# Some new dynamic Gronwall–Bellman–Pachpatte type inequalities with delay on time scales and certain applications

## Abstract

The main objective of the present article is to prove some new delay nonlinear dynamic inequalities of Gronwall–Bellman–Pachpatte type on time scales. We introduce very important generalized results with the help of Leibniz integral rule on time scales. For some specific time scales, we further show some relevant inequalities as special cases: integral inequalities and discrete inequalities. Our results can be used as handy tools for the study of qualitative and quantitative properties of solutions of dynamic equations on time scales. Some examples are provided to demonstrate the applications of the results.

## Introduction

In 1919 Thomas Gronwall  discovered a vital inequality, which can be used as an effective tool in the study of existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of certain nonlinear differential and difference equations. The Gronwall inequality is stated as follows: If u is a continuous function defined on the interval $$D=[a,a +h]$$ and

$$0 \leq u(t) \leq \int ^{t}_{a}\bigl[\zeta u(s)+\xi \bigr]\,ds, \quad \forall t\in D,$$

where a, ξ, ζ, and h are nonnegative constants, then

$$0\leq u(t) \leq \xi he^{\zeta h}, \quad \forall t\in D.$$

In 1943, Richard Bellman  proved the fundamental inequality, named Gronwall–Bellman’s inequality, as a generalization for Gronwall’s inequality. He proved that: If u and f are continuous and nonnegative functions defined on $$[a,b]$$, and let c be a nonnegative constant, then the inequality

$$u(t)\leq c+ \int _{a}^{t}f(s)u(s)\,ds, \quad t\in [a,b],$$
(1.1)

implies that

$$u(t)\leq c\exp \biggl( \int _{a}^{t}f(s)\,ds \biggr),\quad t\in [a,b].$$

As a generalization of (1.1), Bellman himself  proved that: If u, f, a, $$\in C(\mathbb{R}_{+},\mathbb{R}_{+})$$ and a is nondecreasing, then the inequality

$$u(t)\leq a(t)+ \int ^{t}_{0}f(s)u(s)\,ds, \quad \forall t\in \mathbb{R}_{+},$$
(1.2)

implies

$$u(t)\leq a(t)\exp \biggl( \int _{0}^{t}f(s)\,ds \biggr), \quad \forall t\in \mathbb{R}_{+}.$$

The discrete version of (1.2) was studied by Pachpatte in . In particular, he proved that: If $$\Omega (n)$$, $$f(n)$$, $$\gamma (n)$$ are nonnegative sequences defined for $$n\in \mathbb{N}_{0}$$, and $$f(n)$$ is nondecreasing for $$n\in \mathbb{N}_{0}$$, then

$$\Omega (n)\leq f(n)+\sum_{s=0}^{n-1} \gamma (s)\Omega (s),\quad n \in \mathbb{N}_{0},$$
(1.3)

implies

$$\Omega (n)\leq f(n)\prod_{s=0}^{n-1}\bigl[1+ \gamma (s)\bigr], \quad n\in \mathbb{N}_{0}.$$

In , Pachpatte studied the following inequalities:

\begin{aligned}& \Omega ^{p}(t)\leq c^{p}(t)+b(t) \int _{0}^{t}\bigl[g(s)\Omega ^{p}(s)+h(s) \Omega (s)\bigr]\,ds, \quad t\in \mathbb{R}_{+}, \end{aligned}
(1.4)
\begin{aligned}& \Omega ^{p}(t)\leq a(t)+b(t) \int _{0}^{t}k(t,s)\bigl[g(s)\Omega ^{p}(s)+h(s) \Omega (s)\bigr]\,ds, \quad t\in \mathbb{R}_{+}, \end{aligned}
(1.5)

and

\begin{aligned} \Omega ^{p}(t)\leq a(t)+b(t) \int _{0}^{t}f\bigl(s,\Omega (s)\bigr)\,ds, \quad t \in \mathbb{R}_{+}, \end{aligned}
(1.6)

where Ω, a, b, g, h and $$c\in \mathcal{C}(\mathbb{R}\mathbbm{_{+}},\mathbb{R}\mathbbm{_{+}})$$, $$k(t,s)$$ and its partial derivative $$\frac{\partial k(t,s)}{\partial t}$$ are real-valued nonnegative continuous functions for $$0\leq s\leq t\leq \infty$$, $$f:\mathbb{R}_{+}\times \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$$ is a continuous function, and $$p>1$$ is a constant.

On the other hand, also in , Pachpatte investigated the following discrete analogues of (1.4), (1.5), and (1.6):

\begin{aligned}& \Omega ^{p}(n) \leq c^{p}(n)+b(n)\sum _{s=n_{0}}^{n-1}\bigl[g(s)\Omega ^{p}(s)+h(s) \Omega (s)\bigr], \quad n\in \mathbb{N}_{0}, \\& \Omega ^{p}(n) \leq a(n)+b(n)\sum_{s=n_{0}}^{n-1}k(n,s) \bigl[g(s)\Omega ^{p}(s)+h(s) \Omega (s)\bigr],\quad n\in \mathbb{N}_{0}, \\& \Omega ^{p}(n) \leq a(n)+b(n)\sum_{s=n_{0}}^{n-1}F \bigl(s,\Omega (s)\bigr), \quad n\in \mathbb{N}_{0}, \end{aligned}

where $$\Omega (n)$$, $$a(n)$$, $$b(n)$$, $$g(n)$$, $$h(n)$$, and $$c(n)$$ are real-valued nonnegative sequences, $$F: \mathbb{N}_{0}\times \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$$, and $$k(n,s)$$, $$\Delta _{1}k(n,s)$$ are real-valued nonnegative functions for $$n_{0}\leq s\leq n$$, $$n\in \mathbb{N}_{0}$$.

In 2014, El-Owaidy et al.  proved the following new form:

\begin{aligned} \Omega (t) \leq& \gamma (t)+ \int _{a}^{\alpha _{1}(t)}\epsilon (s)w_{1}\bigl( \Omega (s)\bigr)\,ds \\ &{}+ \int _{a}^{\alpha _{2}(t)} \varepsilon (s)w_{2}\bigl( \Omega (s)\bigr)\,ds \quad \text{for all } t\in I_{1}=[a,b]. \end{aligned}
(1.7)

In the same paper , the authors also studied the following inequality:

$$\Omega (t)\leq \gamma (t)+ \int _{a}^{\alpha (t)}\varepsilon (s)w\bigl( \Omega (s)\bigr) \,ds+ \int _{a}^{\alpha (t)} k(t,s)w\bigl(\Omega (s)\bigr)\,ds \quad \text{for all } t \in I_{1},$$

where Ω, ϵ, $$\varepsilon \in \mathcal{C}(I_{1},\mathbb{R}\mathbbm{_{+}})$$, $$\alpha ,\in \mathcal{C}^{1}(I_{1},I_{1})$$ are nondecreasing functions, with $$\alpha _{i}(t)\leq t$$, $$\alpha _{i}(a)=a$$, $$\alpha _{i}^{\prime }(t)\geq 0$$, $$i=1,2$$, and $$w_{i}\in (\mathbb{R}_{+},\mathbb{R}_{+})$$ is a nondecreasing function, and $$k(t,s)\in \mathcal{C}(I_{1}\times I_{1},\mathbb{R}\mathbbm{_{+}})$$ with $$\frac{\partial k(t,s)}{\partial t}\in \mathcal{C}(I_{1}\times I_{1}, \mathbb{R}\mathbbm{_{+}})$$.

In 2015, Abdeldaim and El-Deeb  discussed the new form:

$$\Omega (t)\leq \Omega _{0} + \int _{0}^{\alpha (t)}\gamma (s)\varphi \bigl( \Omega (s) \bigr){ \biggl[\varphi \bigl(\Omega (s)\bigr)+ \int _{0}^{s}\epsilon ( \lambda )\varphi \bigl( \Omega (\lambda )\bigr)\,d\lambda \biggr]}\,ds\quad \text{for all } t\in \mathbb{R}_{+},$$

where γ, $$\epsilon \in \mathcal{C}(\mathbb{R}_{+},\mathbb{R}_{+})$$ and φ, $$\varphi '$$, $$\alpha \in \mathcal{C}^{1}(\mathbb{R}_{+},\mathbb{R}_{+})$$ are increasing functions, with $$\varphi '(t)\leq k$$, $$\varphi >0$$, $$\alpha (t)\leq t$$, $$\alpha (0)=0$$ and k, $$\Omega _{0}$$ are positive constants.

In the same paper , by using the composite function, the authors introduced a new inequality with a different kernel as follows:

$$\varphi _{1}\bigl(\Omega (t)\bigr)\leq \Omega _{0} + \int _{0}^{\alpha (t)} \gamma (s)\varphi _{2} \bigl(\Omega (s)\bigr){ \biggl[\Omega (s)+ \int _{0}^{s} \epsilon (\lambda )\varphi _{1}\bigl(\Omega (\lambda )\bigr)\,d\lambda \biggr]^{p}}\,ds\quad \text{for all } t\in \mathbb{R}_{+},$$

where $$\varphi _{1}$$, $$\varphi _{2}$$, $$\alpha \in \mathcal{C}^{1}(\mathbb{R}_{+},\mathbb{R}_{+})$$ are increasing functions with $$\alpha (t)\leq t$$, $$\varphi _{i}(t)>0$$, $$i=1,2$$, $$\alpha (0)=0$$ and $$\varphi _{1}'(t)=\varphi _{2}(t)$$, $$p\geq 1$$ and $$\Omega _{0}$$ are positive constants.

