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Reconstruction of nonlinear integral inequalities associated with time scales calculus

Abstract

In this paper, we build up some generalizations of nonlinear integral inequalities and recreate the results of some Pachpatte’s inequalities on time scales. We not just settle new estimated bounds of a particular class of nonlinear retarded dynamic inequalities, but additionally determine and unify continuous analogs alongside a subjective time scale \(\mathbb{T}\). We demonstrate applications of the treated inequalities to reflect the benefits of our work. The key effects will be proven by using the analysis procedure and the standard time-scale comparison theorem technique.

Introduction

A dynamic system containing discrete and continuous times is an important tool for modeling real-world problems. It is fair to check if a structure can be given that helps us to integrate all dynamic systems simultaneously to gain some perspective and a superior comprehension of the contrasts between discrete and continuous domains. To counter this, a concept was composed by Hilger [1]. The primary target of dynamic equations on time scales is that they construct a connection between continuous and discrete situations. A while later, this perception was evolved by many researchers [24].

Over the most recent couple of years, great efforts have been made to unify and expand integral inequalities on time scales [512]. These essential inequalities are promoted in numerous classifications for the boundedness, uniqueness, and the solutions of various dynamic equations [1316].

Linear and nonlinear versions of Pachpatte’s inequalities on time scale have been a matter of conversation for quite a while. These inequalities were advanced by means of several authors [1722]. Bohner has planned an assortment of dynamic inequalities, which are basically founded on the inequality of Gronwall. Originally, Bohner et al. [23] unify the continuous-type Gronwall inequality as follows

$$ x(l)\leq b(l)+ \int _{l_{0}}^{l} j(l_{1})x(l_{1}) \Delta l_{1},\quad l\in \mathbb{T}, $$

where j is a right-dense continuous function, \(x\geq 0\) is a regressive right-dense continuous function, and \(\mathbb{T}\) is a time scale. Bohner et al. [24] further suggested the integral inequality on time scales

$$ x(l)\leq a(l)+p(l) \int _{l_{0}}^{l} \bigl[b(l_{1})x(l_{1})+q(l_{1}) \bigr]\Delta l_{1}. $$

After that in 2010, Li [25] considered the nonlinear integral inequality of one independent variable associated with time scales

$$ x^{\gamma }(l)\leq a(l)+c(l) \int _{l_{0}}^{l} \bigl[f(l_{1})x\bigl(\rho (l_{1})\bigr)+n(l_{1}) \bigr]\Delta l_{1} $$

for \(l\in l_{0}\) with initial conditions \(x(l)=\varOmega (l)\), \(l\in [\beta ,l_{0}]\cap \mathbb{T}\), \(\varUpsilon (\rho (l))\leq (a(l))^{\frac{1}{\gamma }}\) for \(l\in l_{0}\), \(\rho (l)\leq l_{0}\), where \(\gamma \geq 1\) is a constant, \(\rho (l)\leq l\), \(-\infty <\beta =\inf \{ \rho (l),l\in \mathbb{T}_{0} \} \leq l_{0}\), and \(\varOmega (l)\in C_{rd}([\beta ,l_{0}]\cap \mathbb{T},\mathbb{R}_{+})\). Meanwhile, Pachpatte [26] stepped forward to discover the extension of the integral inequality of the form

$$ x(l)\leq a(l)+ \int _{l_{0}}^{l} [f(l_{1}) \biggl[x(l_{1})+ \int _{l_{0}}^{l_{1}}m(l_{1},h_{1})x(h_{1}) \Delta h_{1} \biggr]\Delta l_{1} $$

such that \(m(l_{1},h_{1})\geq 0\), \(m^{\Delta }(l_{1},h_{1})\geq 0\) for \(l,h_{1}\in \mathbb{T}\) and \(h_{1}\leq l\). Later, Meng et al. [27] inquired the expansion of the nonlinear integral inequality on time scales as follows:

$$ x(l)\leq x_{0}+ \int _{l_{0}}^{l}f(l_{1}) \biggl[x(l_{1})+ \int _{l_{0}}^{l_{1}}j(h_{1})x(h_{1}) \Delta h_{1} \biggr] \Delta l_{1}+ \int _{l_{0}}^{\alpha }s(l_{1})x(l_{1}) \Delta l_{1} $$

with \(\alpha >l_{0}\). Recently, in 2017, Haidong [28] proved the retarded Volterra–Fredholm integral inequality on time scales

$$\begin{aligned} x(l)\leq{}& a(l)+b(l) \int _{\rho (l_{0})}^{\rho (l)} \biggl[f_{1}(l_{1})x(l_{1})+f_{2}(l_{1}) \int _{\rho (l_{0})}^{l_{1}}j(h_{1})x(h_{1}) \Delta h_{1} \biggr] \Delta l_{1} \\ &{}+\lambda b(T) \int _{\rho (l_{0})}^{\rho (T)} \biggl[f_{1}(l_{1})x(l_{1})+f_{2}(l_{1}) \int _{\rho (l_{0})}^{l_{1}}j(h_{1})x(h_{1}) \Delta h_{1} \biggr] \Delta l_{1}, \end{aligned}$$

where \(\lambda \geq 0\). To delineate the hypothetical theorems, it has been demonstrated that the acquired inequalities can be utilized as significant apparatuses in the investigation of specific properties of dynamic equations on time scales.

Moreover, Nasser et al. [29] introduced some new generalizations and rectifications of many known results of Pachpatte kind, consolidating two nonlinear integral terms on time scales. These acquired consequences played a crucial role in reading a few lessons of integral and integro-differential equations.

Often, the previously noted inequalities are not practical directly in the evaluation of certain retarded differential and integral equations. Therefore it is alluring to discover a few new estimates in which the nonretarded term l is changed to the retarded argument \(\rho (l)\) in specific circumstances. To overcome this hollow, primarily based on the expertise of the research mentioned, in this text, we are able to seek the nonlinear dynamic inequalities constructed up for the solution of the integral inequalities and unifying some known results in the literature.

At the point when we want to examine certain properties of a differential equation, these types of inequalities have many applications (see[3032]). Around the completion of this paper, we discuss several applications to investigate the uniqueness and global existence of solutions of nonlinear delay dynamic integral equations.

The remaining portions of the document are structured as follows. In Sect. 2, we describe major realities and fundamental lemmas that are key devices for our primary results. Theoretical conversations on nonlinear dynamic Pachpatte’s inequalities on general time scales with some finishing remarks are committed in Sect. 3. The final section accomplishes the applications of the abstract results.

Preliminaries on time scales

A time scale \(\mathbb{T}\) is a nonempty closed subset of the real line \(\mathbb{R}\). For \(l\in \mathbb{T}\), the forward jump operator \(\sigma:\mathbb{T}\rightarrow \mathbb{R}\) is defined by \(\sigma(l)=\inf\{ n\in \mathbb{T}: n> l\}\), the backward jump operator \(\varsigma:\mathbb{T}\rightarrow \mathbb{R}\) by \(\varsigma(l)=\sup\{ n\in \mathbb{T}: n< l\}\) and the graininess function \(\psi:\mathbb{T}\rightarrow [0,\infty )\) by \(\psi(l)=\sigma(l)-l\). An element \(l\in \mathbb{T}\) is said to be right-dense if \(\sigma(l)=l\) and right-scattered if \(\sigma(l)>l\), left-dense if \(\varsigma(l)=l\) and left-scattered if \(\varsigma(l)< l\). The set \(\mathbb{T}^{k}\) is defined to be \(\mathbb{T}\) if it has a left-scattered maximum g, then \(\mathbb{T}^{k}=\mathbb{T}- \{ g \} \) otherwise, \(\mathbb{T}^{k}= \mathbb{T}\). is the set of all regressive and rd-continuous functions, and \(\Re ^{+}= \{ y\in \Re: 1+\psi (l)y(l)>0, l\in \mathbb{T} \} \).

On time scales, the reader is supposed to be acquainted with the skills and basic ideas about the analytics given by Bohner [3]. Next, we give some basic lemmas on time scales which will be required in the evidence of the exhibited paper.

