A Nonlinear Evolution Equation in an Ordered Banach Space, Reconsidered

TL;DR

This paper extends previous results on nonlinear evolution equations from Lebesgue spaces to more general ordered Banach spaces, correcting a technical error and broadening the theoretical framework.

Contribution

It generalizes existence results for nonlinear evolution equations to ordered Banach spaces without requiring a Banach lattice structure.

Findings

Existence of solutions is established in a broader setting.

A technical error in prior work is identified and corrected.

The results apply to kinetic theory models in more general spaces.

Abstract

Results of a previous paper [Commun. Contemp. Math., 09 (2007) 217-251] on the existence of solutions to a nonlinear evolution equation in an abstract Lebesgue space, arising from kinetic theory, are re-obtained in the more general setting of a real ordered Banach space, with additive norm on the positive cone, which is not necessarily a (Banach) lattice. In addition, an easily correctable technical error in the aforementioned paper is pointed out, and repaired.

Equations

$$\frac{d}{dt}f(t)=Q^{+}(t,f(t))-Q^{-}(t,f(t)),f(0)=f_0\in X_{+},t\ge 0,$$

$$\frac{d}{dt}f(t)=Af+Q^{+}(t,f(t))-Q^{-}(t,f(t)),f(0)=f_0\in X_{+},t\ge 0,$$

$$\Vert g+h\Vert =\Vert g\Vert +\Vert h\Vert g,h\in X_{+}.$$

$$\forall g,h\in \mathcal{D}(\Gamma ),g\le h⟹\Gamma g\le \Gamma h.$$

$$0\le h\le g\Rightarrow \Vert h\Vert \le \Vert g\Vert .$$

$$g\le h\Rightarrow g\in \mathcal{M}.$$

$${\int }_{\mathfrak{S}}g(s)ds\in X_{+},$$

$$\Vert {\int }_{\mathfrak{S}}g(s)ds\Vert ={\int }_{\mathfrak{S}}\Vert g(s)\Vert ds,$$

$$\Gamma {\int }_{\mathfrak{S}}g(s)ds={\int }_{\mathfrak{S}}\Gamma g(s)ds.$$

$${\int }_0^t{S}^{s}gds\in \mathcal{D}(G^{k+1}),\forall g\in \mathcal{D}(G^k)$$

$$\mathcal{D}(G^{\infty })\ni g_n:=n{\int }_0^{\infty }\phi (nt)S^{t}gdt,n=1,2,....$$

$$\gamma g\le Gg,\forall g\in {\mathcal{D}}_{+}(G).$$

$$0\le S^{t}g\le \exp (-\gamma t)g,\forall t\ge 0.$$

$${\lambda }_{0}g\le \Lambda g,\forall g\in {\mathcal{D}}_{+}(\Lambda ).$$

$$0\le Q^{-}(t,g)\le a(\Vert \Lambda g\Vert )\Lambda g,\forall g\in {\mathcal{D}}_{+}(\Lambda ),$$

$$\Delta (t,g):=\Vert \Lambda Q^{-}(t,g)\Vert -\Vert \Lambda Q^{+}(t,g)\Vert \ge 0,\forall g\in {\mathcal{D}}_{+}({\Lambda }^2),$$

$$\Vert {\Lambda }_1{Q}^{-}(t,g)\Vert -\Vert {\Lambda }_1{Q}^{+}(t,g)\Vert \ge 0,\forall g\in {\mathcal{D}}_{+}({\Lambda }^2),$$

$$\Vert {\Lambda }^2{Q}^{+}(t,g)\Vert \le \Vert {\Lambda }^2{Q}^{-}(t,g)\Vert +\rho (\Vert {\Lambda }_{1}g\Vert )\Vert {\Lambda }^{2}g\Vert ,\forall g\in {\mathcal{D}}_{+}({\Lambda }^3).$$

$$a(x)=a(0)+{\int }_0^x{a}^{\prime }(\xi )d\xi ,x\in {\mathbb{R}}_{+}.$$

$${\Lambda }^{k-1}{Q}^{-}(t,g)\le a(\Vert \Lambda g\Vert ){\Lambda }^{k}g,\forall g\in {\mathcal{D}}_{+}({\Lambda }^k)$$

$$0\le \Delta (t,g)\le \Vert \Lambda Q^{-}(t,g)\Vert \le a(\Vert \Lambda g\Vert )\Vert {\Lambda }^{2}g\Vert ,\forall g\in {\mathcal{D}}_{+}({\Lambda }^2).$$

$$\Vert g\Vert \le {\lambda }_0^{-1}\Vert \Lambda g\Vert ,\forall g\in {\mathcal{D}}_{+}(\Lambda ).$$

$$\Vert Q^{\pm }(t,g)\Vert \le {\lambda }_0^{-1}\Vert \Lambda Q^{\pm }(t,g)\Vert \le {\lambda }_0^{-1}\Vert \Lambda Q^{-}(t,g)\Vert \le a(\Vert \Lambda g\Vert ){\lambda }_0^{-1}\Vert {\Lambda }^{2}g\Vert$$

$$Q^{\pm }(t,0)=0and\Delta (t,0)=0a.e.on{\mathbb{R}}_{+}.$$

$$f(t)=f_0+{\int }_0^t[Q^{+}(s,f(s))-Q^{-}(s,f(t))]ds,\forall t\ge 0$$

$$f(t)=U^t{f}_0+{\int }_0^t{U}^{t-s}[Q^{+}(s,f(s))-Q^{-}(s,f(s))]ds,t\ge 0,$$

$$\Vert \Lambda f(t)\Vert +{\int }_0^t\Delta (s,f(s))ds=\Vert \Lambda f_0\Vert ,\forall t\ge 0,$$

$$\Vert {\Lambda }^{2}f(t)\Vert \le \Vert {\Lambda }^2{f}_0\Vert \exp (\rho (\Vert {\Lambda }_1{f}_0\Vert )t),\forall t\ge 0.$$

$$\frac{d}{dt}F(t)=Q_U^{+}(t,F)-Q_U^{-}(t,F(t)),F(0)=f_0,t\ge 0,a.e.,$$

$$\frac{d}{dt}f(t)+a(\Vert \Lambda f_0\Vert )\Lambda f=B(t,f,f),f(0)=f_0\in X_{+},t\ge 0,$$

Full text

In addition, an easily correctable technical error in the aforementioned paper is pointed out, and repaired.

† ”Gheorghe Mihoc - Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, P.O.Box 1-24, Bucharest, Romania E-mail address: grunfeld@ismma.ro; grunfeld@spacescience.ro

Keywords. nonlinear evolution equation; ordered Banach space; abstract state space; positive semigroup; kinetic equation; Boltzmann-like model. MSC2010. 47J35, 47H07, 82C40

1. Introduction

This note is concerned with the Cauchy problems

[TABLE]

and

[TABLE]

in a separable, ordered real Banach space $X$, whose positive cone, $X_{+}$, is closed and generating, and whose norm is additive on $X_{+}$111 In some papers, such a Banach space is called ”abstract state space” (see, e.g., [4, p. 30], [5], [6].. In (1) and (2), $f$ is defined from ${\mathbb{R}}_{+}:=[0,\infty)$ to $X_{+}$. Here, $Q^{+}$ and $Q^{-}$ are (nonlinear) mappings from ${\mathbb{R}}_{+}\times{\mathcal{D}}$ to $X_{+}$, for some ${\mathcal{D}}\subset X_{+}$ dense in $X_{+}$. It is assumed that for almost all (a.a.) $t\in{\mathbb{R}}_{+}$, with respect to the Lebesgue measure on ${\mathbb{R}}_{+}$, the operators $Q^{\pm}(t,\cdot)$ are isotone (i.e., order preserving) from ${\mathcal{D}}$ to $X_{+}$. Moreover, in (2), $A$ is the infinitesimal generator of a $C_{0}$ group of positive linear isometries on $X_{+}$.

