Non-adherence to anti-retroviral therapy (cART) among people living with HIV/AIDS (PLWHA) is complex, multifactorial, and leads to increases in viral load, immunosuppression, and HIV burden related to direct CNS virulence and cART re-initiation (i.e., immune reconstitution syndrome). Among behavioral disturbances, which may become long-lasting without proper treatment, major depressive disorder, generalized anxiety disorder, schizophrenia, and bipolar spectrum disorders are frequent, as well as exacerbation of other premorbid underlying psychiatric conditions, such as post-traumatic stress disorder (PTSD) and substance use and related conditions, not to mention neurocognitive disorders that are encompassed under the umbrella term of HIV-Associated Neurocognitive Disorders (HAND). In this review, we summarized the neuropsychiatric sequelae of medication non-adherence in PLWHA by utilizing two clinical vignettes to illustrate how syndemic factors may interact and lead to unique presentations.
Citation: Debora Moreira, Rida Khan, Paulo Marcelo Gondim Sales. Neuropsychiatric sequelae of medication non-adherence in people living with HIV[J]. AIMS Medical Science, 2023, 10(2): 162-173. doi: 10.3934/medsci.2023014
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Non-adherence to anti-retroviral therapy (cART) among people living with HIV/AIDS (PLWHA) is complex, multifactorial, and leads to increases in viral load, immunosuppression, and HIV burden related to direct CNS virulence and cART re-initiation (i.e., immune reconstitution syndrome). Among behavioral disturbances, which may become long-lasting without proper treatment, major depressive disorder, generalized anxiety disorder, schizophrenia, and bipolar spectrum disorders are frequent, as well as exacerbation of other premorbid underlying psychiatric conditions, such as post-traumatic stress disorder (PTSD) and substance use and related conditions, not to mention neurocognitive disorders that are encompassed under the umbrella term of HIV-Associated Neurocognitive Disorders (HAND). In this review, we summarized the neuropsychiatric sequelae of medication non-adherence in PLWHA by utilizing two clinical vignettes to illustrate how syndemic factors may interact and lead to unique presentations.
AIDS Dementia Complex;
Acquired immunodeficiency syndrome;
Asymptomatic neurocognitive impairment;
Adenosine triphosphate;
combination anti-retroviral therapy;
Central nervous system;
Generalized anxiety disorder;
HIV-associated dementia;
HIV-associated neurocognitive disorders;
Hypomania checklist 32;
Human Immunodeficiency Virus;
Interleukin-1beta;
Kynurenine-to-tryptophan ratio;
Montreal Cognitive Assessment;
Minor neurocognitive disorder;
Nonnucleoside reverse transcriptase inhibitor;
Nucleoside analog reverse transcriptase inhibitors;
Protease inhibitor;
People living with HIV/AIDS;
Patient Health Questionnaire;
Post Traumatic Stress Disorder;
Tumor necrosis factor-alpha;
Thyroid-stimulating hormone
In 1960, Opial [1] presented an inequality involving integral of a function and its derivative as follows
Theorem 1.1. Let x∈C1[0,h] be such that x(t)>0 in (0,h). Then, the following inequalities hold:
i) If x(0)=x(h)=0, then
∫h0|x(t)x′(t)|dt≤h4∫h0|x′(t)|2dt. | (1.1) |
ii) If x(0)=0, then
∫h0|x(t)x′(t)|dt≤h2∫h0|x′(t)|2dt. | (1.2) |
In (1.1), the constant h4 is the best possible.
Since then, the study of generalizations, extensions and discretizations for inequalities (1.1) and (1.2) of Opial type inequalities has grown into a substantial field, with many important applications in theory of differential equations, approximations and probability, among others. For more details, we cite the readers to [2,3,4,5,6,7] and the references therein.
