Abstract
In this paper, we will state and prove some weighted dynamic inequalities of Opial-type involving integrals of powers of a function and of its derivative on time scales which not only extend some results in the literature but also improve some of them. The main results will be proved by using some algebraic inequalities, the Hölder inequality and a simple consequence of Keller’s chain rule on time scales. As special cases of the obtained dynamic inequalities, we will get some continuous and discrete inequalities.
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1 Introduction
In 1960, the Polish Mathematician Opial [36] proved an inequality involving integrals of functions and their derivatives;
where x is an absolutely differentiable continuous function on \([a,b]\), \(x(a)=x(b)=0\), \(x(t)>0\), and the constant \(\frac{b-a}{4}\) is sharp, in the sense that \(\frac{b-a}{4}\) cannot be replaced by a smaller constant.
Since the publication of the above result in 1960, numerous papers with new evidence, different speculations, and augmentations have showed up in the literature. Inequalities which involve integrals of functions and their derivatives are of great importance in mathematics with applications in the theory of differential equations, approximations and probability [1,2,3,4, 7, 17, 18, 21, 22, 28, 29, 34].
As a generalization of (1.1), Beesack [10] proved that: If x is an absolutely continuous function on \([a,b]\) with \(x(a)=0\), then
where r is a positive and continuous function with \(\int _{a}^{b} \frac{dt}{r(t)}<\infty \).
Yang [44] simplified the Beesack proof and extended the inequality (1.2) as follows: If x is an absolutely continuous function on \((a, b)\) with \(x(a)=0\), then
where r is a positive and continuous function with \(\int _{a}^{b} \frac{dt}{r(t)}<\infty \) and q is a positive, bounded, and nonincreasing function on \([a, b]\).
Recently, the theory of time scales, which has been initiated by Stefen Hilger in his Ph.D. thesis [30] in order to unify discrete and continuous analysis, has gained a lot of attention. During the previous decade, an impressive number of dynamic imbalances have been given by numerous creators who were inspired by certain applications (see [5, 6, 9, 12, 13, 16, 19, 20, 23,24,25,26,27, 31, 35, 37, 39, 41]). The general thought is to demonstrate a result for a dynamic inequality where the domain of the unknown function is a so-called time scale \(\mathbb{T}\), which is an arbitrary nonempty closed subset of real numbers. The three best-known time scales are \(\mathbb{T}=\mathbb{R}\), \(\mathbb{T}=\mathbb{Z}\) and \(\mathbb{T}=\overline{q^{\mathbb{Z}}}=\{q ^{z}:z\in \mathbb{Z}\}\cup \{0\}\) where \(q > 1\). The books [14] and [15] organize and summarize much of time scales calculus.
In [11], Bohner and Kaymakçalan introduced a dynamic Opial inequality which extended the continuous version inequality (1.1) to a general time scale and studied if \(x:[a,b] \cap \mathbb{T}\longrightarrow \mathbb{R}\) is delta differentiable with \(x(a)=0\), then
Dynamic Opial’s inequalities on time scales got a lot of consideration and numerous papers have been composed; see [11, 33, 38, 40, 42, 43] and the references cited therein.
Also in [11] the authors extended the inequality (1.3) of Yang and proved that: If r and q are positive rd-continuous functions on \([a,b]_{\mathbb{T}}\), \(\int _{a}^{b}\frac{\Delta t}{r(t)}< \infty \), q is nonincreasing and \(x:[a, b]\cap \mathbb{T}\longrightarrow \mathbb{R}\) is delta differentiable with \(x(a)=0\), then
Karpuz et al. [33] established the same inequality as in (1.5) by replacing \(q^{\sigma }\) with q of the form
where q is a positive rd-continuous function on \([a,b]_{\mathbb{T}}\), \(x:[a,b]\cap \mathbb{T}\longrightarrow \mathbb{R}\) is delta differentiable with \(x(a)=a\), and
For \(p\geq 1\), Karpuz and Özkan [32] proved that: If \(y:[a,\tau ]\cap \mathbb{T}\rightarrow \mathbb{R}^{+}\) is delta differentiable with \(y(a)=0\) and \(y^{\Delta }\) does not change sign in \((a,\tau )_{\mathbb{T}}\), then we have
where
p, q are positive real numbers such that \(p\geq 1\), and r, s are nonnegative rd-continuous functions on \((a,\tau )_{\mathbb{T}}\) such that \(\int _{a}^{\tau }r^{\frac{-1}{{p+q-1}}}(t)\Delta t<\infty \).
In the same paper, the authors proved that: If \(y:[\tau, b]\cap \mathbb{T}\rightarrow \mathbb{R}^{+}\) is delta differentiable with \(y(b)=0\) and \(y^{\Delta }\) does not change sign in \((\tau, b)_{ \mathbb{T}}\), then we have
where
p, q are positive real numbers such that \(p\geq 1\), and r, s are nonnegative rd-continuous functions on \((\tau, b)_{\mathbb{T}}\) such that \(\int _{\tau }^{b}r^{\frac{-1}{{p+q-1}}}(t)\Delta t<\infty \).