Lemma 2.1

([20])

If\(j,h\)are delta differentiable atl, thenjhis also delta differentiable atl, and

$$ (jh)^{\Delta }(l)=j^{\Delta }(l)h(l)+j\bigl(\sigma (l) \bigr)h^{\Delta }(l). $$

Lemma 2.2

([19])

Let\(l_{0}\in \mathbb{T}^{k}\), and let\(j:\mathbb{T}\times \mathbb{T}^{k} \rightarrow \mathbb{R}\)be continuous at\((l,l)\), where\(l>l_{0}\)and\(l\in \mathbb{T}^{k}\). Assume that\(j^{\Delta }(l,\cdot )\)is rd-continuous on\([l_{0},\sigma (l)]_{\mathbb{T}}\). Suppose that, for every\(\epsilon >0\), there exists a neighborhoodΩofl, independent of\(\eta \in [l_{0},\sigma (l)]_{\mathbb{T}}\), such that

$$ \bigl\vert \bigl[j\bigl(\sigma (l),\eta \bigr)-j(l_{1},\eta ) \bigr]-j^{\Delta }(l,\eta )\bigl[\sigma (l)-l_{1}\bigr] \bigr\vert \leq \epsilon \bigl\vert \sigma (l)-l_{1} \bigr\vert ,\quad l_{1}\in \varOmega , $$

where\(j^{\Delta }\)be the derivative ofjwith respect to the first variable. Then\(x(l)=\int _{l_{0}}^{l}j(l,\eta )\Delta \eta \)yields

$$ x^{\Delta }(l)= \int _{l_{0}}^{l}j^{\Delta }(l,\eta )\Delta \eta +j \bigl( \sigma (l),l\bigr). $$

Lemma 2.3

([23])

Chain Rule 1: Let\(j:\mathbb{R}\rightarrow \mathbb{R}\)be differentiable and suppose that\(h:\mathbb{T}\rightarrow \mathbb{R}\)is delta differentiable. Then\(j\circ h:\mathbb{T}\rightarrow \mathbb{R}\)is delta differentiable, and

$$ (j\circ h)^{\Delta }(l)= \biggl\{ \int _{0}^{1}\bigl[j^{\prime }\bigl(h(l) \bigr)+y\psi (l)h^{ \Delta }(l)\bigr]\,d y \biggr\} h^{\Delta }(l). $$

Chain Rule 2: Assume that\(j:\mathbb{T}\rightarrow \mathbb{R}\)is strictly increasing and\(\mathbb{T}^{\ast }=j(\mathbb{T})\)is a time scale. Let\(v:\mathbb{T}^{\ast }\rightarrow \mathbb{R}\)and\(j^{\Delta }(l)\), \(v^{\Delta }(j(l))\)exist for\(l\in \mathbb{T}^{k}\). Then

$$ (j\circ v)^{\Delta }=\bigl(j^{\Delta }\circ v\bigr)v^{\Delta }. $$

Lemma 2.4

([32])

Let\(j\in C_{rd}\)and\(l\in \mathbb{T}^{k}\). Then

$$ \int _{l}^{\sigma (l)}j(\tau )\Delta \tau = \psi (l)j(l)=j(l) \bigl(\sigma (l)-l\bigr). $$

Lemma 2.5

([19])

If\(j\in \Re \)and\(l\in \mathbb{T}\), then the exponential function\(e_{j}(l,l_{0})\)is the unique solution of the initial value problem

$$ \textstyle\begin{cases} x^{\Delta }(l)= j(l)x(l), \\ x(l_{0})=1. \end{cases} $$

Results and discussion

Without compromising nonspecific statements, throughout in this task, we denote \(\mathbb{R}_{+}=[0,\infty )\) and \(l_{0}\in \mathbb{T}\), \(l_{0}\geq 0\), \(\mathbb{T}_{0}=[l_{0},\infty )\cap \mathbb{T}\).

To demonstrate our elementary results, we first rundown the accompanying suppositions:

  1. (P1)

    The functions \(j(l,l_{1})\), \(j^{\Delta }(l,l_{1})\), \(h(l,l_{1})\), \(h^{\Delta }(l,l_{1})\), \(m(l,l_{1})\), \(m^{\Delta }(l,l_{1})\in C_{rd}(\mathbb{T}_{0}\times \mathbb{T}_{0}, \mathbb{R}_{+})\).

  2. (P2)

    \(x\in C_{rd}(\mathbb{T}_{0}, \mathbb{R}_{+})\).

  3. (P3)

    \(\varrho _{i}\in C_{rd}(\mathbb{R}_{+},\mathbb{R}_{+})\), \(i=1,2\), are continuous nondecreasing functions with \(\varrho _{i}(l)>0\) for \(l>0\).

  4. (P4)

    The function \(\rho \in C_{rd}(\mathbb{T}_{0},\mathbb{R}_{+})\) is strictly increasing.

  5. (P5)

    \(b\in C_{rd}(\mathbb{T}_{0}, \mathbb{R}_{+})\).

We now present the principle lemma and theorems.

Lemma 3.1

Let\(a\in C_{rd}\), \(l\in \mathbb{T}_{k}\), and let\(\rho (l)\in C_{rd}\)be a strictly increasing function for\(l\in \mathbb{T}\). Then

$$ \int _{\sigma (l)}^{\rho (\sigma (l))}a\bigl(\sigma (l),\lambda \bigr)\Delta \lambda = a\bigl(\sigma (l),\rho (l)\bigr) \bigl(\rho \bigl(\sigma (l)\bigr)-\rho (l) \bigr). $$

Proof

If A is the antiderivative of a and \(A^{\Delta \lambda }(\sigma (l),\lambda )=a(\sigma (l),\lambda )\), then

$$\begin{aligned} \int _{\sigma (l)}^{\rho (\sigma (l))}a\bigl(\sigma (l),\lambda \bigr)\Delta \lambda &=A\bigl(\sigma (l),\rho \bigl(\rho (l)\bigr)\bigr)-A\bigl(\sigma (l), \rho (l)\bigr) \\ &= \frac{A(\sigma (l),\rho (\rho (l)))-A(\sigma (l),\rho (l))}{\rho (\sigma (l))-\rho (l)} \bigl(\rho \bigl(\sigma (l)\bigr)-\rho (l) \bigr) \\ &=A^{\Delta \lambda }\bigl(\sigma (l),\lambda \bigr)|_{\lambda =\rho (l)} \bigl( \rho \bigl(\sigma (l)\bigr)-\rho (l) \bigr) \\ &=a\bigl(\sigma (l),\rho (l)\bigr) \bigl(\rho \bigl(\sigma (l)\bigr)-\rho (l) \bigr). \end{aligned}$$

 □

Theorem 3.2

Suppose that suppositions (P1)–(P5) with\(\varrho ^{\Delta }_{1}(l)=\varrho _{2}(l) \)and the inequality

$$\begin{aligned} \varrho _{1}\bigl(x(l)\bigr)\leq{}& b(l)+ \int _{l_{0}}^{\rho (l)}j(l,l_{1}) \varrho _{2}\bigl(x(l_{1})\bigr) \\ &{}\times \biggl[x(l_{1})+ \int _{l_{0}}^{\rho (l_{1})}h(l_{1},q_{1}) \varrho _{1}\bigl(x(q_{1})\bigr)\Delta q_{1} \int _{l_{0}}^{\rho (l_{1})}m(l_{1},q_{1}) \varrho _{1}\bigl(x(q_{1})\bigr)\Delta q_{1} \biggr]^{\xi }\Delta l_{1}, \\ &\quad l \in \mathbb{T}_{0}, \end{aligned}$$
(1)

are satisfied. Then

$$\begin{aligned} x(l)\leq{}& \varrho _{1}^{-1}\bigl(b(l)\bigr)+ \int _{l_{0}}^{\rho (l)}j(l,l_{1}) \biggl\{ \varLambda ^{-1} \biggl(\varLambda \biggl[\varrho _{1}^{\xi -1} \bigl(b(l)\bigr) \\ &{}+(1-\xi ) \int _{l_{0}}^{\rho (l_{1})}j(l_{1},q_{1}) \Delta q_{1} \biggr]^{\frac{1}{1-\xi }}+ \int _{l_{0}}^{\rho (l_{1})}\bigl(h(l_{1},q_{1})m(l_{1},q_{1}) \bigr) \Delta q_{1} \biggr) \biggr\} ^{\xi }\Delta l_{1}, \end{aligned}$$
(2)

where\(\xi \neq 1\),

$$ \varLambda (v)= \int _{v_{0}}^{v}\frac{\Delta p}{\varrho _{1}^{2}(p)},\quad v\geq v_{0}>0, \varLambda (+\infty )=+\infty , $$
(3)

\(\varLambda ^{-1}\)is the inverse function ofΛ, and\(L_{1}\)is the largest number for all\(l< L_{1}\)with

$$\begin{aligned} & \varLambda \biggl[\varrho _{1}^{\xi -1} \bigl(b(l)\bigr) +(1-\xi ) \int _{l_{0}}^{ \rho (l)}j(l,l_{1})\Delta l_{1} \biggr]^{\frac{1}{1-\xi }} \\ &\quad + \int _{l_{0}}^{ \rho (l)}\bigl(h(l,l_{1})m(l,l_{1}) \bigr)\Delta l_{1} \in \operatorname{Dom}\bigl(\varLambda ^{-1}\bigr). \end{aligned}$$
(4)

Proof

Fix an arbitrary \(l^{\ast }\in \mathbb{T}_{0}\) for \(l\in [l_{0}, l^{\ast }]\cap \mathbb{T}\) and denote by \(\varrho _{1}(J(l))\) the function on the right side of (1), which is nonnegative and nondecreasing. Therefore

$$\begin{aligned} & \varrho _{1}\bigl(J(l)\bigr) \\ &\quad =b\bigl(l^{\ast } \bigr)+ \int _{l_{0}}^{\rho (l)}j(l,l_{1}) \varrho _{2}\bigl(x(l_{1})\bigr) \\ &\qquad{}\times \biggl[x(l_{1})+ \int _{l_{0}}^{\rho (l_{1})}h(l_{1},q_{1}) \varrho _{1}\bigl(x(q_{1})\bigr)\Delta q_{1} \int _{l_{0}}^{\rho (l_{1})}m(l_{1},q_{1}) \varrho _{1}\bigl(x(q_{1})\bigr)\Delta q_{1} \biggr]^{\xi }\Delta l_{1} \end{aligned}$$
(5)

and

$$ J(l_{0})=\varrho _{1}^{-1}\bigl(b \bigl(l^{\ast }\bigr)\bigr), $$
(6)

so that by (1)

$$ x(l)\leq J(l),\quad l\in \mathbb{T}_{0}. $$
(7)