Under the additional assumption that $X$ is an abstract Lebesgue space (AL-space), i.e. a Banach lattice whose lattice norm is additive on the positive cone, problems of the form (1) and (2) were studied in Ref. [1], in the case of an abstract nonlinear evolution equation arising from collisional kinetic theory. By construction, the model introduced in [1] summarizes common monotonicity properties (with respect to the order) of various Boltzmann-like kinetic equations and related models, and satisfies a generalization of the so-called Povzner inequality [2], [3].

The main results of [1], Theorem 3.1, and Corollary 3.1, consisted in proving existence and uniqueness of solutions to the above Cauchy problems. In addition, Ref. [1] included applications to the Boltzmann equation with inelastic collisions and chemical reactions, a Povzner-like model with dissipative collisions, and Smoluchowski’s coagulation equation.

The present note is based on the observation that the lattice condition assumed in [1] was needed only at one point of the analysis developed therein, specifically, in the proof of [1, Lemma 2.1]. Therefore, extending the validity of the lemma, by removing the lattice assumption, could allow generalizing results of [1]. Here, our goal is to prove that the main statements of [1] remain valid in the more general setting detailed in the beginning of this section, without imposing that $X$ should be a (Banach) lattice. This may be of interest in applications to problems involving Banach spaces which are not necessarily lattices, e.g., the (real) space of self-adjoint trace class operators over a separable Hilbert space (endowed with the trace norm, and ordered by the natural order of bounded self-adjoint operators), encountered in quantum kinetic modeling.

Once results analogous to those of [1, Lemma 2.1] are proved in our new setting, the rest of the analysis of [1] can be adapted to obtain the generalizations of [1, Theorem 3.1 and Corollary 3.1]. Nevertheless, for the sake of self-containment, in the present note, we outline the entire argument behind the above generalizations, including some clarifications and simplifications with respect to the presentation of [1].

Here we should emphasize the fact222not mentioned elsewhere in literature, as far as we are aware. that, although Theorem 3.1 in Ref. [1] is true (as stated therein), the proof provided in [1] for assertion (a) of the theorem is incomplete, because of an error, fortunately, easily correctable in the context of Ref. [1], as is explained in the last section and in the Appendix to the current paper.

Besides this Introduction, our note contains four more sections and the above mentioned Appendix. In Section 2, we detail the general setting of the analysis to be developed in the rest of the paper, and we extend results of [1, Lemma 2.1]. In Section 3, we state our main results, which have similar formulations as in Ref. [1]. Section 4 includes proofs. In the last section, Conclusions, we briefly compare the results of the present note with those of [1]. Finally, we comment on the error above referred to, leaving its correction to the ”Corrigendum” placed in Appendix.

2. General Setting

As mentioned in Introduction, in this paper, $X$ is a separable, ordered real Banach space, with order and norm denoted by $\leq$ and $\|\cdot\|$, respectively. We assume that the positive cone $X_{+}:=\{g:g\in X;0\leq g\}$ of $X$ is closed and generating (i.e., $X=X_{+}-X_{+}$), and the norm $\|\cdot\|$ is additive on $X_{+}$, i.e.,

[TABLE]

The above conditions are satisfied by AL - spaces, but also by ordered Banach spaces which are not necessarily lattices, e.g. the anti-lattice of self-adjoint trace class operators with the trace norm, mentioned in Introduction (for other examples, see [4, p. 30 - 31]).

By (3), $X_{+}$ satisfies the strong Levi property, i.e., every increasing (i.e., $\leq$ directed) norm-bounded positive sequence of $X_{+}$ is norm convergent (to some element of $X_{+}$, because $X_{+}$ is closed).

Recall some usual definitions. A mapping $\Gamma:{\mathcal{D}}(\Gamma)\subset X\mapsto X$, with ${\mathcal{D}}(\Gamma)\cap X_{+}\neq\emptyset$, is called positive if $0\leq\Gamma g$ for all $0\leq g\in{\mathcal{D}}(\Gamma)$. Further, $\Gamma:{\mathcal{D}}(\Gamma)\subset X\mapsto X$ is called isotone (or monotone) if it preserves the order, i.e.,

[TABLE]

Similar definitions are introduces for mappings between two different ordered (Banach) spaces, in particular between $X$ and ${\mathbb{R}}$ (endowed with the usual order).

Property (3) implies that the norm is monotone, i.e.,

[TABLE]

We will also use the following two definitions of [1]. A set $\emptyset\neq{\mathcal{M}}\subset X$ is called positively saturated (p-saturated) if, for all $h\in{\mathcal{M}}$ and $g\in X_{+}$,

[TABLE]

An operator $\Gamma:{\mathcal{D}}(\Gamma)\subset X\mapsto X$ is called closed with respect to the order (o-closed) if for every increasing sequence $\left\{g_{n}\right\}\subset{\mathcal{D}}(\Gamma)$ converging (in symbols, $\nearrow$) to $g$ in $X$, such that $\Gamma g_{n}\rightarrow h\in X$, as $n\rightarrow\infty$, one has $g\in{\mathcal{D}}(\Gamma)$ and $\Gamma g=h$. Obviously, a closed isotone mapping is also o-closed.

Recall that if the set ${\mathfrak{S}}\subset{\mathbb{R}}$ is (Lebesgue) measurable and $g:{\mathfrak{S}}\mapsto X_{+}$ is Bochner integrable, then

[TABLE]

where $ds$ is the Lebesgue measure on the real line.

In our setting, if $g:{\mathfrak{S}}\mapsto X_{+}$ is Bochner integrable, then, in view of (3),

[TABLE]

$\forall$ ${\mathfrak{S}}\subset{\mathbb{R}}$ measurable (the integral of the right hand side (r.h.s.) of (5) being in the sense of Lebesgue).

According to Hille’s theorem [7, Theorem 3.7.12, p. 83 ], if $\Gamma:{\mathcal{D}}(\Gamma)\subset X\mapsto X$ is a closed linear operator, ${\mathfrak{S}}\subset{\mathbb{R}}$ is measurable, $g:{\mathfrak{S}}\mapsto{\mathcal{D}}(\Gamma)$ is Bochner integrable, and $\Gamma g$ is also Bochner integrable, then

[TABLE]

We denote by $L^{1}({\mathbb{R}}_{+};X_{+})$ ($L^{1}_{loc}({\mathbb{R}}_{+};X_{+})$) the space of equivalent classes of Bochner integrable (locally Bochner integrable) maps from ${\mathbb{R}}_{+}$ to $X_{+}$. Also, $L^{\infty}_{loc}({\mathbb{R}}_{+};X_{+})$ denotes the space of equivalent classes of Lebesgue measurable maps $g:{\mathbb{R}}_{+}\mapsto X_{+}$ such that $||g(\cdot)||$ is locally essentially bounded on ${\mathbb{R}}_{+}$.

Recall that if $\left\{S^{t}\right\}_{t\geq 0}$ is a $C_{0}$ semigroup on $X$, then its infinitesimal generator $G:{\mathcal{D}}(G)\subset X\mapsto X$ is a closed linear operator, with the domain ${\mathcal{D}}(G)$ dense in $X$. The same is true for the positive integral powers $G^{k}$ (defined by $G^{1}:=G$, ${\mathcal{D}}(G^{k}):=\{g:g\in{\mathcal{D}}(G^{k-1}),\;G^{k-1}g\in{\mathcal{D}}(G)\}$, $G^{k}g:=G(G^{k-1}g)$, $k=2,3...$).

