New Principles of Non-Linear Integral Inequalities on Time Scales

For a quarter century, the theory of time scales has played an important role in the representation of differential calculus and integral inequalities. The concept of time scales was introduced by Stefan Hilger in 1988 [1]. Later, this theory was studied by many authors. They have demonstrated various aspects of integral inequalities [2,3,4,5,6,7,8,9,10,11,12,13]. Dynamic equations and inequalities have many applications to quantum mechanics, phsical problems, wave equations, heat transfer and economic problems [26, 27, 28, 29]. For example; Aly R. Seadawy et al. have done a lot of research on the applications of dynamic equations in physics. As a result of these studies, they achieved good results [30]. The most important examples of time scale studies are differential calculus and inequalities [12]. Wong et al. [6, 7] expressed some time scale integral inequalities. Yang [13] obtained a generalization of the ⋄_α-integral Hölder's inequality in time scales. Recently, Li Yin and Feng Qi [24] have introduced some non-linear integral inequalities under certain conditions.

Our aim of this article is to demonstrate new principles of non-linear integral inequalities in time scales via the ∇-integral and the ⋄_α-integral.

Auxiliary Statements and Definitions

Now, let us briefly give information about time scales and give the necessary definitions and notations for our article. For more details, we refer the reader to the articles [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25].

Let ω be a weight function on R, i.e., ω is a non-negative, almost everywhere positive on R and ∫_R ω(y)∇y < ∞.

Let σ(t) be the forward jump operator and let ρ(t) be the backward jump operator in T (T is time scale) for t ∈ T. Respectively, they are defined by $σ (t) = \inf {s \in T : s > t}$ \sigma (t) = \inf \{ s \in T:s > t\} and $ρ (t) = sup {s \in T : s > t} .$ \rho (t) = \sup \{ s \in T:s > t\} .

If σ : T → T, σ(t) > t, then t is right-scattered. If ρ : T → T, ρ(t) < t, then t is left-scattered. And, if σ : T → T,σ(t) = t, then t is called right-dense, and if ρ : T → T, ρ(t) = t, then t is called left-dense. Let two mappings μ,υ : T → R⁺ such that μ(t) = σ(t) − t,υ (t) = t − ρ(t) are called graininess mappings. If T has a left-scattered maximum u ∈ R, then T^k = T /u. If not T^k = T. Briefly

If supT < ∞, then T^k = [ρ supT,supT ] and if supT = ∞, then T^k = T. By the same way

If |infT | < ∞, then T^k = [infT,σ infT ] and if infT = −∞, then T^k = T. Let f : T → R and f ^σ : T → R by f ^σ (t) = f (σ(t)) for ∀t ∈ T, i.e., f ^σ = f ○ σ. And let f : T → R and f ^ρ : T → R by f ^ρ(t) = f (ρ(t)) for ∀t ∈ T, i.e., f ^ρ = f ○ ρ.

Assume that h : T → R,t ∈ T^k(t ≠ minT).

Let h is Δ-differentiable at point t and h is continuous at point t.

Let h is left continuous at point t. t is right-scattered and h is Δ-differentiable at point t, $h^{Δ} (t) = (h^{σ} (t) - h (t)) / μ (t)$ {h^\Delta }(t) = ({h^\sigma }(t) - h(t))/\mu (t)

Let t is right-dense, h is Δ-differentiable at point t and $lim_{s \to t} \frac{h (t) - h (s)}{t - s}$ \mathop {\lim }\limits_{s \to t} {{h(t) - h(s)} \over {t - s}} then $h^{Δ} (t) = lim_{s \to t} \frac{h (t) - h (s)}{t - s}$ {h^\Delta }(t) = \mathop {\lim }\limits_{s \to t} {{h(t) - h(s)} \over {t - s}}

Let h is Δ-differentiable at point t, then $h^{σ} (t) = h (t) + μ (t) h^{Δ} (t) .$ {h^\sigma }(t) = h(t) + \mu (t){h^\Delta }(t).

Definition 2.1

[12] H : T → R is called a Δ-antiderivative of h : T → R. H^Δ = h(t) holds for ∀s,t ∈ T. We define the Δ-integral of h by $\int_{s}^{t} h (τ) Δ τ = H (t) - H (s)$ \int_s^t h(\tau )\Delta \tau = H(t) - H(s) for s,t ∈ T.