Equation (5) by Lemma 2.2 and delta derivative with respect to l imply that

$$\begin{aligned} \varrho _{1}^{\Delta }\bigl(J(l)\bigr)J^{\Delta }(l)= {}&\biggl\{ \int _{l_{0}}^{\rho (l)}j^{ \Delta }(l,l_{1}) \varrho _{2}\bigl(x(l_{1})\bigr)\Delta l_{1}+j \bigl(\sigma (l),\rho (l)\bigr) \rho ^{\Delta }(l)\varrho _{2} \bigl(x\bigl(\rho (l)\bigr)\bigr) \biggr\} \\ &{}\times \biggl[x(l)+ \int _{l_{0}}^{\rho (l)}h(l,q_{1})\varrho _{1}\bigl(x(q_{1})\bigr) \Delta q_{1} \int _{l_{0}}^{\rho (l)}m(l,q_{1})\varrho _{1}\bigl(x(q_{1})\bigr) \Delta q_{1} \biggr]^{\xi } \\ \leq {}&\biggl\{ \int _{l_{0}}^{\rho (l)}j(l,l_{1})\Delta l_{1} \biggr\} ^{ \Delta }\varrho _{2}\bigl(J(l)\bigr) \\ &{}\times \biggl[J(l)+ \int _{l_{0}}^{\rho (l)}h(l,q_{1})\varrho _{1}\bigl(J(q_{1})\bigr) \Delta q_{1} \int _{l_{0}}^{\rho (l)}m(l,q_{1})\varrho _{1}\bigl(J(q_{1})\bigr) \Delta q_{1} \biggr]^{\xi }, \end{aligned}$$

since \(\varrho ^{\Delta }_{1}(J(l))=\varrho _{2}(J(l)) \). This inequality becomes

$$\begin{aligned} J^{\Delta }(l)\leq{}& \biggl\{ \int _{l_{0}}^{\rho (l)}j(l,l_{1})\Delta l_{1} \biggr\} ^{\Delta } \\ &{}\times \biggl[J(l)+ \int _{l_{0}}^{\rho (l)}h(l,q_{1}) \varrho _{1}\bigl(J(q_{1})\bigr)\Delta q_{1} \int _{l_{0}}^{\rho (l)}m(l,q_{1}) \varrho _{1}\bigl(J(q_{1})\bigr)\Delta q_{1} \biggr]^{\xi } \\ \leq{}& \biggl\{ \int _{l_{0}}^{\rho (l)}j(l,l_{1})\Delta l_{1} \biggr\} ^{ \Delta }W^{\xi }(l), \end{aligned}$$
(8)

where

$$ W(l)=J(l)+ \int _{l_{0}}^{\rho (l)}h(l,q_{1})\varrho _{1}\bigl(J(q_{1})\bigr) \Delta q_{1} \int _{l_{0}}^{\rho (l)}m(l,q_{1})\varrho _{1}\bigl(J(q_{1})\bigr) \Delta q_{1}. $$
(9)

Delta differentiating (9) and utilizing \(J(l)\leq W(l)\) and (8), we derive that

$$\begin{aligned} W^{\Delta }(l)={}& J^{\Delta }(l)+ \biggl[ \int _{l_{0}}^{\rho (l)}h(l,q_{1}) \varrho _{1}\bigl(J(q_{1})\bigr)\Delta q_{1} \int _{l_{0}}^{\rho (l)}m(l,q_{1}) \varrho _{1}\bigl(J(q_{1})\bigr)\Delta q_{1} \biggr]^{\Delta } \\ \leq{}& \biggl\{ \int _{l_{0}}^{\rho (l)}j(l,l_{1})\Delta l_{1} \biggr\} ^{ \Delta }W^{\xi }(l) \\ &{}+ \biggl[ \int _{l_{0}}^{\rho (l)}h(l,q_{1})\varrho _{1}\bigl(W(q_{1})\bigr) \Delta q_{1} \int _{l_{0}}^{\rho (l)}m(l,q_{1})\varrho _{1}\bigl(W(q_{1})\bigr) \Delta q_{1} \biggr]^{\Delta }. \end{aligned}$$
(10)

Consider

$$\begin{aligned} & \biggl[ \int _{l_{0}}^{\rho (l)}h(l,q_{1})\varrho _{1}\bigl(W(q_{1})\bigr) \Delta q_{1} \int _{l_{0}}^{\rho (l)}m(l,q_{1})\varrho _{1}\bigl(W(q_{1})\bigr) \Delta q_{1} \biggr]^{\Delta } \\ &\quad= \biggl\{ \int _{l_{0}}^{\rho (l)}h^{\Delta }(l,q_{1}) \varrho _{1}\bigl(W(q_{1})\bigr) \Delta q_{1}+h \bigl(\sigma (l),\rho (l)\bigr)\rho ^{\Delta }(l)\varrho _{1} \bigl(W\bigl( \rho (l)\bigr)\bigr) \biggr\} \\ &\qquad{}\times\biggl( \int _{l_{0}}^{\rho (l)}m(l,q_{1})\varrho _{1}\bigl(W(q_{1})\bigr) \Delta q_{1} \biggr) \\ &\qquad{} + \biggl\{ \int _{l_{0}}^{\rho (l)}m^{\Delta }(l,q_{1}) \varrho _{1}\bigl(W(q_{1})\bigr) \Delta q_{1}+m \bigl(\sigma (l),\rho (l)\bigr)\rho ^{\Delta }(l)\varrho _{1} \bigl(W\bigl( \rho (l)\bigr)\bigr) \biggr\} \\ &\qquad{}\times \biggl( \int _{l_{0}}^{\rho (\sigma (l))}h\bigl(\sigma (l),q_{1} \bigr) \varrho _{1}\bigl(J(q_{1})\bigr)\Delta q_{1} \biggr) \\ &\quad\leq \varrho _{1}^{2}\bigl(W(q_{1})\bigr) \biggl\{ \int _{l_{0}}^{\rho (l)}h^{ \Delta }(l,q_{1}) \Delta q_{1}+h\bigl(\sigma (l),\rho (l)\bigr)\rho ^{\Delta }(l) \biggr\} \biggl( \int _{l_{0}}^{\rho (l)}m(l,q_{1})\Delta q_{1} \biggr) \\ &\qquad{}+\varrho _{1}\bigl(W(q_{1})\bigr) \biggl\{ \int _{l_{0}}^{\rho (l)}m^{\Delta }(l,q_{1}) \Delta q_{1}+m\bigl(\sigma (l),\rho (l)\bigr)\rho ^{\Delta }(l) \biggr\} \\ &\qquad{}\times \biggl( \int _{l_{0}}^{\rho (\sigma (l))}h\bigl(\sigma (l),q_{1} \bigr)\varrho _{1}\bigl(W(q_{1})\bigr) \Delta q_{1} \biggr). \end{aligned}$$
(11)

It is easy to observe from Lemma 3.1 that

$$\begin{aligned} &\int _{l_{0}}^{\rho (\sigma (l))}h\bigl(\sigma (l),q_{1} \bigr)\varrho _{1}\bigl(W(q_{1})\bigr) \Delta q_{1} \\ &\quad= \int _{l_{0}}^{\rho (l)}h\bigl(\sigma (l),q_{1} \bigr)\varrho _{1}\bigl(W(q_{1})\bigr) \Delta q_{1}+ \int _{\rho (l)}^{\rho (\sigma (l))}h\bigl(\sigma (l),q_{1} \bigr) \varrho _{1}\bigl(W(q_{1})\bigr)\Delta q_{1} \\ &\quad\leq \varrho _{1}\bigl(W(l)\bigr) \int _{l_{0}}^{\rho (l)}h\bigl(\sigma (l),q_{1} \bigr) \Delta q_{1}+h\bigl(\sigma (l),\rho (l)\bigr)\varrho _{1}\bigl(\rho (l)\bigr) \bigl(\rho \bigl( \sigma (l)\bigr)-\rho (l) \bigr) \\ &\quad\leq \varrho _{1}\bigl(W(l)\bigr) \biggl[ \int _{l_{0}}^{\rho (l)}h\bigl(\sigma (l),q_{1} \bigr) \Delta q_{1}+ \int _{l_{0}}^{\rho (\sigma (l))}h\bigl(\sigma (l),q_{1} \bigr) \Delta q_{1} \biggr] \\ &\quad\leq \varrho _{1}\bigl(W(l)\bigr) \int _{l_{0}}^{\rho (\sigma (l))}h\bigl(\sigma (l),q_{1} \bigr) \Delta q_{1}. \end{aligned}$$
(12)

By substituting (12) into (11) we have

$$\begin{aligned} &\biggl[ \int _{l_{0}}^{\rho (l)}h(l,q_{1})\varrho _{1}\bigl(W(q_{1})\bigr) \Delta q_{1} \int _{l_{0}}^{\rho (l)}m(l,q_{1})\varrho _{1}\bigl(W(q_{1})\bigr) \Delta q_{1} \biggr]^{\Delta } \\ &\quad\leq \varrho _{1}^{2}\bigl(W(l)\bigr) \biggl\{ \int _{l_{0}}^{ \rho (l)} \bigl(h(l,q_{1})m(l,q_{1}) \bigr)\Delta q_{1} \biggr\} ^{\Delta }. \end{aligned}$$
(13)