It is known that for every real number $t>0$, and each $k=0,1,...$,

[TABLE]

(with the notations $G^{0}:=I$, ${\mathcal{D}}(G^{0}):=X$, where $I$ is the identity operator on $X$).

One also knows that ${\mathcal{D}}(G^{\infty}):=\cap_{k=1}^{\infty}{\mathcal{D}}(G^{k})$ is dense in $X$. Indeed, following, e.g., [7, Theorem 10.3.4, p. 308]), let $\varphi:{\mathbb{R}}_{+}\mapsto{\mathbb{R}}$, indefinitely differentiable on $(0,\infty)$, with compact support, and satisfying, $\int_{0}^{\infty}\varphi(t)dt=1.$ Then every $g\in X$ can be approximated by a sequence ${g}_{n}\rightarrow g$ as $n\rightarrow\infty$, where

[TABLE]

We recall that a positive $C_{0}$ semigroup on $X$ is a $C_{0}$ semigroup on $X$, which leaves the cone $X_{+}$ invariant. If $G$ denotes its infinitesimal generator, then ${\mathcal{D}}_{+}(G^{\infty}):={\mathcal{D}}(G^{\infty})\cap X_{+}$ is dense in $X_{+}$, as can be immediately seen by choosing $\varphi\geq 0$ and $g\in X_{+}$ in (8). In particular, ${\mathcal{D}}_{+}(G^{k}):={\mathcal{D}}(G^{k})\cap X_{+}$ is dense in $X_{+}$ for all $k=1,2,...$.

The next lemma includes simple but useful facts stated in [1, Lemma 2.1], which actually hold under more general conditions than those assumed in the lattice setting of Ref [1].

Lemma 1**.**

Let $\{S^{t}\}_{t\geq 0}$ be a positive semigroup on $X$ with infinitesimal generator $(-G)$. Suppose that there exists some number $\gamma>0$ such that

[TABLE]

Then:

(a) For each $g\in X_{+}$,

[TABLE]

(b) For each $g\in X_{+}$, there exists an increasing sequence $\left\{g_{n}\right\}\subset{\mathcal{D}}_{+}(G^{\infty})$, such that $g_{n}\nearrow g$ as $n\rightarrow\infty$.

(c) Let $p$ be a positive integer. If $\{g_{n}\}\subset{\mathcal{D}}_{+}(G^{p})$ is increasing and $\{G^{p}g_{n}\}$ is norm bounded, then there exists $g\in{\mathcal{D}}_{+}(G^{p})$ such that $G^{k}g_{n}\nearrow G^{k}g$ for all $k=0,1,..,p$.

(d) The sets ${\mathcal{D}}_{+}(G^{k})$, $k=1,2,...$, and ${\mathcal{D}}_{+}(G^{\infty})$ are p-saturated.

Proof.

(a) If $g\in{\mathcal{D}}_{+}(G)$, then $\frac{d}{dt}S^{t}g=-GS^{t}g$. But (9) implies $\frac{d}{dt}S^{t}g\leq-\gamma S^{t}g$, which yields (10) (in the case $g\in{\mathcal{D}}_{+}(G)$). Since ${\mathcal{D}}_{+}(G)$ is dense in $X_{+}$, it follows that for every $g\in X_{+}$, there is some sequence ${\mathcal{D}}_{+}(G)\ni{g}_{n}\rightarrow g$, as $n\rightarrow\infty$, with the property $0\leq S^{t}g_{n}\leq\exp(-\gamma t)g_{n}$ for all $n$. Then the limit satisfies (10), because $X_{+}$ is closed.

(b) Let $\varphi$ be positive in (8). Then a simple computation starting from (8), and making use of (10), implies easily $0\leq g_{n}\leq g_{n+1}$ for all $n=1,2,...$.

(c) By Levi’s property, there exists $u_{p}\in X_{+}$ such that $G^{p}g_{n}\nearrow u_{p}$, as $\rightarrow\infty$. Observe that, by (9) and (4), one has $\|G^{k}g_{n}\|\leq\gamma^{k-p}\|G^{p}g_{n}\|$ for all $k=0,1,2,...,p$. Then, for each $k=0,1,2,...,p$, the sequence $\{G^{k}g_{n}\}$ is norm bounded. Therefore, by Levi’s property, there exists $u_{k}\in X_{+}$ such that $u_{k,n}:=G^{k}g_{n}\nearrow u_{k}$ as $n\rightarrow\infty$. For $k=0$ and $k=1$, we have $g_{n}\nearrow u_{0}$ and $u_{1,n}=Gg_{n}\nearrow u_{1}$, respectively, as $n\rightarrow\infty$. But $G$ is closed. Consequently, $u_{0}\in{\mathcal{D}}_{+}(G)$ and $Gu_{0}=u_{1}$. To complete the proof of (c), we proceed inductively, using that $G^{k}$ is closed, $k=2,3,...$.

(d). Let $k=1,2,..,$ be fixed, and $0\leq g\leq h\in{\mathcal{D}}_{+}(G^{k})$. By virtue of (b), there is a sequence ${\mathcal{D}}_{+}(G^{\infty})\ni g_{n}\nearrow g\leq h$. Therefore, $G^{k}g_{n}$ is increasing and $\|G^{k}g_{n}\|\leq\|G^{k}h\|$ for all $k=1,2,...$. Then (c) applies, hence $g\in{\mathcal{D}}_{+}(G^{k})$. ∎

We remark that the above lemma does not use the additivity of the norm. Moreover, Levi’s property is not needed in the proof of (a) and (b).

3. Model and main result

In the setting detailed in the previous sections, we investigate (1) and (2), by assuming the hypotheses of the model introduced in Ref. [1], as follows.

I. General assumptions:

$\bullet$ The mappings ${\mathbb{R}}_{+}\ni t\mapsto Q^{\pm}(t,g(t))\in X_{+}$ are (Lebesgue) measurable333We do not distinguish between strong and weak measurability, because $X$ is separable. for every measurable $g:{\mathbb{R}}_{+}\mapsto X_{+}$ which satisfies $g(t)\in{\mathcal{D}}$ almost everywhere (a.e.) on ${\mathbb{R}}_{+}$.

$\bullet$ For a.a. $t\geq 0$, the positive operators $Q^{\pm}(t,\cdot)$ are isotone and o-closed, and their common domain ${\mathcal{D}}$ is p-saturated.

II. Specific assumptions:

Assumption ${\bf(A)}$.

There exists a linear operator $\Lambda:{\mathcal{D}}(\Lambda)\subset X\mapsto X$ such that $(-\Lambda)$ is the infinitesimal generator of a positive $C_{0}$ semigroup on $X$, with the properties ${\mathcal{D}}_{+}(\Lambda)\subset{\mathcal{D}}$, $Q^{\pm}(t,{\mathcal{D}}_{+}(\Lambda^{k}))\subset{\mathcal{D}}_{+}(\Lambda^{k-1})$, $t\geq 0$ a.e., $k=2,3$, and:

Assumption ${\bf(A_{0})}$.

There exists a number $\lambda_{0}>0$ such that

[TABLE]

Assumption ${\bf(A_{1})}$.

There exists a positive, non-decreasing, convex function $a:{\mathbb{R}}_{+}\mapsto{\mathbb{R}}_{+}$, such that for a.a. $t\geq 0$,

[TABLE]

and the mapping ${\mathcal{D}}_{+}(\Lambda)\ni g\mapsto a(\left\|\Lambda g\right\|)\Lambda g-Q^{-}(t,g)$ $\in X$ is isotone.

Assumption ${\bf(A_{2})}$.

For a.a. $t\geq 0$,

[TABLE]

and the map ${\mathcal{D}}_{+}(\Lambda^{2})\ni g\mapsto\Delta(t,g)\in{\mathbb{R}}_{+}$ is isotone.

