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.
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1 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 [2–4].
Over the most recent couple of years, great efforts have been made to unify and expand integral inequalities on time scales [5–12]. These essential inequalities are promoted in numerous classifications for the boundedness, uniqueness, and the solutions of various dynamic equations [13–16].
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 [17–22]. 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
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
After that in 2010, Li [25] considered the nonlinear integral inequality of one independent variable associated with time scales
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
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:
with \(\alpha >l_{0}\). Recently, in 2017, Haidong [28] proved the retarded Volterra–Fredholm integral inequality on time scales
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[30–32]). 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.
2 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, \(\mathbb{T}^{k}= \mathbb{T}\). ℜ 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
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
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
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
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
Lemma 2.4
([32])
Let\(j\in C_{rd}\)and\(l\in \mathbb{T}^{k}\). Then
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
3 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:
-
(P1)
The functions \(j(l,l_{1})\), \(j^{\Delta }(l,l_{1})\), \(h(l,l_{1})\), \(h^{\Delta }(l,l_{1})\), \(m(l,l_{1})\), \(m^{\Delta }(l,l_{1})\in C_{rd}(\mathbb{T}_{0}\times \mathbb{T}_{0}, \mathbb{R}_{+})\).
-
(P2)
\(x\in C_{rd}(\mathbb{T}_{0}, \mathbb{R}_{+})\).
-
(P3)
\(\varrho _{i}\in C_{rd}(\mathbb{R}_{+},\mathbb{R}_{+})\), \(i=1,2\), are continuous nondecreasing functions with \(\varrho _{i}(l)>0\) for \(l>0\).
-
(P4)
The function \(\rho \in C_{rd}(\mathbb{T}_{0},\mathbb{R}_{+})\) is strictly increasing.
-
(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
Proof
If A is the antiderivative of a and \(A^{\Delta \lambda }(\sigma (l),\lambda )=a(\sigma (l),\lambda )\), then
□
Theorem 3.2
Suppose that suppositions (P1)–(P5) with\(\varrho ^{\Delta }_{1}(l)=\varrho _{2}(l) \)and the inequality
are satisfied. Then
where\(\xi \neq 1\),
\(\varLambda ^{-1}\)is the inverse function ofΛ, and\(L_{1}\)is the largest number for all\(l< L_{1}\)with
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
and
so that by (1)
Equation (5) by Lemma 2.2 and delta derivative with respect to l imply that
since \(\varrho ^{\Delta }_{1}(J(l))=\varrho _{2}(J(l)) \). This inequality becomes
where
Delta differentiating (9) and utilizing \(J(l)\leq W(l)\) and (8), we derive that
Consider
It is easy to observe from Lemma 3.1 that
By substituting (12) into (11) we have
or, equivalently,
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
also,
Define the function \(R^{1-\xi }(l)\) as the right-hand side of (15). Since \(R(l)\) is nondecreasing, we have
From (16) by delta differentiating \(R^{1-\xi }(l)\) with respect to l we get that
which leads to
In comparison, for \(l\in [l_{0},\mathbb{T}]\cap \mathbb{T}\), if \(\sigma (l)>l\), then
If \(\sigma (l)=l\), then we have
where μ lies between \(R(l_{1})\) and \(R(l)\). Together (18) and (19) produce
Inequalities (17) and (20) turn out into
From the definition of Λ in (3) by integration (20) from \(l_{0}\) to l we get
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
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
with\(u\in C_{rd}(\mathbb{T}, \mathbb{R}_{+})\)and conditions (P1)–(P5) are fulfilled. Then
where
\(\varUpsilon ^{-1}\), \(\varTheta ^{-1}\)are the inverses ofϒ, Θ, and\(L_{1}\)is the largest number for all\(l< L_{1}\)with
Proof
Fixing \(l^{\ast }\in \mathbb{T}_{0}\) for \(l\in [l_{0}, l^{\ast }]\cap \mathbb{T}\) and denoting the nondecreasing function
Delta differentiating (27) and applying the same analysis from (11)–(13), Lemmas 2.1 and 2.2, and (28), we notice that
which can be transformed into
Integrating (29) from \(l_{0}\) to l and using \(\varUpsilon (l_{0})=b(l^{\ast })\) and \(J_{1}(l)>0\), from (24) we acquire
where
and
From the definition of \(Z(l)\) with (30) we have
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
Then
where
\(\Delta ^{-1}\), \(\varPi ^{-1}\)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
By delta differentiating (37), using Lemma 2.1, (38), and similar steps from (11)0-(13), we obtain
or
Integrating over \([l_{0},l]\), from (35) we have
Equation (39) with \(J_{2}(l_{0})=b(l^{\ast })\) gives
where
Differentiation of \(Z_{1}(l)\) with respect to l and (40) imply that
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
whereξ, b, ϖare constants, \(b\geq 0\), and\(\xi >\varpi >0\). Then
where
Proof
Defining
(42) can be restated as
Obviously, \(N(l)\) is nondecreasing. The definition of \(N(l)\) in (45) with Lemma 2.2 and (46) yields
or
where
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
which yields
with \(G_{2}\) given in (44). From (47) and (49) we claim
We can notice from Theorem 1.90 of [23] and \(N^{\Delta }(l)\geq 0\) that
which, together with (50), implies that
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
Then
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. □
4 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:
The global existence on the solutions of (51) can be explored by the following corollary.
Corollary 4.1
Assume that
and
for\(l\in \mathbb{T}_{0}\), \(x,y\in \mathbb{R}\). If\(x(l)\)is a solution of (4), then
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
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
Then (51) has at most one solution.
Proof
Two solutions \(x_{1}(l)\), \(x_{2}(l)\) of (51) are equivalent to
which by hypotheses (56) and (57) leads to
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
Therefore \(x_{1}(l)=x_{2}(l)\). Along these lines the delay dynamic equation (51) has one positive solution. □
Example 4.3
If
and
then the solution \(x(l)\) indicates
Proof
Equation (51) with (58), (59), and \(\xi =3\) can be reconstructed as
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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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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DOI: https://doi.org/10.1186/s13662-020-02842-3