Adding (1.7) and (1.8), Karpus and Özkan proved that: If \(y:[a, b]\cap \mathbb{T}\rightarrow \mathbb{R}^{+}\) is delta differentiable with \(y(a)=y(b)=0\), then
where
For \(p\leq 1\), Karpuz and Özkan [32] proved that: If \(y:[a,\tau ]\cap \mathbb{T}\rightarrow \mathbb{R}^{+}\) is delta differentiable with \(y(a)=0\) and \(y^{\Delta }\) does not change sign in \((a,\tau )_{\mathbb{T}}\), then we have
where
p, q are positive real numbers such that \(p\leq 1\), \(p+q>1\) and r, s are nonnegative rd-continuous functions on \((a,\tau )_{ \mathbb{T}}\) such that \(\int _{a}^{\tau }r^{\frac{-1}{{p+q-1}}}(t) \Delta t<\infty \).
Also, in the same paper, the authors proved that: If \(y:[\tau, b] \cap \mathbb{T}\rightarrow \mathbb{R}^{+}\) is delta differentiable with \(y(b)=0\) and \(y^{\Delta }\) does not change sign in \((\tau, b)_{ \mathbb{T}}\), then we have
where
p, q are positive real numbers such that \(p\leq 1\), \(p+q>1\) and r, s are nonnegative rd-continuous functions on \((\tau, b)_{ \mathbb{T}}\) such that \(\int _{\tau }^{b}r^{\frac{-1}{{p+q-1}}}(t) \Delta t<\infty \).
Combining (1.10) and (1.11), Karpus and Özkan proved that: If \(y:[a, b]\cap \mathbb{T}\rightarrow \mathbb{R}^{+}\) is delta differentiable with \(y(a)=y(b)=0\), then
where
In this article, motivated by the above inequalities, we will explore some dynamic Opial-type inequalities on time scales, which generalize inequalities (1.7)–(1.12). After each result, we will study the special cases when \(\mathbb{T}=\mathbb{R}\) and \(\mathbb{T} = \mathbb{N}\) to obtain some continuous and discrete results.
2 Basics of time scales
Firstly, we recall some essentials of time scales, and some universal symbols that will be used in the present paper. From now on, \(\mathbb{R}\) and \(\mathbb{Z}\) are the set of real numbers and the set of integers, respectively.
A time scale \(\mathbb{T}\) is an arbitrary nonempty closed subset of the set of real numbers \(\mathbb{R}\). Throughout the article, we assume that \(\mathbb{T}\) has the topology that it inherits from the standard topology on \(\mathbb{R}\). We define the forward jump operator \(\sigma: \mathbb{T}\rightarrow \mathbb{T}\) for any \(t\in \mathbb{T}\) by
and the backward jump operator \(\rho: \mathbb{T}\rightarrow \mathbb{T}\) for any \(t\in \mathbb{T}\) by
In the preceding two definitions, we set \(\inf \emptyset =\sup \mathbb{T}\) (i.e., if t is the maximum of \(\mathbb{T}\), then \(\sigma (t)=t\)) and \(\sup \emptyset =\inf \mathbb{T}\) (i.e., if t is the minimum of \(\mathbb{T}\), then \(\rho (t)=t\)), where ∅ denotes the empty set.
A point \(t\in \mathbb{T}\) with \(\inf \mathbb{T}< t<\sup \mathbb{T}\) is said to be right-scattered if \(\sigma (t)>t\), right-dense if \(\sigma (t)=t\), left-scattered if \(\rho (t)< t\), and left-dense if \(\rho (t)=t\). Points that are simultaneously right-dense and left-dense are said to be dense points. Points that are simultaneously right-scattered and left-scattered are said to be isolated points.
The forward graininess function \(\mu:\mathbb{T}\rightarrow [0,\infty )\) is defined for any \(t \in \mathbb{T}\) by \(\mu (t):= \sigma (t)-t\) and the backward graininess function \(\nu:\mathbb{T}\rightarrow [0, \infty )\) is defined for any \(t \in \mathbb{T}\) by \(\nu (t):= t- \rho (t)\).
If \(f: \mathbb{T} \rightarrow \mathbb{R}\) is a function, then the function \(f^{\sigma }: \mathbb{T} \rightarrow \mathbb{R}\) is defined by \(f^{\sigma }(t)=f(\sigma (t)), \forall t\in \mathbb{T}\), that is, \(f^{\sigma }=f \circ \sigma \). Similarly, the function \(f^{\rho }: \mathbb{T} \rightarrow \mathbb{R}\) is defined by \(f^{\rho }(t)=g( \rho (t)), \forall t\in \mathbb{T}\), that is, \(f^{\rho }=f\circ \rho \).
The sets \(\mathbb{T}^{\kappa }\), \(\mathbb{T}_{\kappa }\) and \(\mathbb{T}_{\kappa }^{\kappa }\) are introduced as follows: If \(\mathbb{T}\) has a left-scattered maximum \(t_{1}\), then \(\mathbb{T} ^{\kappa } = \mathbb{T}-\{t_{1}\}\), otherwise \(\mathbb{T}^{\kappa } = \mathbb{T}\). If \(\mathbb{T}\) has a right-scattered minimum \(t_{2}\), then \(\mathbb{T}^{\kappa } = \mathbb{T}-\{t_{2}\}\), otherwise \(\mathbb{T}_{\kappa } = \mathbb{T}\). Finally, we have \(\mathbb{T}^{ \kappa }_{\kappa }=\mathbb{T}^{\kappa }\cap \mathbb{T}_{\kappa }\).