Recently, Mirkovi et.al [8] established a new integral inequality of the q-Opial type as follows:
Theorem 1.2. Let f∈C1[0,1] be q-decreasing function with f(bq0)=0. Then, for any p≥0,
∫ba|Dqf(x)||f(x)|pdqx≤(b−a)p∫ba|Dqf(x)|p+1dqx. | (1.3) |
In [9], Alp et al. gave the following q-Opial type inequality for quantum integral :
Theorem 1.3. Let x(t)∈C1[0,h] be such that x(0)=x(h)=0, and x(t)>0 in (0,h). Then, the following inequality holds :
∫h0|(x(t)+x(qt))Dqx(t)|dqt≤h1+q∫h0|Dqx(t)|2dqt. | (1.4) |
The (p,q)-calculus is known as two parameter quantum calculus, a generalization of q-calculus, whose applications play important roles in physics, chemistry, orthogonal polynomials and number theory [10,11]. In [12], Mursaleen et al. applied (p,q)-calculus in approximation theory and investigated first (p,q)-analogue of Bernstein operators. Recently, Sadjang in [13] studied the (p,q)-derivative, the (p,q)-integration, and obtained some of their properties and the fundamental theorem of (p,q)-calculus. About the recent results of (p,q)-calculus, please see [14,15,16,17,18].
Inspired by the above mentioned works [8,9,13], in this paper, we will establish some (p,q)-Opial type inequalities by using (p,q)-calculus and analysis technique. If p=1 and q→1−, then all the results we have obtained in this paper reduce to the classical cases.
In what follows, p,q are two real numbers satisfying 0<q<p≤1, and (p,q)-bracket is defined as
[n]p,q=pn−qnp−q=pn−1+qpn−2+⋯+qn−1,n∈N. |
Definition 2.1. ([13]) The (p,q)-derivative of the function f is defined as
Dp,qf(x)=f(px)−f(qx)(p−q)x,x≠0, |
and (Dp,qf)(0)=limx→0Dp,qf(x)=f′(0), privided that f is differentiable at 0.
Definition 2.2. ([13]) Let f be an arbitrary function and a be a positive real number, the (p,q)-integral of f from 0 to a is defined by
∫a0f(x)dp,qx=(p−q)a∞∑i=0qipi+1f(qipi+1a). |
Also for two nonnegative numbers such that a<b, we have
∫baf(x)dp,qx=∫b0f(x)dp,qx−∫a0f(x)dp,qx. |
Lemma 2.1. ([13]) The (p,q)-derivative fulfills the following product rules
Dp,q(f(x)g(x))=f(px)Dp,qg(x)+g(qx)Dp,qf(x), |
Dp,q(f(x)g(x))=g(px)Dp,qf(x)+f(qx)Dp,qg(x). |
Theorem 2.2. ([13]) (Fundamental theorem of (p,q)-calculus). If F(x) is a (p,q)-antiderivative of f(x) and F(x) is continuous at x=0, we have
∫baf(x)dp,qx=F(b)−F(a), |
where 0≤a<b≤∞.
