABLE]

such that $\displaystyle X$ is a separable reflexive Banach space, $\displaystyle H$ is a separable Hilbert space identified with its dual (pivot space), and $\displaystyle X$ is continuously embedded into $\displaystyle H$. We say that $\displaystyle(X,H,X^{*})$ is an evolution triple of Hilbert spaces, if all three spaces are Hilbert.

Evidently, $\displaystyle H^{*}=H$ is continuously and densely embedded into $\displaystyle X^{*}$. By $\displaystyle||\cdot||$ (resp $\displaystyle|\cdot|,\ ||\cdot||_{*}$), we denote the norm of $\displaystyle X$ (resp. of $\displaystyle H,X^{*}$). We have

[TABLE]

We denote by $\displaystyle\langle\cdot,\cdot\rangle$ the duality brackets for the pair $\displaystyle(X^{*},X)$ and by $\displaystyle(\cdot,\cdot)$ the inner product of $\displaystyle H$. We have

[TABLE]

Given an evolution triple $\displaystyle(X,H,X^{*})$ and $\displaystyle 1<p<\infty$, we can define the following Banach space:

[TABLE]

In this definition, $\displaystyle\frac{1}{p}+\frac{1}{p^{\prime}}=1$ and the derivative $\displaystyle u^{\prime}$ of $\displaystyle u$ is understood in the sense of vectorial distributions. A function $\displaystyle u\in W_{p}(0,b)$ viewed as a function with values in $\displaystyle X^{*}$, is absolutely continuous and so

[TABLE]

Also, we know that $\displaystyle L^{p}(T,X^{*})^{*}=L^{p^{\prime}}(T,X).$ The space $\displaystyle W_{p}(0,b)$ is continuously and densely embedded into $\displaystyle C(T,H)$ and its elements satisfy the following integration by parts formula.

Proposition 2**.**

If $\displaystyle(X,H,X^{*})$ is an evolution triple and $\displaystyle u,v\in W_{p}(0,b)$ $\displaystyle(1<p<\infty)$, then the mapping $\displaystyle t\mapsto(u(t),v(t))$ is absolutely continuous and

[TABLE]

If $\displaystyle(X,H,X^{*})$ is an evolution triple and $\displaystyle X$ is compactly embedded into $\displaystyle H$, then $\displaystyle H^{*}=H$ is compactly embedded into $\displaystyle X^{*}$ (Schauder’s theorem) and $\displaystyle W_{p}(0,b)$ is compactly embedded into $\displaystyle L^{p}(T,H)$. For details, see Gasinski & Papageorgiou [7].

We will use the following notions from set-valued analysis (see [9]).

• (a)

If $\displaystyle V,W$ are Hausdorff topological spaces and $\displaystyle G:V\rightarrow 2^{W}\backslash\{\emptyset\}$ is a multivalued map, then we say that $\displaystyle G(\cdot)$ is “upper semicontinuous” (“usc” for short), if for every $\displaystyle C\subseteq W$ closed, the set $\displaystyle G^{-}(C)=\{v\in V:G(v)\cap C\neq\emptyset\}$ is closed.

• (b)

If $\displaystyle T=[0,b]$, $\displaystyle Y$ is a separable Banach space and $\displaystyle G:T\rightarrow 2^{Y}\backslash\{\emptyset\}$ is a multivalued map, then we say that $\displaystyle G(\cdot)$ is “graph measurable” if

[TABLE]

with $\displaystyle\mathcal{L}_{T}$ being the Lebesgue $\displaystyle\sigma$-field of $\displaystyle T$ and $\displaystyle B(Y)$ the Borel $\displaystyle\sigma$-field on $\displaystyle Y$.