The interval \([a,b]\) in \(\mathbb{T}\) is defined by
We define the open intervals and half-closed intervals similarly.
Assume \(f: \mathbb{T} \rightarrow \mathbb{R}\) is a function and \(t\in \mathbb{T}^{\kappa }\). Then \(f^{\Delta }(t)\in \mathbb{R}\) is said to be the delta derivative of f at t if for any \(\varepsilon > 0\) there exists a neighborhood U of t such that, for every \(s\in U\), we have
Moreover, f is said to be delta differentiable on \(\mathbb{T}^{ \kappa }\) if it is delta differentiable at every \(t\in \mathbb{T}^{ \kappa }\).
Similarly, we say that \(f^{\nabla }(t)\in \mathbb{R}\) is the nabla derivative of f at t if for any \(\varepsilon > 0\) there exists a neighborhood V of t such that for all \(s\in V\)
Furthermore, f is said to be nabla differentiable on \(\mathbb{T} _{\kappa }\) if it is nabla differentiable at each \(t\in \mathbb{T} _{\kappa }\).
A function \(f: \mathbb{T}\rightarrow \mathbb{R}\) is said to be right-dense continuous (rd-continuous) if f is continuous at all right-dense points in \(\mathbb{T}\) and its left-sided limits exist at all left-dense points in \(\mathbb{T}\).
In a similar manner, a function \(f: \mathbb{T}\rightarrow \mathbb{R}\) is said to be left-dense continuous (ld-continuous) if f is continuous at all left-dense points in \(\mathbb{T}\) and its right-sided limits exist at all right-dense points in \(\mathbb{T}\).
The delta integration by parts on time scales is given by the following formula:
whereas the nabla integration by parts on time scales is given by
We will use the following crucial relations between calculus on time scales \(\mathbb{T}\) and either differential calculus on \(\mathbb{R}\) or difference calculus on \(\mathbb{Z}\). Note that:
- (i)
If \(\mathbb{T}=\mathbb{R}\), then
$$ \begin{aligned} &\sigma (t)=\rho (t)=t, \\ &\mu (t)=\nu (t)=0, \\ &f^{\Delta }(t)=f^{\nabla }(t)=f'(t), \\ &\int _{a}^{b}f(t)\Delta t= \int _{a}^{b}f(t)\nabla t= \int _{a}^{b}f(t)\,dt. \end{aligned} $$(2.4) - (ii)
If \(\mathbb{T}=\mathbb{Z}\), then
$$ \begin{aligned} &\sigma (t)=t+1, \\ &\rho (t)=t-1, \\ &\mu (t)=\nu (t)=1, \\ &f^{\Delta }(t)=\Delta f(t), \\ &f^{\nabla }(t)=\nabla f(t), \\ &\int _{a}^{b}f(t)\Delta t=\sum _{t=a}^{b-1}f(t), \\ &\int _{a}^{b}f(t)\nabla t=\sum _{t=a+1}^{b}f(t), \end{aligned} $$(2.5)where Δ and ∇ are the forward and backward difference operators, respectively.
3 Main results
In this section, we will state and prove our main results.
First, we present the basic theorems that will be needed in the proof of our main results.
Theorem 3.1
(Chain rule on time scales [14])
Assume \(g:\mathbb{R}\rightarrow \mathbb{R}\) is continuous, \(g: \mathbb{T}\rightarrow \mathbb{R}\) is delta differentiable on \(\mathbb{T^{\kappa }}\), and \(f:\mathbb{R}\rightarrow \mathbb{R}\) is continuously differentiable. Then there exists \(c\in [t,\sigma (t)]\) with
Theorem 3.2
(Chain rule on time scales [14])
Let \(f: \mathbb{R}\rightarrow \mathbb{R}\) be continuously differentiable and suppose \(g: \mathbb{T}\rightarrow \mathbb{R}\) is delta differentiable. Then \(f\circ g: \mathbb{T}\rightarrow \mathbb{R}\) is delta differentiable and the formula
holds.
Theorem 3.3
(Dynamic Hölder inequality [14])
Let \(a,b\in \mathbb{T}\) and f, \(g\in C_{rd}([a,b]_{\mathbb{T}},[0, \infty ))\). If p, \(q>1\) with \(\frac{1}{p}+\frac{1}{q}=1\), then
Also, the main results here will be proved by employing the inequalities (see [8], page 51)
Next, we enlist the following assumptions for the proofs of our main results:
- (A1)
\(\mathbb{T}\) be a time scale with (i) \(a,\tau \in \mathbb{T}\); (ii) \(\tau,b\in \mathbb{T}\); (iii) \(a,b\in \mathbb{T}\).
- (A2)
p, q be positive real numbers such that (i) \(p\geq 1\); (ii) \(p\leq 1\); (iii) \(p+q>1\).
- (A3)
r, s be nonnegative rd-continuous functions on (i) \((a,\tau )_{\mathbb{T}}\) provided that \(\int _{a}^{\tau }r^{\frac{-1}{ {p+q-1}}}(t)\Delta t<\infty \); (ii) \((\tau,b)_{\mathbb{T}}\) such that \(\int _{\tau }^{b}r^{\frac{-1}{ {p+q-1}}}(t)\Delta t<\infty \); (iii) \((a,b)_{\mathbb{T}}\) with \(\int _{a}^{b}r^{\frac{-1}{{p+q-1}}}(t) \Delta t<\infty \).