Definition 2.2

[14] Let h : T_k → R is called a ∇-differentiable at t ∈ T_k,h^∇(t), if ɛ > 0 then there exists a neighborhood V of t such that $| h (ρ (t)) - h (s) - h^{\nabla} (t) (ρ (t) - s) | \leq ε | ρ (t) - s |$ |h(\rho (t)) - h(s) - {h^\nabla }(t)(\rho (t) - s)| \le \varepsilon |\rho (t) - s| for ∀s ∈ V.

Assume that h : T → R,t ∈ T^k (t ≠ maxT).

Let h is ∇-differentiable at point t and h is continuous at point t.

Let h is right continuous at point t. t is leftt-scattered and h is ∇-differentiable at point t, $h^{\nabla} (t) = \frac{h (t) - h^{ρ} (t)}{ν (t)}$ {h^\nabla }(t) = {{h(t) - {h^\rho }(t)} \over {\nu (t)}}

Let t is left-dense, h is ∇-differentiable at point t and $lim_{s \to t} \frac{h (t) - h (s)}{t - s}$ \mathop {\lim }\limits_{s \to t} {{h(t) - h(s)} \over {t - s}} then $h^{\nabla} (t) = lim_{s \to t} \frac{h (t) - h (s)}{t - s}$ {h^\nabla }(t) = \mathop {\lim }\limits_{s \to t} {{h(t) - h(s)} \over {t - s}}

Let h is ∇-differentiable at point t, then $h^{ρ} (t) = h (t) - ϑ (t) h^{\nabla} (t) .$ {h^\rho }(t) = h(t) - \vartheta (t){h^\nabla }(t).

Definition 2.3

[14] H : T → R is called a ∇-antiderivative of h : T → R. H^∇ = h(t) holds for ∀s,t ∈ T. Then, we define the ∇-integral of h by $\int_{s}^{t} h (τ) \nabla τ = H (t) - H (s)$ \int_s^t h(\tau )\nabla \tau = H(t) - H(s) for s,t ∈ T.

Let h(t) be differentiable on T. And let b,t ∈ T. Then, $h^{⋄_{a}} (t) = {bh}^{Δ} (t) + (1 - b) h^{\nabla} (t), 0 \leq b \leq 1 .$ {h^{{\diamondsuit _a}}}(t) = b{h^\Delta }(t) + (1 - b){h^\nabla }(t),\quad \quad 0 \le b \le 1.

Proposition 2.4

[15] If we get f, g : T → R, ⋄_a-differentiable at t ∈ T, then

(f + g)^⋄_a (t) = f ^⋄_a (t) + g^⋄_a (t)

If c ∈ R, then (c f )^⋄_a (t) = c f ^⋄_a (t).

(f g)^⋄_a (t) = f ^⋄_a (t)g(t) + b f ^σ (t)g^Δ(t) + (1 − b) f ^ρ(t)g^∇(t).

Definition 2.5

[15] If we get b,t ∈ T, f : T → R, then $\int_{b}^{t} f (γ) ⋄_{a} γ = b \int_{b}^{t} f (γ) Δ γ + (1 - b) \int_{b}^{t} f (γ) \nabla γ, 0 \leq b \leq 1 .$ \int_b^t f(\gamma ){\diamondsuit _a}\gamma = b\int_b^t f(\gamma )\Delta \gamma + (1 - b)\int_b^t f(\gamma )\nabla \gamma ,\quad \quad 0 \le b \le 1.

Proposition 2.6

[15] Let u,v,t ∈ T, c ∈ R and if f (γ),g(γ) are ⋄_a-integrable functions on [u,v]_T, then the following statements are valid.

$\int_{u}^{t} [f (γ) + g (γ)] ⋄_{a} γ = \int_{u}^{t} f (γ) ⋄_{a} γ + \int_{u}^{t} g (γ) ⋄_{a} γ,$ \int_u^t [f(\gamma ) + g(\gamma )]{\diamondsuit _a}\gamma = \int_u^t f(\gamma ){\diamondsuit _a}\gamma + \int_u^t g(\gamma ){\diamondsuit _a}\gamma ,

$\int_{u}^{t} cf (γ) ⋄_{a} γ = c \int_{u}^{t} f (γ) ⋄_{a} γ,$ \int_u^t cf(\gamma ){\diamondsuit _a}\gamma = c\int_u^t f(\gamma ){\diamondsuit _a}\gamma ,