Assumption ${\bf(A_{3})}$.

There exist a positive non-decreasing function $\rho:{\mathbb{R}}_{+}\mapsto{\mathbb{R}}_{+}$ and a closed, positive, linear operator $\Lambda_{1}:{\mathcal{D}}(\Lambda_{1})\subset X_{+}\mapsto X_{+}$, with ${\mathcal{D}}_{+}(\Lambda)\subset{\mathcal{D}}(\Lambda_{1})$, such that for a.a. $t\geq 0$,

[TABLE]

[TABLE]

Some comments and remarks are in order.

The convexity assumed in $(A_{1})$ implies that the function $a$ is continuous on $(0,\infty)$, and its derivative is a.e. defined, positive and non-decreasing on ${\mathbb{R}}_{+}$. In our case, $a(0)=a(0+)$, because $a$ is positive and non-decreasing. So, the usual representation of $a$, in terms of its derivative, takes the form

[TABLE]

By the above assumptions, for each $k=1,2,...$, the linear operator $\Lambda^{k}$ is positive, closed, and densely defined. Besides, ${\mathcal{D}}_{+}(\Lambda^{k})$, $k=1,2....$, and ${\mathcal{D}}_{+}(\Lambda^{\infty}):={\mathcal{D}}_{+}(\Lambda^{\infty})\cap X_{+}$ are p-saturated, because $\Lambda$ fulfills the conditions of Lemma 1.

Note here that, since ${\mathcal{D}}_{+}(\Lambda^{k})$ is p-saturated (for all $k$), we get from (12)

Remark 1**.**

For each $k=1,2,..$, and for a.a. $t\geq 0$, one has $Q^{-}(t,{\mathcal{D}}_{+}(\Lambda^{k})))$ $\subset$ ${\mathcal{D}}_{+}(\Lambda^{k-1}))$, and

[TABLE]

(where $\Lambda^{0}=I$). In particular, $Q^{-}(t,{\mathcal{D}}_{+}(\Lambda^{\infty}))\subset{\mathcal{D}}_{+}(\Lambda^{\infty})$ a.e. on ${\mathbb{R}}_{+}$.

Remark 1 shows that the inclusion conditions on $Q^{\pm}(t,{\mathcal{D}}_{+}(\Lambda^{k}))$ imposed in the beginning of this section are implicitly fulfilled by $Q^{-}$, in the context of Assumption ($A_{1}$).444However, we kept those conditions, in order to have a priori well-defined statements in Assumptions $A_{2}$ and $A_{3}$.

The above model assumptions indicate some control on $Q^{\pm}$, in terms of powers of $\Lambda$. Specifically, assumption ($A_{1}$) shows that, although $Q^{+}$ might exhibit a highly nonlinear behavior, it remains somehow controlled by the linear operator $\Lambda$ on each set $\{g:\in{\mathcal{D}}_{+}(\Lambda):\|\Lambda g\|=constant\}$. By ($A_{2}$) one controls $\Delta$, in terms of $\Lambda$ and $\Lambda^{2}$. Indeed, by (13) for a.a. $t\geq 0$,

[TABLE]

Observe that (11) implies immediately

[TABLE]

By applying (19) to $Q^{\pm}(t,g)$, and using (18), one finds

[TABLE]

a.e on ${\mathbb{R}}_{+}$ for all $g\in{\mathcal{D}}_{+}(\Lambda^{2})$.

Property (20) entails

[TABLE]

It is useful to introduce the following spaces of ”abstract moments of $\Lambda$ ”. For $k=1,2,...$, let $L^{1}_{k,loc}({\mathbb{R}}_{+};X_{+})$ denote the space (of equivalent classes) of measurable maps $g:{\mathbb{R}}_{+}\mapsto\mathcal{D}(\Lambda^{k})$, satisfying $\Lambda^{k}g$ $\in$ $L^{1}_{loc}({\mathbb{R}}_{+};X_{+})$. We set $L^{1}_{0,loc}({\mathbb{R}}_{+};X_{+})$ $:=$ $L^{1}_{loc}({\mathbb{R}}_{+};X_{+})$ and $L^{1}_{\infty,loc}({\mathbb{R}}_{+};X_{+})$ $:=$ $\cap_{k=1}^{\infty}L^{1}_{k,loc}({\mathbb{R}}_{+};X_{+})$. Also, by $L^{\infty}_{2,loc}({\mathbb{R}}_{+};X_{+})$, we denote the space of measurable maps $g:{\mathbb{R}}_{+}\mapsto\mathcal{D}(\Lambda^{2})$, satisfying $\Lambda^{2}g$ $\in$ $L^{\infty}_{loc}({\mathbb{R}}_{+};X_{+})$.

We consider the above spaces, with the natural order of $X_{+}$, (i.e., induced by the order $\leq$ of $X$).

Due to (11) and (4), clearly, $L^{\infty}_{2,loc}({\mathbb{R}}_{+};X_{+})$ $\subseteq$ $L^{1}_{2,loc}({\mathbb{R}}_{+};X_{+})$, and

Remark 2**.**

If $k,p=0,1,2,...,\infty$, and $k\leq p$, then $L^{1}_{p,loc}({\mathbb{R}}_{+};X_{+})$ $\subseteq$ $L^{1}_{k,loc}({\mathbb{R}}_{+};X_{+})$.

Inequality (13) puts into an abstract form common elements of various conservation/dissipation properties of kinetic models (for details, see Ref. [1]). In addition, (15) can be regarded as an abstract correspondent to the Povzner inequality [2, 3] (see also [1]) .

Our results are concerned with the existence and uniqueness of global (in time), strong solutions to Eq. (1) and mild solutions to Eq. (2).

Here we say that $f$ is a (strong) solution to Eq. (1) on ${\mathbb{R}}_{+}$ if it is (strongly) absolutely continuous on ${\mathbb{R}}_{+}$, (strongly) differentiable a.e. on ${\mathbb{R}}_{+}$, satisfies Eq. (1) a.e. on ${\mathbb{R}}_{+}$, and $f(0)=f_{0}$.

Note that $f$ is a strong solution of Problem (1) on ${\mathbb{R}}_{+}$ iff

[TABLE]

(where the integral is in the sense of Bochner).

One can similarly define strong solutions to Eq. (2).

Let $\{U^{t}\}_{t\in{\mathbb{R}}}$ denote the $C_{0}$ group of positive linear isometries, defined by the the infinitesimal generator $A$ introduced in (2). Recall that any strong solution of Eq. (2) satisfies

[TABLE]

but the converse is not generally true.

Let $C({\mathbb{R}}_{+};X_{+})$ be the space of continuous functions from ${\mathbb{R}}_{+}$ to $X_{+}$. We say that $f\in C({\mathbb{R}}_{+};X_{+})$ is a mild solution of Eq. (2) on ${\mathbb{R}}_{+}$, if it satisfies Eq. (23) for all $t\geq 0$.

A statement similar to [1, Theorem 3.1] can be proved in the more general setting of this paper.

Theorem 1**.**

Suppose $f_{0}\in{\mathcal{D}}_{+}(\Lambda^{2})$, in (1). Let either of the following hold:

(a) $Q^{+}(t,{\mathcal{D}}_{+}(\Lambda^{\infty}))$ $\subset$ ${\mathcal{D}}_{+}(\Lambda^{\infty})$, $t\geq 0$ a.e., and $\Lambda^{k}Q^{+}(\cdot,{\mathcal{D}}_{+}(\Lambda^{\infty}))$ $\subset$ $L^{1}_{loc}({\mathbb{R}}_{+};X_{+})$, $k=1,2,...$.

(b) The operators $Q^{\pm}$ do not depend explicitly on $t$.