- (A4)
\(y:[a,\tau ]\cap \mathbb{T}\rightarrow \mathbb{R}^{+}\) is delta differentiable with \(y(a)\neq 0\) and \(y^{\Delta }\) does not change sign in \((a,\tau )_{\mathbb{T}}\).
- (A5)
\(y:[\tau,b]\cap \mathbb{T}\rightarrow \mathbb{R}^{+}\) be delta differentiable such that \(y(b)\neq 0\) and \(y^{\Delta }\) does not change sign in \((\tau,b)_{\mathbb{T}}\).
- (A6)
\(y:[a,b]\cap \mathbb{T}\rightarrow \mathbb{R}^{+}\) is delta differentiable and \(y(a)\neq 0\), \(y(b)\neq 0\). Also \(y^{\Delta }\) does not change sign in \((a,b)_{\mathbb{T}}\).
- (A7)
\(\{r_{i}\}_{0\leq i\leq N}\) and \(\{s_{i}\}_{0\leq i \leq N}\) are nonnegative real sequences.
- (A8)
\(\{y_{i}\}_{0\leq i\leq N}\) is a sequence of real numbers with (i) \(y(a)\neq 0\); (ii) \(y(b)\neq 0\); (iii) \(y(a)\neq 0\) and \(y(b)\neq 0\).
Now, we are ready to state and prove the first result, which generalizes many inequalities in the literature.
Theorem 3.4
Let (A1)(i), (A2)(i), (A3)(i) and (A4) be satisfied.
- (a)
Then
$$\begin{aligned} & \int _{a}^{\tau }s(x) \bigl\vert y(x)+y^{\sigma }(x) \bigr\vert ^{p} \bigl\vert y^{\Delta }(x) \bigr\vert ^{q} \Delta x \\ &\quad \leq K_{5}(a,\tau,p,q) \int _{a}^{\tau }r(x) \bigl\vert y^{\Delta }(x) \bigr\vert ^{p+q} \Delta x +2^{3p-2} \bigl\vert y(a) \bigr\vert ^{p} \int _{a}^{\tau }s(x) \bigl\vert y^{\Delta }(x) \bigr\vert ^{q} \Delta x, \end{aligned}$$(3.6)where
$$\begin{aligned} &K_{5}(a,\tau,p,q) \\ &\quad =2^{3p-2} \biggl(\frac{q}{p+q} \biggr)^{\frac{q}{p+q}} \biggl[ \int _{a}^{\tau }s^{\frac{p+q}{p}}(x) \biggl( \frac{1}{r(x)} \biggr)^{ \frac{q}{p}} \biggl( \int _{a}^{x}\frac{1}{r^{\frac{1}{p+q-1}}(t)}\Delta t \biggr)^{p+q-1}\Delta x \biggr]^{\frac{p}{p+q}} \\ &\qquad{}+2^{p-1}\max_{a\leq x\leq \tau } \biggl(\frac{\mu ^{p}(x)s(x)}{r(x)} \biggr). \end{aligned}$$ - (b)
If \(r=s\), then
$$\begin{aligned} & \int _{a}^{\tau }r(x) \bigl\vert y(x)+y^{\sigma }(x) \bigr\vert ^{p} \bigl\vert y^{\Delta }(x) \bigr\vert ^{q} \Delta x \\ &\quad \leq K_{6}(a,\tau,p,q) \int _{a}^{\tau }r(x) \bigl\vert y^{\Delta }(x) \bigr\vert ^{p+q} \Delta x +2^{3p-2} \bigl\vert y(a) \bigr\vert ^{p} \int _{a}^{\tau }r(x) \bigl\vert y^{\Delta }(x) \bigr\vert ^{q} \Delta x, \end{aligned}$$(3.7)where
$$\begin{aligned} K_{6}(a,\tau,p,q) ={}&2^{3p-2} \biggl( \frac{q}{p+q} \biggr)^{\frac{q}{p+q}} \biggl[ \int _{a}^{\tau } r(x) \biggl( \int _{a}^{x}\frac{1}{r^{ \frac{1}{p+q-1}}(t)}\Delta t \biggr)^{p+q-1}\Delta x \biggr]^{ \frac{p}{p+q}} \\ &{}+2^{p-1}\max_{a\leq x\leq \tau } \bigl(\mu ^{p}(x) \bigr). \end{aligned}$$(3.8) - (c)
Let \(r=1\). Then
$$\begin{aligned} & \int _{a}^{\tau } \bigl\vert y(x)+y^{\sigma }(x) \bigr\vert ^{p} \bigl\vert y^{\Delta }(x) \bigr\vert ^{q}\Delta x \\ &\quad\leq K_{7}(a,\tau,p,q) \int _{a}^{\tau } \bigl\vert y^{\Delta }(x) \bigr\vert ^{p+q}\Delta x +2^{3p-2} \bigl\vert y(a) \bigr\vert ^{p} \int _{a}^{\tau } \bigl\vert y^{\Delta }(x) \bigr\vert ^{q}\Delta x, \end{aligned}$$(3.9)where
$$ K_{7}(a, \tau, p, q)= \biggl(2^{3p-2}\frac{q^{\frac{q}{p+q}}}{p+q}( \tau -a)^{p} +2 ^{p-1}\max_{a\leq x\leq \tau } \bigl(\mu ^{p}(x) \bigr) \biggr). $$
Proof
(a) Since \(y^{\Delta }\) does not change sign in \((a,\tau )_{ \mathbb{T}}\), we have
From (3.10), we get
Now, since r is nonnegative on \((a,\tau )_{\mathbb{T}}\), it follows from the Hölder inequality (3.3) with indices \(\frac{p+q}{p+q-1}\) and \(p+q\), and with
that
Since \(p\geq 1\), by taking the power p for both sides of (3.11), we have
Applying the inequality (3.4) on the right-hand side of (3.12), we deduce
Since \(y^{\sigma }=y +\mu y^{\Delta }\), we have
Obviously, \(p\geq 1\). Taking power p for both sides of (3.13) and using the inequality (3.4), we deduce
Setting
we see that \(z(a)=0\), and
From (3.16), we get
From (3.14), (3.17) and since s is nonnegative on \((a, \tau )_{\mathbb{T}}\), we have
Integrating the above inequality from a to τ, we get
By applying Hölder inequality (3.3) with \((p+q)/p\) and \((p+q)/q\) on the right side of integral of the above inequality, we have
From (3.1), we obtain
Since \(z^{\Delta }(x) \geq 0\) and \(x\leq c\), we get
Substituting (3.19) into (3.18) and since \(z(a)=0\), we have
The above inequality, (3.15) and (3.16) imply that
which is the desired inequality (3.6).