Given a Banach space, we will use the following notation

[TABLE]

Also, if $\displaystyle C\subseteq Y$, then we define

[TABLE]

Let $\displaystyle Y$ be a reflexive Banach space and $\displaystyle A:Y\rightarrow 2^{Y^{*}}$ a multivalued map. We say that $\displaystyle A(\cdot)$ is pseudo-monotone, if the following conditions are satisfied:

• •

for every $\displaystyle y\in Y,\ A(y)$ is nonempty, closed, and convex;

• •

$\displaystyle A(\cdot)$ is bounded (that is, maps bounded sets to bounded sets);

• •

if $\displaystyle y_{n}\xrightarrow{w}y$ in $\displaystyle Y$, $\displaystyle y^{*}_{n}\xrightarrow{w}y^{*}$ in $\displaystyle Y^{*}$ with $\displaystyle y^{*}_{n}\in A(y_{n})$ for all $\displaystyle n\in\mathbb{N}$ and

[TABLE]

then $\displaystyle y^{*}\in A(y)$ and $\displaystyle\langle y^{*}_{n},y_{n}\rangle_{Y^{*}Y}\rightarrow\langle y^{*},y\rangle_{Y^{*}Y}$.

Any maximal monotone map $\displaystyle A:Y\rightarrow 2^{Y^{*}}\backslash\{\emptyset\}$ is pseudo-monotone (see Gasinski & Papageorgiou [7, pp. 331-332]). As in the case of maximal monotone maps, pseudo-monotone operators exhibit nice surjectivity properties. In particular, a pseudo-monotone coercive (that is, $\displaystyle\frac{\inf\{\langle y^{*},y\rangle_{Y^{*}Y}:y^{*}\in A(y)\}}{||y||_{Y}}\rightarrow+\infty$ as $\displaystyle||y||_{Y}\rightarrow+\infty$) map is surjective (see Gasinski & Papageorgiou [7, p. 326]).

For dynamic problems (evolution equations), we have the following variant of the notion of pseudo-monotonicity.

Definition 3**.**

Let $\displaystyle Y$ be a reflexive Banach space, $\displaystyle L:D(L)\subseteq Y\rightarrow Y^{*}$ be a linear, maximal monotone operator and $\displaystyle A:Y\rightarrow 2^{Y^{*}}$ a multivalued map. We say that $\displaystyle A(\cdot)$ is “L-pseudo-monotone”, if the following conditions hold:

• (i)

for every $\displaystyle y\in Y$, $\displaystyle A(y)\subseteq Y^{*}$ is nonempty, $\displaystyle w$-compact, and convex;

• (ii)

$\displaystyle A:Y\rightarrow 2^{Y^{*}}\backslash\{\emptyset\}$* is usc from every finite dimensional subspace of $\displaystyle Y$ into $\displaystyle Y^{*}$ furnished with the weak topology;*

• (iii)

if $\displaystyle\{y_{n}\}_{n\geqslant 1}\subseteq D(L)$, $\displaystyle y_{n}\xrightarrow{w}y\in D(L)$ in $\displaystyle Y$, $\displaystyle L(y_{n})\xrightarrow{w}L(y)$ in $\displaystyle Y^{*}$, $\displaystyle y^{*}_{n}\in A(y_{n})$ for all $\displaystyle n\in\mathbb{N}$, $\displaystyle y^{*}_{n}\xrightarrow{w}y^{*}$ in $\displaystyle Y^{*}$ and $\displaystyle\limsup_{n\rightarrow\infty}\langle y^{*}_{n},y_{n}-y\rangle\leqslant 0$, then $\displaystyle y^{*}\in A(y)$ and $\displaystyle\langle y^{*}_{n},y_{n}\rangle_{Y^{*}Y}\rightarrow\langle y^{*},y\rangle_{Y^{*}Y}$.

These operators have nice surjectivity properties. The following result can be found in Papageorgiou, Papalini & Renzacci [12] (the single-valued version of this property is due to Lions [10]).

Theorem 4**.**

If $\displaystyle Y$ is a strictly convex reflexive Banach space, $\displaystyle L:D(L)\subseteq Y\rightarrow Y^{*}$ is a linear, maximal monotone operator, and $\displaystyle A:Y\rightarrow 2^{Y^{*}}$ is bounded, L-pseudo-monotone, and coercive, then $\displaystyle L+A$ is surjective.

