New Principles of Non-Linear Integral Inequalities on Time Scales

[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 ∫sth(τ)Δτ=H(t)−H(s) \int_s^t h(\tau )\Delta \tau = H(t) - H(s) for s,t ∈ T.

[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∇(t)(ρ(t)−s)|≤ε|ρ(t)−s| |h(\rho (t)) - h(s) - {h^\nabla }(t)(\rho (t) - s)| \le \varepsilon |\rho (t) - s| for ∀s ∈ V.

[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 ∫sth(τ)∇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)=bhΔ(t)+(1−b)h∇(t), 0≤b≤1. {h^{{\diamondsuit _a}}}(t) = b{h^\Delta }(t) + (1 - b){h^\nabla }(t),\quad \quad 0 \le b \le 1.

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

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

[15] If we get b,t ∈ T, f : T → R, then ∫btf(γ)⋄aγ=b∫btf(γ)Δγ+(1−b)∫btf(γ)∇γ, 0≤b≤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.

[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.

∫ut[f(γ)+g(γ)]⋄aγ=∫utf(γ)⋄aγ+∫utg(γ)⋄aγ, \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 ,

[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 ∫uvh(γ)⋄aγ≥0 \int_u^v h(\gamma ){\diamondsuit _a}\gamma \ge 0 .

(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 ∫uv[g(y)]p[h(y)]p/qΔy≥(∫uvg(y)Δy)p(∫uvh(y)Δy)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.

Let h,w ∈ C_rd(T,R) and let w be a weight function and let w(y),h(y) > 0, ∫uvh(y)w(y)∇y<∞ \int_u^v h(y)w(y)\nabla y < \infty and p > 1 or q < 0, while 1/p + 1/q = 1. If ∫uvw(y)h(y)≥(v−u)p−1 \int_u^v w(y)h(y) \ge {(v - u)^{p - 1}} , while u,v ∈ T, then (1) ∫uv[h(y)w(y)]p∇y≥(∫uvh(y)w(y)∇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}}.

Using Lemma 2.8, we obtain ∫uv[h(y)w(y)]p∇y=∫uv[h(y)w(y)]p1p−1∇y≥[∫uvh(y)w(y)∇y]p[∫uv1∇y]p−1≥(∫uvh(y)w(y)∇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}}.

Let w be a weight function, w(y),g(y) > 0, ∫uvg(y)w(y)∇y<∞ \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)]′≥(ε−φ)φ1/(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) ∫uv[g(y)w(y)]ε∇y≥[∫uvg(y)w(y)∇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 }.

If we use Cauchy's Mean Value Theorem consecutively for δ ∈ (u,v) and θ ∈ (u,δ), then we obtain (4) [∫uvg(y)w(y)∇y]φ∫uv[g(y)]ε∇y=φ[∫uδg(y)w(y)∇y]φ−1w(y)g(δ)[w(y)g(δ)]ε={φ1/(φ−1)∫uδw(y)g(y)∇y[w(y)g(δ)](ε−1)/(φ−1)}φ−1={φ1/(φ−1)w(θ)g(θ)ε−1φ−1[w(θ)g(θ)]ε−φφ−1(wg)′(θ)}φ−1={(ε−φ)φ1/(φ−1)/(φ−1)[(wg)(ε−φ)/(φ−1)(θ)]}φ−1. \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.

Let w be a weight function, w(y),g(y) > 0, ∫uvg(y)w(y)∇y<∞ \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.

For ϕ = 1, inequality (3) reduced to (5) ∫uv[w(y)g(y)]ε∇y≥∫uvw(y)g(y)∇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) ∫uv[w(y)g(y)]ε∇y∫uvw(y)g(y)∇y=[w(δ)g(δ)]εw(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}}.

Let w be a weight function, ∫uvg(y)w(y)∇y<∞ \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)≥[(y−ε)φ−1φφ−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.

If we use Cauchy's Mean Value Theorem together with the condition given in the theorem, we get the following. (7) ∫uv[w(y)g(y)]ε∇y[∫uvw(y)g(y)∇y]φ=[w(c1)g(c1)]ε−1φ[∫uc1w(y)g(y)∇y]φ−1 u<c1<v≥[(c1−u)w(c1g(c1))]φ−1/φφ−2φ[∫uc1w(y)g(y)∇y]φ−1=[(c1−ε)w(c1)g(c1)φ∫uc1w(y)g(y)∇y]φ−1. \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) (c1−ε)w(c1)g(c1)φ∫uc1w(y)g(y)∇y=1+(c2−ε)(wg)′(c2)w(c2)g(c2) u<c2<c1...=m+(cm+1−ε)(wg)(m)(cm+1)(wg)(m−1)(cm+1) u<cm+1<cm. \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)≥(wg)(m−1)(k)>0. {(wg)^{(m)}}(k)(k - u) \ge {(wg)^{(m - 1)}}(k) > 0.

Applying (9) to (8) yields (10) (c1−ε)g(c1)∫uc1g(y)∇y≥m+1. {{({c_1} - \varepsilon )g({c_1})} \over {\int_u^{{c_1}} g(y)\nabla y}} \ge m + 1.

Hence ∫uv[g(y)]ε∇y[∫uvg(y)∇y]φ≥(m+1φ)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.

Suppose that w be a weight function ∫uvw(y)g(y)∇y<∞ \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.

If w(y)g(y)≥[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 )}} .

If w(y)g(y)≥[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 ∫uv[w(y)g(y)]ε∇y[∫uvw(y)g(y)∇y]φ≥[(c1−ε)w(c1)g(c1)(φ−1)∫uc1w(y)g(y)∇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 (c1−ε)w(c1)g(c1)∫uc1w(y)g(y)∇y≥m {{({c_1} - \varepsilon )w({c_1})g({c_1})} \over {\int_u^{{c_1}} w(y)g(y)\nabla y}} \ge m .

Let w be a weight function, h(y),w(y) > 0, ∫uvw(y)h(y)∇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 ∫uvw(y)h(y)⋄ay≥(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) ∫uv[w(y)h(y)]p⋄ay≥[∫uvw(y)h(y)⋄ay]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}}.

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

Let g,w ∈ C_rd(T,R),⋄_a-differantiabla on (u,v), and let w be a weight function, ∫uvw(y)h(y)⋄ay \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)]′≥(ε−φ)φ1/(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) ∫uv[w(y)g(y)]ε⋄ay≥[∫uvw(y)g(y)⋄ay]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 }.

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

Let a ∈ R, w be a weight function, ∫uvw(y)g(y)⋄y<∞ \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) ∫uv[w(y)g(y)]ε⋄ay≥[∫uvw(y)g(y)⋄ay]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 }.

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

Suppose that w be a weight function, ∫uvw(y)g(y)⋄y<∞ \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)≥[(y−ε)φ−1φφ−2]1/(ε−φ), w(y)g(y) \ge {\left[ {{{{{(y - \varepsilon )}^{\varphi - 1}}} \over {{\varphi ^{\varphi - 2}}}}} \right]^{1/(\varepsilon - \varphi )}}, then (15) ∫uv[w(y)g(y)]ε⋄ay≥[∫uvw(y)g(y)⋄ay]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.

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

Suppose that w be a weight function, ∫uvw(y)g(y)⋄y<∞ \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)≥[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) ∫uv[w(y)g(y)]ε⋄ay≥[∫uvw(y)g(y)⋄ay]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.

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