$\int_{u}^{t} f (γ) ⋄_{a} γ = - \int_{t}^{u} f (γ) ⋄_{a} γ,$ \int_u^t f(\gamma ){\diamondsuit _a}\gamma = - \int_t^u f(\gamma ){\diamondsuit _a}\gamma ,

$\int_{u}^{t} f (γ) ⋄_{a} γ = \int_{u}^{v} f (γ) ⋄_{a} γ + \int_{v}^{t} f (γ) ⋄_{a} γ,$ \int_u^t f(\gamma ){\diamondsuit _a}\gamma = \int_u^v f(\gamma ){\diamondsuit _a}\gamma + \int_v^t f(\gamma ){\diamondsuit _a}\gamma ,

$\int_{u}^{u} f (γ) ⋄_{a} γ = 0 .$ \int_u^u f(\gamma ){\diamondsuit _a}\gamma = 0.

Lemma 2.7

[15] Let u,v,t ∈ T with u < v. Suppose that h(γ) and g(γ) are ⋄_a-integrable functions on [u,v]_T, then the following statements are valid.

If h(γ) ≥ 0 for ∀γ ∈ [u,v]_T, then $\int_{u}^{v} h (γ) ⋄_{a} γ \geq 0$ \int_u^v h(\gamma ){\diamondsuit _a}\gamma \ge 0 .

If h(γ) ≤ g(γ) for ∀γ ∈ [u,v]_T, then $\int_{u}^{v} h (γ) ⋄_{a} γ \leq \int_{u}^{v} g (γ) ⋄_{a} γ$ \int_u^v h(\gamma ){\diamondsuit _a}\gamma \le \int_u^v g(\gamma ){\diamondsuit _a}\gamma .

If h(γ) ≥ 0 for ∀γ ∈ [u,v]_T, then h(γ) = 0 iff $\int_{u}^{v} h (γ) ⋄_{a} γ = 0$ \int_u^v h(\gamma ){\diamondsuit _a}\gamma = 0 .

Lemma 2.8

(For details, Lemma 2.5 in [24]) Let p > 1 or q < 0, while 1/p + 1/q = 1, if g,h ∈ C_rd(T,R), g(y) > 0,h(y) > 0, while u,v ∈ T, then $\int_{u}^{v} \frac{{[g (y)]}^{p}}{{[h (y)]}^{p / q}} Δ y \geq \frac{{(\int_{u}^{v} g (y) Δ y)}^{p}}{{(\int_{u}^{v} h (y) Δ y)}^{p / q}} .$ \int_u^v {{{{[g(y)]}^p}} \over {{{[h(y)]}^{p/q}}}}\Delta y \ge {{{{(\int_u^v g(y)\Delta y)}^p}} \over {{{(\int_u^v h(y)\Delta y)}^{p/q}}}}.

Qi F. et al. [25] proved some inequalities under the condition of Δ-differentiable. In the next section, we will prove these inequalities under the conditions of the ∇-differentiable and the ⋄_a-differentiable.

Main Result

In this section, we will prove non-linear ∇-differentiable weighted integral inequalities under certain conditions. Later, we will prove their ⋄_a-differentiable extensions. We have listed these studies in the references of the article for the relevant readers.

Theorem 3.1

Let h,w ∈ C_rd(T,R) and let w be a weight function and let w(y),h(y) > 0, $\int_{u}^{v} h (y) w (y) \nabla y < \infty$ \int_u^v h(y)w(y)\nabla y < \infty and p > 1 or q < 0, while 1/p + 1/q = 1. If $\int_{u}^{v} w (y) h (y) \geq {(v - u)}^{p - 1}$ \int_u^v w(y)h(y) \ge {(v - u)^{p - 1}} , while u,v ∈ T, then (1) $\int_{u}^{v} {[h (y) w (y)]}^{p} \nabla y \geq {(\int_{u}^{v} h (y) w (y) \nabla y)}^{p - 1} .$ \int_u^v {[h(y)w(y)]^p}\nabla y \ge {\left( {\int_u^v h(y)w(y)\nabla y} \right)^{p - 1}}.