3. Periodic solutions

In what follows, $\displaystyle T=[0,b]$ and $\displaystyle(X,H,X^{*})$ is an evolution triple of Hilbert spaces. We assume that $\displaystyle X$ is compactly embedded into $\displaystyle H$ (hence so is $\displaystyle H^{*}=H$ into $\displaystyle X^{*}$). The hypotheses on the data of (1) are the following:

$\displaystyle H(B)$: $\displaystyle B\in\mathcal{L}(X,X^{*})$ and is symmetric and monotone.

$\displaystyle H(A)$: $\displaystyle A:T\times X\rightarrow P_{f_{c}}(X^{*})$ is a multivalued map such that

• (i)

for all $\displaystyle x\in X$, the mapping $\displaystyle t\mapsto A(t,x)$ is graph measurable;

• (ii)

for almost all $\displaystyle t\in T$, the mapping $\displaystyle x\mapsto A(t,x)$ is pseudo-monotone;

• (iii)

for almost all $\displaystyle t\in T$ and all $\displaystyle x\in X$, we have

[TABLE]

with $\displaystyle c_{1}\in L^{p^{\prime}}(T),2\leqslant p<\infty$ and $\displaystyle c_{2}>0$;

• (iv)

for almost all $\displaystyle t\in T$ and all $\displaystyle x\in X$, we have

[TABLE]

with $\displaystyle c_{3}>0$ and $\displaystyle c_{4}\in L^{1}(T)$.

Let $\displaystyle J:X\rightarrow X^{*}$ be the duality (Riesz) map on the Hilbert space $\displaystyle X$. We know that $\displaystyle J(\cdot)$ is an isometric isomorphism (the Riesz-Fréchet theorem) which is monotone. Hence for every $\displaystyle\epsilon>0$ we have $\displaystyle(\epsilon J+B)^{-1}\in\mathcal{L}(X^{*},X)$. Then on $\displaystyle X^{*}$ we consider the following bilinear form

[TABLE]

Hypotheses $\displaystyle H(B)$ imply that $\displaystyle(\cdot,\cdot)_{*}$ is an inner product on $\displaystyle X^{*}$. Let $\displaystyle|\cdot|_{*}$ denote the norm corresponding to this inner product. Clearly, $\displaystyle|\cdot|_{*}$ and $\displaystyle||\cdot||_{*}$ are equivalent norms on $\displaystyle X^{*}$. So, if $\displaystyle V^{*}$ denotes the space $\displaystyle X^{*}$ equipped with the norm $\displaystyle|\cdot|_{*}$, then $\displaystyle V^{*}$ is a Hilbert space. Using the Riesz-Fréchet theorem, we identify $\displaystyle V^{*}$ with its dual.

Let $\displaystyle A_{\epsilon}:T\times V^{*}\rightarrow P_{f_{c}}(V^{*})$ be defined by

[TABLE]

Then we introduce the multivalued Nemitsky map $\displaystyle\hat{A}_{\epsilon}:L^{p}(T,V^{*})\rightarrow 2^{L^{p^{\prime}}(T,V^{*})}$ corresponding to $\displaystyle A_{\epsilon}(\cdot,\cdot)$, defined by

[TABLE]

Consider the function space

[TABLE]

We know that $\displaystyle W^{per}_{p}((0,b),V^{*})\hookrightarrow C(T,V^{*})$ and so the evaluations of $\displaystyle u$ at $\displaystyle t=0$ and $\displaystyle t=b$ make sense. Let $\displaystyle L:W^{per}_{p}((0,b),V^{*})\subseteq L^{p}(T,V^{*})\rightarrow L^{p^{\prime}}(T,V^{*})$ be defined by

[TABLE]

We know that $\displaystyle L(\cdot)$ is linear and maximal monotone (see Hu & Papageorgiou [9, p. 419] and Zeidler [16, p. 855]).