Then the Cauchy problem (1) has a unique positive strong solution $f$ on ${\mathbb{R}}_{+}$, such that $f(t)\in{\mathcal{D}}_{+}(\Lambda^{2})$ for all $t\geq 0$, and $\|\Lambda^{2}f(\cdot)\|$ is locally bounded on ${\mathbb{R}}_{+}$. Moreover, $\Lambda f\in C({\mathbb{R}}_{+};X_{+})$. Furthermore, $f$ satisfies

[TABLE]

and

[TABLE]

In Theorem 1, conditions (a) and (b) do not mutually exclude each other.

Formula (24) generalizes a priori ”conservation/ dissipation” estimates considered in, e.g., [8] (For more details, the reader is referred to [1]).

In applications, one may have $\Lambda_{1}=\Lambda$, when some conditions of ($A_{3}$) become redundant (see [1]).

The above theorem has an immediate consequence, with a similar statement as [1, Corollary 3.1].

Suppose that $U^{t}{\mathcal{D}}(\Lambda)={\mathcal{D}}(\Lambda)$ and $U^{t}{\mathcal{D}}(\Lambda_{1})={\mathcal{D}}(\Lambda_{1})$. Also assume that for each $t>0$, $U^{t}\Lambda=\Lambda U^{t}$ on ${\mathcal{D}}(\Lambda)$ and $U^{t}\Lambda_{1}=\Lambda_{1}U^{t}$ on ${\mathcal{D}}(\Lambda_{1})$.

Corollary 1**.**

Suppose $f_{0}\in{\mathcal{D}}_{+}(\Lambda^{2})$, in (2). Let $Q^{+}(t,{\mathcal{D}}_{+}(\Lambda^{\infty}))\subset{\mathcal{D}}_{+}(\Lambda^{\infty})$ a.e. on ${\mathbb{R}}_{+}$, and $\Lambda^{k}Q^{+}(\cdot,U^{\cdot}g)$ $\in L_{loc}^{1}({\mathbb{R}}_{+};X_{+})$ for all $g\in{\mathcal{D}}_{+}^{\infty}$, $k=1,2,....$. Then the Cauchy problem (2) has a unique positive global mild solution $f$ on ${\mathbb{R}}_{+}$, such that $f(t)\in{\mathcal{D}}_{+}(\Lambda^{2})$ for all $t\geq 0$, and $\|\Lambda^{2}f(\cdot)\|$ is locally bounded on ${\mathbb{R}}_{+}$. Moreover, $\Lambda f\in C({\mathbb{R}}_{+};X_{+})$. Furthermore, $f$ satisfies Eq. (24) and inequality (25).

Proof.

(see [1, Corollary 3.1]). The transformation $F(t):=U^{-t}f(t)$, simply reduces (23) to

[TABLE]

where $Q^{\pm}_{U}(t,g):=U^{-t}Q^{\pm}(t,U^{t}g)$ for all $g\in{\mathcal{D}}$, and a.a. $t\geq 0$. Then it is sufficient to check that Theorem 1 applies with $Q^{\pm}$ replaced by $Q^{\pm}_{U}$. ∎

4. Technical proofs

As in Ref. [1], we reduce (1) to an equivalent problem for an equation more suitable for monotone iteration. To this end, consider the problem

[TABLE]

and its associated integral form

[TABLE]

where $a$ is given by ($A_{1}$), and $B$ is formally defined a.e. on $t$ $\in$ $R_{+}$, by

[TABLE]

for, say, any $g,h$ $\in$ $L^{\infty}_{2,loc}(\mathbb{R}_{+};X_{+})$.

Proposition 1**.**

If $f$ is a positive strong solution to (1) such that $f(t)\in{\mathcal{D}}_{+}(\Lambda^{2})$ for all $t\geq 0$, and $\|\Lambda^{2}f(\cdot)\|$ is locally bounded on ${\mathbb{R}}_{+}$, then $\Lambda f\in C({\mathbb{R}}_{+};X_{+})$ and $f$ satisfies (24).

Proof.

a) From (20) and the assumptions on $f$, we get $\Lambda f$, $\Lambda Q^{\pm}(\cdot,f)$ $\in$ $L^{1}_{loc}({\mathbb{R}}_{+};X_{+})$. Then we simply find that $\Lambda f\in C({\mathbb{R}}_{+};X_{+})$ by applying $\Lambda$ to (22), and using (6). Moreover, playing conveniently with the terms in the resulting equality, we apply (3) and (5), and, finally, take advantage of (13), to obtain (24). ∎

Proposition 2**.**

Let ${\mathbb{R}}_{+}\ni t\mapsto f(t)\in{\mathcal{D}}_{+}(\Lambda^{2})$ be such that $\|\Lambda^{2}f(\cdot)\|$ is locally bounded on ${\mathbb{R}}_{+}$. Then $f$ is a strong solution to (1) iff it is a strong solution to (27).

Proof.

Under the conditions of the proposition, if $f$ is a strong solution to (1), then it fulfills the conditions of Proposition 1, so that, in view of (24), $f$ is also a strong solution to (27). Conversely, suppose that $f$ is a strong solution of (27), Then $f$ satisfies (28), where applying (6) (with $\Gamma=\Lambda$), and writing conveniently the resulting equality, we get

[TABLE]

Further, applying (3) and (5) in the above equality we obtain

[TABLE]

where $\psi(t):=\left\|\Lambda f_{0}\right\|-\left\|\Lambda f(t)\right\|-\int_{0}^{t}\Delta(s,f(s))ds$. Consequently,

[TABLE]

Fix an arbitrary $0<T<\infty$. Since $\|\Lambda^{2}f(\cdot)\|$ is locally bounded on ${\mathbb{R}}_{+}$, by (11) and the positivity of $\Delta$, we can write $\left\|\Lambda f(t)\right\|+\int_{0}^{t}\Delta(\tau,f(\tau))d\tau\leq\alpha_{T,f}$ $\forall$ $t\in[0,T]$, where $\alpha_{T,f}$:=$\lambda^{-1}[\operatorname*{ess\,sup}_{t\in[0,T]}\left\|\Lambda^{2}f(t)\right\|]+\int_{0}^{T}\Delta(\tau,f(\tau))d\tau<\infty$ depends on $T$ and $f$. In particular, $\|\Lambda f_{0}\|\leq\alpha_{T,f}$. But the convexity (or, simply (16)) implies that $a$ is Lipschitz on $[0,\alpha_{T,f}]$. Then there is a number $\beta_{T,f}>0$ (depending on $T$ and $f$) such that

[TABLE]

which can be introduced in (31), and combined with the local boundedness of $\|\Lambda^{2}f\|$, to obtain

[TABLE]

where $\gamma_{T,f}>0$ is also a number depending only on $T$ and $f$. Finally, the Gronwall’s inequality yields $\psi(t)=0$ for all $0\leq t\leq T$. This concludes the proof, because $T$ is arbitrary. ∎

Thus proving Theorem 1 is equivalent to demonstrating the existence and uniqueness of positive solutions to (27), having the property (25).

Before proceeding to the study of (27) in the general case, observe that from (21), (24), and the properties of $\Lambda$ and $\Delta$, it results that, in the class of solutions considered in Theorem 1, $f\equiv 0$ is the only solution of Problem (1) with initial datum $f_{0}=0$. Moreover, if $0\neq f_{0}\in{\mathcal{D}}_{+}(\Lambda^{2})$, $a(\|\Lambda f_{0}\|)=0$, and if $f$ is strong solution of (1), with $f(0)=f_{0}$, and properties as in Theorem 1, then $\|\Lambda f(t)\|\leq\|\Lambda f_{0}\|$ on ${\mathbb{R}}_{+}$, due to (20). But $a$ is positive and non-decreasing. Therefore, $a(\|\Lambda f(t)\|)=0$ on ${\mathbb{R}}_{+}$. Consequently, (20) implies $Q^{\pm}(t,f(t))\equiv 0$ a.e. on ${\mathbb{R}}_{+}$, which, introduced in (1), yields $f(t)\equiv f_{0}$ on ${\mathbb{R}}_{+}$.