Theorem 3.1. Let f∈C1[0,h] be such that f(t)>0 in (0,h). If f(0)=0 and m∈N. Then, the following inequality holds:
∫h0|m∑j=0fm−j(px)fj(qx)||Dp,qf(x)|dp,qx≤hm∫h0|Dp,qf(x)|m+1dp,qx. | (3.1) |
Proof. Choosing g(x) as
g(x)=∫x0|Dp,qf(t)|dp,qt, | (3.2) |
such that
|Dp,qf(x)|=Dp,qg(x), | (3.3) |
and for x∈[0,h], we have
|f(x)|=|∫x0Dp,qf(t)dp,qt|≤∫x0|Dp,qf(t)|dp,qt=g(x), | (3.4) |
|f(px)|=|∫px0Dp,qf(t)dp,qt|≤∫px0|Dp,qf(t)|dp,qt=g(px), | (3.5) |
and
|f(qx)|=|∫qx0Dp,qf(t)dp,qt|≤∫qx0|Dp,qf(t)|dp,qt=g(qx). | (3.6) |
By Lemma 2.1, we can obtain that
Dp,qfm+1(x)=m∑j=0fm−j(px)fj(qx)Dp,qf(x). | (3.7) |
Thus, by (3.2)–(3.7), it follows that
∫h0|m∑j=0fm−j(px)fj(qx)Dp,qf(x)|dp,qx≤∫h0m∑j=0|f(px)|m−j|f(qx)|j|Dp,qf(x)|dp,qx≤∫h0m∑j=0gm−j(px)gj(qx)Dp,qg(x)dp,qx=∫h0Dp,qgm+1(x)dp,qx=gm+1(h). | (3.8) |
By using the H¨older's inequality and (3.8) with (3.2) for (p,q)-integral with indices m+1 and m+1m, we obtain
∫h0|m∑j=0fm−j(px)fj(qx)Dp,qf(x)|dp,qx≤gm+1(h)=[∫h0|Dp,qf(x)|dp,qx]m+1≤[(∫h0dp,qx)mm+1(∫h0|Dp,qf(x)|m+1dp,qx)1m+1]m+1=hm∫h0|Dp,qf(x)|m+1dp,qx, |
which proves the theorem.
Remark 3.1. If p=1 and m=1, then Theorem 3.1 reduces to Theorem 3.3 in [9]. If p=1 and m=α, then Theorem 3.1 reduces to Theorem 3.9 in [9]. Moreover, if p=1, m=1, and q→1−, then (3.1) reduces to (1.2) in Theorem 1.1.
Theorem 3.2. Let f∈C1[0,h] be such that f(t)>0 in (0,h). If f(0)=f(h)=0 and m∈N. Then, the following inequality holds :
∫h0|m∑j=0fm−j(px)fj(qx)||Dp,qf(x)|dp,qx≤(h1+q)m∫h0|Dp,qf(x)|m+1dp,qx. | (3.9) |
Proof. Let g(x) be as in (3.2) and w(x) be as follows
w(x)=∫hx|Dp,qf(t)|dp,qt. | (3.10) |
Then, we obtain by the condition f(h)=0 that
|Dp,qf(x)|=−Dp,qw(x), | (3.11) |
and for x∈[0,h], we have
|f(x)|=|∫hxDp,qf(t)dp,qt|≤∫hx|Dp,qf(t)|dp,qt=w(x), | (3.12) |
|f(px)|=|∫hpxDp,qf(t)dp,qt|≤∫hpx|Dp,qf(t)|dp,qt=w(px), | (3.13) |
and
|f(qx)|=|∫hqxDp,qf(t)dp,qt|≤∫hqx|Dp,qf(t)|dp,qt=w(qx). | (3.14) |
From (3.8), one has
∫h1+q0|m∑j=0fm−j(px)fj(qx)||Dp,qf(x)|dp,qx≤gm+1(h1+q). | (3.15) |
Similarly, by (3.10)–(3.14), we can get that
∫hh1+q|m∑j=0fm−j(px)fj(qx)||Dp,qf(x)|dp,qx≤−∫hh1+qm∑j=0wm−j(px)wj(qx)Dp,qw(x)dp,qx=wm+1(h1+q). | (3.16) |
Adding (3.15) to (3.16), we obtain
∫h0|m∑j=0fm−j(px)fj(qx)|Dp,qf(x)dp,qx≤gm+1(h1+q)+wm+1(h1+q). | (3.17) |
Using the H¨older's inequality with indices m+1 and m+1m, we can write that
gm+1(h1+q)=[∫h1+q0|Dp,qf(t)|dp,qt]m+1≤[(∫h1+q0dp,qt)mm+1(∫h1+q0|Dp,qf(t)|m+1dp,qt)1m+1]m+1=(h1+q)m∫hp+q0|Dp,qf(t)|m+1dp,qt. | (3.18) |
Similarly, we obtain
wm+1(h1+q)=[∫hh1+q|Dp,qf(t)|dp,qt]m+1≤[(∫hh1+qdp,qt)mm+1(∫hh1+q|Dp,qf(t)|m+1dp,qt)1m+1]m+1=(hq1+q)m∫hhp+q|Dp,qf(t)|m+1dp,qt≤(h1+q)m∫hhp+q|Dp,qf(t)|m+1dp,qt. | (3.19) |
Therefore, (3.17)–(3.19) imply that (3.9) holds.