In , one of the new generalizations of Gronwall type inequalities has been proved by Abdeldaim and El-Deeb, and it can be written as follows:

$$\varphi _{1}\bigl(\Omega (t)\bigr)\leq \Omega _{0} + \int _{0}^{\alpha (t)} \epsilon (s)\varphi _{1} \bigl(\Omega (s)\bigr)\,ds+ \int _{0}^{\alpha (t)} \varepsilon (s)\varphi _{2} \bigl(\Omega (s)\bigr)\,ds\quad \text{for all } t \in \mathbb{R}_{+},$$

with $$\alpha (t)\leq t$$, $$\varphi _{i}(t)>0$$, $$i=1,2$$, $$\alpha (0)=0$$, $$\varphi _{1}'(t)=\varphi _{2}(t)$$, and $$\varphi _{1}^{-1}(t)$$ is a submultiplicative function and $$\Omega _{0}$$ is a positive constant.

Recently, in 2017, El-Deeb and Ahmed  studied the following inequality with retardation $$\alpha (t)\leq t$$:

$$\Omega ^{p}(t)\leq c(t)+ \int _{a}^{\alpha (t)}\gamma (s)\Omega (s)\,ds+ \int _{a}^{b}\epsilon (s)\Omega ^{p}(s) \,ds \quad \text{for all } t \in [a,b],$$

where Ω, γ, $$\epsilon \in \mathcal{C}([a,b],\mathbb{R}_{+})$$ and α, $$c\in \mathcal{C}^{1}([a,b],\mathbb{R}_{+})$$ with $$\alpha (t)\leq t$$, $$\alpha (a)=0$$ and $$p\geq 1$$ is a constant.

Lately, in 2019, Li and Wang  established the following inequality:

$$\Omega (t)\leq a(t)+ \int _{t_{0}}^{\alpha (t)}\gamma (s) \biggl[ \Omega ^{m}(s)+ \int _{t_{0}}^{s}\epsilon (\tau )\Omega ^{n}(\tau )\,d\tau \biggr]^{p}\,ds \quad \text{for all } t\in [t_{0},+\infty ),$$

where Ω, a, γ, $$\epsilon \in \mathcal{C}(\mathbb{R}_{+},\mathbb{R}_{+})$$ and α is a continuously differentiable nondecreasing function on $$[t_{0},+\infty )$$ with $$\alpha (t)\leq t$$, $$\alpha (t_{0})=0$$ and p, m, $$n\in (0,1]$$ are positive constants.

Many generalizations, refinements, and extensions of Gronwall–Bellman type inequalities can be found in .

Stefan Hilger was the first to discover the theory of time scales which he demonstrated in his PhD thesis . For further information and details on the time scales, we refer the reader to books [19, 20]. Many dynamic inequalities have been investigated by different authors during the past decade (see  and the references cited therein). Throughout this paper, knowledge and understanding of the time scales notion and time scale calculus are assumed.

In [35, Theorem 6.4, page 256], Bohner and Peterson introduced a dynamic inequality on a time scale $$\mathbb{T}$$ which unifies the continuous version inequality (1.2) and the discrete version inequality (1.3) as follows: If Ω, ζ are right dense continuous functions and $$\gamma \geq 0$$ is a regressive and right-dense continuous function, then

$$\Omega (t)\leq \zeta (t)+ \int _{t_{0}}^{t}\Omega (\eta )\gamma (\eta ) \Delta \eta\quad \text{for all } t\in \mathbb{T}$$

implies

$$\Omega (t)\leq \zeta (t)+ \int _{t_{0}}^{t}e_{\gamma }\bigl(t,\sigma (\eta ) \bigr) \zeta (\eta )\gamma (\eta )\Delta \eta \quad \text{for all } t \in \mathbb{T}.$$

In this paper, motivated by the above-mentioned inequalities, we prove some new delay dynamic inequalities of Gronwall–Bellman–Pachpatte type on time scales. Some special cases of our results contain continuous Gronwall type inequalities and their discrete analogues. We also present some application examples to illustrate our results at the end. The paper is organized as follows: Sect. 2 contains the main results of this paper. In Sect. 3, an application to study some qualitative properties of the solutions of certain retarded dynamic equations are demonstrated. In Sect. 4, we state the conclusion.

Before we arrive at the main results in the next section, we need the following lemmas and essential relations on some time scales such as $$\mathbb{R}$$, $$\mathbb{Z}$$, $$h \mathbb{Z}$$, and $$\overline{q^{\mathbb{Z}}}$$. Note that:

1. (i)

If $$\mathbb{T}=\mathbb{R}$$, then

\begin{aligned} &\sigma (\tau )=\tau , \qquad \mu (\tau )=0,\qquad f^{\Delta }(\tau )=f'( \tau ), \\ &\int _{a}^{b}f(\tau )\Delta \tau = \int _{a}^{b}f(\tau )\,d\tau ; \end{aligned}
(1.8)
2. (ii)

If $$\mathbb{T}=\mathbb{Z}$$, then

\begin{aligned} &\sigma (\tau )=\tau +1, \qquad \mu (\tau )=1,\qquad f^{\Delta }(\tau )= \Delta f(\tau ), \\ &\int _{a}^{b}f(\tau )\Delta \tau =\sum _{\tau =a}^{b-1}f( \tau ); \end{aligned}
(1.9)
3. (iii)

If $$\mathbb{T}=\overline{q^{\mathbb{Z}}}=\{q^{k}: k\in \mathbb{Z}\}\cup \{0\}$$, $$q>1$$, then

\begin{aligned} &\sigma (\tau )=q\tau ,\qquad \mu (\tau )=(q-1)\tau , \\ &\int _{a}^{b}f( \tau )\Delta \tau =(q-1)\sum _{k=\log _{q}(a)}^{\log _{q}(b)-1}q^{k}f \bigl(q^{k}\bigr), \quad \forall a,b \in q^{\mathbb{N}_{0}}. \end{aligned}
(1.10)

If $$\lambda \in C_{rd}(\mathbb{T})$$ (see ), then the Cauchy integral $$\Lambda (\tau ) :=\int _{\tau _{0}}^{\tau } \lambda (s)\Delta s$$ exists, $$\tau _{0} \in \mathbb{T}$$, and satisfies $$\Lambda ^{\Delta }(\tau ) = \lambda (\tau )$$, $$\tau \in \mathbb{T}$$. An infinite integral follows

$$\int _{a}^{\infty }\Omega (\tau )\Delta \tau =\lim _{b\to \infty } \int _{a}^{b} \Omega (\tau )\Delta \tau .$$

The function $$\eta : \mathbb{T}\rightarrow \mathbb{R}$$ is called regressive provided $$1+\mu (t)\eta (t)\neq 0$$ for all $$t\in \mathbb{T}^{\kappa }$$. The set of all positively regressive elements of is $$\Re ^{+}=\{\eta \in \Re : 1+\mu (t)\eta (t)>0,\forall t\in \mathbb{T}\}$$. We form an Abelian group under the addition by the set of all regressive functions on a time scale $$\mathbb{T}$$ by $$\eta \oplus \zeta =\eta +\zeta +\mu \eta \zeta$$. If $$\eta \in \Re$$, then the exponential function is defined by

$$e_{\eta }(t,s)=\exp \biggl( \int _{s}^{t}\hat{\xi }_{\mu (\tau )}\bigl(\eta ( \tau )\bigr)\Delta \tau \biggr), \quad s, t \in \mathbb{T},$$

where $$\hat{\xi }_{\hat{h}}(z)$$ is the cylinder transformation, which is defined by

$$\hat{\xi }_{\hat{h}}(z)=\textstyle\begin{cases} \frac{\operatorname{Log}(1+\hat{h}z)}{\hat{h}}, & \hat{h}\neq 0, \\ z, & \hat{h}=0. \end{cases}$$

If $$\eta \in \Re$$, then $$e_{\eta }(\tau ,s)$$ is real-valued and nonzero on $$\mathbb{T}$$. If $$\eta \in \Re ^{+}$$, then $$e_{\eta }(\tau ,\tau _{0})$$ is always positive.

Note that:

• If $$\mathbb{T}=\mathbb{R}$$, then

$$e_{b}(\tau ,\tau _{0})=\exp \biggl( \int _{\tau _{0}}^{\tau }b(s)\,ds \biggr).$$
(1.11)
• If $$\mathbb{T}=\mathbb{Z}$$, then

$$e_{b}(\tau ,\tau _{0})=\prod _{s=\tau _{0}}^{\tau -1} \bigl(1+b(s) \bigr).$$
(1.12)
• If $$\mathbb{T}=q^{\mathbb{N}_{0}}$$, then

$$e_{b}(\tau ,\tau _{0})=\prod _{s=\tau _{0}}^{\tau -1} \bigl(1+(q-1)sb(s) \bigr).$$
(1.13)

### Lemma 1.1

()

If $$\eta \in \Re$$ and a, b, $$d \in \mathbb{T}$$, then

1. 1.

$$e_{\eta }(\tau ,\tau )=1$$ and $$e_{0}(\tau ,s)=1$$;

2. 2.

$$e_{\eta }(\sigma (\tau ),s)=(1+\mu (\tau )\eta (\tau ))e_{\eta }(\tau ,s)$$;

3. 3.