Therefore, in the following, we assume $f_{0}\neq 0$ and $a(\|\Lambda f_{0}\|)\neq 0$.

The proof of Theorem 1 is close to the main argument of [1], and relies on the fact that any positive strong solution of (27) satisfies

[TABLE]

(the integral being in the sense of Bochner), where $\left\{V^{t}\right\}_{t\geq 0}$ is the positive $C_{0}$ semigroup on $X$ with infinitesimal generator $L:=-a(\left\|\Lambda f_{0}\right\|)\Lambda$. Thus the positive strong solutions of Problem (1) can be found among the positive solutions of Eq. (33), which satisfy (24).

Note that $\left\{V^{t}\right\}_{t\geq 0}$ fulfills the conditions of Lemma 1. Then $\forall$ $g\in X_{+}$,

[TABLE]

Formally, $B(t,g,h)$ defined by (29) is positive. Indeed, applying (12) in (29), and using the monotonicity of $a$, and the positivity of $\Lambda$ and $\Delta$, we get $0\leq$ $Q^{+}(t,g(t))+\left(a\left(\|\Lambda g(t)\|+\int_{0}^{t}\Delta(s,h(s))ds\right)-a(\|\Lambda g(t)\|\right)\Lambda g(t)$ $\leq B(t,g,h)$. By (20) and the properties of $a$ (cf. ($A_{1}$)), expression (29) defines a mapping

[TABLE]

Lemma 2**.**

The mapping (35) is isotone555in the sense that if $(g_{i},h_{i})$ $\in$ $L^{\infty}_{2,loc}({\mathbb{R}}_{+};X_{+})\times L^{\infty}_{2,loc}({\mathbb{R}}_{+};X_{+})$, $i=1,2$, and $g_{1}(t)\leq g_{2}(t)$, $h_{1}(t)\leq h_{2}(t)$ a.e. on ${\mathbb{R}}_{+}$, then $B(t,g_{1},h_{1})\leq B(t,g_{2},h_{2})$ a.e. on ${\mathbb{R}}_{+}$..

Proof.

(see [1, Lemma 4.1]) If $g,h$ $\in$ $L^{\infty}_{2,loc}({\mathbb{R}}_{+};X_{+})$, then $B(t,g,h)$ given by (29) can be written as

[TABLE]

where $F(x,y)$ $:=$ $a(x+y)-a(x)$, $x,y\geq 0$, and $\chi(t,h)$ $:=$ $\int_{0}^{t}\Delta(s,h(s))ds$. By (16), $F(x,y)=\int_{0}^{y}a^{\prime}(x+\xi)d\xi$. Recall that $a^{\prime}$ is a.e. defined, positive and non-decreasing on ${\mathbb{R}}_{+}$. Thus $F(x_{2},y_{2})$ $\geq$ $F(x_{1},y_{1})\geq 0$, $\forall$ $x_{2}\geq x_{1}\geq 0$ and $y_{2}\geq y_{1}\geq 0$. This and the isotonicity properties of $\Lambda$ and $\Delta$ imply that $(g,h)\mapsto F(\|\Lambda g\|,\chi(\cdot,h))\Lambda g$ is isotone from $L^{\infty}_{2,loc}({\mathbb{R}}_{+};X_{+})\times L^{\infty}_{2,loc}({\mathbb{R}}_{+};X_{+})$ to $L_{1,loc}^{1}({\mathbb{R}}_{+};X_{+})$. To complete the proof, it is sufficient to remark that $g\mapsto Q^{+}(\cdot,g)+a(\|\Lambda g\|)\Lambda g-Q^{-}(t,g)$ is isotone from $L^{\infty}_{2,loc}({\mathbb{R}}_{+};X_{+})$ to $L_{1,loc}^{1}({\mathbb{R}}_{+};X_{+})$, by virtue of the isotonicity of $Q^{+}$ and assumption $(A_{1})$. ∎

It appears that Eq. (33) could be solved by monotone iteration, Levi’s property being useful to prove the convergence of the iteration. To this end, we introduce a sequence of approximation solutions to (33), following ideas of [1].666We proceed as in Step 1 of the proof of [1, Theorem 3.1], with the difference that, here, we also approximate the initial datum $f_{0}$ of (33). Specifically, for $f_{0}\in{\mathcal{D}}_{+}(\Lambda^{2})$ in (33), we apply Lemma 1(b) and choose an increasing sequence ${\mathcal{D}}_{+}(\Lambda^{\infty})\ni f_{0,n}\nearrow f_{0}$, as $n\rightarrow\infty$, where the first term of the sequence is $f_{0,1}=0$. Then our approximating sequence is formally given by

[TABLE]

where

[TABLE]

From (35) and the presence of the integral in the r.h.s of (37), we have

Remark 3**.**

If $g$, $h$ $\in$ $L^{\infty}_{2,loc}({\mathbb{R}}_{+};X_{+})$ then $S(\cdot,g,h)\in C({\mathbb{R}}_{+};X_{+})$.

The next three lemmas serve to show that $f_{n}$ is well defined and has useful integrability and regularity properties.

For each $k=1,2,...,\infty$, let ${\mathcal{M}}_{k}$ be the family of those $g$ $\in$ $C({\mathbb{R}}_{+};X_{+})$ with the property that $\forall$ $0<T<\infty$, there is $g_{T}$ $\in$ ${\mathcal{D}}_{+}(\Lambda^{k})$, which may depend only on $g$ and $T$, such that $g(t)$ $\leq$ $g_{T}$ on $[0,T]$. Obviously, if $k,p=1,2,,..,\infty$, and $k\leq p$, then ${\mathcal{M}}_{p}$ $\subset$ $L_{p,loc}^{1}({\mathbb{R}}_{+};X_{+})$ $\subseteq$ $L_{k,loc}^{1}({\mathbb{R}}_{+};X_{+})$. In particular, ${\mathcal{M}}_{\infty}$ \subset$$L_{\infty,loc}^{1}({\mathbb{R}}_{+};X_{+}).

Lemma 3**.**

(a) Under conditions (a) of Theorem 1, $S(\cdot,{\mathcal{M}}_{\infty},{\mathcal{M}}_{\infty})$ $\subset$ ${\mathcal{M}}_{\infty}$.

(b) Under conditions (b) of Theorem 1, $S(\cdot,{\mathcal{M}}_{3},{\mathcal{M}}_{3})$ $\subset$ ${\mathcal{M}}_{3}$.

Proof.

If $g$, $h$ $\in$ ${\mathcal{M}}_{\infty}$, in case (a) ($g$, $h$ $\in$ ${\mathcal{M}}_{3}$ in case (b)), then $S(\cdot,g,h)\in C({\mathbb{R}}_{+};X)$, by virtue of Remark 3. Moreover, for some fixed (arbitrary) $0<T<\infty$, there is $g_{T}$ $\in$ ${\mathcal{D}}_{+}(\Lambda^{\infty})$, in case (a) ($g_{T}$ $\in$ $\mathcal{D}(\Lambda^{3})\cap X_{+}$, in case (b)) such that $g(t)\leq g_{T}$ for all $t\in[0,T]$. Consequently, using Lemma 2, the monotonicity assumptions of ($A_{0}$) and ($A_{1}$), and the obvious inequality $\int_{0}^{t}\Delta(s,h(s))ds$ $\leq\int_{0}^{T}\Delta(s,h(s))ds$, we obtain from (29)

[TABLE]

where $u_{g,h,T}(t)$ $:=$ Q^{+}(t,g_{T})$$+$$a\left(\|\Lambda g_{T}\|+\int_{0}^{T}\Delta(s,h(s))ds\right)\Lambda g_{T}.