Remark 3.2. If p=1 and m=1, then Theorem 3.2 reduces to Theorem 3.1 in [9]. In Theorem 3.2 if we take p=1, m=1, and q→1−, we recapture the inequality (1.1).
Theorem 3.3. Let m>0. Assume that μ(t) is a nonnegative and continuous function on [0,h], m∈N, and f∈C1[0,1] with f(0)=0. Then, the following inequality holds:
∫h0μ(x)|f(x)|m|Dp,qf(x)|dp,qx≤hm2m+1(∫h0μ(x)m+1mdp,qx)mm+1∫h0|Dp,qf(x)|m+1dp,qx. | (3.20) |
Proof. Using the H¨older's inequality for (p,q)-integral with indices m+1m and m+1, we obtain
∫h0μ(x)|f(x)|m|Dp,qf(x)|dp,qx≤(∫h0μ(x)m+1m|f(x)|m+1dp,qx)mm+1⋅(∫h0|Dp,qf(x)|m+1dp,qx)1m+1. | (3.21) |
By using (3.4) and from the H¨older's inequality for (p,q)-integral with indices m+1m and m+1, we get
∫h0μ(x)m+1m|f(x)|m+1dp,qx≤∫h0μ(x)m+1m[∫h0|Dp,qf(t)|dp,qt]m+1dp,qx≤∫h0μ(x)m+1mdp,qx⋅[(∫h0dp,qt)mm+1(∫h0|Dp,qf(t)|m+1dp,qt)1m+1]m+1=hm∫h0μ(x)m+1mdp,qx∫h0|Dp,qf(x)|m+1dp,qx. | (3.22) |
Substituting (3.22) into (3.21), we have
∫h0μ(t)|f(x)|m|Dp,qf(x)|dp,qx≤hm2m+1(∫h0μ(x)m+1mdp,qx)mm+1∫h0|Dp,qf(x)|m+1dp,qx. |
This completes the proof.
Theorem 3.4. If f and g are absolutely continuous functions on [0,h], and f(0)=g(0)=0, then
∫h0[f(px)Dp,qg(x)+g(qx)Dp,qf(x)]dp,qx≤h2∫h0[(Dp,qf(x))2+(Dp,qg(x))2]dp,qx. | (3.23) |
Proof. As in (3.4), we have
|f(x)|≤∫x0|Dp,qf(t)|dp,qt,|g(x)|≤∫x0|Dp,qg(t)|dp,qt. | (3.24) |
By using Lemma 2.1, Theorem 2.2, (3.24) and Cauchy inequality, we get
∫h0[f(px)Dp,qg(x)+g(qx)Dp,qf(x)]dp,qx |
=∫h0Dp,q(f(x)g(x))dp,qx=f(h)g(h)−f(0)g(0) |
=f(h)g(h)≤|f(h)||g(h)|≤∫h0|Dp,qf(x)|dp,qx⋅∫h0|Dp,qg(x)|dp,qx |
≤12[(∫h0|Dp,qf(x)|dp,qx)2+(∫h0|Dp,qg(x)|dp,qx)2] |
≤12[∫h0dp,qx⋅∫h0|Dp,qf(x)|2dp,qx+∫h0dp,qx⋅∫h0|Dp,qg(x)|2dp,qx] |
=h2∫h0[(Dp,qf(x))2+(Dp,qg(x))2]dp,qx. |
Theorem 3.5. Let m,r>0. Assume that f and g are absolutely continuous functions on [0,h], f(0)=g(0)=0, f(h)=g(h)=0, and f(t),g(t)>0 in (0,h). Then
∫h0|f(x)|m|g(x)|rdp,qx≤(h2)m+rm+r(m∫h0|Dp,qf(x)|m+rdp,qx+r∫h0|Dp,qg(x)|m+rdp,qx). | (3.25) |
Proof. Let
y(x)=∫x0|Dp,qf(t)|dp,qt,z(x)=∫hx|Dp,qf(t)|dp,qt,t∈[0,h]. | (3.26) |
Then, we have
|Dp,qf(x)|=Dp,qy(x)=−Dp,qz(x),|f(x)|=|∫x0Dp,qf(t)dp,qt|≤∫x0|Dp,qf(t)|dp,qt=y(x), | (3.27) |
and
|f(x)|=|∫hxDp,qf(t)dp,qt|≤∫hx|Dp,qf(t)|dp,qt=z(x). | (3.28) |
By (3.26)–(3.28), we obtain
|f(x)|≤y(x)+z(x)2=12∫h0|Dp,qf(t)|dp,qt. | (3.29) |
Similarly, we have