If $$\eta \in \Re ^{+}$$, then $$e_{\eta }(\tau ,\tau _{0})>0$$, $$\forall \tau \in \mathbb{T}$$;

4. 4.

$$\int _{a}^{b}\eta (\tau )e_{\eta }(d,\sigma (\tau ))\Delta \tau =- \int _{a}^{b}[e_{\eta }(d,\cdot )]^{\Delta }\Delta \tau =e_{\eta }(d,a)-e_{\eta }(d,b)$$.

### Lemma 1.2

(See )

Let $$\chi :\mathbb{T}\rightarrow \mathbb{R}$$ be a delta differentiable function. If $$\eta \in \Re$$ and fix $$t_{0}\in \mathbb{T}$$, then the exponential function $$e_{\eta }(t,t_{0})$$ is the unique solution of the following initial value problem:

$$\textstyle\begin{cases} \chi ^{\Delta }(t)=\eta (t)\chi (t), \\ \chi (t_{0})= 1. \end{cases}$$
(1.14)

### Lemma 1.3

(See )

Let $$t_{0}\in \mathbb{T}^{\kappa }$$ and $$\varsigma :\mathbb{T} \times \mathbb{T}^{\kappa }\rightarrow \mathbb{R}$$ be continuous at $$(t,t)$$, where $$t>t_{0}$$ and $$t\in \mathbb{T}^{\kappa }$$. Assume that $$\varsigma ^{\Delta }(t,\cdot )$$ is rd-continuous on $$[t_{0},\sigma (t)]_{\mathbb{T}}$$. If for any $$\varepsilon > 0$$ there exists a neighborhood U of t, independent of $$\lambda \in [t_{0},\sigma (t)]_{\mathbb{T}}$$, such that

$$\bigl\vert \bigl[\varsigma \bigl(\sigma (t),\lambda \bigr)-\varsigma (s, \lambda )\bigr]-\varsigma ^{\Delta }(t,\lambda )\bigl[\sigma (t)-s\bigr] \bigr\vert \leq \varepsilon \bigl\vert \sigma (t)-s \bigr\vert , \quad \forall s \in U,$$

where $$\varsigma ^{\Delta }$$ denotes the derivative of ς with respect to the first variable, then

$$\chi (t)= \int _{t_{0}}^{t}\varsigma (t,\lambda )\Delta \lambda$$

implies

$$\chi ^{\Delta }(t)= \int _{t_{0}}^{t}\varsigma ^{\Delta }(t,\lambda ) \Delta \lambda +\varsigma \bigl(\sigma (t),t\bigr).$$

### Lemma 1.4

()

Suppose χ, $$b\in C_{rd}$$, $$a\in \Re ^{+}$$, then

$$\chi ^{\Delta }(t)\leq a(t)\chi (t)+b(t),\quad t\geq t_{0}, t\in \mathbb{T}^{ \kappa },$$

implies

$$\chi (t)\leq \chi (t_{0})e_{a}(t,t_{0})+ \int _{t_{0}}^{t}e_{a}\bigl(t, \sigma (\tau ) \bigr)b(\tau )\Delta \tau ,\quad t\geq t_{0}, t\in \mathbb{T}^{ \kappa }.$$

### Lemma 1.5

()

If $$x\geq 0$$ and $$p\geq 1$$, then

$$x^{1/p}\leq m_{1}x+m_{2},$$
(1.15)

where $$m_{1}=\frac{1}{p}K^{(1-p)/p}$$, $$m_{2}=\frac{p-1}{p}K^{1/p}$$, and $$K>0$$.

### Theorem 1.6

([43, Leibniz integral rule on time scales])

In the following, by $$f^{\Delta }(t,s)$$ we mean the delta derivative of $$f(t,s)$$ with respect to t. Similarly, $$f^{\nabla }(t,s)$$ is understood. If f, $$f^{\Delta }$$, and $$f^{\nabla }$$ are continuous and $$u,h:\mathbb{T}\rightarrow \mathbb{T}$$ are delta differentiable functions, then the following formulas hold $$\forall t\in \mathbb{T^{\kappa }}$$:

1. (i)

$$[ \int ^{h(t)}_{u(t)}f(t,s)\Delta s ]^{\Delta }= \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\Delta s + h^{\Delta }(t)f(\sigma (t),h(t))- u^{\Delta }(t)f(\sigma (t),u(t))$$;

2. (ii)

$$[ \int ^{h(t)}_{u(t)}f(t,s)\Delta s ]^{\nabla }= \int ^{h(t)}_{u(t)}f^{\nabla }(t,s)\Delta s + h^{\nabla }(t)f(\rho (t),h(t))- u^{\nabla }(t)f(\rho (t),u(t))$$;

3. (iii)

$$[ \int ^{h(t)}_{u(t)}f(t,s)\nabla s ]^{\Delta }= \int ^{h(t)}_{u(t)}f^{\Delta }(t,s)\nabla s + h^{\Delta }(t)f(\sigma (t),h(t))- u^{\Delta }(t)f(\sigma (t),u(t))$$;

4. (iv)

$$[ \int ^{h(t)}_{u(t)}f(t,s)\nabla s ]^{\nabla }= \int ^{h(t)}_{u(t)}f^{\nabla }(t,s)\nabla s + h^{\nabla }(t)f(\rho (t),h(t))- u^{\nabla }(t)f(\rho (t),u(t))$$.

## Main results

In this section, the authors state and justify the main results and investigate some dynamic Gronwall–Bellman inequalities on time scales.

### Theorem 2.1

Let $$a,b\in \mathbb{T}^{k}$$ with $$a< b$$, and let , f, g, $$c \in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R}\mathbbm{_{+}})$$ and $$\alpha :\mathbb{T}\rightarrow \mathbb{T}$$. Furthermore, assume that α and c are delta-differentiable on $$\mathbb{T}$$ with $$c^{\Delta }(t)\geq 0$$, $$\alpha ^{\Delta }(t)\geq 0$$, $$\alpha (t)\leq t$$ and $$\alpha (a)=a$$. For any constant $$p\geq 1$$, if

$$\Im ^{p} (t)\leq c(t)+ \int _{a}^{\alpha (t)}g(s) \Im (s)\Delta s + \int _{a}^{b} f(s)\Im ^{p}(s)\Delta s \quad \textit{for all } t \in [a,b]_{\mathbb{T}},$$
(2.1)

then

$$\Im (t)\leq \biggl\{ \Lambda _{1}e_{\ell _{1}}(t,a)+ \int _{a}^{t}e_{ \ell _{1}}\bigl(t,\sigma (s) \bigr)\Gamma _{1}(s)\Delta s \biggr\} ^{1/p} \quad \textit{for all } t \in [a,b]_{\mathbb{T}},$$
(2.2)

where

$$\Lambda _{1}= \frac{c(a)+\int _{a}^{b}f(s) (\int _{a}^{s}e_{\ell _{1}}(s,\sigma (\lambda ))\Gamma _{1}(\lambda )\Delta \lambda ) \Delta s}{1-\int _{a}^{b}f(s)e_{\ell _{1}}(s,a)\Delta s},$$
(2.3)

such that

$$\int _{a}^{b}f(s)e_{\ell _{1}}(s,a)\Delta s< 1,$$

and

\begin{aligned}& \Gamma _{1}(t)=c^{\Delta }(t)+m_{2} \alpha ^{\Delta }(t)g\bigl(\alpha (t)\bigr), \end{aligned}
(2.4)
\begin{aligned}& \ell _{1}(t)=m_{1}\alpha ^{\Delta }(t)g \bigl(\alpha (t)\bigr), \end{aligned}
(2.5)

where $$m_{1}$$, $$m_{2}$$ are defined as in Lemma 1.5.