From (10) and (13) we obtain

$$ W^{\Delta }(l)\leq \biggl\{ \int _{l_{0}}^{\rho (l)}j(l,l_{1})\Delta l_{1} \biggr\} ^{\Delta }W^{\xi }(l)+\varrho _{1}^{2}\bigl(W(l)\bigr) \biggl\{ \int _{l_{0}}^{ \rho (l)} \bigl(h(l,q_{1})m(l,q_{1}) \bigr)\Delta q_{1} \biggr\} ^{\Delta } $$

or, equivalently,

$$ \frac{W^{\Delta }(l)}{W^{\xi }(l)}\leq \biggl\{ \int _{l_{0}}^{\rho (l)}j(l,l_{1}) \Delta l_{1} \biggr\} ^{\Delta }+ \frac{\varrho _{1}^{2}(W(l))}{W^{\xi }(l)} \biggl\{ \int _{l_{0}}^{\rho (l)} \bigl(h(l,q_{1})m(l,q_{1}) \bigr)\Delta q_{1} \biggr\} ^{\Delta }. $$
(14)

Integrating both sides of (14) from \(l_{0}\) to l and using \(W(l_{0})=\varrho _{1}^{-1}(b(l^{\ast }))\) and \(W(l)>0\) yield the estimate

$$\begin{aligned} W^{1-\xi }(l)\leq{}& \varrho _{1}^{\xi -1}\bigl(b \bigl(l^{\ast }\bigr)\bigr)+(1-\xi ) \int _{l_{0}}^{ \rho (l)}j(l,l_{1})\Delta l_{1} \\ &{}+(1-\xi ) \int _{l_{0}}^{l} \frac{\varrho _{1}^{2}(W(l_{1}))}{W^{\xi }(l_{1})} \biggl\{ \int _{l_{0}}^{ \rho (l)} \bigl(h(l_{1},q_{1})m(l_{1},q_{1}) \bigr)\Delta q_{1} \biggr\} ^{ \Delta }\Delta l_{1},\quad \forall l\in \mathbb{T}_{0}; \end{aligned}$$

also,

$$\begin{aligned} W^{1-\xi }(l)\leq{}& \varrho _{1}^{\xi -1} \bigl(b\bigl(l^{\ast }\bigr)\bigr)+(1-\xi ) \int _{l_{0}}^{ \rho (l^{\ast })}j(l,l_{1})\Delta l_{1} \\ &{}+(1-\xi ) \int _{l_{0}}^{l} \frac{\varrho _{1}^{2}(W(l_{1}))}{W^{\xi }(l_{1})} \biggl\{ \int _{l_{0}}^{ \rho (l)} \bigl(h(l_{1},q_{1})m(l_{1},q_{1}) \bigr)\Delta q_{1} \biggr\} ^{ \Delta }\Delta g. \end{aligned}$$
(15)

Define the function \(R^{1-\xi }(l)\) as the right-hand side of (15). Since \(R(l)\) is nondecreasing, we have

$$ W(l)\leq R(l),\quad \forall l< L. $$
(16)

From (16) by delta differentiating \(R^{1-\xi }(l)\) with respect to l we get that

$$ R^{1-\xi }(l)R^{\Delta }(l)=\frac{\varrho _{1}^{2}(W(l))}{W^{\xi }(l)} \biggl\{ \int _{l_{0}}^{\rho (l)} \bigl(h(l,q_{1})m(l,q_{1}) \bigr)\Delta q_{1} \biggr\} ^{\Delta }, $$

which leads to

$$ \frac{R^{\Delta }(l)}{\varrho _{1}^{2}(R(l))}\leq \biggl\{ \int _{l_{0}}^{ \rho (l)} \bigl(h(l,q_{1})m(l,q_{1}) \bigr)\Delta q_{1} \biggr\} ^{\Delta },\quad \forall l< L. $$
(17)

In comparison, for \(l\in [l_{0},\mathbb{T}]\cap \mathbb{T}\), if \(\sigma (l)>l\), then

$$\begin{aligned} \bigl[\varLambda \bigl(R(l)\bigr)\bigr]^{\Delta }&= \frac{\varLambda (R(\sigma (l)))-\varLambda (R(l))}{\sigma (l)-l}= \frac{1}{\sigma (l)-l} \int _{R(l)}^{R(\sigma (l))} \frac{1}{{\varrho }_{1}^{2}(v)}\Delta v \\ &\leq \frac{R(\sigma (l))-R(l)}{\sigma (l)-l} \frac{1}{{\varrho }_{1}^{2}(R(l))}= \frac{R^{\Delta }(l)}{\varrho _{1}^{2}(R(l))}. \end{aligned}$$
(18)

If \(\sigma (l)=l\), then we have

$$\begin{aligned} \bigl[\varLambda \bigl(R(l)\bigr)\bigr]^{\Delta }&= \lim _{g\rightarrow l} \frac{\varLambda (R(l))-\varLambda (R(l_{1}))}{l-l_{1}}=\lim_{g \rightarrow l} \frac{1}{l-g} \int _{R(l_{1})}^{R(l)} \frac{1}{\varrho _{1}^{2}(v)}\Delta v \\ &=\lim_{l_{1}\rightarrow l}\frac{R(l)-R(l_{1})}{l-l_{1}} \frac{1}{\varrho _{1}^{2}(\mu )}= \frac{R^{\Delta }(l)}{\varrho _{1}^{2}(R(l))}, \end{aligned}$$
(19)

where μ lies between \(R(l_{1})\) and \(R(l)\). Together (18) and (19) produce

$$ \bigl[\varLambda \bigl(R(l)\bigr)\bigr]^{\Delta }\leq \frac{R^{\Delta }(l)}{\varrho _{1}^{2}(R(l))}. $$
(20)

Inequalities (17) and (20) turn out into

$$ \bigl[\varLambda \bigl(R(l)\bigr)\bigr]^{\Delta }\leq \biggl\{ \int _{l_{0}}^{\rho (l)} \bigl(h(l,q_{1})m(l,q_{1}) \bigr)\Delta q_{1} \biggr\} ^{\Delta }. $$

From the definition of Λ in (3) by integration (20) from \(l_{0}\) to l we get

$$ \varLambda \bigl(R(l)\bigr)-\varLambda \bigl(R(l_{0})\bigr)\leq \int _{l_{0}}^{\rho (l)}\bigl(h(l,q_{1})m(l,q_{1}) \bigr) \Delta q_{1}. $$

Since Λ is increasing and \(R(l_{0})= [\varrho _{1}^{\xi -1}(b(l^{\ast }))+(1-\xi )\int _{l_{0}}^{ \rho (l^{\ast })}j(l,l_{1})\Delta l_{1} ]^{\frac{1}{1-\xi }}\), the last inequality takes the form

$$\begin{aligned} R(l)\leq{}& \varLambda ^{-1} \biggl(\varLambda \biggl[ \varrho _{1}^{\xi -1}\bigl(b\bigl(l^{ \ast }\bigr) \bigr)+(1-\xi ) \int _{l_{0}}^{\rho (l^{\ast })}j(l,l_{1})\Delta l_{1} \biggr]^{\frac{1}{1-\xi }} \\ &{}+ \int _{l_{0}}^{\rho (l)}\bigl(h(l,q_{1})m(l,q_{1}) \bigr) \Delta q_{1} \biggr). \end{aligned}$$
(21)

The conclusion in (2) can be achieved by the arbitrariness of \(l^{\ast }\), inserting (21) into (16) and (8) simultaneously, integrating the resulting inequality, and taking the benefit of (6) and (7). Explanations are discarded. □

Essential comments on Theorem 3.2 are listed underneath.

Remark 3.3

By taking \(\varrho _{1}(x(l))=u(t)\), \(\rho (l)\leq t\), \(b(l)=a(t)\), \(\varrho _{2}=1\), \(h=0\), \(\xi =1\), \(j(l,l_{1})=k(t,s)\) and \(x(l)=u(t)\) Theorem 3.2 changes into Corollary 3.9 of [24].

Remark 3.4

It is very amazing to realize that, as a distinctive case, Theorem 3.2 diminishes into [7, Theorem 3.2] by setting \(\varrho _{1}(x(l))=u(t)\), \(\rho (l)\leq t\), \(h=0\), \(\xi =1\), \(b(l)=c\), \(c\geq 0\), \(j(l,l_{1})=f(t)p(t)\), \(\varrho _{2}(x(l))=1\), and \(x(l)=u(t)+f(t)q(t)\).