(b) The proof follows from (a) by setting \(r=s\).
(c) It is noted from the chain rule on time scales (3.2) that
so that
From (3.7) and (3.8) (by taking \(r(t)=1\)) and using (3.20), we get
which is the desired inequality (3.9). This completes the proof. □
Based on Theorem 3.4, we obtain the following result by replacing \([a, \tau ]_{\mathbb{T}}\) by \([\tau, b]_{\mathbb{T}}\) and \(\vert y(x) \vert =\int _{x}^{b} \vert y^{\Delta }(t) \vert \Delta t + |y(b)|\).
Theorem 3.5
Let (A1)(ii), (A2)(i), (A3)(ii) and (A5) hold. Then
where
Let \(K^{\star }_{2}(p, q)=K_{7}(a, \tau, p, q)=K_{8}(\tau, b, p, q) < \infty \) such that \(K_{7}(a, \tau, p, q)\) and \(K_{8}(\tau, b, p, q)\) are given in Theorems 3.4 and 3.5 and τ is the unique solution of the equation \(K_{7}(a, \tau, p, q)= K_{8}(\tau, b, p, q)\). Therefore,
So combining Theorems 3.4 and 3.5 gives the following result.
Theorem 3.6
Let (A1)(iii), (A2)(i), (A3)(iii) and (A6) be fulfilled.
- (a)
Then
$$\begin{aligned} & \int _{a}^{b}s(x) \bigl\vert y(x)+y^{\sigma }(x) \bigr\vert ^{p} \bigl\vert y^{\Delta }(x) \bigr\vert ^{q}\Delta x \\ &\quad \leq K^{\star }_{2}(p,q) \int _{a}^{b}r(x) \bigl\vert y^{\Delta }(x) \bigr\vert ^{p+q} \Delta x \\ &\qquad{} +2^{3p-2} \bigl( \bigl\vert y(a) \bigr\vert ^{p}+ \bigl\vert y(b) \bigr\vert ^{p} \bigr)\int _{a}^{b}s(x) \bigl\vert y^{\Delta }(x) \bigr\vert ^{q}\Delta x. \end{aligned}$$(3.22) - (b)
By applying (3.9) for \([a, \tau ]\) and \([\tau, b]\) and choosing \(\tau =\frac{a+b}{2}\in \mathbb{T}\), therefore
$$\begin{aligned} & \int _{a}^{b} \bigl\vert y(x)+y^{\sigma }(x) \bigr\vert ^{p} \bigl\vert y^{\Delta }(x) \bigr\vert ^{q}\Delta x \\ &\quad \leq K_{9}(a, b, p, q) \int _{a}^{b} \bigl\vert y^{\Delta }(x) \bigr\vert ^{p+q}\Delta x \\ &\qquad{}+2^{3p-2} \bigl( \bigl\vert y(a) \bigr\vert ^{p}+ \bigl\vert y(b) \bigr\vert ^{p} \bigr) \int _{a}^{b} \bigl\vert y^{\Delta }(x) \bigr\vert ^{q} \Delta x, \end{aligned}$$(3.23)where
$$ K_{9}(a, b, p, q)= \biggl(2^{3p-2}\frac{q^{\frac{q}{p+q}}}{p+q} \biggl( \frac{b-a}{2} \biggr)^{p} + 2^{p-1}\max _{a\leq x\leq \tau } \bigl(\mu ^{p}(x) \bigr) \biggr). $$ - (c)
Setting \(p=q=1\) in (3.23), hence
$$\begin{aligned} & \int _{a}^{b} \bigl\vert y(x)+y^{\sigma }(x) \bigr\vert \bigl\vert y^{\Delta }(x) \bigr\vert \Delta x \\ &\quad \leq \biggl(\frac{b-a}{2}+\max_{a\leq x\leq b}\mu (x) \biggr) \int _{a}^{b} \bigl\vert y^{\Delta }(x) \bigr\vert ^{2}\Delta x \\ &\qquad{}+2 \bigl( \bigl\vert y(a) \bigr\vert + \bigl\vert y(b) \bigr\vert \bigr) \int _{a}^{b} \bigl\vert y^{\Delta }(x) \bigr\vert \Delta x. \end{aligned}$$(3.24)
Listed below are some remarks on particular cases of Theorem 3.4, Theorem 3.5 and Theorem 3.6:
Remark 3.7
If we take \(y(a)=0\), the inequality (3.6) reduces to the inequality (1.7).