Proof

Using Lemma 2.8, we obtain $\int_{u}^{v} {[h (y) w (y)]}^{p} \nabla y = \int_{u}^{v} \frac{{[h (y) w (y)]}^{p}}{1^{p - 1}} \nabla y \geq \frac{{[\int_{u}^{v} h (y) w (y) \nabla y]}^{p}}{{[\int_{u}^{v} 1 \nabla y]}^{p - 1}} \geq {(\int_{u}^{v} h (y) w (y) \nabla y)}^{p - 1} .$ \int_u^v {[h(y)w(y)]^p}\nabla y = \int_u^v {{{{[h(y)w(y)]}^p}} \over {{1^{p - 1}}}}\nabla y \ge {{{{[\int_u^v h(y)w(y)\nabla y]}^p}} \over {{{[\int_u^v 1\nabla y]}^{p - 1}}}} \ge {\left( {\int_u^v h(y)w(y)\nabla y} \right)^{p - 1}}.

Theorem 3.2

Let w be a weight function, w(y),g(y) > 0, $\int_{u}^{v} g (y) w (y) \nabla y < \infty$ \int_u^v g(y)w(y)\nabla y < \infty for y ∈ (u,v]and g,w ∈ C([u,v],R),∇-differantiable in (u,v),Let ɛ,ϕ be positive real numbers such that 1 < ϕ < ɛ. If (2) ${[{(wg)}^{(ε - φ) / (φ - 1)} (y)]}^{'} \geq \frac{(ε - φ) φ^{1 / (φ - 1)}}{ε - 1}$ {\left[ {{{(wg)}^{(\varepsilon - \varphi )/(\varphi - 1)}}(y)} \right]}^\prime \ge {{(\varepsilon - \varphi ){\varphi ^{1/(\varphi - 1)}}} \over {\varepsilon - 1}} for ∀y ∈ (u,v), then (3) $\int_{u}^{v} {[g (y) w (y)]}^{ε} \nabla y \geq {[\int_{u}^{v} g (y) w (y) \nabla y]}^{φ} .$ \int_u^v {[g(y)w(y)]^\varepsilon }\nabla y \ge {\left[ {\int_u^v g(y)w(y)\nabla y} \right]^\varphi }.

Proof

If we use Cauchy's Mean Value Theorem consecutively for δ ∈ (u,v) and θ ∈ (u,δ), then we obtain (4) $\begin{matrix} \frac{{[\int_{u}^{v} g (y) w (y) \nabla y]}^{φ}}{\int_{u}^{v} {[g (y)]}^{ε} \nabla y} = \frac{φ {[\int_{u}^{δ} g (y) w (y) \nabla y]}^{φ - 1} w (y) g (δ)}{{[w (y) g (δ)]}^{ε}} = {\frac{φ^{1 / (φ - 1)} \int_{u}^{δ} w (y) g (y) \nabla y}{{[w (y) g (δ)]}^{(ε - 1) / (φ - 1)}}}^{φ - 1} \\ = {\frac{φ^{1 / (φ - 1)} w (θ) g (θ)}{\frac{ε - 1}{φ - 1} {[w (θ) g (θ)]}^{\frac{ε - φ}{φ - 1}} (wg)^{'} (θ)}}^{φ - 1} = {\frac{(ε - φ) φ^{1 / (φ - 1)} / (φ - 1)}{[(wg)^{(ε - φ) / (φ - 1)} (θ)]}}^{φ - 1} . \end{matrix}$ \matrix{ {{{{{\left[ {\int_u^v g(y)w(y)\nabla y} \right]}^\varphi }} \over {\int_u^v {{[g(y)]}^\varepsilon }\nabla y}} = {{\varphi {{\left[ {\int_u^\delta g(y)w(y)\nabla y} \right]}^{\varphi - 1}}w(y)g(\delta )} \over {{{[w(y)g(\delta )]}^\varepsilon }}} = {{\left\{ {{{{\varphi ^{1/(\varphi - 1)}}\int_u^\delta w(y)g(y)\nabla y} \over {{{[w(y)g(\delta )]}^{(\varepsilon - 1)/(\varphi - 1)}}}}} \right\}}^{\varphi - 1}}} \cr { = {{\left\{ {{{{\varphi ^{1/(\varphi - 1)}}w(\theta )g(\theta )} \over {{{\varepsilon - 1} \over {\varphi - 1}}{{[w(\theta )g(\theta )]}^{{{\varepsilon - \varphi } \over {\varphi - 1}}}}(wg)'(\theta )}}} \right\}}^{\varphi - 1}} = {{\left\{ {{{(\varepsilon - \varphi ){\varphi ^{1/(\varphi - 1)}}/(\varphi - 1)} \over {[(wg{)^{(\varepsilon - \varphi )/(\varphi - 1)}}(\theta )]}}} \right\}}^{\varphi - 1}}.} \cr } thus, (3) inequality holds.