### Proof

Define a function $$\chi _{1}(t)$$ by

$$\chi _{1}(t)=c(t)+ \int _{a}^{\alpha (t)}g(s) \Im (s)\Delta s + \int _{a}^{b} f(s)\Im ^{p}(s)\Delta s.$$
(2.6)

We notice that $$\chi _{1}(t)\geq 0$$ and nondecreasing on $$[a,b]_{\mathbb{T}}$$. Since $$\alpha (a)=a$$, we get that

$$\chi _{1}(a)=c(a)+ \int _{a}^{b} f(s)\Im ^{p}(s)\Delta s.$$
(2.7)

Then from (2.1), (2.6) and by using the monotonicity of $$\chi _{1}(t)$$, we get

$$\Im (t) \leq \chi _{1}^{1/p}(t),$$

which implies

$$\Im \bigl(\alpha (t)\bigr) \leq \chi _{1}^{1/p} \bigl(\alpha (t)\bigr)\leq \chi _{1}^{1/p}(t).$$
(2.8)

From (2.6), (2.8) and using Theorem 1.6, we have

$$\chi _{1}^{\Delta }(t) =c^{\Delta }(t)+\alpha ^{\Delta }(t)g\bigl(\alpha (t)\bigr) \Im \bigl(\alpha (t)\bigr)\leq c^{\Delta }(t)+\alpha ^{\Delta }(t)g\bigl(\alpha (t)\bigr) \chi _{1}^{1/p}(t).$$
(2.9)

Therefore, using (2.9) and Lemma 1.5, we get that

\begin{aligned} \chi _{1}^{\Delta }(t) \leq &c^{\Delta }(t)+ m_{1}\alpha ^{\Delta }(t)g\bigl( \alpha (t)\bigr) \chi _{1}(t)+m_{2}g(t) \\ =&m_{1}\alpha ^{\Delta }(t)g\bigl(\alpha (t)\bigr) \chi _{1}(t)+\bigl[c^{\Delta }(t)+m_{2} \alpha ^{\Delta }(t)g\bigl(\alpha (t)\bigr)\bigr] \\ =&\ell _{1} \chi _{1}(t)+\Gamma _{1}(t), \end{aligned}
(2.10)

where $$\Gamma _{1}(t)$$ and $$\ell _{1}(t)$$ are defined as in (2.4) and (2.5), respectively.

Now an application of Lemma 1.4 to (2.10) yields

$$\chi _{1}(t)\leq \chi _{1}(a)e_{\ell _{1}}(t,a)+ \int _{a}^{t}e_{\ell _{1}}\bigl(t, \sigma (s) \bigr)\Gamma _{1}(s)\Delta s.$$
(2.11)

From (2.8) and (2.11), we get that

$$\Im ^{p}(t)\leq \chi _{1}(a)e_{\ell _{1}}(t,a)+ \int _{a}^{t}e_{\ell _{1}}\bigl(t, \sigma (s) \bigr)\Gamma _{1}(s)\Delta s.$$
(2.12)

From (2.7) and (2.12), we have

\begin{aligned} \chi _{1}(a) =&c(a)+ \int _{a}^{b} f(s)\Im ^{p}(s)\Delta s \\ \leq &c(a)+ \int _{a}^{b} f(s) \biggl[\chi _{1}(a)e_{\ell _{1}}(s,a)+ \int _{a}^{s}e_{\ell _{1}}\bigl(s,\sigma ( \lambda )\bigr)\Gamma _{1}(\lambda ) \Delta \lambda \biggr]\Delta s \\ \leq &c(a)+\chi _{1}(a) \int _{a}^{b} f(s)e_{\ell _{1}}(s,a)\Delta s \\ &{}+ \int _{a}^{b} f(s) \biggl( \int _{a}^{s}e_{\ell _{1}}\bigl(s,\sigma ( \lambda )\bigr)\Gamma _{1}(\lambda )\Delta \lambda \biggr)\Delta s. \end{aligned}
(2.13)

Thus, from (2.13), we obtain

$$\chi _{1}(a)\leq \Lambda _{1},$$
(2.14)

where $$\Lambda _{1}$$ is defined as in (2.3).

Then we get the desired inequality (2.2) by combining (2.12) and (2.14). This completes the proof. □

### Remark 2.2

If we take $$\mathbb{T}=\mathbb{R}$$, $$\alpha (t)=t$$, and $$p=1$$, then, using relations (1.8), Theorem 2.1 reduces to [44, Theorem 1.5.1].

### Remark 2.3

If we take $$\mathbb{T}=\mathbb{R}$$ and $$\alpha (t)=t$$, then, using relations (1.8), Theorem 2.1 reduces to [45, Theorem 2.1].

### Remark 2.4

If we take $$\mathbb{T}=\mathbb{R}$$, then, using relations (1.8), Theorem 2.1 reduces to [9, Theorem 2.1].

As a special case of Theorem 2.1, if we take $$\mathbb{T}=\mathbb{Z}$$ and the delay function $$\alpha (n)=n-\tau$$, where $$\tau >0$$, and so $$\Delta \alpha (n)=1>0$$, then, using relations (1.9) and (1.12), we obtain the following completely new discrete result.

### Corollary 2.5

Assume that $$\Im (n)$$, $$g(n)$$, $$c(n)$$, and $$f(n)$$ are nonnegative sequences defined for $$n\in \mathbb{N}_{0}$$, with $$\Delta c(n)\geq 0$$ for $$n\in \mathbb{N}_{0}$$. If $$\Im (n)$$ satisfies the following delay discrete inequality:

\begin{aligned} \Im ^{p}(n) \leq & c(n)+\sum_{s=a}^{n-\tau -1}g(s) \Im (s)+\sum_{s=a}^{b-1}f(s) \Im ^{p}(s), \end{aligned}

then

\begin{aligned} \Im (n) \leq & \Biggl\{ \tilde{\Lambda }_{1}\prod _{s=a}^{n-1} \bigl(1+ \hat{\ell }_{1}(s) \bigr)+\sum_{s=a}^{n-1} \hat{\Gamma }_{1}(s)\prod_{ \lambda =s+1}^{n-1} \bigl(1+ \hat{\ell }_{1}(\lambda ) \bigr) \Biggr\} ^{1/p}, \end{aligned}

where

$$\hat{\Lambda }_{1}= \frac{c(a)+\sum_{s=a}^{b-1}f(s) [\sum_{\lambda =a}^{s-1}\hat{\Gamma }_{1} (\lambda )\prod_{\upsilon =\lambda +1}^{s-1}(1+\hat{\ell }_{1}(\upsilon )) ]}{1-\sum_{s=a}^{b-1}f(s)\prod_{\lambda =a}^{s-1}(1+\hat{\ell }_{1}(\lambda ))},$$

such that

$$\sum_{s=a}^{b-1}f(s)\prod _{\lambda =a}^{s-1} \bigl(1+\hat{\ell }_{1}( \lambda ) \bigr)< 1,$$

and

\begin{aligned}& \hat{\Gamma }_{1}(n) = \Delta c(n)+m_{2}\Delta (n-\tau )g(n-\tau ) \\& \hphantom{\hat{\Gamma }_{1}(n)} = c(n+1)-c(n)+m_{2}g(n-\tau ), \\& \hat{\ell }_{1}(n) = m_{1}\Delta (n-\tau )g(n-\tau ) \\& \hphantom{\hat{\ell }_{1}(n)}= m_{1}g(n-\tau ). \end{aligned}

### Theorem 2.6

Let $$a,b\in \mathbb{T}^{k}$$ with $$a< b$$, and let , $$g, c \in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R}\mathbbm{_{+}})$$ and $$\alpha :\mathbb{T}\rightarrow \mathbb{T}$$. Further, assume that α and c are delta-differentiable on $$\mathbb{T}$$ with $$c^{\Delta }(t)\geq 0$$, $$\alpha ^{\Delta }(t)\geq 0$$, $$\alpha (t)\leq t$$, and $$\alpha (a)=a$$. Moreover, assume that $$k(t,s)$$, $$k^{\Delta }(t,s) \in C_{rd}([a,b]_{\mathbb{T}}\times [a,b]_{\mathbb{T}}, \mathbb{R}_{+})$$ for $$a\leq s\leq t\leq b$$. For any constant $$p\geq 1$$, if

$$\Im ^{p} (t)\leq c(t)+ \int _{a}^{\alpha (t)}k(t,s) \Im (s)\Delta s + \int _{a}^{b} g(s)\Im ^{p}(s)\Delta s \quad \textit{for all } t\in [a,b]_{ \mathbb{T}},$$
(2.15)

then

$$\Im (t)\leq \biggl\{ \Lambda _{2}e_{\ell _{2}}(t,a)+ \int _{a}^{t} \Gamma _{2}(s)e_{\ell _{2}} \bigl(t,\sigma (s)\bigr)\Delta s \biggr\} ^{1/p} \quad \textit{for all } t \in [a,b]_{\mathbb{T}},$$
(2.16)

where

$$\Lambda _{2}= \frac{c(a)+\int _{a}^{b} g(s) (\int _{a}^{s}e_{\ell _{2}}(s,\sigma (\tau ))\Gamma _{2}(\tau )\Delta \tau )\Delta s}{1-\int _{a}^{b} g(s)e_{\ell _{2}}(s,a)\Delta s},$$

such that

$$\int _{a}^{b}g(s)e_{\ell _{2}}(s,a)\Delta s< 1,$$

and

\begin{aligned}& \Gamma _{2}(t)=c^{\Delta }(t)+m_{2} \biggl[\alpha ^{\Delta }(t)k\bigl(\sigma (t), \alpha (t)\bigr) + \int _{a}^{\alpha (t)}k^{\Delta }(t,\tau )\Delta \tau \biggr], \end{aligned}
(2.17)
\begin{aligned}& \ell _{2}(t)= m_{1} \biggl[\alpha ^{\Delta }(t)k\bigl(\sigma (t),\alpha (t)\bigr) + \int _{a}^{\alpha (t)}k^{\Delta }(t,\tau )\Delta \tau \biggr], \end{aligned}
(2.18)

where $$m_{1}$$, $$m_{2}$$ are defined as in Lemma 1.5.