Proposition 5**.**

If hypotheses $\displaystyle H(B),H(A)$ hold and $\displaystyle\epsilon>0$, then for every $\displaystyle u\in L^{p}(T,V^{*})$, $\displaystyle\hat{A}_{\epsilon}(u)\subseteq L^{p^{\prime}}(T,V^{*})$ is nonempty, $\displaystyle w$-compact and convex, and the mapping $\displaystyle u\mapsto\hat{A}_{\epsilon}(u)$ is $\displaystyle L$-pseudo-monotone.

Proof.

It is clear that $\displaystyle\hat{A}_{\epsilon}(u)$ is closed, convex and bounded, thus $\displaystyle w$-compact in $\displaystyle L^{p^{\prime}}(T,V^{*})$. We need to show that $\displaystyle\hat{A}_{\epsilon}(\cdot)$ has nonempty values. Note that hypotheses $\displaystyle H(A)(i),(ii)$ do not imply the graph measurability of $\displaystyle(t,x)\mapsto A_{\epsilon}(t,x)$ (see Hu & Papageorgiouo [9, p. 227]). To show the nonemptiness of $\displaystyle\hat{A}_{\epsilon}(u)$ we proceed as follows. Let $\displaystyle\{s_{n}\}_{n\geqslant 1}\subseteq L^{p}(T,V^{*})$ be step functions such that

[TABLE]

On account of hypothesis $\displaystyle H(A)(i)$, for every $\displaystyle n\in\mathbb{N}$ the mapping

[TABLE]

is graph measurable. So, we can apply the Yankov-von Neumann-Aumann selection theorem (see Hu & Papageorgiou [9, p. 158]) and obtain that $\displaystyle v_{n}:T\rightarrow V^{*}$ is measurable and $\displaystyle v_{n}(t)\in A_{\epsilon}(t,s_{n}(t))$ for almost all $\displaystyle t\in T,n\in\mathbb{N}$. Evidently, $\displaystyle v_{n}\in L^{p^{\prime}}(T,V^{*})$ and $\displaystyle\{v_{n}\}_{n\geqslant 1}\subseteq L^{p^{\prime}}(T,V^{*})$ is bounded. So, by passing to a suitable subsequence if necessary we may assume that

[TABLE]

Note that the pseudo-monotonicity of $\displaystyle A_{\epsilon}(t,\cdot)$ (see hypothesis $\displaystyle H(A)(ii)$) implies that $\displaystyle\mbox{Gr}\,A_{\epsilon}(t,\cdot)$. is demiclosed (that is, sequentially closed in $\displaystyle V^{*}\times V^{*}_{w}$, where $\displaystyle V^{*}_{w}$ denotes the Hilbert space $\displaystyle V^{*}$ furnished with the weak topology). So, by (3) and Proposition 3.9 of Hu & Papageorgiou [9, p. 694], we have

[TABLE]

Next, we will prove the $\displaystyle L$-pseudo-monotonicity of $\displaystyle\hat{A}_{\epsilon}$. So, let $\displaystyle((\cdot,\cdot))_{*}$ denote the duality brackets for the pair $\displaystyle(L^{p^{\prime}}(T,V^{*}),L^{p}(T,V^{*}))$, that is,

[TABLE]

Consider a sequence $\displaystyle\{u_{n}\}_{n\geqslant 1}\subseteq W^{per}_{p}((0,b),V^{*})$ such that

[TABLE]

We have

[TABLE]

Let $\displaystyle y_{n}(t)=(\epsilon J+B)^{-1}u_{n}(t),\ y(t)=(\epsilon J+B)^{-1}u(t)$. Then $\displaystyle y_{n},y\in L^{p}(T,X)$ and we have

[TABLE]

with $\displaystyle v_{n}(t)\in A(t,y_{n}(t))$ for almost all $\displaystyle t\in T$, all $\displaystyle n\in\mathbb{N}$. Evidently,

[TABLE]

Also, we have

[TABLE]

It follows from (7) and (8) that

[TABLE]

So, we may assume that

[TABLE]

Evidently, we have $\displaystyle y=(\epsilon J+B)^{-1}u$ and so

[TABLE]

If we denote by $\displaystyle((\cdot,\cdot))$ the duality brackets for the pair $\displaystyle(L^{p^{\prime}}(T,X^{*}),L^{p}(T,X))$, that is,