Theorem 3.3

Let w be a weight function, w(y),g(y) > 0, $\int_{u}^{v} g (y) w (y) \nabla y < \infty$ \int_u^v g(y)w(y)\nabla y < \infty for y ∈ (u,v],g,w ∈ C([u,v],R), ɛ ∈ R. If ϕ = 1 and [w(y)g(y)]^1−ϕ ≤ 1 for ∀y ∈ (u,v), then (3) holds.

Proof

For ϕ = 1, inequality (3) reduced to (5) $\int_{u}^{v} {[w (y) g (y)]}^{ε} \nabla y \geq \int_{u}^{v} w (y) g (y) \nabla y .$ \int_u^v {[w(y)g(y)]^\varepsilon }\nabla y \ge \int_u^v w(y)g(y)\nabla y.

If we use Cauchy's Mean Value Theorem, we obtain the following equation (6) $\frac{\int_{u}^{v} {[w (y) g (y)]}^{ε} \nabla y}{\int_{u}^{v} w (y) g (y) \nabla y} = \frac{{[w (δ) g (δ)]}^{ε}}{w (δ) g (δ)} = [w (δ) g (δ {)]}^{ε - 1} .$ {{\int_u^v {{[w(y)g(y)]}^\varepsilon }\nabla y} \over {\int_u^v w(y)g(y)\nabla y}} = {{{{[w(\delta )g(\delta )]}^\varepsilon }} \over {w(\delta )g(\delta )}} = [w(\delta )g(\delta {)]^{\varepsilon - 1}}.

Theorem 3.4

Let w be a weight function, $\int_{u}^{v} g (y) w (y) \nabla y < \infty$ \int_u^v g(y)w(y)\nabla y < \infty for y ∈ (u,v], m ∈ N and 1 ≤ ϕ ≤ m + 1, there exist (wg)^m(y) derivative of the m-th order on [u,v] and (wg)^m(y) is increasing, then g^(m)(y) ≥ 0,g⁽ⁱ⁾(u) = 0 for 0 ≤ j ≤ m − 1. If $w (y) g (y) \geq {[\frac{{(y - ε)}^{φ} - 1}{φ^{φ - 2}}]}^{1 / (ε - φ)}$ w(y)g(y) \ge {\left[ {{{{{(y - \varepsilon )}^\varphi } - 1} \over {{\varphi ^{\varphi - 2}}}}} \right]^{1/(\varepsilon - \varphi )}} , then (3) holds.

Proof

If we use Cauchy's Mean Value Theorem together with the condition given in the theorem, we get the following. (7) $\begin{matrix} \frac{\int_{u}^{v} {[w (y) g (y)]}^{ε} \nabla y}{{[\int_{u}^{v} w (y) g (y) \nabla y]}^{φ}} = \frac{{[w (c_{1}) g (c_{1})]}^{ε - 1}}{φ {[\int_{u}^{c_{1}} w (y) g (y) \nabla y]}^{φ - 1}} u < c_{1} < v \\ \geq \frac{{[(c_{1} - u) w (c_{1} g (c_{1}))]}^{φ - 1} / φ^{φ - 2}}{φ {[\int_{u}^{c_{1}} w (y) g (y) \nabla y]}^{φ - 1}} = {[\frac{(c_{1} - ε) w (c_{1}) g (c_{1})}{φ \int_{u}^{c_{1}} w (y) g (y) \nabla y}]}^{φ - 1} . \end{matrix}$ \matrix{ {{{\int_u^v {{[w(y)g(y)]}^\varepsilon }\nabla y} \over {{{[\int_u^v w(y)g(y)\nabla y]}^\varphi }}} = {{{{[w({c_1})g({c_1})]}^{\varepsilon - 1}}} \over {\varphi {{[\int_u^{{c_1}} w(y)g(y)\nabla y]}^{\varphi - 1}}}}\quad \quad \quad \quad u < {c_1} < v} \cr { \ge {{{{[({c_1} - u)w({c_1}g({c_1}))]}^{\varphi - 1}}/{\varphi ^{\varphi - 2}}} \over {\varphi {{[\int_u^{{c_1}} w(y)g(y)\nabla y]}^{\varphi - 1}}}} = {{\left[ {{{({c_1} - \varepsilon )w({c_1})g({c_1})} \over {\varphi \int_u^{{c_1}} w(y)g(y)\nabla y}}} \right]}^{\varphi - 1}}.} \cr }