### Proof

Define a function $$\chi _{2}(t)$$ by

$$\chi _{2}(t)=c(t)+ \int _{a}^{\alpha (t)}k(t,s) \Im (s)\Delta s + \int _{a}^{b} g(s)\Im ^{p}(s)\Delta s.$$
(2.19)

Clearly, $$\chi _{2}(t)$$ is nonnegative nondecreasing on $$[a,b]_{\mathbb{T}}$$. As $$\alpha (a)=a$$, we have

$$\chi _{2}(a)=c(a)+ \int _{a}^{b} f(s)\Im ^{p}(s)\Delta s.$$
(2.20)

Then from (2.15), (2.19) and by using the monotonicity of $$\chi _{2}(t)$$, we obtain

$$\Im (t) \leq \chi _{2}^{1/p}(t),\qquad \Im \bigl( \alpha (t)\bigr) \leq \chi _{2}^{1/p}\bigl( \alpha (t)\bigr) \leq \chi _{2}^{1/p}(t).$$
(2.21)

Using Theorem 1.6 to delta differentiating (2.19) and from (2.21), we get

\begin{aligned} \chi _{2}^{\Delta }(t) =&c^{\Delta }(t)+ \alpha ^{\Delta }(t)k\bigl(\sigma (t), \alpha (t)\bigr)\Im \bigl(\alpha (t) \bigr) + \int _{a}^{\alpha (t)}k^{\Delta }(t,\tau ) \Im (\tau ) \Delta \tau \\ \leq &c^{\Delta }(t)+\alpha ^{\Delta }(t)k\bigl(\sigma (t),\alpha (t) \bigr)\chi _{2}^{1/p}(t) + \int _{a}^{\alpha (t)}k^{\Delta }(t,\tau )\chi _{2}^{1/p}(\tau ) \Delta \tau \\ \leq &c^{\Delta }(t)+ \biggl[\alpha ^{\Delta }(t)k\bigl(\sigma (t), \alpha (t)\bigr) + \int _{a}^{\alpha (t)}k^{\Delta }(t,\tau )\Delta \tau \biggr]\chi _{2}^{1/p}(t). \end{aligned}
(2.22)

Using Lemma 1.5, inequality (2.22) can be rewritten as

\begin{aligned} \chi _{2}^{\Delta }(t) \leq &c^{\Delta }(t)+ m_{1} \biggl[\alpha ^{ \Delta }(t)k\bigl( \sigma (t),\alpha (t)\bigr) + \int _{a}^{\alpha (t)}k^{\Delta }(t, \tau )\Delta \tau \biggr]\chi _{2}(t) \\ &{}+m_{2} \biggl[\alpha ^{\Delta }(t)k\bigl(\sigma (t),\alpha (t) \bigr) + \int _{a}^{ \alpha (t)}k^{\Delta }(t,\tau )\Delta \tau \biggr] \\ =&\ell _{2}(t) \chi _{2}(t)+\Gamma _{2}(t), \end{aligned}
(2.23)

where $$\ell _{2}(t)$$ and $$\Gamma _{2}(t)$$ are defined as in (2.18) and (2.17), respectively.

Now, applying Lemma 1.4 to (2.23) yields

$$\chi _{2}(t)\leq \chi _{2}(a)e_{\ell _{2}}(t,a)+ \int _{a}^{t}e_{\ell _{2}}\bigl(t, \sigma (\tau ) \bigr)\Gamma _{2}(s)\Delta \tau .$$
(2.24)

From (2.21) and (2.24), we get that

$$\Im ^{p}(t)\leq \chi _{2}(a)e_{\ell _{2}}(t,a)+ \int _{a}^{t}e_{\ell _{2}}\bigl(t, \sigma (\tau ) \bigr)\Gamma _{2}(\tau )\Delta \tau .$$
(2.25)

From (2.20) and (2.25), we have

\begin{aligned} \chi _{2}(a) =&c(a)+ \int _{a}^{b} g(s)\Im ^{p}(s)\Delta s \\ \leq &c(a)+ \int _{a}^{b} g(s) \biggl[\chi _{2}(a)e_{\ell _{2}}(s,a)+ \int _{a}^{s}e_{\ell _{2}}\bigl(s,\sigma (\tau ) \bigr)\Gamma _{2}(\tau )\Delta \tau \biggr]\Delta s \\ \leq &c(a)+\chi _{2}(a) \int _{a}^{b} g(s)e_{\ell _{2}}(s,a)\Delta s \\ &{}+ \int _{a}^{b} g(s) \biggl( \int _{a}^{s}e_{\ell _{2}}\bigl(s,\sigma ( \tau )\bigr)\Gamma _{2}(\tau )\Delta \tau \biggr)\Delta s. \end{aligned}
(2.26)

Thus, from (2.26) we obtain

$$\chi _{2}(a)\leq \frac{c(a)+\int _{a}^{b} g(s) (\int _{a}^{s}e_{\ell _{2}}(s,\sigma (\tau ))\Gamma _{2}(\tau )\Delta \tau )\Delta s}{1-\int _{a}^{b} g(s)e_{\ell _{2}}(s,a)\Delta s}= \Lambda _{2}.$$
(2.27)

Our desired result (2.16) follows directly from (2.25) and (2.27). This concludes the proof. □

### Remark 2.7

If we take $$\mathbb{T}=\mathbb{R}$$, $$\alpha (t)=t$$, and $$p=1$$, then, using relations (1.8), Theorem 2.6 reduces to [44, Theorem 1.5.2 $$(b_{1})$$].

### Remark 2.8

If we take $$\mathbb{T}=\mathbb{R}$$ and $$\alpha (t)=t$$, then, using relations (1.8), Theorem 2.6 reduces to [45, Theorem 2.3].

### Remark 2.9

If we take $$\mathbb{T}=\mathbb{R}$$, then, using relations (1.8), Theorem 2.6 reduces to [9, Theorem 2.2].

As a special case of Theorem 2.6, if we take $$\mathbb{T}=\mathbb{Z}$$ and the delay function $$\alpha (n)=n-\tau$$, where $$\tau >0$$, and so $$\Delta \alpha (n)=1>0$$, then, using relations (1.9) and (1.12), we obtain the following completely new discrete result.

### Corollary 2.10

Assume that $$\Im (n)$$, $$g(n)$$, $$c(n)$$ $$\alpha (n)$$, and $$f(n)$$ are nonnegative sequences defined for $$t\in \mathbb{N}_{0}$$, with $$\Delta c(n)\geq 0$$ and $$k(n,s)$$, $$\Delta k(n,s)$$ are nonnegative sequences defined on $$E=\{(m,n)\in \mathbb{N}_{0}^{2}: 0\leq n \leq m< \infty \}$$. If $$\Im (n)$$ satisfies the following delay discrete inequality

\begin{aligned} \Im ^{p}(n) \leq & c(n)+\sum_{s=a}^{n-\tau -1}k(n,s) \Im (s)+\sum_{s=a}^{b-1}g(s) \Im ^{p}(s), \end{aligned}

then

\begin{aligned} \Im (n) \leq & \Biggl\{ \hat{\Lambda }_{2}\prod _{s=a}^{n-1} \bigl(1+ \hat{\ell }_{2}(s) \bigr)+\sum_{s=a}^{n-1}\hat{\Gamma }_{2}(s) \prod_{ \tau =s+1}^{n-1} \bigl(1+ \hat{\ell }_{2}(\tau ) \bigr) \Biggr\} ^{1/p}, \end{aligned}

where

$$\hat{\Lambda }_{2}= \frac{c(a)+\sum_{s=a}^{b-1}g(s) [\sum_{\lambda =a}^{s-1}\hat{\Gamma }_{2}(\lambda ) \prod_{\tau =\lambda +1}^{s-1}(1+\hat{\ell }_{2}(\tau )) ]}{1-\sum_{s=a}^{b-1}g(s)\prod_{\lambda =\upsilon }^{s-1}(1+\hat{\ell }_{2}(\lambda ))},$$

such that

$$\sum_{s=a}^{b-1}g(s)\prod _{\lambda =a}^{s-1} \bigl(1+\hat{\ell }_{2}( \lambda ) \bigr)< 1,$$

and

\begin{aligned}& \hat{\Gamma }_{2}(t) = \Delta c(n)+m_{2} \Biggl[\Delta (n-\tau )k(n+1,n- \tau )+\sum_{s=a}^{n-\tau -1} \Delta k(n,s) \Biggr] \\& \hphantom{\hat{\Gamma }_{2}(t)} = c(n+1)-c(n)+m_{2} \Biggl\{ k(n+1,n-\tau ) \\& \hphantom{\hat{\Gamma }_{2}(t)=} {}+\sum_{s=a}^{n-\tau -1}\bigl[k(n+1,s)-k(n,s) \bigr] \Biggr\} , \\& \hat{\ell }_{2}(t) = m_{1} \Biggl[\Delta (n-\tau )k(n+1,n-\tau )+ \sum_{s=a}^{n-\tau -1}\Delta k(n,s) \Biggr] \\& \hphantom{\hat{\ell }_{2}(t)} = m_{1} \Biggl\{ k(n+1,n-\tau )+\sum_{s=a}^{n-\tau -1} \bigl[k(n+1,s)-k(n,s)\bigr] \Biggr\} . \end{aligned}