Theorem 3.5

Suppose that the relation

$$\begin{aligned} \varrho _{1}\bigl(x(l)\bigr)\leq {}&b(l)+ \int _{l_{0}}^{\rho (l)} \bigl[j(l,l_{1}) \varrho _{2}\bigl(x(l_{1})\bigr)\varrho _{1} \bigl(x(l_{1})\bigr)+u(l_{1})\varrho _{2} \bigl(x(l_{1})\bigr) \bigr]\Delta l_{1} \\ &{}+ \int _{l_{0}}^{\rho (l)}h(l,l_{1})\varrho _{2}\bigl(x(l_{1})\bigr)\Delta l_{1} \int _{l_{0}}^{\rho (l)}m(l,l_{1})\varrho _{1}\bigl(x(l_{1})\bigr)\Delta l_{1},\quad l\in \mathbb{T}_{0}, \end{aligned}$$
(22)

with\(u\in C_{rd}(\mathbb{T}, \mathbb{R}_{+})\)and conditions (P1)–(P5) are fulfilled. Then

$$\begin{aligned} x(l)\leq{}& \varrho _{1}^{-1} \biggl[ \varUpsilon ^{-1} \biggl\{ \varTheta ^{-1} \biggl(\varTheta \biggl[\varUpsilon \bigl(b(l)\bigr)+ \int _{l_{0}}^{\rho (l)}u(l_{1}) \Delta l_{1} \\ &{}+ \int _{l_{0}}^{\rho (l)} \bigl(j(l,l_{1})+h(l,l_{1})m(l,l_{1}) \bigr)\Delta l_{1} \biggr] \biggr) \biggr\} \biggr], \end{aligned}$$
(23)

where

$$\begin{aligned} &\varUpsilon (v)= \int _{v_{0}}^{v} \frac{\Delta r}{\varrho _{2}(\varrho _{1}^{-1}(r))},\quad v\geq v_{0}>0, \varUpsilon (+\infty )=+\infty , \end{aligned}$$
(24)
$$\begin{aligned} &\varTheta (s)= \int _{s_{0}}^{s}\frac{\Delta k}{\varUpsilon ^{-1}(k)},\quad s \geq s_{0} >0, \varTheta (+\infty )=+\infty , \end{aligned}$$
(25)

\(\varUpsilon ^{-1}\), \(\varTheta ^{-1}\)are the inverses ofϒ, Θ, and\(L_{1}\)is the largest number for all\(l< L_{1}\)with

$$ \varTheta \biggl[\varUpsilon \bigl(b(l)\bigr)+ \int _{l_{0}}^{\rho (l)}u(l_{1})\Delta l_{1}+ \int _{l_{0}}^{\rho (l)} \bigl(j(l,l_{1})+h(l,l_{1})m(l,l_{1}) \bigr) \Delta l_{1} \biggr] \in \operatorname{Dom}\bigl(\varTheta ^{-1}\bigr). $$
(26)

Proof

Fixing \(l^{\ast }\in \mathbb{T}_{0}\) for \(l\in [l_{0}, l^{\ast }]\cap \mathbb{T}\) and denoting the nondecreasing function

$$\begin{aligned} J_{1}(l)={}& b\bigl(l^{\ast }\bigr)+ \int _{l_{0}}^{\rho (l)} \bigl[j(l,l_{1}) \varrho _{2}\bigl(x(l_{1})\bigr)\varrho _{1} \bigl(x(l_{1})\bigr)+u(l_{1})\varrho _{2} \bigl(x(l_{1})\bigr) \bigr]\Delta l_{1} \\ &{}+ \int _{l_{0}}^{\rho (l)}h(l,l_{1})\varrho _{2}\bigl(x(l_{1})\bigr)\Delta l_{1} \int _{l_{0}}^{\rho (l)}m(l,l_{1})\varrho _{1}\bigl(x(l_{1})\bigr)\Delta l_{1}, \end{aligned}$$
(27)

from (22) and (27), we obtain

$$ x(l)\leq \varrho _{1}^{-1} \bigl(J_{1}(l)\bigr), \quad l\in \mathbb{T}_{0}. $$
(28)

Delta differentiating (27) and applying the same analysis from (11)–(13), Lemmas 2.1 and 2.2, and (28), we notice that

$$\begin{aligned} J_{1}^{\Delta }(l)={}&u(l)\varrho _{2}\bigl(x(l)\bigr) \rho ^{\Delta }(l) \\ &{}+ \biggl\{ \int _{l_{0}}^{\rho (l)}j^{\Delta }(l,l_{1}) \varrho _{2}\bigl(x(l_{1})\bigr) \varrho _{1} \bigl(x(l_{1})\bigr)\Delta l_{1}+j\bigl(\sigma (l),\rho (l) \bigr)\rho ^{ \Delta }(l)\varrho _{2}\bigl(x(l)\bigr)\varrho _{1}\bigl(x(l)\bigr) \biggr\} \\ &{} + \biggl[ \int _{l_{0}}^{\rho (l)}h(l,l_{1})\varrho _{2}\bigl(x(l_{1})\bigr) \Delta l_{1} \int _{l_{0}}^{\rho (l)}m(l,l_{1})\varrho _{1}\bigl(x(l_{1})\bigr) \Delta l_{1} \biggr]^{\Delta }, \\ \leq{}& u(l)\varrho _{2}\bigl(\varrho _{1}^{-1} \bigl(J_{1}(l)\bigr)\bigr)\rho ^{\Delta }(l) \\ &{}+\varrho _{2}\bigl(\varrho _{1}^{-1} \bigl(J_{1}(l)\bigr)\bigr) \biggl[ \biggl\{ \int _{l_{0}}^{ \rho (l)}j(l,l_{1})\Delta l_{1} \biggr\} ^{\Delta }+ \biggl\{ \int _{l_{0}}^{ \rho (l)} \bigl(h(l,l_{1})m(l,l_{1}) \bigr)\Delta l_{1} \biggr\} ^{\Delta } \biggr]J_{1}(l), \end{aligned}$$

which can be transformed into

$$ \frac{J_{1}^{\Delta }(l)}{\varrho _{2}(\varrho _{1}^{-1}(J_{1}(l)))} \leq u(l)\rho ^{\Delta }(l)+ \biggl\{ \int _{l_{0}}^{\rho (l)} \bigl(j(l,l_{1})+h(l,l_{1})m(l,l_{1}) \bigr)\Delta l_{1} \biggr\} ^{\Delta }J_{1}(l). $$
(29)

Integrating (29) from \(l_{0}\) to l and using \(\varUpsilon (l_{0})=b(l^{\ast })\) and \(J_{1}(l)>0\), from (24) we acquire

$$\begin{aligned} J_{1}(l)\leq{}& \varUpsilon ^{-1} \biggl[ \varUpsilon \bigl(b\bigl(l^{\ast }\bigr)\bigr)+ \int _{l_{0}}^{l}u(l_{1}) \rho ^{\Delta }(l_{1})\Delta l_{1} \\ &{}+ \int _{l_{0}}^{l}J_{1}(l_{1}) \biggl\{ \int _{l_{0}}^{\rho (l_{1})} \bigl(j(l_{1},q_{1})+h(l_{1},q_{1})m(l_{1},q_{1}) \bigr)\Delta q_{1} \biggr\} ^{\Delta }\Delta l_{1} \biggr] \\ \leq{}& \varUpsilon ^{-1} \biggl[\varUpsilon \bigl(b \bigl(l^{\ast }\bigr)\bigr)+ \int _{l_{0}}^{ \rho (l^{\ast })}u(l_{1})\Delta l_{1} \\ &{}+ \int _{l_{0}}^{l}J_{1}(l_{1}) \biggl\{ \int _{l_{0}}^{\rho (l_{1})} \bigl(j(l_{1},q_{1})+h(l_{1},q_{1})m(l_{1},q_{1}) \bigr)\Delta q_{1} \biggr\} ^{\Delta }\Delta l_{1} \biggr] \\ \leq{}& \varUpsilon ^{-1}\bigl(Z(l)\bigr), \end{aligned}$$
(30)

where

$$\begin{aligned} Z(l)={}&\varUpsilon \bigl(b\bigl(l^{\ast }\bigr)\bigr)+ \int _{l_{0}}^{\rho (l^{\ast })}u(l_{1}) \Delta l_{1} \\ &{}+ \int _{l_{0}}^{l}J_{1}(l_{1}) \biggl\{ \int _{l_{0}}^{ \rho (l_{1})} \bigl(j(l_{1},q_{1})+h(l_{1},q_{1})m(l_{1},q_{1}) \bigr) \Delta q_{1} \biggr\} ^{\Delta }\Delta l_{1}, \end{aligned}$$
(31)

and

$$ Z(l_{0})=\varUpsilon \bigl(b\bigl(l^{\ast } \bigr)\bigr)+ \int _{l_{0}}^{\rho (l^{\ast })}u(l_{1}) \Delta l_{1}. $$
(32)

From the definition of \(Z(l)\) with (30) we have

$$\begin{aligned} &Z^{\Delta }(l)= \biggl\{ \int _{l_{0}}^{\rho (l)} \bigl(j(l,l_{1})+h(l,l_{1})m(l,l_{1}) \bigr)\Delta l_{1} \biggr\} ^{\Delta }J_{1}(l), \\ &\frac{Z^{\Delta }(l)}{\varUpsilon ^{-1}(Z(l))}\leq \biggl\{ \int _{l_{0}}^{ \rho (l)} \bigl(j(l,l_{1})+h(l,l_{1})m(l,l_{1}) \bigr)\Delta l_{1} \biggr\} ^{\Delta }. \end{aligned}$$

The desired bound in (23) can be carried out by integrating over \([l_{0},l]\), using (25) and (32), setting \(l=l^{\ast }\), and simultaneously putting the resultant inequality into (30) and (28). The proof is completed. □

Remark 3.6

If \(h=0\), \(\rho (l)\leq t\), \(b(l)=u_{0}\), which is a constant, \(j(l,l_{1})=f(t)\), \(\varrho _{1}(x(l))=u(t)\), \(\varrho _{2}(x(l))=W(u(t))\), and \(u(l_{1})=h(t)\), then Theorem 3.5 becomes [7, Theorem 3.4] by Pachpatte with \(g(t)=1\).