If we use Cauchy's Mean Value Theorem consecutively in (7), we obtain (8) $\begin{matrix} \frac{(c_{1} - ε) w (c_{1}) g (c_{1})}{φ \int_{u}^{c_{1}} w (y) g (y) \nabla y} = 1 + \frac{(c_{2} - ε) (wg)^{'} (c_{2})}{w (c_{2}) g (c_{2})} u < c_{2} < c_{1} \\ ... \\ = m + \frac{(c_{m + 1} - ε) (wg)^{(m)} (c_{m + 1})}{{(wg)}^{(m - 1)} (c_{m + 1})} u < c_{m + 1} < c_{m} . \end{matrix}$ \matrix{ {{{({c_1} - \varepsilon )w({c_1})g({c_1})} \over {\varphi \int_u^{{c_1}} w(y)g(y)\nabla y}} = 1 + {{({c_2} - \varepsilon )(wg)'({c_2})} \over {w({c_2})g({c_2})}}\quad \quad \quad \quad u < {c_2} < {c_1}} \cr {...} \cr { = m + {{({c_{m + 1}} - \varepsilon )(wg{)^{(m)}}({c_{m + 1}})} \over {{{(wg)}^{(m - 1)}}({c_{m + 1}})}}\quad \quad \quad \quad \quad \quad u < {c_{m + 1}} < {c_m}.} \cr }

But (wg)^(m−1)(k) = (wg)^(m−1)(k)−(wg)^(m−1)(u) = (k −u)(wg)^m(k₁) for k₁ ∈ (u,k). If (wg)^m(k₁) ≤ g^m(k), then (wg)^(m)(y) is increasing.

Hence (9) ${(wg)}^{(m)} (k) (k - u) \geq {(wg)}^{(m - 1)} (k) > 0 .$ {(wg)^{(m)}}(k)(k - u) \ge {(wg)^{(m - 1)}}(k) > 0.

Applying (9) to (8) yields (10) $\frac{(c_{1} - ε) g (c_{1})}{\int_{u}^{c_{1}} g (y) \nabla y} \geq m + 1 .$ {{({c_1} - \varepsilon )g({c_1})} \over {\int_u^{{c_1}} g(y)\nabla y}} \ge m + 1.

Hence $\frac{\int_{u}^{v} {[g (y)]}^{ε} \nabla y}{{[\int_{u}^{v} g (y) \nabla y]}^{φ}} \geq {(\frac{m + 1}{φ})}^{φ - 1}$ {{\int_u^v {{[g(y)]}^\varepsilon }\nabla y} \over {{{[\int_u^v g(y)\nabla y]}^\varphi }}} \ge {\left( {{{m + 1} \over \varphi }} \right)^{\varphi - 1}} for 1 ≤ ɛ ≤ m + 1.

Theorem 3.5

Suppose that w be a weight function $\int_{u}^{v} w (y) g (y) \nabla y < \infty$ \int_u^v w(y)g(y)\nabla y < \infty for y ∈ (u,v), m ∈ N, 1 < ϕ ≤ m + 1, there exist (wg)^(m)(y) derivative of the m-th order on [u,v] and (wg)^(m)(y) is increasing, then (wg)^(m)(y) ≥ 0 and g^(j)(u) = 0 for m − 1 ≥ j ≥ 0.

Proof

If $w (y) g (y) \geq {[\frac{φ {(y - ε)}^{(φ - 1)}}{{(φ - 1)}^{(φ - 2)}}]}^{1 / (ε - φ)}$ w(y)g(y) \ge {\left[ {{{\varphi {{(y - \varepsilon )}^{(\varphi - 1)}}} \over {{{(\varphi - 1)}^{(\varphi - 2)}}}}} \right]^{1/(\varepsilon - \varphi )}} , (6) becomes $\frac{\int_{u}^{v} {[w (y) g (y)]}^{ε} \nabla y}{{[\int_{u}^{v} w (y) g (y) \nabla y]}^{φ}} \geq {[\frac{(c_{1} - ε) w (c_{1}) g (c_{1})}{(φ - 1) \int_{u}^{c_{1}} w (y) g (y) \nabla y}]}^{φ - 1} .$ {{\int_u^v {{[w(y)g(y)]}^\varepsilon }\nabla y} \over {{{[\int_u^v w(y)g(y)\nabla y]}^\varphi }}} \ge {\left[ {{{({c_1} - \varepsilon )w({c_1})g({c_1})} \over {(\varphi - 1)\int_u^{{c_1}} w(y)g(y)\nabla y}}} \right]^{\varphi - 1}}.