### Theorem 2.11

Assume that $$a, b\in \mathbb{T}^{k}$$ with $$a< b$$, and let , α, and c be defined as in Theorem 2.6. Further, suppose that $$k_{1}(t,s)$$, $$k_{2}(t,s)$$, $$k_{1}^{\Delta }(t,s)$$, and $$k_{2}^{\Delta }(t,s) \in C_{rd}([a,b]_{\mathbb{T}}\times [a,b]_{\mathbb{T}}, \mathbb{R}\mathbbm{_{+}})$$ for $$a\leq s\leq t\leq b$$. For any constant $$p\geq 1$$, if

$$\Im ^{p} (t)\leq c(t)+ \int _{a}^{\alpha (t)}k_{1}(t,s) \Im (s)\Delta s + \int _{a}^{b} k_{2}(t,s)\Im ^{p}(s)\Delta s \quad \textit{for all } t \in [a,b]_{\mathbb{T}},$$
(2.28)

then

$$\Im (t)\leq \biggl\{ \Lambda _{3}e_{\ell _{3}}(t,a)+ \int _{a}^{t} \Gamma _{3}(s)e_{\ell _{3}} \bigl(t,\sigma (s)\bigr)\Delta s \biggr\} ^{1/p} \quad \textit{for all } t \in [a,b]_{\mathbb{T}},$$
(2.29)

where

$$\Lambda _{3}= \frac{c(a)+\int _{a}^{b} k_{2}(a,s) (\int _{a}^{s}\Gamma _{3}(\lambda )e_{\ell _{3}}(s,\sigma (\lambda ))\Delta \lambda )\Delta s}{1-\int _{a}^{b} k_{2}(a,s)e_{\ell _{3}}(s,a)\Delta s},$$
(2.30)

such that

$$\int _{a}^{b} k_{2}(s,a)e_{\ell _{3}}(s,a) \Delta s< 1,$$
(2.31)

and

\begin{aligned}& \ell _{3}(t)=m_{1} \biggl[\alpha ^{\Delta }(t)k_{1} \bigl(\sigma (t),\alpha (t)\bigr)+ \int _{a}^{\alpha (t)}k_{1}^{\Delta }(t,s) \Delta s \biggr] + \int _{a}^{b}k_{2}^{\Delta }(t,s) \Delta s, \\& \Gamma _{3}(t)=c^{\Delta }(t)+m_{2} \biggl[\alpha ^{\Delta }(t)k_{1}\bigl( \sigma (t),\alpha (t)\bigr)+ \int _{a}^{\alpha (t)}k_{1}^{\Delta }(t,s) \Delta s \biggr], \end{aligned}

where $$m_{1}$$, $$m_{2}$$ are defined as in Lemma 1.5.

### Proof

Define a function $$\chi _{3}(t)$$ by

$$\chi _{3}(t)=c(t)+ \int _{a}^{\alpha (t)}k_{1}(t,s) \Im (s)\Delta s + \int _{a}^{b}k_{2}(t,s)\Im ^{p}(s)\Delta s.$$
(2.32)

We notice that $$\chi _{3}(t)$$ is nonnegative nondecreasing on $$[a,b]_{\mathbb{T}}$$. Since $$\alpha (a)=a$$, we get that

$$\chi _{3}(a)=c(a)+ \int _{a}^{b} f(s)\Im ^{p}(s)\Delta s.$$
(2.33)

Then from (2.28), (2.32) and by using the monotonicity of $$\chi _{1}(t)$$, we obtain

$$\Im (t) \leq \chi _{3}^{1/p}(t),$$

which implies

$$\Im \bigl(\alpha (t)\bigr) \leq \chi _{3}^{1/p} \bigl(\alpha (t)\bigr)\leq \chi _{3}^{1/p}(t).$$
(2.34)

From (2.32), (2.34) and by using Theorem 1.6, we have

\begin{aligned} \chi _{3}^{\Delta }(t) =&c^{\Delta }(t)+ \alpha ^{\Delta }(t)k_{1}\bigl(\sigma (t), \alpha (t)\bigr) \Im \bigl(\alpha (t)\bigr)+ \int _{a}^{\alpha (t)}k_{1}^{\Delta }(t,s) \Im (s)\Delta s \\ &{}+ \int _{a}^{b}k_{2}^{\Delta }(t,s) \Im ^{p}(s)\Delta s \\ \leq &c^{\Delta }(t)+\alpha ^{\Delta }(t)k_{1}\bigl(\sigma (t),\alpha (t)\bigr) \chi _{3}^{1/p}(t)+ \int _{a}^{\alpha (t)}k_{1}^{\Delta }(t,s)\chi _{3}^{1/p}(s) \Delta s \\ &{}+ \int _{a}^{b}k_{2}^{\Delta }(t,s) \chi _{3}(s)\Delta s \\ \leq &c^{\Delta }(t)+ \biggl[\alpha ^{\Delta }(t)k_{1} \bigl(\sigma (t), \alpha (t)\bigr)+ \int _{a}^{\alpha (t)}k_{1}^{\Delta }(t,s) \Delta s \biggr] \chi _{3}^{1/p}(t) \\ &{}+ \biggl( \int _{a}^{b}k_{2}^{\Delta }(t,s) \Delta s \biggr)\chi _{3}(t). \end{aligned}
(2.35)

By applying Lemma 1.5 to (2.35), we get

\begin{aligned} \chi _{3}^{\Delta }(t) \leq & c^{\Delta }(t)+ m_{1} \biggl[\alpha ^{ \Delta }(t)k_{1} \bigl(\sigma (t),\alpha (t)\bigr)+ \int _{a}^{\alpha (t)}k_{1}^{\Delta }(t,s) \Delta s \biggr]\chi _{3}(t) \\ &{}+ m_{2} \biggl[\alpha ^{\Delta }(t)k_{1}\bigl(\sigma (t),\alpha (t)\bigr)+ \int _{a}^{ \alpha (t)}k_{1}^{\Delta }(t,s) \Delta s \biggr] \\ &{}+ \biggl( \int _{a}^{b}k_{2}^{\Delta }(t,s) \Delta s \biggr)\chi _{3}(t) \\ \leq & \biggl\{ m_{1} \biggl[\alpha ^{\Delta }(t)k_{1} \bigl(\sigma (t), \alpha (t)\bigr)+ \int _{a}^{\alpha (t)}k_{1}^{\Delta }(t,s) \Delta s \biggr] \\ &{}+ \int _{a}^{b}k_{2}^{\Delta }(t,s) \Delta s \biggr\} \chi _{3}(t) \\ &{}+c^{\Delta }(t)+ m_{2} \biggl[\alpha ^{\Delta }(t)k_{1} \bigl(\sigma (t), \alpha (t)\bigr)+ \int _{a}^{\alpha (t)}k_{1}^{\Delta }(t,s) \Delta s \biggr] \\ =&\ell _{3}(t)\chi _{3}(t)+\Gamma _{3}(t). \end{aligned}
(2.36)

Therefore, using Lemma (1.4) in (2.36), we get that

$$\chi _{3}(t)\leq \chi _{3}(a)e_{\ell _{3}}(t,a)+ \int _{a}^{t}\Gamma _{3}(s)e_{ \ell _{3}} \bigl(t,\sigma (s)\bigr)\Delta s.$$
(2.37)

Combining (2.34) and (2.37) yields

$$\Im ^{p}(t)\leq \chi _{3}(a)e_{\ell _{3}}(t,a)+ \int _{a}^{t}\Gamma _{3}(s)e_{ \ell _{3}} \bigl(t,\sigma (s)\bigr)\Delta s.$$
(2.38)

From (2.33) and (2.38), we have

\begin{aligned} \chi _{3}(a) \leq &c(a)+ \int _{a}^{b} k_{2}(a,s) \biggl[\chi _{3}(a)e_{ \ell _{3}}(s,a) \\ &{}+ \int _{a}^{s}\Gamma _{3}(\lambda )e_{\ell _{3}}\bigl(s,\sigma (\lambda )\bigr) \Delta \lambda \biggr]\Delta s \\ \leq &c(a)+\chi _{3}(a) \int _{a}^{b} k_{2}(a,s)e_{\ell _{3}}(s,a) \Delta s \\ &{}+ \int _{a}^{b} k_{2}(a,s) \biggl( \int _{a}^{s}\Gamma _{3}(\lambda )e_{ \ell _{3}}\bigl(s, \sigma (\lambda )\bigr)\Delta \lambda \biggr)\Delta s. \end{aligned}
(2.39)

Therefore, from (2.39) we obtain

$$\chi _{3}(a)\leq \Lambda _{3},$$
(2.40)

where $$\Lambda _{3}$$ is defined as in (2.30).

We obtain the desired inequality (2.29) by combining (2.38) and (2.40). The proof is complete. □

### Remark 2.12

If we take $$\mathbb{T}=\mathbb{R}$$, $$\alpha (t)=t$$, $$p=1$$, then, using relations (1.8), Theorem 2.6 reduces to [44, Theorem 1.5.2 $$(b_{2})$$].