[TABLE]

then we have

[TABLE]

Recall that $\displaystyle W_{p}(0,b)$ is continuously embedded in $\displaystyle C(T,H)$. So, from (9) we have

[TABLE]

Let $\displaystyle\vartheta_{n}(t)=\langle v_{n}(t),y_{n}(t)-y(t)\rangle$ and let $\displaystyle N\subseteq T$ be the Lebesgue-null set outside of which hypotheses $\displaystyle H(A)(ii),\,(iii)\,(iv)$ hold. Then for $\displaystyle t\in T\backslash N$, we have

[TABLE]

Let $\displaystyle E=\{t\in T:\liminf_{n\rightarrow\infty}\vartheta_{n}(t)<0\}$. This is a Lebesgue measurable set. Suppose that $\displaystyle\lambda^{1}(E)>0$ ($\displaystyle\lambda^{1}(\cdot)$ denotes the Lebesgue measure on $\displaystyle\mathbb{R}$). From (11), we see that $\displaystyle\{y_{n}(t)\}_{n\geqslant 1}\subseteq X$ is bounded for all $\displaystyle t\in E\cap(T\backslash N)$. So, on account of (10) we obtain that $\displaystyle y_{n}(t)\xrightarrow{w}y(t)$ in $\displaystyle X$. Fix $\displaystyle t\in E\cap(T\backslash N)$ and choose a suitable subsequence (depending on $\displaystyle t$) such that $\displaystyle\liminf_{n\rightarrow\infty}\vartheta_{n}(t)=\lim_{k\rightarrow\infty}\vartheta_{n_{k}}(t)$. The pseudo-monotonicity of $\displaystyle A(t,\cdot)$ (see hypothesis $\displaystyle H(A)(ii)$), implies that

[TABLE]

a contradiction since $\displaystyle t\in E$. Therefore $\displaystyle\lambda^{1}(E)=0$ and so we have

[TABLE]

Invoking Fatou’s lemma, we have

[TABLE]

We have $\displaystyle|\vartheta_{n}|=\vartheta^{+}_{n}+\vartheta^{-}_{n}=\vartheta_{n}+2\vartheta^{-}_{n}$ and $\displaystyle\vartheta^{-}_{n}(t)\rightarrow 0$ for almost all $\displaystyle t\in T$ (see (12)). Also, from (11) we have

[TABLE]

and $\displaystyle\{\gamma_{n}\}_{n\geqslant 1}\subseteq L^{1}(T)$ is uniformly integrable. We have

[TABLE]

Applying the extended dominated convergence theorem (see, for example, Gasinski & Papageorgiou [7, p. 901]), we have

[TABLE]

So, by passing to a subsequence if necessary, we may assume that

[TABLE]

Since $\displaystyle v_{n}(t)\in A(t,y_{n}(t))$ for almost all $\displaystyle t\in T$ and for all $\displaystyle n\in\mathbb{N}$, on account of the pseudo-monotonicity of $\displaystyle A(t,\cdot)$ (see hypothesis $\displaystyle H(A)(ii)$), we have

[TABLE]

and $\displaystyle v_{n}(t)\xrightarrow{w}v(t)$ in $\displaystyle X^{*}$, $\displaystyle\langle v_{n}(t),y_{n}(t)\rangle\rightarrow\langle v(t),y(t)\rangle$ for almost all $\displaystyle t\in T$.

By the dominated convergence theorem, we have

[TABLE]

Finally, using Proposition 2.23 of Hu & Papageorgiou [9, p. 43], we easily see that $\displaystyle\hat{A}_{\epsilon}(\cdot)$ is usc from finite dimensional subspaces of $\displaystyle L^{p}(T,V^{*})$ into $\displaystyle L^{p^{\prime}}(T,V^{*})_{w}$.