If all terms of (8) are positive, then $\frac{(c_{1} - ε) w (c_{1}) g (c_{1})}{\int_{u}^{c_{1}} w (y) g (y) \nabla y} \geq m$ {{({c_1} - \varepsilon )w({c_1})g({c_1})} \over {\int_u^{{c_1}} w(y)g(y)\nabla y}} \ge m .

Now let's consider the ⋄_a-integral in time scales.

Theorem 3.6

Let w be a weight function, h(y),w(y) > 0, $\int_{u}^{v} w (y) h (y) \nabla y$ \int_u^v w(y)h(y)\nabla y for y ∈ (u,v), p > 1 or q < 0, while 1/p + 1/q = 1 and h,w ∈ C_rd(T,R). If $\int_{u}^{v} w (y) h (y) ⋄_{a} y \geq {(v - u)}^{p - 1}$ \int_u^v w(y)h(y){\diamondsuit _a}y \ge {(v - u)^{p - 1}} for u,v ∈ T, then (11) $\int_{u}^{v} {[w (y) h (y)]}^{p} ⋄_{a} y \geq {[\int_{u}^{v} w (y) h (y) ⋄_{a} y]}^{p - 1} .$ \int_u^v {[w(y)h(y)]^p}{\diamondsuit _a}y \ge {\left[ {\int_u^v w(y)h(y){\diamondsuit _a}y} \right]^{p - 1}}.

Proof

See proof of Theorem 3.1. Moreover, when α = 0, (11) reduce to (1).

Theorem 3.7

Let g,w ∈ C_rd(T,R),⋄_a-differantiabla on (u,v), and let w be a weight function, $\int_{u}^{v} w (y) h (y) ⋄_{a} y$ \int_u^v w(y)h(y){\diamondsuit _a}y for y ∈ (u,v],w(y)g(y) > 0, and let ɛ,ϕ be positive real numbers such that 1 < ϕ < ɛ. If (12) ${[{(wg)}^{(ε - φ) / (φ - 1)} (y)]}^{'} \geq \frac{(ε - φ) φ^{1 / (φ - 1)}}{ε - 1} .$ {\left[ {{{(wg)}^{(\varepsilon - \varphi )/(\varphi - 1)}}(y)} \right]}^\prime \ge {{(\varepsilon - \varphi ){\varphi ^{1/(\varphi - 1)}}} \over {\varepsilon - 1}}. for ∀y ∈ (u,v), then (13) $\int_{u}^{v} {[w (y) g (y)]}^{ε} ⋄_{a} y \geq {[\int_{u}^{v} w (y) g (y) ⋄_{a} y]}^{φ} .$ \int_u^v {[w(y)g(y)]^\varepsilon }{\diamondsuit _a}y \ge {\left[ {\int_u^v w(y)g(y){\diamondsuit _a}y} \right]^\varphi }.

Proof

See proof of Theorem 3.2. Moreover, (13) inequality is an extension of (3) inequality. When α = 0, (13) reduce to (3).

Theorem 3.8

Let a ∈ R, w be a weight function, $\int_{u}^{v} w (y) g (y) ⋄_{y} < \infty$ \int_u^v w(y)g(y){\diamondsuit _y} < \infty for y ∈ (u,v), g(y),w(y) > 0, g,w ∈ C([u,v],R) and [w(y),g(y)], ⋄_a-differantiable on (u,v). If ϕ = 1 and [w(y)g(y)]^1−ϕ ≤ 1 for ∀y ∈ (u,v), then (14) $\int_{u}^{v} {[w (y) g (y)]}^{ε} ⋄_{a} y \geq {[\int_{u}^{v} w (y) g (y) ⋄_{a} y]}^{φ} .$ \int_u^v {[w(y)g(y)]^\varepsilon }{\diamondsuit _a}y \ge {\left[ {\int_u^v w(y)g(y){\diamondsuit _a}y} \right]^\varphi }.

Proof

See proof of Theorem 3.3. Moreover, (14) inequality is an extension of (3) inequality. When α = 0, (14) reduce to (3).