Remark 3.7

If \(h=0\), \(b(l)=a(t)\), \(\varrho _{1}(x(l))=\varrho _{1}(x(l))=u(t)\), \(j(l,l_{1})=b(t)\), \(u(l_{1})=0\), and \(\rho (l)\leq t\), then from Theorem 3.5 we are able to get Theorem 3.6 in [24].

Theorem 3.8

Under (P1)–(P5), assume that

$$\begin{aligned} \varrho _{1}\bigl(x(l)\bigr)\leq{}& b(l)+ \int _{l_{o}}^{\rho (l)}j(l,l_{1}) \varrho _{1}\bigl(x(l_{1})\bigr)\Delta l_{1} \\ &{}+ \int _{l_{o}}^{\rho (l)}h(l,l_{1}) \varrho _{2}\bigl(x(l_{1})\bigr)\Delta l_{1} \int _{l_{o}}^{\rho (l)}m(l,l_{1}) \varrho _{2}\bigl(x(l_{1})\bigr)\Delta l_{1},\quad l\in \mathbb{T}_{0}. \end{aligned}$$
(33)

Then

$$\begin{aligned} x(l)\leq{}& \varrho _{1}^{-1} \biggl[\Delta ^{-1} \biggl\{ \varPi ^{-1} \biggl( \varPi \biggl[\Delta \bigl(b(l)\bigr)+ \int _{l_{0}}^{\rho (l)} \bigl(h(l,l_{1})m(l,l_{1}) \bigr)\Delta l_{1} \biggr] \\ &{}+ \int _{l_{0}}^{\rho (l)}j(l,l_{1})\Delta l_{1} \biggr) \biggr\} \biggr], \end{aligned}$$
(34)

where

$$\begin{aligned} & \Delta (v)= \int _{v_{0}}^{v} \frac{\Delta r}{\varrho _{2}^{2}(\varrho _{1}^{-1}(r))}, \quad v\geq v_{0}>0, \Delta (+\infty )=+\infty , \end{aligned}$$
(35)
$$\begin{aligned} &\varPi (s)= \int _{s_{0}}^{s} \frac{{\varrho _{2}^{2}(\varrho _{1}^{-1}(\Delta ^{-1}(k)))}}{\Delta ^{-1}(k)} \Delta k,\quad s\geq s_{0} >0, \varPi (+\infty )=+\infty , \end{aligned}$$
(36)

\(\Delta ^{-1}\), \(\varPi ^{-1}\)are the inverses of Δ, Π, respectively, and\(\varPi [\Delta (b(l))+\int _{l_{0}}^{\rho (l)} (h(l,l_{1})m(l,l_{1}) )\Delta l_{1} ]+\int _{l_{0}}^{\rho (l)}j(l,l_{1})\Delta l_{1}\)is in the domain ofΠ.

Proof

Let \(l^{\ast }\in \mathbb{T}_{0}\) for \(l\in [l_{0}, l^{\ast }]\cap \mathbb{T}\) be fixed. Define the nondecreasing function

$$\begin{aligned} J_{2}(l)={}& b\bigl(l^{\ast }\bigr)+ \int _{l_{o}}^{\rho (l)}j(l,l_{1})\varrho _{1}\bigl(x(l_{1})\bigr) \Delta l_{1} \\ &{}+ \int _{l_{o}}^{\rho (l)}h(l,l_{1})\varrho _{2}\bigl(x(l_{1})\bigr) \Delta l_{1} \int _{l_{o}}^{\rho (l)}m(l,l_{1})\varrho _{2}\bigl(x(l_{1})\bigr) \Delta l_{1}. \end{aligned}$$
(37)

From (33) and (37) we get

$$ x(l)\leq \varrho _{1}^{-1} \bigl(J_{2}(l)\bigr). $$
(38)

By delta differentiating (37), using Lemma 2.1, (38), and similar steps from (11)0-(13), we obtain

$$ J_{2}^{\Delta }(l)\leq \biggl\{ \int _{l_{0}}^{\rho (l)}j(l,l_{1}) \Delta l_{1} \biggr\} ^{\Delta }J_{2}(l)+ \biggl\{ \int _{l_{0}}^{\rho (l)} \bigl(h(l,l_{1})m(l,l_{1}) \bigr)\Delta l_{1} \biggr\} ^{\Delta }\varrho _{2}^{2} \bigl( \varrho _{1}^{-1}\bigl(J_{2}(l)\bigr)\bigr) $$

or

$$ \frac{J_{2}^{\Delta }(l)}{\varrho _{2}^{2}(\varrho _{1}^{-1}(J_{2}(l)))} \leq \biggl\{ \int _{l_{0}}^{\rho (l)}j(l,l_{1})\Delta l_{1} \biggr\} ^{ \Delta }\frac{J_{2}(l)}{\varrho _{2}^{2}(\varrho _{1}^{-1}(J_{2}(l)))}+ \biggl\{ \int _{l_{0}}^{\rho (l)} \bigl(h(l,l_{1})m(l,l_{1}) \bigr) \Delta l_{1} \biggr\} ^{\Delta }. $$

Integrating over \([l_{0},l]\), from (35) we have

$$\begin{aligned} \Delta J_{2}(l)={}&\Delta J_{2}(l_{0})+ \int _{l_{0}}^{\rho (l)} \bigl(h(l,l_{1})m(l,l_{1}) \bigr)\Delta l_{1} \\ &{}+ \int _{l_{0}}^{l} \frac{J_{2}(l_{1})}{\varrho _{2}^{2}(\varrho _{1}^{-1}(J_{2}(l_{1})))} \biggl\{ \int _{l_{0}}^{\rho (l_{1})}j(l_{1},q_{1}) \Delta q_{1} \biggr\} ^{ \Delta }\Delta l_{1}. \end{aligned}$$
(39)

Equation (39) with \(J_{2}(l_{0})=b(l^{\ast })\) gives

$$\begin{aligned} J_{2}(l)\leq{}& \Delta ^{-1} \biggl[\Delta \bigl(b\bigl(l^{\ast }\bigr)\bigr)+ \int _{l_{0}}^{ \rho (l)} \bigl(h(l,l_{1})m(l,l_{1}) \bigr)\Delta l_{1} \\ &{}+ \int _{l_{0}}^{l} \frac{J_{2}(l_{1})}{\varrho _{2}^{2}(\varrho _{1}^{-1}(J_{2}(l_{1})))} \biggl\{ \int _{l_{0}}^{\rho (l_{1})}j(l_{1},q_{1}) \Delta q_{1} \biggr\} ^{ \Delta }\Delta l_{1} \biggr] \\ \leq{}& \Delta ^{-1}\bigl(Z_{1}(l)\bigr), \end{aligned}$$
(40)

where

$$ Z_{1}(l_{0})=\Delta \bigl(b \bigl(l^{\ast }\bigr)\bigr)+ \int _{l_{0}}^{\rho (l^{\ast })} \bigl(h(l,l_{1})m(l,l_{1}) \bigr)\Delta l_{1}. $$
(41)

Differentiation of \(Z_{1}(l)\) with respect to l and (40) imply that

$$ \frac{\varrho _{2}^{2}(\varrho _{1}^{-1}(\Delta ^{-1}(Z_{1}(l))))Z_{1}^{\Delta }(l)}{\Delta ^{-1}(Z_{1}(l))} \leq \biggl\{ \int _{l_{0}}^{\rho (l)}j(l,l_{1})\Delta l_{1} \biggr\} ^{ \Delta }, $$

integration the prior inequality from \(l_{0}\) to l, use (36), (41) and \(l^{\ast }\in \mathbb{T}_{0}\) is chosen. The resultant inequality, (38), (40) yield the required bound in (34). Details are omitted. □

Theorem 3.9

Under the assumptions (P1), (P2), and (P4), suppose that

$$\begin{aligned} x^{\xi }(l)\leq{}& b^{\frac{\xi }{\xi -\varpi }} +\frac{\xi }{\xi -\varpi } \int _{l_{0}}^{\rho (l)}j(l,l_{1})x^{\varpi }(l_{1}) \\ &{}\times \biggl[x^{\xi -\varpi }(l_{1})+ \int _{l_{0}}^{\rho (l_{1})}h(l_{1},q_{1})x^{ \varpi }(q_{1}) \Delta q_{1} \int _{l_{0}}^{\rho (l_{1})}m(l_{1},q_{1})x^{ \xi -\varpi }(q_{1}) \Delta q_{1} \biggr]\Delta l_{1}, \\ & l\in \mathbb{T}_{0}, \end{aligned}$$
(42)

whereξ, b, ϖare constants, \(b\geq 0\), and\(\xi >\varpi >0\). Then

$$\begin{aligned} x(l)&\leq \biggl[b \biggl(1+ \int _{l_{0}}^{\rho (l)}j(l,l_{1})e_{G_{2}}(l_{1},l_{0}) \Delta l_{1} \biggr) \biggr]^{\frac{1}{\xi -\varpi }}, \end{aligned}$$
(43)

where

$$ G_{2}(l)= \biggl\{ \int _{l_{0}}^{\rho (l)} \bigl(j(l,l_{1})+h(l,l_{1})m(l,l_{1}) \bigr)\Delta l_{1} \biggr\} ^{\Delta }. $$
(44)