(a) To prove (a), set

[TABLE]

Using (34), (37), and (38), we get $S(t,g,h)\leq s_{T,a}$ a.e. on $[0,T]$. But, in case (a), $g_{T}$, $\Lambda g_{T}$ $\in$ ${\mathcal{D}}_{+}(\Lambda^{\infty})$. Besides, $\Lambda^{k}Q^{+}(\cdot,g_{T})\in L^{1}(0,T;X_{+})$ for all $k=0,1,2,...$, due to assumptions (a) of Theorem 1. Thus we have $\Lambda^{k}u_{g,h,T}$ $\in$ $L^{1}(0,T;X_{+})$ for all $k=0,1,2,...$. Then, by virtue of (6), for each $k=1,2,...$, the operator $\Lambda^{k}$ can be applied to (39), and interchanged with the integral therein. Therefore, $s_{T,a}\in\mathcal{D}_{+}(\Lambda^{\infty})$, concluding the proof of (a), because $T$ is arbitrary.

(b) In case (b), recall that $Q^{+}$ does not depend explicitly of $t$, because of the assumptions (b) of Theorem 1. Therefore, $u_{g,h,T}$ is $t$ - independent. Then, from (37) and (38), we get $S(t,g,h)$ $\leq$ $\int_{0}^{t}V^{t-s}u_{g,T}ds$ $=$ $\int_{0}^{t}V^{s}u_{g,h,T}ds$ $\leq$ $s_{T,b}$ for all $0\leq t\leq T$, where

[TABLE]

Since $g_{T}$ $\in$ $\mathcal{D}(\Lambda^{3})\cap X_{+}$, the domain assumptions on $Q^{+}$ and $\Lambda$ imply $u_{g,h,T}$ $\in$ $\mathcal{D}(\Lambda^{2})\cap X_{+}$. Then by applying (7) to (40), we get $s_{T,b}$ $\in$ $\mathcal{D}(\Lambda^{3})\cap X_{+}$, which completes the proof of (b). ∎

Lemma 4**.**

(a) Assume conditions (a) of Theorem 1 hold. Then for each $n=1,2,...$, one has $f_{n}$ $\in$ ${\mathcal{M}}_{\infty}$.

(b) Assume conditions (b) of Theorem 1 hold. Then for each $n=1,2,...$, one has $f_{n}$ $\in$ ${\mathcal{M}}_{3}$.

(c) In both cases, (a) and (b), for each $n=1,2,...$, $f_{n}$ is a.e. differentiable on ${\mathbb{R}}_{+}$.

Proof.

From (36), it follows that the assertions (a) and (b) are trivially checked for $f_{1}$ and $f_{2}$, by setting g_{1,T}$$:=[math] and (due to (34)) $g_{2,T}$ :=$$f_{0,2}, respectively. Since, in (36), obviously, $V^{(\cdot)}f_{0,n}\in{\mathcal{M}}_{\infty}$ the proof of (a) and (b) can be completed by a straightforward induction based on the application of Lemma 3 to (36).

(c) Cases $n=1$ and $n=2$ are trivial. Let $n=3,4,...$. Fix an arbitrary $0<T<\infty$. By (a), (b), and, say, Lemma 2, we have $B(\cdot,f_{n-1},f_{n-2})$, $\Lambda B(\cdot,f_{n-1},f_{n-2})$ $\in$ $L^{1}(0,T;X_{+})$. But $\int_{0}^{t}V^{t-s}B(s,f_{n-1},f_{n-2})ds$ $=$ $f_{n}(t)$ $-$ $V^{t}f_{0,n}$ $\in$ ${\mathcal{D}}(\Lambda^{3})$ $\subset$ ${\mathcal{D}}(\Lambda)$ on $[0,T]$. Thus, since $L:=-a(\left\|\Lambda f_{0}\right\|)\Lambda$ is the infinitesimal generator of the semigroup $V^{t}$, the proof can be easily concluded by a standard argument (see, e.g., [9, Ch.4 $\S$ 4.2]), and, finally recalling that $T$ is arbitrary. ∎

Lemma 5**.**

Suppose the conditions of Theorem 1 are fulfilled. Then for each $n=1,2,3,...$,

(a) $f_{n}$ $\in$ ${\mathcal{M}}_{3}$. In particular, $f_{n}\in L_{k,loc}^{1}({\mathbb{R}}_{+};X_{+})$, $k=0,1,2,3$;

(b) $\Lambda_{1}f_{n}$ $\in$ $L_{loc}^{1}({\mathbb{R}}_{+};X_{+})$;

(c) $Q^{\pm}(\cdot,f_{n}(\cdot))\in L_{k,loc}^{1}({\mathbb{R}}_{+};X_{+})$ for $k=0,1,2$, and $\Lambda_{1}Q^{\pm}(\cdot,f_{n}(\cdot))$ $\in$ $L_{loc}^{1}({\mathbb{R}}_{+};X_{+})$.

Proof.

(a) is immediate from Lemma 4 and Remark 2.

(b) follows from (a) and the property \Lambda_{1}({\mathcal{M}}_{3})$$\subset $L^{1}_{loc}({\mathbb{R}}_{+};X_{+})$ (due to $(A_{3})$).

(c) By (a) and (12), clearly, $Q^{-}(\cdot,f_{n}(\cdot))$ $\in L_{2,loc}^{1}({\mathbb{R}}_{+};X_{+})$. Moreover, (12) yields $\Lambda_{1}Q^{-}(t,f_{n}(t))$ $\leq$ $a(\|\Lambda f_{n}(t)\|)\Lambda_{1}\Lambda f_{n}(t)$ for a.a. $t\geq 0$. But $\Lambda({\mathcal{M}}_{3})$ $\subset$ ${\mathcal{M}}_{2}$. Besides, $\Lambda_{1}({\mathcal{M}}_{2})$ $\subset$ $L^{1}_{loc}({\mathbb{R}}_{+};X_{+})$, by virtue of $(A_{3})$. Consequently, $\Lambda_{1}Q^{-}(\cdot,f_{n}(\cdot))$ $\in$ $L_{loc}^{1}({\mathbb{R}}_{+};X_{+})$. So $\Lambda_{1}Q^{+}(\cdot,f_{n})$ $\in$ $L_{loc}^{1}({\mathbb{R}}_{+};X_{+})$, by virtue of (14). Also, from (15), we get $Q^{+}(\cdot,f_{n}(\cdot))$ $\in L_{2,loc}^{1}({\mathbb{R}}_{+};X_{+})$. Remark 2 completes the proof of (c). ∎

The next lemmas are needed to establish the convergence of the approximating sequence $\{f_{n}(t)\}$ defined by (36).

Lemma 6**.**

The sequence $\{f_{n}(t)\}$ is positive and increasing for all $t\geq 0$.

Proof.

This result follows from (36) by a straightforward induction which uses the positivity and monotonicity of $\{f_{0.n}\}$, the positivity of the linear semigroup $V^{t}$, and applies Lemma 2. ∎

Based on Lemma 4(c), we can differentiate (36), a.e. on ${\mathbb{R}}_{+}$. We obtain

[TABLE]

Integrating again Eq. (41), and using (29), we obtain for $\ n\geq 3$,

[TABLE]

which is useful to prove the following property.

Lemma 7**.**

For all $n=2,3,....$,

[TABLE]

and

[TABLE]

Proof.