Therefore we conclude that $\displaystyle\hat{A}_{\epsilon}$ is indeed $\displaystyle L$-pseudo-monotone. ∎

We consider the following auxiliary approximate periodic problem:

[TABLE]

Proposition 6**.**

If hypotheses $\displaystyle H(B),H(A)$ hold and $\displaystyle\epsilon>0$, then problem (14) has a solution $\displaystyle u_{\epsilon}\in W^{per}_{p}((0,b),V^{*})$.

Proof.

We rewrite (14) as the following abstract operator inclusion

[TABLE]

Let $\displaystyle v\in\hat{A}_{\epsilon}(u)$. We have

[TABLE]

Let $\displaystyle y=(\epsilon J+B)^{-1}u$. Then $\displaystyle v\in\hat{A}(y)$ and so, using hypothesis $\displaystyle H(A)(iv)$, we have

[TABLE]

(recall that $\displaystyle|\cdot|_{*}$ and $\displaystyle||\cdot||_{*}$ are equivalent norms on $\displaystyle X^{*}$). It follows that $\displaystyle\hat{A}_{\epsilon}(\cdot)$ is coercive. Clearly it is bounded (see hypothesis $\displaystyle H(A)(iii)$). Also, from Proposition 5 we know that $\displaystyle\hat{A}_{\epsilon}(\cdot)$ is L-pseudo-monotone. Since $\displaystyle L(\cdot)$ is maximal monotone, we can use Theorem 4 and find $\displaystyle u_{\epsilon}\in W^{per}_{p}((0,b),V^{*})=D(L)$ such that it solves (15). Evidently, this is a solution of problem (14). ∎

Next, we will let $\displaystyle\epsilon\downarrow 0$ to produce a solution of problem (1).

Theorem 7**.**

If hypotheses $\displaystyle H(B),H(A)$ hold, then problem (1) has a solution $\displaystyle y\in L^{p}(T,X)$ which satisfies $\displaystyle(By)^{\prime}\in L^{p^{\prime}}(T,X^{*})$.

Proof.

For each $\displaystyle\epsilon>0$, let $\displaystyle u_{\epsilon}\in W^{per}_{p}((0,b),V^{*})$ be a solution of the approximate problem (14) (see Proposition 6). We have

[TABLE]

We take the inner product in $\displaystyle V^{*}$ with $\displaystyle u_{\epsilon}(t)$. Then

[TABLE]

with $\displaystyle v_{\epsilon}\in L^{p^{\prime}}(T,V^{*}),v_{\epsilon}(t)\in A_{\epsilon}(t,u_{\epsilon}(t))$ for almost all $\displaystyle t\in T$. Integrating on $\displaystyle T$ and using (16) and the periodic conditions, we obtain

[TABLE]

We set $\displaystyle y_{\epsilon}(t)=(\epsilon J+B)^{-1}u_{\epsilon}(t)$. Then

[TABLE]

On account of hypothesis $\displaystyle H(A)(iii)$, we have

[TABLE]

Then it follows from (17), (19) and (20) that

[TABLE]

This together with (18) implies that

[TABLE]

Now let $\displaystyle\epsilon_{n}=\frac{1}{n},u_{n}=u_{\epsilon_{n}},y_{n}=y_{\epsilon_{n}},v_{n}=v_{\epsilon_{n}}$ for all $\displaystyle n\in\mathbb{N}$. Note that

[TABLE]

We have

[TABLE]

Note that

[TABLE]

Also, on account of (19), (21) and (22), we may assume that

[TABLE]

We know that $\displaystyle W^{1,p^{\prime}}((0,b),V^{*})\hookrightarrow C(T,V^{*})$ continuously. Hence by (18), up to a subsequence, we have

[TABLE]

On the first equation in (22) we act with $\displaystyle(y_{n}-y)(t)$ and then integrate over $\displaystyle T$. We obtain

[TABLE]

We obtain

[TABLE]

Note that

[TABLE]

Also, we have

[TABLE]

So, if we return to (28) and use (29), (30) we obtain

[TABLE]

If we use (31) in (27), we get

[TABLE]

Invoking Proposition 5, we have

[TABLE]

Thus, we obtain from (22) taking the limit as $\displaystyle n\rightarrow\infty$

[TABLE]