Remark 3.8
If we take \(y(a)=0\) and \(r=s\), the inequality (3.7) reduces to the inequality [6, (3.3.16), page 126].
Remark 3.9
The inequality (3.9) changes to the inequality [6, (3.3.19), page 126] by putting \(y(a)=0\) and \(r=s=1\).
Remark 3.10
If we take \(y(b)=0\), the inequality (3.21) reduces to the inequality (1.8).
Remark 3.11
If we take \(y(a)=0\) and \(y(b)=0\), the inequality (3.22) reduces to the inequality (1.9).
Remark 3.12
If we take \(y(a)=0\) and \(y(b)=0\), \(r=s=1\) and choose \(\tau = \frac{a+b}{2}\in \mathbb{T}\), the inequality (3.23) reduces to the inequality [6, (3.3.20), page 126].
Remark 3.13
If we take \(y(a)=0\) and \(y(b)=0\) the inequality (3.24) reduces to the inequality [6, (3.3.21), page 127].
Now, we give some integral and discrete inequalities as special cases from Theorems 3.4, 3.5 and 3.6, respectively:
Corollary 3.14
When \(\mathbb{T}=\mathbb{R}\) in Theorem 3.4, and using Eqs. (2.4), the inequality (3.6) reduces to
where
Corollary 3.15
When \(\mathbb{T}=\mathbb{R}\), in Theorem 3.5, and using Eqs. (2.4), the inequality (3.21) reduces to
where
Corollary 3.16
When \(\mathbb{T}=\mathbb{R}\), in Theorem 3.6, and using Eqs. (2.4), the inequality (3.22) reduces to
where \(K^{\star }_{3}(p, q)=K_{10}(a, \tau, p, q)=K_{11}(\tau, b, p, q) < \infty \) such that \(K_{10}(a, \tau, p, q)\) and \(K_{11}(\tau, b, p, q)\) are given in Corollaries 3.14 and 3.15 and τ is the unique solution of the equation \(K_{10}(a, \tau, p, q)=K _{11}(\tau, b, p, q)\).
Corollary 3.17
If \(\mathbb{T}=\mathbb{N}\) in Theorem 3.4 and (A2)(i), (A7), (A8)(i) are satisfied, and using Eqs. (2.5), then
where
Corollary 3.18
If \(\mathbb{T}=\mathbb{N}\) in Theorem 3.5 and (A2)(i), (A7), (A8)(ii) hold, and using Eqs. (2.5), then
where
Corollary 3.19
If \(\mathbb{T}=\mathbb{N}\) in Theorem 3.6, and (A2)(i), (A7), (A8)(iii) are satisfied, and using Eqs. (2.5), then
where \(K^{\star }_{4}(p, q)=K_{12}(a, \tau, p, q)=K_{13}(\tau, b, p, q) < \infty \) such that \(K_{12}(a, \tau, p, q)\) and \(K_{13}(\tau, b, p, q)\) are given in Corollaries 3.17 and 3.18 and τ is the unique solution of the equation \(K_{12}(a, \tau, p, q)=K _{13}(\tau, b, p, q)\).
Now we study the case of some weighted dynamic Opial inequalities on time scales of the type
where p, q be positive real numbers such that \(p\leq 1\), \(p+q>1\).
Our next results, which will be proved by using inequality (3.5), generalize the inequalities (1.10), (1.11) and (1.12).
Theorem 3.20
Assume (A1)(i), (A2)((ii), (iii)), (A3)(i) and (A4) are fulfilled.