Proof

Defining

$$\begin{aligned} N(l)={}& b^{\frac{\xi }{\xi -\varpi }} +\frac{\xi }{\xi -\varpi } \int _{l_{0}}^{ \rho (l)}j(l,l_{1})x^{\varpi }(l_{1}) \\ &{}\times \biggl[x^{\xi -\varpi }(l_{1})+ \int _{l_{0}}^{\rho (l_{1})}h(l_{1},q_{1})x^{ \varpi }(q_{1}) \Delta q_{1} \\ &{}\times \int _{l_{0}}^{\rho (l_{1})}m(l_{1},q_{1})x^{ \xi -\varpi }(q_{1}) \Delta q_{1} \biggr]\Delta l_{1}, \end{aligned}$$
(45)

(42) can be restated as

$$ x^{\xi }(l)\leq N(l)\quad\Rightarrow \quad x(l)\leq N^{\frac{1}{\xi }}(l), l \in \mathbb{T}_{0}. $$
(46)

Obviously, \(N(l)\) is nondecreasing. The definition of \(N(l)\) in (45) with Lemma 2.2 and (46) yields

$$\begin{aligned} N^{\Delta }(l)\leq{} &\frac{\xi }{\xi -\varpi } \biggl\{ \int _{l_{0}}^{\rho (l)}j(l,l_{1}) \Delta l_{1} \biggr\} ^{\Delta }N^{\frac{\varpi }{\xi }}(l) \\ &{}\times \biggl[N^{\frac{\xi -\varpi }{\xi }}(l)+ \int _{l_{0}}^{\rho (l_{1})}h(l_{1},q_{1})N^{ \frac{\varpi }{\xi }}(q_{1}) \Delta q_{1} \int _{l_{0}}^{\rho (l_{1})}m(l_{1},q_{1})N^{ \frac{\xi -\varpi }{\xi }}(q_{1}) \Delta q_{1} \biggr], \end{aligned}$$

or

$$ N^{-\frac{\varpi }{\xi }}(l)N^{\Delta }(l)\leq \frac{\xi }{\xi -\varpi } \biggl\{ \int _{l_{0}}^{\rho (l)}j(l,l_{1})\Delta l_{1} \biggr\} ^{\Delta }N_{1}(l), $$
(47)

where

$$ N_{1}(l)= N^{\frac{\xi -\varpi }{\xi }}(l)+ \int _{l_{0}}^{\rho (l)}h(l,l_{1})N^{ \frac{\varpi }{\xi }}(l_{1}) \Delta l_{1} \int _{l_{0}}^{\rho (l)}m(l,l_{1})N^{ \frac{\xi -\varpi }{\xi }}(l_{1}) \Delta l_{1}. $$
(48)

Taking the delta derivative of (48) and using (47), (11)–(13), \(N(l)\leq N_{1}(l)\), and \(N_{1}(l_{0})=b\), we infer that

$$\begin{aligned} N_{1}^{\Delta }(l)&=\frac{\xi -\varpi }{\xi }N^{-\frac{\varpi }{\xi }}(l)N^{\Delta }(l)+ \biggl[ \int _{l_{0}}^{\rho (l)}h(l,l_{1})\Delta l_{1} \int _{l_{0}}^{\rho (l)}m(l,l_{1})\Delta l_{1} \biggr]^{\Delta }N(l) \\ &\leq \biggl[ \int _{l_{0}}^{\rho (l)} \bigl[j(l,l_{1})+h(l,l_{1})m(l,l_{1}) \bigr]\Delta l_{1} \biggr]^{\Delta }N_{1}(l) \\ &\leq G_{2}(l)N_{1}(l), \end{aligned}$$

which yields

$$ N_{1}(l)\leq b e_{G_{2}}(l,l_{0}), $$
(49)

with \(G_{2}\) given in (44). From (47) and (49) we claim

$$ N^{-\frac{\varpi }{\xi }}(l)N^{\Delta }(l)\leq b \frac{\xi }{\xi -\varpi } \biggl\{ \int _{l_{0}}^{\rho (l)}j(l,l_{1})\Delta l_{1} \biggr\} ^{\Delta }e_{G_{2}}(l_{1},l_{0}). $$
(50)

We can notice from Theorem 1.90 of [23] and \(N^{\Delta }(l)\geq 0\) that

$$\begin{aligned} \biggl[\frac{\xi }{\xi -\varpi }N^{\frac{\xi -\varpi }{\xi }}(l) \biggr]^{\Delta }&=N^{\Delta }(l) \int _{0}^{1} \bigl[N(l)+h\psi (l)N^{\Delta }(l) \bigr]^{- \frac{\varpi }{\xi }}\,dh \\ & =\frac{N^{\Delta }(l)}{N^{\frac{\varpi }{\xi }}(l)} \int _{0}^{1} \biggl[1+h \psi (l)\frac{N^{\Delta }(l)}{N(l)} \biggr]^{-\frac{\varpi }{\xi }}\,dh\leq \frac{N^{\Delta }(l)}{N^{\frac{\varpi }{\xi }}(l)}, \end{aligned}$$

which, together with (50), implies that

$$ \bigl[N^{\frac{\xi -\varpi }{\xi }}(l) \bigr]^{\Delta }\leq b \biggl\{ \int _{l_{0}}^{ \rho (l)}j(l,l_{1})\Delta l_{1} \biggr\} ^{\Delta }e_{G_{2}}(l_{1},l_{0}). $$

Integrate this inequality and using \(N(l_{0})=b\) and (46), we get the required inequality (43). □

Remark 3.10

Theorem 3.9 becomes Theorem 2.1 of [21] by letting \(\xi =1\), \(\varpi =0\), \(\rho (l)\leq t\), \(j(l,l_{1})=k(x,t)\), \(b=g(x)\), \(x(l)=u(t)\) with \(\mathbb{T}=\mathbb{R}\).

Remark 3.11

As a particular case of delta derivative on time scales, if \(\xi =1\), \(\varpi =0\), \(\rho (l)\leq b\), \(h=0\), \(j(l,l_{1})=n(t)\), \(b=m(t)\), and \(x(l)=u(t)\) in Theorem 3.9, then it reduces to Lemma 3.1 due to Boukerrioua et al. [4] with \(l(t)=1\).

Corollary 3.12

Let (P2), (P4), and\(j(l,l_{1})\), \(j^{\Delta }(l,l_{1})\), \(h(l,l_{1})\), \(h^{\Delta }(l,l_{1})\in C_{rd}(\mathbb{T}_{0}\times \mathbb{T}_{0}, \mathbb{R}_{+})\). Suppose that

$$ x^{\xi }(l)\leq b + \int _{l_{0}}^{\rho (l)}j(l,l_{1})x^{\xi }(l_{1}) \Delta l_{1} \int _{l_{0}}^{\rho (l)}h(l,l_{1})x(l_{1}) \Delta l_{1},\quad l\in \mathbb{T}_{0}. $$

Then

$$\begin{aligned} x(l)&\leq \frac{b^{\frac{1}{\xi }}}{ [1-\frac{1}{\xi }b^{\frac{1}{\xi }}\int _{l_{0}}^{\rho (l)} (j(l,l_{1})h(l,l_{1}) )\Delta l_{1} ]}, \end{aligned}$$

where\(\xi \neq 0\)and\(b\ge 0\)are constants.

Proof

The proof of Corollary 3.12 is the same that of Theorem 3.9 with appropriate alterations. □

Application

This segment indicates a prompt use of Theorem 3.9 for analyzing the boundedness and uniqueness of the delay integral equations on time scales. Consider the following class of nonlinear delay dynamic integral equations:

$$\begin{aligned} \textstyle\begin{cases} (x^{\xi }(l))^{\Delta }= M (l,l_{1},x(l), \int _{l_{0}}^{l}E(l,l_{1},x(l_{1}) \Delta l_{1} ), \\ x(l_{0})= b^{\frac{1}{\xi -\varpi }}. \end{cases}\displaystyle \end{aligned}$$
(51)

The global existence on the solutions of (51) can be explored by the following corollary.

Corollary 4.1

Assume that

$$ \bigl\vert M(l,l_{1},x,y) \bigr\vert \leq j(l,l_{1}) \vert x \vert ^{\xi -1}\bigl[ \vert x \vert +h(l,l_{1}) \vert x \vert ^{\xi -1} \vert y \vert \bigr] $$
(52)

and

$$ \bigl\vert E(l,l_{1},x) \bigr\vert \leq m(l,l_{1}) \vert x \vert $$
(53)

for\(l\in \mathbb{T}_{0}\), \(x,y\in \mathbb{R}\). If\(x(l)\)is a solution of (4), then

$$ \bigl\vert x(l) \bigr\vert \leq b \biggl(1+ \int _{l_{0}}^{l}j(l,l_{1}) e_{G_{2}}(l_{1},l_{0}) \Delta l_{1} \biggr), $$
(54)

where\(M\in C_{rd}([\beta ,l_{0}]\cap {\mathbb{T}}\times \mathbb{R}^{2}, \mathbb{R})\), \(E\in C_{rd}([\beta ,l_{0}]\cap {\mathbb{T}}\times \mathbb{R}, \mathbb{R})\), j, h, m, x, ρare defined as in (P1), (P2), (P4), band\(\xi >1\)are constants, and\(G_{2}\)is as in (44).