(See the proof of Lemma 4.3 in [1].) We proceed by induction. Using (34), one finds from (36) that $0=f_{1}(t)\leq f_{2}(t)\leq f_{0}$. But $\Delta(t,0)=0$ a.e., because of (21). Therefore, (43) and (44) are trivially verified for $n=2$.

Suppose that (43) and (44) are true for $n=2,3,...,q$. Since $0\leq\Lambda f_{q}(t)\leq\Lambda f_{q+1}(t)$ (as a consequence of Lemma 6) and $a$ is non-decreasing, we get

[TABLE]

which can be applied to (42) in the case $n=q+1$. Thus we obtain

[TABLE]

concluding the induction argument for the validity of (43). Further, based on Lemma 5, we can apply $\Lambda$ to (45) and use (6). Then, applying, (4), (3), (5) and (13), we finally obtain (43) for $n=q+1$, which concludes the proof of the lemma. ∎

Lemma 8**.**

Under the conditions of Theorem 1,

[TABLE]

Proof.

(See the proof of Lemma 4.4 in [1].) Cases $n=1$ is trivial. For $n\geq 2$, based on Lemma 5, we apply $\Lambda_{1}$ to (43) and use (4), (3), and (5). We get

[TABLE]

for all $n=2,3,...$, and $t\geq 0$. But in the above inequality, the integrand is positive, because of (14). Thus

[TABLE]

Based on Lemma 5, we apply (6) (with $\Gamma=\Lambda^{2}$) to (43). Then, using (4), (3), (5), and (15), we obtain

[TABLE]

($n=2,3,....$). But $\rho((\left\|\Lambda_{1}f_{n-1}(s)\right\|)\leq\rho((\left\|\Lambda_{1}f_{0}\right\|)$, by virtue of (47). Therefore,

[TABLE]

where the use of Gronwall’s inequality concludes the proof. ∎

By Lemmas 6 and 8, for every $t\geq 0$, the positive sequence $\{\Lambda^{2}f_{n}(t)\}$ is increasing, and norm bounded, respectively. Then Lemma 1(c) applies. Therefore, $\exists$ $f:{\mathbb{R}}_{+}\mapsto{\mathcal{D}}_{+}(\Lambda^{2})$ measurable, such that, $\forall t\geq 0$,

[TABLE]

In particular, $\Lambda^{k}f_{0,n}\nearrow\Lambda^{k}f_{0}$, $k=1,2$. Then taking the limit in (46), we find that $f$ satisfies (25). This and Remark 2 imply $f$ $\in$ $L^{1}_{k,loc}({\mathbb{R}}_{+};X_{+})$, $k=0,1,2$. Moreover, by (20), we find that $Q^{\pm}(\cdot,f)$, $\Lambda Q^{\pm}(\cdot,f)$, and $\Delta f$ are in $L^{1}_{loc}({\mathbb{R}}_{+};X_{+})$. Further, from (20) and (48), it follows that the increasing sequences $\{\Lambda^{k}Q^{\pm}(t,f_{n})\}$, $k=0,1$, are bounded in norm. Therefore, Levi’s property implies that they are convergent a.e. on ${\mathbb{R}}_{+}$. But $Q^{+}(t,\cdot)$ and $Q^{-}(t,\cdot)$ are o-closed. Then, $Q^{\pm}(t,f_{n})\nearrow Q^{\pm}(t,f)$ as $n\rightarrow\infty$ a.e. on ${\mathbb{R}}_{+}$. Since $\Lambda$ is closed, it also follows that $\Lambda Q^{\pm}(t,f_{n})\nearrow\Lambda Q^{\pm}(t,f)$ as $n\rightarrow\infty$, a.e. on ${\mathbb{R}}_{+}$. Consequently, $\Delta(t,f_{n})\nearrow\Delta(t,f)$ as $n\rightarrow\infty$, a.e. on ${\mathbb{R}}_{+}$. Thus, applying the Lebesgue’s dominated convergence theorem, we obtain $\int_{0}^{s}\Delta(\tau,f_{n}(\tau))d\tau$ $\nearrow$ $\int_{0}^{s}\Delta(\tau,f(\tau))d\tau$ as $n\rightarrow\infty$, $\forall s>0$. By the above considerations, and using that $a$ is non-decreasing and continuous, we are enabled to apply conveniently the dominated convergence theorem in (42) and (44). It follows that $f$ is solution to (28), and satisfies (25). Finally, Proposition 2 concludes the existence part of Theorem 1.

The uniqueness of the solution follows by the same argument as in Ref. [1] (inspired from [3])). In detail, since $f$ is the limit of $\{f_{n}\}$, by applying the dominated convergence theorem to (36), we find that $f$ is also solution to Eq. (33). Let $F$ be another continuous, positive solution of (1), with the properties stated in Theorem 1. Thus $F$ satisfies (24), as $f$ does. Obviously, $F$ is also a solution to (33), and a simple induction implies $f_{n}(t)\leq F(t)$ for all $t\geq 0$, and $n=1,2...$. Consequently, $0\leq f(t)\leq F(t)$ for all $t\geq 0$. Thus, if $\exists$ $t_{*}>0$ such that $F(t_{*})\neq f(t_{*})$, then $f(t_{*})<F(t_{*})$, hence $\|\Lambda f(t_{*})\|<\|\Lambda F(t_{*})\|$. Since $\Delta$ is isotone, we get

[TABLE]

in contradiction with the fact that $F$ satisfies (24). $\Box$

5. Conclusions

In the present work, we have revised and generalized the main results of Ref. [1]. If the setting of the current note reduces to an AL - space, then, in essence, Theorem 1 and Corollary 1 reduce to [1, Theorem 3.1] and [1, Corollary 3.1], respectively. (In fact, Theorem 1(b) slightly improves the result of [1, Theorem 3.1(b)], by relaxing the conditions on the initial datum.) An analogue of [1, Proposition 3.1] can be also obtained by adapting directly its proof to the present context.

The analysis of this paper is close to the argument of [1], but technically there are some differences. Indeed, in Ref. [1], (Theorem 3.1(a)) was obtained by a two-step demonstration. In the first step (”Step 1”) the theorem was proved for an initial datum in ${\mathcal{D}}_{+}(\Lambda^{\infty})$ (in the setting of an AL-space)777${\mathcal{D}}_{+}(\Lambda^{\infty})$ appears in [1] as ${\mathcal{D}}^{\infty}_{+}(\Lambda)$. . This was done by approximating the solution of [1, Eq. (1.1)] by a sequence similar to that defined by (36), in the previous section, but keeping the initial datum fixed in ${\mathcal{D}}_{+}(\Lambda^{\infty})$. The purpose of the second step (”Step 2”) was to extend the result of ”Step 1”, by considering an initial datum $f_{0}$ $\in{\mathcal{D}}_{+}(\Lambda^{2})$. Thus the solution of [1, Eq. (1.1)] was approximated by a sequence, denoted $\{F_{i}\}$ in [1], of solutions of the same equation, but corresponding to an increasing sequence of initial data in ${\mathcal{D}}_{+}(\Lambda^{\infty})$, converging to the original $f_{0}$. Then ”Step 2” was concluded, based on the assertion that $\{F_{i}\}$ is increasing. Unfortunately, the monotonicity of $\{F_{i}\}$ has been erroneously justified in Ref. [1], so the proof of Theorem 3.1(a) is incomplete therein. However, the error can be easily corrected by reconstructing $\{F_{i}\}$ to approximate [1, Eq. (3.15)] instead of [1, Eq. (1.1)], as was done in [1] (see the Appendix below).

Finally, it should be emphasized that in the present note, the two-step proof of [1, Theorem 3.1(a)] has been reduced to a modified version of ”Step 1” of that proof. This was done by introducing (36) as a diagonalization, in some sense, of the main approximation scheme used in [1].