- (a)
Then
$$\begin{aligned} & \int _{a}^{\tau }s(x) \bigl\vert y(x)+y^{\sigma }(x) \bigr\vert ^{p} \bigl\vert y^{\Delta }(x) \bigr\vert ^{q} \Delta x \\ &\quad \leq K_{14}(a,\tau,p,q) \int _{a}^{\tau }r(x) \bigl\vert y^{\Delta }(x) \bigr\vert ^{p+q} \Delta x +2^{p} \bigl\vert y(a) \bigr\vert ^{p} \int _{a}^{\tau }s(x) \bigl\vert y^{\Delta }(x) \bigr\vert ^{q} \Delta x, \end{aligned}$$(3.25)where
$$\begin{aligned} &K_{14}(a,\tau,p,q) \\ &\quad=2^{p} \biggl(\frac{q}{p+q} \biggr)^{\frac{q}{p+q}} \biggl[ \int _{a}^{\tau }s^{\frac{p+q}{p}}(x) \biggl( \frac{1}{r(x)} \biggr)^{ \frac{q}{p}} \biggl( \int _{a}^{x}\frac{1}{r^{\frac{1}{p+q-1}}(t)}\Delta t \biggr)^{p+q-1}\Delta x \biggr]^{\frac{p}{p+q}} \\ &\qquad{}+\max_{a\leq x\leq \tau } \biggl(\frac{\mu ^{p}(x)s(x)}{r(x)} \biggr). \end{aligned}$$ - (b)
For \(r=s\), we obtain
$$\begin{aligned} & \int _{a}^{\tau }r(x) \bigl\vert y(x)+y^{\sigma }(x) \bigr\vert ^{p} \bigl\vert y^{\Delta }(x) \bigr\vert ^{q} \Delta x \\ &\quad \leq K_{15}(a,\tau,p,q) \int _{a}^{\tau }r(x) \bigl\vert y^{\Delta }(x) \bigr\vert ^{p+q} \Delta +2^{p} \bigl\vert y(a) \bigr\vert ^{p} \int _{a}^{\tau }r(x) \bigl\vert y^{\Delta }(x) \bigr\vert ^{q}\Delta x, \end{aligned}$$(3.26)where
$$\begin{aligned} K_{15}(a,\tau,p,q) ={}&2^{p} \biggl( \frac{q}{p+q} \biggr)^{\frac{q}{p+q}} \biggl[ \int _{a}^{\tau } r(x) \biggl( \int _{a}^{x}\frac{1}{r^{ \frac{1}{p+q-1}}(t)}\Delta t \biggr)^{p+q-1}\Delta x \biggr]^{ \frac{p}{p+q}} \\ &{}+\max_{a\leq x\leq \tau } \bigl(\mu ^{p}(x) \bigr). \end{aligned}$$(3.27) - (c)
Setting \(r=1\) in (3.26) and (3.27), then
$$\begin{aligned} & \int _{a}^{\tau } \bigl\vert y(x)+y^{\sigma }(x) \bigr\vert ^{p} \bigl\vert y^{\Delta }(x) \bigr\vert ^{q}\Delta x \\ &\quad \leq K_{16}(a,\tau,p,q) \int _{a}^{\tau } \bigl\vert y^{\Delta }(x) \bigr\vert ^{p+q} \Delta +2^{p} \bigl\vert y(a) \bigr\vert ^{p} \int _{a}^{\tau } \bigl\vert y^{\Delta }(x) \bigr\vert ^{q}\Delta x, \end{aligned}$$(3.28)where
$$ K_{16}(a, \tau, p, q)= \biggl(2^{p}\frac{q^{\frac{q}{p+q}}}{p+q}( \tau -a)^{p} +\max_{a\leq x\leq \tau } \bigl(\mu ^{p}(x) \bigr) \biggr). $$
Proof
(a) Since \(y^{\Delta }(t)\) does not change sign in \((a,\tau )_{ \mathbb{T}}\), we have
From (3.29), we get
Now, since r is nonnegative on \((a,\tau )_{\mathbb{T}}\), then it follows from the Hölder inequality (3.3) with indices \(\frac{p+q}{p+q-1}\) and \(p+q\), and with
that
Since \(p\leq 1\), by taking the power p for both sides of (3.30), we have
Applying the inequality (3.5) on the right-hand side of (3.31), we deduce
Since \(y^{\sigma }=y + \mu y^{\Delta }\), we have
Since \(p\leq 1\), by taking the power p of both sides of (3.32) and applying again the inequality (3.5), we deduce
Setting
using the fact that \(z(a)=0\), and
This implies
From (3.33) and (3.36), since s is nonnegative on \((a, \tau )_{\mathbb{T}}\), we have
Integrating the above inequality from a to τ, we get
Applying the Hölder inequality (3.3), with indices \((p+q)/p\) and \((p+q)/q\) on the first integral of the right-hand side of the above inequality, we have
From the chain rule (3.1), we obtain
Since \(z^{\Delta }(x) \geq 0\) and \(x\leq c\), we get
Substituting (3.38) into (3.37) and by \(z(a)=0\), we have
The last inequality, (3.34) and (3.35) imply that
which is the required inequality (3.25).
The proof of (b) and (c) follows by a similar argument to the proof of (a) with suitable changes. This completes the proof. □
Based on Theorem 3.20, we obtain the following result by replacing \([a, \tau ]_{\mathbb{T}}\) by \([\tau, b]_{\mathbb{T}}\) and \(\vert y(x) \vert =\int _{x}^{b} \vert y^{\Delta }(t) \vert \Delta t + |y(b)|\).
Theorem 3.21
Assume (A1)(ii), (A2)((ii), (iii)), (A3)(ii), and (A5) are satisfied. Then we have
where
In the following, we assume that \(K^{\star }_{5}(p,q)=K_{14}(a, \tau, p, q)=K_{17}(\tau, b, p, q) < \infty \), where \(K_{14}(a, \tau, p, q)\) and \(K_{17}(\tau, b, p, q)\) are defined as in Theorems 3.20 and 3.21 and τ is the unique solution of the equation \(K_{14}(a, \tau, p, q)= K_{17}(\tau, b, p, q)\). Therefore,
So combining Theorems 3.20 and 3.21 gives the following result.
Theorem 3.22
Assume (A1)(iii), (A2)((ii), (iii)), (A3)(iii), and (A6) are satisfied.