Proof

Clearly, equation (51) by employing (52) and (53) transforms into

$$\begin{aligned} \bigl\vert x^{\xi }(l) \bigr\vert \leq{}& \vert b \vert + \int _{l_{0}}^{l}|M (l,l_{1},x(l_{1}), \int _{l_{0}}^{l_{1}}E\bigl(l_{1},q_{1},x(q_{1}) \Delta q_{1} \bigr)|\Delta l_{1} \\ \leq{}& \vert b \vert + \int _{l_{0}}^{l}j(l,l_{1}) \bigl\vert x(l_{1}) \bigr\vert ^{\xi -1} \\ &{}\times \biggl[ \bigl\vert x(l_{1}) \bigr\vert + \int _{l_{0}}^{l_{1}}h(l_{1},q_{1}) \bigl\vert x(q_{1}) \bigr\vert ^{ \xi -1}\Delta q_{1} \int _{l_{0}}^{l_{1}}m(l_{1},q_{1}) \bigl\vert x(q_{1}) \bigr\vert \Delta q_{1} \biggr] \Delta l_{1}. \end{aligned}$$
(55)

We argue as in the case of Theorem 3.9 with \(\varpi =\xi -1\) in order to get (54) from (55). The proof is done. □

Next, we look at the delay dynamic equation (4) with \(x(l_{0})=x_{0}\) and \(\xi =3\).

Example 4.2

Let

$$\begin{aligned} &\bigl\vert M(l,l_{1},x_{1},y_{1})-M(l,l_{1},x_{2},y_{2}) \bigr\vert \\ &\quad \leq j(l,l_{1}) \bigl\vert x_{1}^{3}-x_{2}^{3} \bigr\vert \\ &\qquad{}\times \bigl[ \bigl\vert x_{1}^{2}-x_{1}^{2} \bigr\vert +h(l,l_{1}) \vert x_{1}-x_{2} \vert \vert y_{1}-y_{2} \vert \bigr], \end{aligned}$$
(56)
$$\begin{aligned} &\bigl\vert E(l,l_{1},x_{1})-E(l,l_{1},x_{2}) \bigr\vert \leq m(l,l_{1}) \bigl\vert x_{1}^{2}-x_{2}^{2} \bigr\vert . \end{aligned}$$
(57)

Then (51) has at most one solution.

Proof

Two solutions \(x_{1}(l)\), \(x_{2}(l)\) of (51) are equivalent to

$$\begin{aligned} x_{1}^{3}(l)-x_{2}^{3}(l)={}& \int _{l_{0}}^{l}M (l,l_{1},x_{1}(l_{1}), \int _{l_{0}}^{l_{1}}E\bigl(l_{1},q_{1},x_{1}(q_{1}) \Delta q_{1} \bigr) \Delta l_{1} \\ &{}- \int _{l_{0}}^{l}M (l,l_{1},x_{2}(l_{1}), \int _{l_{0}}^{l_{1}}E\bigl(l_{1},q_{1},x_{2}(q_{1}) \Delta q_{1} \bigr)\Delta l_{1}, \end{aligned}$$

which by hypotheses (56) and (57) leads to

$$\begin{aligned} &\bigl\vert x_{1}^{3}(l)-x_{2}^{3}(l) \bigr\vert \\ &\quad \leq \int _{l_{0}}^{l}|M \biggl(l,l_{1},x_{1}(l_{1}), \int _{l_{0}}^{l_{1}}E\bigl(l_{1},q_{1},x_{1}(q_{1}) \bigr)\Delta q_{1} \biggr) \Delta l_{1} \\ &\qquad{}-M (l,l_{1},x_{2}(l_{1}), \int _{l_{0}}^{l_{1}}E\bigl(l_{1},q_{1},x_{2}(q_{1}) \Delta q_{1} \bigr)|\Delta l_{1}, \\ &\quad \leq \int _{l_{0}}^{l}j(l,l_{1}) \bigl\vert x_{1}^{3}(l_{1})-x_{2}^{3}(l_{1}) \bigr\vert \biggl[ \bigl\vert x_{1}(l_{1})-x_{2}(l_{1}) \bigr\vert \\ &\qquad{}+ \int _{l_{0}}^{l_{1}}h(l_{1},q_{1}) \bigl\vert x_{1}(q_{1})-x_{2}(q_{1}) \bigr\vert \Delta q_{1} \int _{l_{0}}^{l_{1}}m(l_{1},q_{1}) \bigl\vert x_{1}^{2}(q_{1})-x_{2}^{2}(q_{1}) \bigr\vert \Delta q_{1} \biggr]\Delta l_{1} \\ &\quad \leq \int _{l_{0}}^{l}j(l,l_{1})\sqrt{ \bigl\vert x_{1}^{3}(l_{1})-x_{2}^{3}(l_{1}) \bigr\vert } [\sqrt{ \bigl\vert x_{1}^{3}(l_{1})-x_{2}^{3}(l_{1}) \bigr\vert }+ \int _{l_{0}}^{l_{1}}h(l_{1},q_{1}) \\ &\qquad{}\times [\sqrt{ \bigl\vert x_{1}^{3}(l_{1})-x_{2}^{3}(l_{1}) \bigr\vert }\Delta q_{1} \int _{l_{0}}^{l_{1}}m(l_{1},q_{1}) \Bigl[\sqrt{ \bigl\vert x_{1}^{3}(l_{1})-x_{2}^{3}(l_{1}) \bigr\vert } \Delta q_{1} \Bigr]\Delta l_{1}. \end{aligned}$$

The earlier inequality by some changes in the method of Theorem 3.9 with \(\xi =1\) and \(\varpi =\frac{1}{2}\) applied to the function \(|x_{1}^{3}(l)-x_{2}^{3}(l)|\) produces

$$ \bigl\vert x_{1}^{3}(l)-x_{2}^{3}(l) \bigr\vert \leq 0,\quad l\in \mathbb{T}_{0}. $$

Therefore \(x_{1}(l)=x_{2}(l)\). Along these lines the delay dynamic equation (51) has one positive solution. □

Example 4.3

If

$$ \bigl\vert M(l,l_{1},x,y) \bigr\vert \leq j(l,l_{1}) \vert x \vert \bigl[ \vert x \vert ^{2}+h(l,l_{1}) \vert x \vert \vert y \vert \bigr] $$
(58)

and

$$ \bigl\vert E(l,l_{1},x) \bigr\vert \leq m(l,l_{1}) \vert x \vert ^{2}, $$
(59)

then the solution \(x(l)\) indicates

$$ \bigl\vert x(l) \bigr\vert \leq \vert x_{0} \vert \sqrt{1+ \int _{l_{0}}^{l}j(l,l_{1})e_{G_{2}}(l_{1},l_{0}) \Delta l_{1}},\quad l\in \mathbb{T}_{0}. $$
(60)

Proof

Equation (51) with (58), (59), and \(\xi =3\) can be reconstructed as

$$\begin{aligned} \bigl\vert x(l) \bigr\vert ^{3}\leq{}& \vert x_{0} \vert ^{3}+ \int _{l_{0}}^{l}|M \biggl(l,l_{1},x(l_{1}), \int _{l_{0}}^{l_{1}}E\bigl(l_{1},q_{1},x(q_{1}) \Delta q_{1} \bigr)| \Delta l_{1} \\ \leq{}& \vert x_{0} \vert ^{3}+ \int _{l_{0}}^{l}j(l,l_{1})|x(l_{1}) \biggr)| \biggl[ \bigl\vert x(l_{1}) \bigr\vert ^{2}+ \int _{l_{0}}^{l_{1}}h(l_{1},q_{1}) \bigl\vert x(q_{1}) \bigr\vert \Delta q_{1} \\ &{}\times \int _{l_{0}}^{l_{1}}m(l_{1},q_{1}) \bigl\vert x(q_{1}) \bigr\vert ^{2}\Delta q_{1} \biggr]\Delta l_{1} \\ \leq{} &\bigl\vert x_{0}^{2} \bigr\vert ^{\frac{3}{2}}+ \int _{l_{0}}^{l}j(l,l_{1}) \bigl\vert x(l_{1}) \bigr\vert \biggl[ \bigl\vert x(l_{1}) \bigr\vert ^{2}+ \int _{l_{0}}^{l_{1}}h(l_{1},q_{1}) \bigl\vert x(q_{1}) \bigr\vert \Delta q_{1} \\ &{}\times \int _{l_{0}}^{l_{1}}m(l_{1},q_{1}) \bigl\vert x(q_{1}) \bigr\vert ^{2}\Delta q_{1} \biggr]\Delta l_{1}. \end{aligned}$$
(61)

The required estimate (60) can be retrieved by intently looking at the arguments of Theorem 3.9 with \(\xi =3\) and \(\varpi =1\) and making few modifications to (61). □

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Acknowledgements

The editor and those of the reviewers’s remarks were profoundly instructive and accommodating to extensively improve the quality of the original copy.

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This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.

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Khan, Z.A. Reconstruction of nonlinear integral inequalities associated with time scales calculus. Adv Differ Equ 2020, 380 (2020). https://doi.org/10.1186/s13662-020-02842-3

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MSC

  • 35A23
  • 26E70
  • 34N05

Keywords

  • Integral Pachpatte’s inequalities
  • Time scales
  • Dynamic equations