### Remark 2.13

If we take $$\mathbb{T}=\mathbb{R}$$ and $$\alpha (t)=t$$, then, using relations (1.8), Theorem 2.6 reduces to [45, Theorem 2.3].

### Remark 2.14

If we take $$\mathbb{T}=\mathbb{R}$$, then, using relations (1.8), Theorem 2.6 reduces to [9, Theorem 2.3].

As a special case of Theorem 2.11, if we take $$\mathbb{T}=\mathbb{Z}$$ and the delay function $$\alpha (n)=n-\tau$$, where $$\tau >0$$, and so $$\Delta \alpha (n)=1>0$$, then, using relations (1.9) and (1.12), we obtain the following completely new discrete result.

### Corollary 2.15

Assume that $$\Im (n)$$, $$g(n)$$, $$c(n)$$, and $$\alpha (n)$$ are nonnegative sequences defined for $$t\in \mathbb{N}_{0}$$, with $$\Delta c(n)\geq 0$$ and $$k_{1}(n,s)$$, $$k_{2}(n,s)$$, $$\Delta k_{1}(n,s)$$, $$\Delta k_{2}(n,s)$$ are nonnegative sequences defined on $$E=\{(m,n)\in \mathbb{N}_{0}^{2}: 0\leq n \leq m< \infty \}$$. If $$\Im (n)$$ satisfies the following delay discrete inequality:

\begin{aligned} \Im ^{p}(n) \leq & c(n)+\sum_{s=a}^{n-1}k_{1}(n,s- \tau )\Im (s-\tau )+ \sum_{s=a}^{b-1}k_{2}(n,s) \Im ^{p}(s), \end{aligned}

then

\begin{aligned} \Im (n) \leq & \Biggl\{ \hat{\Lambda }_{3}\prod _{s=a}^{n-1} \bigl(1+ \hat{\ell }_{3}(s) \bigr) )+\sum_{s=a}^{n-1}\hat{\Gamma }_{3}(s) \prod_{ \lambda =s+1}^{n-1} \bigl(1+ \hat{\ell }_{3}(\lambda ) \bigr) \Biggr\} ^{1/p}, \end{aligned}

where

$$\hat{\Lambda }_{3}= \frac{c(a)+\sum_{s=a}^{b-1}k_{2}(a,s) [\sum_{\lambda =a}^{s-1}\hat{\Gamma }_{3}(\lambda ) \prod_{\lambda =\upsilon +1}^{s-1}(1+\hat{\ell }_{3}(\lambda )) ]}{1-\sum_{s=a}^{b-1}k_{2}(a,s)\prod_{\lambda =a}^{s-1}(1+\hat{\ell }_{3}(\lambda )},$$

such that

$$\sum_{s=a}^{b-1}k_{2}(a,s)\prod _{\lambda =a}^{s-1} \bigl(1+ \hat{\ell }_{3}(\lambda ) \bigr)< 1,$$

and

\begin{aligned}& \hat{\ell }_{3}(n) = m_{1} \Biggl[\Delta (n-\tau )k_{1}(n+1,n-\tau )+ \sum_{s=a}^{n-1} \Delta k_{1}(n,s) \Biggr] \\& \hphantom{\hat{\ell }_{3}(n) =}{}+\sum_{s=a}^{b-1}\Delta k_{2}(n,s), , \\& \hphantom{\hat{\ell }_{3}(n) } = m_{1} \Biggl[k_{1}(n+1,n-\tau )+\sum _{s=a}^{n-1}\bigl[k_{1}(n+1,s)-k_{1}(n,s) \bigr] \Biggr] \\& \hphantom{\hat{\ell }_{3}(n) =} {}+\sum_{s=a}^{b-1}k_{2}(n+1,s)-k_{2}(n,s), \\& \hat{\Gamma }_{3}(t) = \Delta c(n)+m_{2} \Biggl[\Delta (n-\tau )k_{1}(n+1,n- \tau )+\sum_{s=a}^{n-1} \Delta k_{1}(n,s) \Biggr] \\& \hphantom{\hat{\Gamma }_{3}(t)} = c(n+1)-c(n)+m_{2} \Biggl[k_{1}(n+1,n-\tau )+\sum _{s=a}^{n-1}\bigl[k_{1}(n+1,s)-k_{1}(n,s) \bigr] \Biggr]. \end{aligned}

## Applications

In this section, by using Theorem 2.11, we demonstrate the global existence of solutions for a class of nonlinear retarded dynamic integral equations of the form

\begin{aligned} &\Im ^{p}(t)=h(t)+\Upsilon \biggl(t, \int _{a}^{\alpha (t)}\Psi _{1}\bigl(s, \Im (s),k_{1}\bigr)\Delta s, \int _{a}^{b}\Psi _{2}\bigl(s,\Im ^{p}(s),k_{2}\bigr) \Delta s \biggr), \\ &\Im ^{p}(a)= \tilde{r}, \end{aligned}
(3.1)

where $$\Upsilon \in C_{rd}([a,b]_{\mathbb{T}}\times \mathbb{R}\mathbbm{_{+}}\times \mathbb{R}\mathbbm{_{+}},\mathbb{R}\mathbbm{_{+}})$$.

Now, in the following theorem, we obtain the explicit estimates for the solution of (3.1).

### Theorem 3.1

Consider the retarded dynamic integral equation (3.1), and assume the following:

\begin{aligned}& \begin{aligned} &\bigl\vert h(t) \bigr\vert \leq c(t), \\ &\bigl\vert \Upsilon _{1}(t,u,\tilde{\nu }) \bigr\vert \leq \vert u \vert + \vert \tilde{\nu } \vert , \\ &\vert \Psi _{1} \vert \leq k_{1}(t,s)\Im (s), \\ &\vert \Psi _{2} \vert \leq k_{2}(t,s)\Im (s), \end{aligned} \end{aligned}
(3.2)

where , c, h, $$\in C_{rd}([a,b]_{\mathbb{T}},\mathbb{R}\mathbbm{_{+}})$$, c is delta-differentiable on $$\mathbb{T}^{k}$$ with $$c^{\Delta }(t)\geq 0$$, $$k_{1}(t,s)$$, $$k_{1}^{\Delta }(t,s)$$, $$k_{2}(t,s)$$, $$k_{2}^{\Delta }(t,s) \in C_{rd}([a,b]_{\mathbb{T}}\times [a,b]_{\mathbb{T}}, \mathbb{R}\mathbbm{_{+}})$$ for $$a\leq s\leq t\leq b$$ and $$p\geq 1$$ is a constant. Then we have the explicit bound estimation for the solution of (3.1) as follows:

$$\Im (t)\leq \biggl\{ \Lambda _{3}e_{\ell _{3}}(t,a)+ \int _{a}^{t} \Gamma _{3}(s)e_{\ell _{3}} \bigl(t,\sigma (s)\bigr)\Delta s \biggr\} ^{1/p},$$
(3.3)

where

$$\Lambda _{3}= \frac{c(a)+\int _{a}^{b} k_{2}(a,s) (\int _{a}^{s}\Gamma _{3}(\lambda )e_{\ell _{3}}(s,\sigma (\lambda ))\Delta \lambda )\Delta s}{1-\int _{a}^{b} k_{2}(a,s)e_{\ell _{3}}(s,a)\Delta s},$$

such that

$$\int _{a}^{b} k_{2}(s,a)e_{\ell _{3}}(s,a) \Delta s< 1,$$

and

\begin{aligned}& \ell _{3}(t)=m_{1} \biggl[\alpha ^{\Delta }(t)k_{1} \bigl(\sigma (t),\alpha (t)\bigr)+ \int _{a}^{\alpha (t)}k_{1}^{\Delta }(t,s) \Delta s \biggr] + \int _{a}^{b}k_{2}^{\Delta }(t,s) \Delta s, \\& \Gamma _{3}(t)=c^{\Delta }(t)+m_{2} \biggl[\alpha ^{\Delta }(t)k_{1}\bigl( \sigma (t),\alpha (t)\bigr)+ \int _{a}^{\alpha (t)}k_{1}^{\Delta }(t,s) \Delta s \biggr], \end{aligned}

where $$m_{1}$$, $$m_{2}$$ are defined as in Lemma 1.5.

### Proof

From (3.1) and (3.2), we have

$$\bigl\vert \Im (t) \bigr\vert ^{p}\leq c(t)+ \int _{a}^{t}k_{1}(t,s) \bigl\vert \Im (s) \bigr\vert \Delta s+ \int _{a}^{b}k_{2}(t,s) \bigl\vert \Im (s) \bigr\vert ^{p}\Delta s.$$
(3.4)

Now, applying Theorem 2.11 to inequality (3.4), we get

$$\Im (t)\leq \biggl\{ \Lambda _{3}e_{\ell _{3}}(t,a)+ \int _{a}^{t} \Gamma _{3}(s)e_{\ell _{3}} \bigl(t,\sigma (s)\bigr)\Delta s \biggr\} ^{1/p},$$

which is the desired estimation in (3.3). This completes the proof. □

## Conclusion

First, we introduced Theorem 1.6 which was needed in the proofs of the rest of results. Second, we generalized a number of Gronwall–Pachpatte type inequalities, in two independent variables, to a general time scale. We applied our results to study the uniqueness and global existence of solutions for a class of nonlinear retarded Volterra–Fredholm dynamic integral equations.