- (a)
Then
$$\begin{aligned} & \int _{a}^{b}s(x) \bigl\vert y(x)+y^{\sigma }(x) \bigr\vert ^{p} \bigl\vert y^{\Delta }(x) \bigr\vert ^{q}\Delta x \\ &\quad\leq K^{\star }_{5}(p, q) \int _{a}^{b}r(x) \bigl\vert y^{\Delta }(x) \bigr\vert ^{p+q} \Delta x +2^{p} \bigl\vert y(a)+y(b) \bigr\vert ^{p} \int _{a}^{b}s(x) \bigl\vert y^{\Delta }(x) \bigr\vert ^{q} \Delta x. \end{aligned}$$(3.40) - (b)
Let \(\tau =\frac{a+b}{2}\in \mathbb{T}\) and apply (3.28) to \([a, \tau ]\) and \([\tau, b]\). Then
$$\begin{aligned} & \int _{a}^{b} \bigl\vert y(x)+y^{\sigma }(x) \bigr\vert ^{p} \bigl\vert y^{\Delta }(x) \bigr\vert ^{q}\Delta x \\ &\quad \leq K_{18}(a, b, p, q) \int _{a}^{b} \bigl\vert y^{\Delta }(x) \bigr\vert ^{p+q}\Delta +2^{p} \bigl( \bigl\vert y(a) \bigr\vert ^{p}+|y(b)^{p} \bigr) \int _{a}^{b} \bigl\vert y^{\Delta }(x) \bigr\vert ^{q}\Delta x, \end{aligned}$$(3.41)where
$$ K_{18}(a, b, p, q)= \biggl(\frac{q^{\frac{q}{p+q}}}{p+q}(b-a)^{p} + \max_{a\leq x\leq \tau } \bigl(\mu ^{p}(x) \bigr) \biggr). $$
Listed below are some remarks on particular cases of Theorem 3.20, Theorem 3.21 and Theorem 3.22:
Remark 3.23
If we take \(y(a)=0\), the inequality (3.25) reduces to the inequality (1.10).
Remark 3.24
If we take \(y(b)=0\), the inequality (3.39) reduces to the inequality (1.11).
Remark 3.25
If we take \(y(a)=0\) and \(y(b)=0\), the inequality (3.40) reduces to the inequality (1.12).
Remark 3.26
If we take \(y(a)=0\) and \(r=s\), the inequality (3.25) reduces to the inequality [6, (3.3.32), page 130].
Remark 3.27
If we take \(y(a)=0\) and \(r=s=1\), the inequality (3.25) reduces to the inequality [6, (3.3.35), page 130].
Remark 3.28
If we take \(y(a)=0\) and \(y(b)=0\), \(r=s=1\) and choose \(\tau = \frac{(a+b)}{2}\), the inequality (3.41) reduces to the inequality [6, (3.3.36), page 131].
Now, we give some integral and discrete inequalities as special cases from Theorems 3.20, 3.21 and 3.22, respectively:
Corollary 3.29
When \(\mathbb{T}=\mathbb{R}\) in Theorem 3.20, and using Eqs. (2.4), the inequality (3.25) reduces to
where
Corollary 3.30
When \(\mathbb{T}=\mathbb{R}\) in Theorem 3.21, and using Eqs. (2.4), the inequality (3.39) reduces to
where
Corollary 3.31
When \(\mathbb{T}=\mathbb{R}\) in Theorem 3.22, and using Eqs. (2.4), the inequality (3.40) reduces to
where \(K^{\star }_{6}(p, q)=K_{19}(a, \tau, p, q)=K_{20}(\tau, b, p, q) < \infty \) such that \(K_{19}(a, \tau, p, q)\) and \(K_{20}(\tau, b, p, q)\) are given in Corollaries 3.29 and 3.30 and τ is the unique solution of the equation \(K_{19}(a, \tau, p, q)=K _{20}(\tau, b, p, q)\).
Corollary 3.32
If \(\mathbb{T}=\mathbb{N}\) in Theorem 3.20, and (A2)(i), (A7), (A8)(i) are satisfied, and using Eqs. (2.5), then
where
Corollary 3.33
If \(\mathbb{T}=\mathbb{N}\) in Theorem 3.21, and (A2)(ii), (A7), (A8)(ii) are satisfied, and using Eqs. (2.5), then
where
Corollary 3.34
If \(\mathbb{T}=\mathbb{N}\) in Theorem 3.22 and (A2)(ii), (A7), (A8)(iii) are satisfied, and using Eqs. (2.5), then
where \(K^{\star }_{7}(p, q)=K_{21}(a, \tau, p, q)=K_{22}(\tau, b, p, q) < \infty \) such that \(K_{21}(a, \tau, p, q)\) and \(K_{22}(\tau, b, p, q)\) are given in Corollaries 3.32 and 3.33 and τ is the unique solution of the equation \(K_{21}(a, \tau, p, q)=K _{22}(\tau, b, p, q)\).
4 Conclusion
In this article, we obtained some weighted dynamic inequalities of Opial-type involving integrals of powers of a function and of its derivative on time scales which not only extend some results in the literature but also improve some of them. Furthermore, we got some continuous and discrete inequalities as special cases of the obtained dynamic inequalities.
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El-Deeb, A.A., Kh, F.M., Ismail, G.A.F. et al. Weighted dynamic inequalities of Opial-type on time scales. Adv Differ Equ 2019, 393 (2019). https://doi.org/10.1186/s13662-019-2325-8
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DOI: https://doi.org/10.1186/s13662-019-2325-8