Therefore $\displaystyle y\in L^{p}(T,X)$ is a solution of (1) with $\displaystyle(By)^{\prime}\in L^{p^{\prime}}(T,X^{*})$. ∎

4. An example

Let $\displaystyle T=[0,b]$ and let $\displaystyle\Omega\subseteq\mathbb{R}^{\mathbb{N}}$ be a bounded domain with a $\displaystyle C^{2}$-boundary $\displaystyle\partial\Omega$. We consider the following initial boundary value problem:

[TABLE]

We impose the following conditions on the data for problem (32):

$\displaystyle H(m)$: $\displaystyle m\in L^{N/2}(\Omega)$ if $\displaystyle N>2$, $\displaystyle m\in L^{r}(\Omega)$ with $\displaystyle r>1$ if $\displaystyle N=2$ and $\displaystyle m\in L^{1}(\Omega)$ if $\displaystyle N=1$, $\displaystyle m(z)\geqslant 0$ for almost all $\displaystyle z\in\Omega$, $\displaystyle m\not\equiv 0$.

$\displaystyle H(a)$: $\displaystyle a\in L^{\infty}(T\times\Omega)$ and $\displaystyle a(t,z)\geqslant a_{0}>0$ for almost all $\displaystyle(t,z)\in T\times\Omega$.

$\displaystyle H(g)$: $\displaystyle g:\mathbb{R}\rightarrow\mathbb{R}$ is a continuous convex function and its subdifferential $\displaystyle\partial g(x)$ satisfies

[TABLE]

Remark 1**.**

For any continuous convex function $\displaystyle g(\cdot)$, we know that $\displaystyle\partial g(x)\neq\emptyset$ for all $\displaystyle x\in\mathbb{R}$ (see Gasinski & Papageorgiou [7, p. 527]).

We introduce the following multifunction

[TABLE]

for all $\displaystyle u\in H^{1}_{0}(\Omega)$. Evidently, $\displaystyle N_{g}(\cdot)$ is maximal monotone.

In this case, the evolution triple consists of the following Hilbert spaces:

[TABLE]

We know that $\displaystyle X\hookrightarrow H$ compactly (by the Sobolev embedding theorem).

Let $\displaystyle A_{1}:T\times X\rightarrow X^{*}$ be the nonlinear map defined by

[TABLE]

Then the mapping $\displaystyle t\mapsto A_{1}(t,u)$ is measurable, whereas $\displaystyle u\mapsto A_{1}(t,u)$ is pseudo-monotone (see, for example, Zeidler [16, p. 591]). We set

[TABLE]

Then $\displaystyle A(t,u)$ satisfies hypotheses $\displaystyle H(A)$ (see $\displaystyle H(a)$ and $\displaystyle H(g)$).

In addition, we let $\displaystyle B\in\mathcal{L}(X,X^{*})$ be defined by

[TABLE]

Clearly, $\displaystyle B(\cdot)$ satisfies $\displaystyle H(B)$.

We can rewrite problem (32) as the following abstract implicit evolution inclusion:

[TABLE]

We can apply Theorem 7 and obtain the following result.

Proposition 8**.**

If hypotheses $\displaystyle H(m),H(a),H(g)$ hold, then problem (32) admits a solution $\displaystyle u\in L^{p}(T,H^{1}_{0}(\Omega))$ with

[TABLE]

Remark 2**.**

Using the methods developed in this paper one can also treat antiperiodic problems (see Gasinski & Papageorgiou [8]), problems with subdifferential terms (see Papageorgiou & Rădulescu [13]), and applications to distributed parameter control systems (see Papageorgiou, Rădulescu & Repovš [14]).

Acknowledgments. The authors thank an anonymous referee for the careful reading of this paper and for useful remarks. This research was supported by the Slovenian Research Agency grants P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083. V.D. Rădulescu acknowledges the support through a grant of the Romanian Ministry of Research and Innovation, CNCS–UEFISCDI, project number PN-III-P4-ID-PCE-2016-0130, within PNCDI III.