Not applicable.

## References

1. Gronwall, T.H.: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. (2) 20(4), 292–296 (1919)

2. Bellman, R.: The stability of solutions of linear differential equations. Duke Math. J. 10, 643–647 (1943)

3. Bellman, R.: Asymptotic series for the solutions of linear differential-difference equations. Rend. Circ. Mat. Palermo 7, 1–9 (1958)

4. Pachpatte, B.G.: On some fundamental integral inequalities and their discrete analogues. J. Inequal. Pure Appl. Math. 2(2), 1–13 (2001)

5. Pachpatte, B.G.: On some new inequalities related to a certain inequality arising in the theory of differential equations. J. Math. Anal. Appl. 251(2), 736–751 (2000)

6. Abdeldaim, A., El-Deeb, A.A.: Some new retarded nonlinear integral inequalities with iterated integrals and their applications in retarded differential equations and integral equations. J. Fract. Calc. Appl. 5(suppl. 3S), 9 (2014)

7. Abdeldaim, A., El-Deeb, A.A.: On some generalizations of certain retarded nonlinear integral inequalities with iterated integrals and an application in retarded differential equation. J. Egypt. Math. Soc. 23(3), 470–475 (2015)

8. Abdeldaim, A., El-Deeb, A.A.: On generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro-differential equations. Appl. Math. Comput. 256, 375–380 (2015)

9. El-Deeb, A.A., Ahmed, R.G.: On some generalizations of certain nonlinear retarded integral inequalities for Volterra–Fredholm integral equations and their applications in delay differential equations. J. Egypt. Math. Soc. 25(3), 279–285 (2017)

10. Li, Z., Wang, W.-S.: Some nonlinear Gronwall–Bellman type retarded integral inequalities with power and their applications. Appl. Math. Comput. 347(2), 839–852 (2019)

11. El-Deeb, A.A.: A variety of nonlinear retarded integral inequalities of Gronwall type and their applications. In: Advances in Mathematical Inequalities and Applications, pp. 143–164. Springer, Berlin (2018)

12. El-Deeb, A.A., Ahmed, R.G.: On some explicit bounds on certain retarded nonlinear integral inequalities with applications. Adv. Inequal. Appl. 2016, Article ID 15 (2016)

13. El-Owaidy, H.M., Ragab, A.A., Eldeeb, A.A., Abuelela, W.M.K.: On some new nonlinear integral inequalities of Gronwall–Bellman type. Kyungpook Math. J. 54(4), 555–575 (2014)

14. Abdeldaim, A., El-Deeb, A.A.: On some new nonlinear retarded integral inequalities with iterated integrals and their applications in integro-differential equations. Br. J. Math. Comput. Sci. 5(4), 479–491 (2015)

15. El-Owaidy, H., Abdeldaim, A., El-Deeb, A.A.: On some new retarded nonlinear integral inequalities and their applications. Math. Sci. Lett. 3(3), 157 (2014)

16. Abdeldaim, A., El-Deeb, A.A.: Some new retarded nonlinear integral inequalities with iterated integrals and their applications in retarded differential equations and integral equations. J. Fract. Calc. Appl. 5(suppl. 3S), 9 (2014)

17. El-Deeb, A.A.: On Integral Inequalities and their Applications Lambert. LAP Lambert Acad. Pub. (2017)

18. Hilger, S.: Analysis on measure chains–a unified approach to continuous and discrete calculus. Results Math. 18(1–2), 18–56 (1990)

19. Bohner, M., Peterson, A.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)

20. Bohner, M., Erbe, L., Peterson, A.: Oscillation for nonlinear second order dynamic equations on a time scale. J. Math. Anal. Appl. 301(2), 491–507 (2005)

21. Saker, S.H., Osman, M.M., O’Regan, D., Agarwal, R.P.: Hardy-type operators with general kernels and characterizations of dynamic weighted inequalities. In: Annales Polonici Mathematici, vol. 126, pp. 55–78. Instytut Matematyczny Polskiej Akademii Nauk (2021)

22. Saker, S.H., Osman, M.M., Anderson, D.R.: On a new class of dynamic Hardy-type inequalities and some related generalizations. Aequ. Math., 1–21 (2021)

23. Saker, S.H., Osman, M.M., O’Regan, D., Agarwal, R.P.: Characterizations of reverse dynamic weighted Hardy-type inequalities with kernels on time scales. Aequ. Math. 95(1), 125–146 (2021)

24. Saker, S.H., Osman, M.M., Anderson, D.R.: Two weighted norm dynamic inequalities with applications on second order half-linear dynamic equations. Qual. Theory Dyn. Syst. 21(1), 1–26 (2022)

25. El-Deeb, A.A., Xu, H., Abdeldaim, A., Wang, G.: Some dynamic inequalities on time scales and their applications. Adv. Differ. Equ. 2019, 130 (2019)

26. El-Deeb, A.A.: On some generalizations of nonlinear dynamic inequalities on time scales and their applications. Appl. Anal. Discrete Math. (to appear)

27. Tian, Y., El-Deeb, A.A., Meng, F.: Some nonlinear delay Volterra–Fredholm type dynamic integral inequalities on time scales. Discrete Dyn. Nat. Soc. 2018, Article ID 5841985 (2018)

28. El-Deeb, A.A., Cheung, W.-S.: A variety of dynamic inequalities on time scales with retardation. J. Nonlinear Sci. Appl. 11(10), 1185–1206 (2018)

29. El-Deeb, A.A., Elsennary, H.A., Nwaeze, E.R.: Generalized weighted Ostrowski, trapezoid and Grüss type inequalities on time scales. Fasc. Math. 60, 123–144 (2018)

30. Abdeldaim, A., El-Deeb, A.A., Agarwal, P., El-Sennary, H.A.: On some dynamic inequalities of Steffensen type on time scales. Math. Methods Appl. Sci. 41(12), 4737–4753 (2018)

31. El-Deeb, A.A.: Some Gronwall–Bellman type inequalities on time scales for Volterra-Fredholm dynamic integral equations. J. Egypt. Math. Soc. 26(1), 1–17 (2018)

32. Saker, S.H., El-Deeb, A.A., Rezk, H.M., Agarwal, R.P.: On Hilbert’s inequality on time scales. Appl. Anal. Discrete Math. 11(2), 399–423 (2017)

33. El-Deeb, A.A., El-Sennary, H.A., Khan, Z.A.: Some Steffensen-type dynamic inequalities on time scales. Adv. Differ. Equ. 2019, 246 (2019)

34. KH, F.M., El-Deeb, A.A., Abdeldaim, A., Khan, Z.A.: On some generalizations of dynamic Opial-type inequalities on time scales. Adv. Differ. Equ. 2019, 323 (2019)

35. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales. Birkhäuser, Boston (2001) An introduction with applications

36. Saker, S.H., O’Regan, D.: Hardy’s type integral inequalities on time scales. Appl. Math. Inf. Sci. 9(6), 2955–2962 (2015)

37. Li, W.N., Han, M.: Bounds for certain nonlinear dynamic inequalities on time scales. Discrete Dyn. Nat. Soc. 2009, Article ID 897087 (2009)

38. Saker, S.H.: Some nonlinear dynamic inequalities on time scales and applications. J. Math. Inequal. 4(4), 561–579 (2010)

39. Li, W.N., Zhang, Q., Qiu, F.: Some nonlinear delay discrete inequalities and their applications. Demonstr. Math. 39(4), 771–782 (2006)

40. Agarwal, R.P., Bohner, M., Peterson, A.: Inequalities on time scales: a survey. Math. Inequal. Appl. 4(4), 535–557 (2001)

41. Li, W.N.: Some Pachpatte type inequalities on time scales. Comput. Math. Appl. 57(2), 275–282 (2009)

42. Mitrinovic, D.S., Vasic, P.M.: Analytic Inequalities, vol. 1. Springer, Berlin (1970)

43. El-Deeb, A.A., Rashid, S.: On some new double dynamic inequalities associated with Leibniz integral rule on time scales. Adv. Differ. Equ. 2021(1), 1 (2021)

44. Pachpatte, B.G.: Integral and Finite Difference Inequalities and Applications. North-Holland, Amsterdam (2006)

45. Kendre, S.D., Latpate, S.G., Ranmal, S.S.: Some nonlinear integral inequalities for Volterra–Fredholm integral equations. Adv. Inequal. Appl. 2014, 11–21 (2014)

Not applicable.

## Funding

Open Access funding enabled and organized by Projekt DEAL.

## Author information

Authors

### Contributions

Resources and methodology, AAE-D and DB; investigation, DB; data curation, AAE-D; writing—original draft preparation, AAE-D; conceptualization, writing—review and editing, DB; administration, project, AAE-D and DB. All authors read and approved the final manuscript.

### Corresponding author

Correspondence to Ahmed A. El-Deeb.

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests.

## Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions 