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Background To support public health measures during the Covid-19 pandemic, oral opioid agonist treatment (OAT) take-home doses were expanded in Western countries with positive results. Injectable OAT (iOAT) take-home doses, which were previously not recommended, were made available for the first time in several sites to align with public health measures. Building upon these temporary risk-mitigating guidelines, a clinic in Vancouver, BC continued to offer two of a possible three daily doses of take-home injectable medications to eligible clients. The present study explores the processes through which take-home iOAT doses improved clients’ quality of life and continuity of care in real-life settings. Methods Three rounds of semi-structured qualitative interviews were conducted over a period of seventeen months beginning in July 2021 with eleven participants receiving iOAT take-home doses at a community clinic in Vancouver, BC. Interviews followed a topic guide that evolved iteratively in response to emerging lines of inquiry. Interviews were recorded, transcribed, and then coded using NVivo 1.6. Results Participants reported that take-home doses granted them the freedom away from the clinic to have daily routines, form plans, and enjoy unstructured time. Participants appreciated the greater privacy, accessibility, and ability to engage in paid work. Furthermore, participants enjoyed greater autonomy to manage their medication and level of engagement with the clinic. These factors contributed to greater quality of life and continuity of care. Participants shared that their dose was too essential to divert and that they felt safe transporting and administering their medication off-site. In the future, participants would like to access longer take-home prescriptions, the ability to pick-up at different and convenient community pharmacies, and a medication delivery service. Conclusions Reducing the number of daily onsite injections from two or three to only one revealed the diversity of rich and nuanced needs that added flexibility and accessibility in iOAT can meet. Urgent action such as policy reform and licencing diverse opioid medications/formulations is necessary to make these measures permanent and meet the varied needs and preferences of OUD clients.

Injectable opioid agonist treatment (iOAT) is effective for opioid use disorder (OUD), yet little is known about client preferences for accessing iOAT (e.g., with diacetylmorphine, hydromorphone, buprenorphine, fentanyl, etc.). Best-worst scaling (BWS) is a preference elicitation method from health economics that has never been applied to addiction care broadly, or iOAT specifically. We describe the stages of developing a BWS scale that assesses iOAT clients' treatment delivery preferences to inform program planning and maximize healthcare efficiency.We underwent several steps to reveal the relevant attributes/levels and design the scale structure. An initial list of potential attributes and levels was established from a literature review and prior qualitative data. Then, we conducted semi-structured interviews and focus groups with clients (n=21) on their iOAT preferences to confirm the attributes and prioritize/include new ones. Next, we conducted semi-structured interviews and focus groups with iOAT experts and stakeholders to receive their input on the draft list of attributes and levels. A BWS profile case design was piloted with iOAT clients (n=18) from different sites during a think aloud interview. After several rounds of revisions, the final version was tested by iOAT clients (n=2) before the scale was launched.We developed a person-centered scale that assesses current and former iOAT clients' most and least wanted aspects of iOAT delivery. The final version yielded 7 unique attributes: choice of medication, choice of dose, convenience, locationspace, schedulingroutines, stafftraining, and types of services offered.This scale can help expand iOAT programs in a way that is person-centered, rapid, and affordable. The methodology is a guide for other regions with similar populations who aim to develop strong quantitative methodologies that prioritize client collaboration.

A class of perturbed discrete nonlinear Schrodinger equations with sublinear terms is studied. The new features are: (1) negative and positive energy homoclinic solutions are obtained; (2) the continuity of solutions in the perturbed terms at zero is studied; (3) the positive-definite condition can be removed.

Many problems arising in science and engineering call for the solving of the Euler–Lagrange equations of functionals. Thus, solving the Euler–Lagrange equations is tantamount to finding critical points of the corresponding functional. An idea that has been very successful is to find appropriate sets that sandwich the functional. This means that the functional is bounded from above on one of the sets and bounded from below on the other. Two sets of the space are said to form a sandwich if they produce a critical sequence whenever they sandwich a functional. If the critical sequence has a convergent subsequence, then that produces a critical point. Finding sets that sandwich a functional is quite easy, but determining whether or not the sets form a sandwich is quite another story. It appears that the only way we can check to see if two sets form a sandwich is to require that one of them be contained in a finite-dimensional subspace. The reason is that in order to verify the definition, we need to invoke the Brouwer fixed point theorem. Our aim is to find a counterpart that holds true when both sets are infinite dimensional. We adjust our definitions to accommodate infinite dimensions. These definitions reduce to the usual when one set is finite dimensional. In order to prove the corresponding theorems, we make adjustments to the topology of the space and introduce infinite dimensional splitting. This allows us to use a form of compactness on infinite dimensional subspaces that does not exist under the usual topology. We lose the Brouwer index, but we are able to replace it with the Leray–Schauder index. We carry out the details in Sections 5, 7, and 8. In Section 6 we solve a system of equations which require infinite dimensional splitting.

We use critical point theory to find solutions of the nonlinear steady state Schrodinger equations arising in the study of photonic lattices.

Using global optimization, we are able to find nontrivial solutions of the nonlinear steady-state Schrodinger equation arising in optics for wide ranges of the parameters. Our results hold in arbitrary dimensions.

In infinite \begin{document}$ m $\end{document}-dimensional lattices, we obtain the existence of two nontrivial solutions for a class of non-periodic Schrödinger lattice systems with perturbed terms, where the potentials are coercive and the nonlinearities are asymptotically linear at infinity. In addition, examples are given to illustrate our results.

In the previous chapter we noted that in our study of semilinear elliptic partial differential equations of the form $$\displaystyle \mathcal A u = f(x,u), u \in D $$ in unbounded domains, we required that the resolvent set of \(\mathcal A\) not be empty. For convenience, we assumed that it contain 0. This allowed us to choose an interval \((a,b) \subset \rho (\mathcal A),\) where a < 0 < b. We let \( D = D(|\mathcal A|{ }^{(1/2)}).\) With the scalar product \((u,v)_D= (|\mathcal A|{ }^{(1/2)}u,|\mathcal A|{ }^{(1/2)}v),\) it became a Hilbert space. We let $$\displaystyle N= E(-\infty , a], \quad M = E[b, \infty ) $$ be the negative and positive invariant subspaces of \(\mathcal A.\) Then $$\displaystyle (\mathcal A v,v) \le a \|v\|{ }^2, \quad v \in D \cap N, $$ and $$\displaystyle (\mathcal A w,w) \ge b\|w\|{ }^2, \quad w \in D \cap M. $$ The hypotheses of our theorems depended on a and b using the fact that 0 was embedded in \(\rho (\mathcal A).\) The purpose of the present chapter is to study the situation when 0 is a boundary point of \(\rho (\mathcal A)\) and the arguments do not work.

We consider the system $$\displaystyle -\ddot x(t)=\; B(t)x(t) +\nabla _xV(t,x(t)), $$ where $$\displaystyle x(t)=(x_1(t),\cdots ,x_n(t)) $$ is a map from I = [0, T] to \(\mathbb R^n\) such that each component xj(t) is a periodic function in H1 with period T, and the function V (t, x) = V (t, x1, ⋯ , xn) is continuous from \(\mathbb R^{n+1}\) to \(\mathbb R\) with $$\displaystyle \nabla _xV(t,x)=(\partial V/\partial x_1,\cdots ,\partial V/ \partial x_n) \in C(\mathbb R^{n+1},\mathbb R^n). $$ For each \(x \in \mathbb R^n,\) the function V (t, x) is periodic in t with period T.

In this chapter we study periodic solutions of the Dirichlet problem for the semilinear wave equation: $$\displaystyle \square u-\mu u:= u_{tt}- u_{rr} -\mu u = p(t,r,u),\quad t \in \mathbb R, \quad 0

In this chapter we study periodic solutions of the Dirichlet problem for the semilinear wave equation $$\displaystyle \square u:= u_{tt}-\Delta u = f(t,x,u),\quad t\in {\mathbb R},\quad x\in \mathcal B_R, $$ $$\displaystyle u(t,x) = 0,\quad t\in {\mathbb R},\quad x\in \partial \mathcal B_R, $$ $$\displaystyle u(t+T,x) = u(t,x),\quad t\in {\mathbb R},\quad x\in \mathcal B_R, $$ where $$\displaystyle \mathcal B_R = \{x\in {\mathbb R}^n:|x

The theory presented in Chap. 2 gives precise criteria for the existence of sandwich pairs, but it does not provide a list of such pairs. There is good reason for this. The criterion $$\displaystyle \sigma (1)\, A \cap B \ne \phi , \quad \forall \sigma \in \Sigma _Q $$ is very difficult to verify in practice, while the corresponding statement for linking pairs is easier. We were able to provide a reasonable list of linking sets at the end of Chap. 1, but we have not yet been able to do so for sandwich sets. In this chapter we shall focus our attention on this matter. It turns out that we can obtain a lot of help from the theory of linking sets.

BACKGROUND Indigenous leaders continue to be concerned about high rates of HIV and barriers to HIV treatment among young Indigenous people involved in substance use. Growing evidence suggests that using mobile phones for health (mHealth) may be a powerful way to support connection with health services, including HIV prevention and treatment. OBJECTIVE This study examined the patterns of mobile phone ownership and use among young Indigenous people who have used drugs living with or vulnerable to HIV and explored the acceptability of mHealth to support access to health care in this population. METHODS The Cedar Project is a cohort study involving young Indigenous people who have used drugs in Vancouver and Prince George, British Columbia. This mixed methods exploratory study involved 131 Cedar Project participants enrolled in our WelTel mHealth program. At enrollment, participants completed a questionnaire related to mobile phone use and interest in mHealth. Data were linked to Cedar Project questionnaires and serodata. We present comparative statistics (quantitative) and results of a rapid thematic analysis (qualitative) related to mobile phone patterns and interest in receiving mHealth. RESULTS Less than half of the participants (59/130; 45.4%) reported owning a phone. Among those with a phone, the majority owned a smartphone (46/59; 78%). Most participants with a phone reported having an unlimited texting plan (39/55; 71%), using the internet on their phone (44/59; 75%), and texting daily (44/55; 80%). A majority reported that using a mobile phone for health would be invaluable (120/130; 92.3%). There were no differences in mHealth acceptance between participants who owned a phone and those who did not (P>.99). All but one participant living with HIV felt using a mobile phone would be helpful for their health, while a small proportion of HIV-negative participants remained unsure (1.9% vs 11.7%; P=.047). In response to open-ended questions asking why using a mobile phone may be helpful for health, participants identified a diverse set of anticipated benefits: (1) connection for emotional, mental, and spiritual support, (2) connection to family, (3) staying in touch and/or being reachable, (4) overcoming current barriers to phone use, (5) convenience, privacy, and safety, and (6) access to health care and emergency services. CONCLUSIONS We observed high acceptance and interest in using mobile phone technology for health despite low rates of personal mobile phone connectivity among young Indigenous people who have used drugs living with and vulnerable to HIV in British Columbia, Canada. Mobile phones were viewed as a way to support connections and relationships that are seen as critical to health and well-being among young Indigenous people in this study. Findings may be useful for health care providers preparing to scale up mHealth programs to support HIV prevention and treatment in this population.

In this paper, we obtain the existence of two non-trivial solutions for a class of non-periodic Schrodinger lattice systems with perturbed terms when the nonlinearities are super-linear at infinity. In addition, several examples are given to illustrate our results. Our theorems appear to be the first such result.

Many problems arising in science and engineering call for the solving of the Euler-Lagrange equations of functionals. Thus, solving the Euler-Lagrange equations is tantamount to finding critical points of the corresponding functional. An idea that has been very successful is to find appropriate sets that separate the functional. This method is called linking. Two sets A , B are said to separate a functional G if the supremum of G on A is less than or equal to the infimum of G on B. Two sets of the space are said to link if they produce a critical sequence whenever they separate a functional. If the critical sequence has a convergent sub-sequence, then that produces a critical point. Finding sets that separate a functional is quite easy, but determining whether or not the sets link is quite another story. It appears that the only way we can check to see if two sets link, is to require that one of them be contained in a finite-dimensional subspace. The reason is that in order to verify the definition, we need to invoke the Brouwer fixed point theorem. Our aim is to find a counterpart of linking that holds true when both sets are infinite dimensional. We adjust our definitions to accommodate infinite dimensions. These definitions reduced to the usual definitions when one set is finite dimensional. In order to prove the corresponding theorems, we make adjustments to the topology of the space and introduce infinite dimensional splitting. This allows us to use a form of compactness on infinite dimensional subspaces which does not exist under the usual topology. We lose the Brouwer index, but we are able to replace it with the Leray-Schauder index. We carry out the details in Sections 5 and 6 . In Section 7 we solve a system of equations which require infinite dimensional splitting.