Theorem 3.9

Suppose that w be a weight function, $\int_{u}^{v} w (y) g (y) ⋄_{y} < \infty$ \int_u^v w(y)g(y){\diamondsuit _y} < \infty for y ∈ (u,v). m ∈ N,1 ≤ ϕ ≤ m+1, there exist (wg)^(m)(y) derivative of the m-th order on [u,v], (wg)^(m)(y) is increasing and [w(y),g(y)],⋄_a-differantiable in (u,v), then (wg)^(m)(y) ≥ 0 and g^j(u) = 0 for 0 ≤ j ≤ m − 1. If $w (y) g (y) \geq {[\frac{{(y - ε)}^{φ - 1}}{φ^{φ - 2}}]}^{1 / (ε - φ)},$ w(y)g(y) \ge {\left[ {{{{{(y - \varepsilon )}^{\varphi - 1}}} \over {{\varphi ^{\varphi - 2}}}}} \right]^{1/(\varepsilon - \varphi )}}, then (15) $\int_{u}^{v} {[w (y) g (y)]}^{ε} ⋄_{a} y \geq {[\int_{u}^{v} w (y) g (y) ⋄_{a} y]}^{φ} .$ \int_u^v {[w(y)g(y)]^\varepsilon }{\diamondsuit _a}y \ge {\left[ {\int_u^v w(y)g(y){\diamondsuit _a}y} \right]^\varphi }. inequality holds.

Proof

See proof of Theorem 3.4. Moreover, (15) inequality is an extension of (3) inequality. When α = 0, (15) reduce to (3).

Theorem 3.10

Suppose that w be a weight function, $\int_{u}^{v} w (y) g (y) ⋄_{y} < \infty$ \int_u^v w(y)g(y){\diamondsuit _y} < \infty for y ∈ (u,v). m ∈ N,1 ≤ ϕ ≤ m + 1, there exist (wg)^(m)(y) derivative of the m-th order on [u,v], (wg)^(m)(y) is increasing and [w(y),g(y)],⋄_a-differantiable in (u,v), then (wg)^(m)(y) ≥ 0 and g^j(u) = 0 for 0 ≤ j ≤ m − 1. If $w (y) g (y) \geq {[\frac{φ {(y - ε)}^{φ - 1}}{{(φ - 1)}^{φ - 2}}]}^{1 / (ε - φ)},$ w(y)g(y) \ge {\left[ {{{\varphi {{(y - \varepsilon )}^{\varphi - 1}}} \over {{{(\varphi - 1)}^{\varphi - 2}}}}} \right]^{1/(\varepsilon - \varphi )}}, then (16) $\int_{u}^{v} {[w (y) g (y)]}^{ε} ⋄_{a} y \geq {[\int_{u}^{v} w (y) g (y) ⋄_{a} y]}^{φ} .$ \int_u^v {[w(y)g(y)]^\varepsilon }{\diamondsuit _a}y \ge {\left[ {\int_u^v w(y)g(y){\diamondsuit _a}y} \right]^\varphi }. inequality holds.

Proof

See proof of Theorem 3.5. Moreover, (16) inequality is an extension of (3) inequality. When α = 0, (16) reduce to (3).

Conclusion

Integral inequalities and dynamic equations are the cornerstones of both time scales and harmonic analysis. Mathematicians proved many integral inequalities on time scales [4,5,6,7,8,9]. And they also showed generalized forms of these inequalities [10, 11, 13, 25]. Time scales theory has also been of interest in different sciences. For example, quantum mechanics, wave equations, physical problems, heat transfer, electrical engineering and economics [26,27,28,29,30]. In this article, we proved non-linear integral inequalities in time scales via the ∇-integral and the ⋄_a-integral. We think that the multidimensional and multivariate cases of the inequalities proved in this article are also worth examining.

eISSN:: 2444-8656
Język:: Angielski

Częstotliwość wydawania:: Volume Open
Dziedziny czasopisma:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

Kanał RSS czasopisma

New Principles of Non-Linear Integral Inequalities on Time Scales

Data publikacji: 18 sty 2021

Zakres stron: 387 - 394

Otrzymano: 10 kwi 2020

Przyjęty: 18 paź 2020

DOI: https://doi.org/10.2478/amns.2021.1.00001

Słowa kluczoweOperator theory, Time scales, Integral inequalities

© 2021 Lutfi Akin, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Słowa kluczowe
Operator theory, Time scales, Integral inequalities