Proposition 3.1

The operator T defined by the formal expression:

Proof

Let us consider the resolvent equation of (3.1):

Next we observe that for $f\in X$,

From this proposition it follows that the operator

Theorem 4.1 [11,17]

Under the above assumptions, there exists a smallest substochastic semigroup$(S_{\tilde{G}}{(t))}_{t⩾0}$generated by an extension$\tilde{G}$of the operator$\tilde{T}+K$. This semigroup, for arbitrary$u\in D(\tilde{G})$and$t>0$, satisfies:

The generator $\tilde{G}$ of $(S_{\tilde{G}}{(t))}_{t⩾0}$ is characterized by:

Definition 4.1

We say that a substochastic semigroup $(S_{\tilde{G}}{(t))}_{t⩾0}$ (generated by an extension $\tilde{G}$ of the operator $\tilde{T}+K$) is honest if c is finite on $D(\tilde{G})$, and, for any $0⩽\stackrel{○}{u}\in D(\tilde{G})$, the solution $u(t)=S_{\tilde{G}}(t)\stackrel{○}{u}$ of (4.2) satisfies:

Remark 4.1

The definition of honesty is not restricted to contractive semigroups and is valid even if c in (4.4) is of undetermined sign. In fact, for the original model (2.1) we shall be using this definition with a positive right-hand side in (4.6). However, for a general c, the existence part of the theory is usually not a trivial matter and this is why we prefer to present a complete theory for substochastic semigroups, and then apply it to a wider class of models that can be transformed to a substochastic case.

Theorem 4.2

For any fixed$\lambda >0$, there is$0⩽{\beta }_{\lambda }\in X^{*}$with$\Vert {\beta }_{\lambda }\Vert ⩽1$such that:

Proposition 4.1

The following holds:

• (i) for any $f\in X_{+}$ , ${\eta }_f$ is a non-positive and a non-increasing function for $t⩾0$ ;
• (ii)

  $$\underset{0}{\overset{\infty }{\int }}{\mathrm{e}}^{-\lambda t}{\eta }_f(t)\mathrm{d}t=-\frac{1}{\lambda }\langle {\beta }_{\lambda }\text{,}f\rangle$$

  hence$(S_{\tilde{G}}{(t))}_{t⩾0}$is honest if and only if${\beta }_{\lambda }\equiv 0$for any (some) $\lambda >0$;
• (iii) if$(S_{\tilde{G}}{(t))}_{t⩾0}$is dishonest, then for some$f\in X_{+}$and any$t>0$:

  $$\Vert S_{\tilde{G}}(t)f\Vert <\Vert f\Vert +\underset{0}{\overset{t}{\int }}c(S_{\tilde{G}}(s)f)\mathrm{d}s$$

Theorem 4.3

The following are equivalent:

• (a) The semigroup $(S_{\tilde{G}}{(t))}_{t⩾0}$ is honest;
• (b) $\tilde{G}=\overline{\tilde{T}+K}$ ;
• (c) For any $u\in R(\lambda \text{,}\tilde{G})X_{+}$ , where $\lambda >0$ is arbitrary, we have:

  $$\underset{\Omega }{\overset{}{\int }}\tilde{G}u\mathrm{d}\mu ⩾-c(u)$$

  (4.10)

Theorem 4.4

Assume that there exists $u\in D{\left(\tilde{\mathcal{G}}\right)}_{+}$ such that

• (i) $[{\tilde{\mathcal{L}}}_{\lambda }(\lambda I-\tilde{\mathcal{T}})u](x)=u(x)$ , a.e., for some $\lambda >0$ ,
• (ii) for some $\lambda >0$ , $\lambda u(x)-[\tilde{\mathcal{G}}u](x)=g(x)\in X_{+}$ ,
• (iii) $c(u)$ is finite and

  $$\underset{\Omega }{\overset{}{\int }}\tilde{\mathcal{G}}u\mathrm{d}\mu <-c(u)$$

  (4.11)

Proposition 5.1

There is an extension G of $T+K$ given by $(G\text{,}D(G))=(\tilde{G}+\tilde{b}I\text{,}D(\tilde{G}))$ that generates a positive semigroup $(S_G{(t))}_{t⩾0}=({\mathrm{e}}^{\tilde{b}t}{S}_{\tilde{G}}{(t))}_{t⩾0}$ . Moreover, the generator G is characterized by:

Proof

The operator $\tilde{T}$ was constructed from T by subtracting the bounded operator $\tilde{b}I$. Let us consider the approximating semigroups $(S_r{(t))}_{t⩾0}$, mentioned in Theorem 4.1. They are generated by $(T-\tilde{b}I+rK\text{,}D(T))$, $0<r<1$ and

Formula (5.2) follows immediately from (4.3) by noting that since $\lambda I-G=(\lambda -\tilde{b})I-\tilde{G}$, we have ${(\lambda I-G)}^{\mathrm{-1}}={({\lambda }^{\prime }I-\tilde{G})}^{\mathrm{-1}}$ for $\lambda >\tilde{b}$ and the same holds for the resolvent of T. □

Proposition 5.2

Any function$u\in D(G)$is continuous on$(0\text{,}\infty )$.

Proof

Let first $f\in X_{+}$ and $\lambda >\tilde{b}$. Since $\lambda I-T$ extends to a positive integral operator ${\mathcal{L}}_{\lambda }$ on $\mathsf{E}$, by (5.2) the element ${\overline{u}}_{+}={(\lambda I-G)}^{\mathrm{-1}}f={\mathcal{L}}_{\lambda }g\text{,}$ where $g={\sum }_{n=0}^{\infty }{\left[K{(\lambda I-T)}^{\mathrm{-1}}\right]}^{n}f$, is a well-defined element of $\mathsf{E}$ as the series is increasing. However, as ${\overline{u}}_{+}\in D(G)\subset X$, it must be finite almost everywhere. From (5.10) we have:

Lemma 5.1

Let $\mathcal{K}$ and ${\mathcal{L}}_{\lambda }$ be the extensions introduced above. If for some $g\in D{\left(\mathcal{L}\right)}_{+}$ , both g and ${\mathcal{KL}}_{\lambda }g$ belong to $L_1([\alpha \text{,}N]\text{,}x\mathrm{d}x)$ , where $0⩽\alpha <N⩽\infty$ , then:

Theorem 5.1

If $u\in D(G)$ , then there are sequences ${\alpha }_k\to 0^{+}$ and $N_k\to \infty$ as $k\to \infty$ such that:

Proof

Using a similar argument to Proposition 5.2 we see that if $g\in {\mathsf{E}}_{+}$ is such that ${\mathcal{L}}_{\lambda }g\in X$, then $g\in L_1([\alpha \text{,}N]\text{,}x\mathrm{d}x)$ for any $0<\alpha <N<\infty$. Following [10 (Corollary 4.1)], we observe that if $u=R(\lambda \text{,}G)f$, $f\in X_{+}$ there is $g\in {\mathsf{E}}_{f\text{,}+}$, constructed as in the proof of Proposition 5.2, such that $u={\mathcal{L}}_{\lambda }g$ and:

Since we know that $u\in L_1([0\text{,}\infty )\text{,}x\mathrm{d}x)\cap C(0\text{,}\infty )$, we have ${\mathrm{lim\hspace{0.17em}inf}}_{x\to \infty }{x}^2|u(x)|=0$. Thus, there is a sequence ${\left(N_k\right)}_{k\in \mathbb{N}}$ converging to ∞ such that ${\lim }_{k\to \infty }{N}_k^2|u(N_k)|=0$. Similarly, we obtain a sequence ${\left({\alpha }_k\right)}_{k\in \mathbb{N}}$ that converges to 0 as $k\to \infty$, such that ${\lim }_{k\to \infty }{\alpha }_k^2|u({\alpha }_k)|=0$. Since $b(x)⩽\tilde{b}x$ for $x>0$, we obtain the thesis.  □

Theorem 5.2

If

Proof

As in the previous proof, it is enough to consider $u=R(\lambda \text{,}G)f$, $f\in X_{+}\text{,}\lambda >\tilde{b}$; for such f we have also $u={\mathcal{L}}_{\lambda }g$ for some $g\in {\mathsf{E}}_{+}$. Since $u\in X$, by (5.10) and Tonelli's theorem, we obtain:

Theorem 5.3

Assume that $b\in C^1([0\text{,}\infty ))$ with ${\inf }_{0⩽x<\infty }{b}^{\prime }(x)>-\infty$ ,