|g(x)|≤12∫h0|Dp,qg(t)|dp,qt. | (3.30) |
On the other hand, the following elementary inequality in [8] holds :
mAm+r+rBm+r−(m+r)AmBr≥0,A,B≥0,m,r>0. | (3.31) |
From (3.29)–(3.31), we get
|f(x)|m|g(x)|r≤(12)m+rm+r[m(∫h0|Dp,qf(t)|dp,qt)m+r+r(∫h0|Dp,qg(t)|dp,qt)m+r]. | (3.32) |
By using H¨older's inequality on the right side of (3.32) with indices m+rm+r−1, m+r, we obtain
|f(x)|m|g(x)|r≤(12)m+rm+r[mhm+r−1∫h0|Dp,qf(t)|m+rdp,qt+rhm+r−1∫h0|Dp,qg(t)|m+rdp,qt]. | (3.33) |
Integrating (3.33) on [0,h], we have
∫h0|f(x)|m|g(x)|rdp,qx≤(h2)m+rm+r(m∫h0|Dp,qf(t)|m+rdp,qt+r∫h0|Dp,qg(t)|m+rdp,qt), |
and the proof is completed.
Remark 3.3. If m=r>0 and f(x)=g(x), then the inequality (3.25) reduces to the following (p,q)-Wirtinger inequality:
∫h0|f(x)|2mdp,qx≤(h2)2m∫h0|Dp,qf(x)|2mdp,qx. |
In the following, we will give an example to illustrate our main result.
Example 4.1. Let p=23, q=12 and m=5. Set μ(t)=t, it is clear that μ(t) is a nonnegative and continuous function on [0,π2]. Set h=π2, and f(x)=sinx, then f∈C1[0,1] with f(0)=0. Thus, by Theorem 3.3, we have
∫π20x|sinx|5|D23,12sinx|d23,12x≤(π2)256(∫π20x65d23,12x)56∫π20|D23,12sinx|6d23,12x. |
It is known that (p,q)-calculus is a generalization of q-calculus. In this paper, we have established 5 new kinds of general Opial type integral inequalities in (p,q)-calculus. The methods we used to establish our results are quite simple and in virtue of some basic observations and applications of some fundamental inequalities and analysis technique. First, we investigated the Opial inequalities in (p,q)-calculus involving one function and its (p,q) derivative. Furthermore, Opial inequalities in (p,q)-calculus involving two functions and two functions with their (p,q) derivatives are given. We also discussed several particular cases. Our results are (p,q)-generalizations of Opial-type integral inequalities and (p,q)-Wirtinger inequality. An example is given to illustrate the effectiveness of our main result.
The authors thanks anonymous referees for their remarkable comments, suggestions which help to improve this paper. This work is supported by Natural Science Foundation of China (11571136).
The authors declare that there are no conflicts of interest.
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