Abstract
In this paper, we study some new retarded Volterra-Fredholm type integral inequalities on time scales, which provide explicit bounds on unknown functions. These inequalities generalize and extend some known inequalities and can be used as tools in the qualitative theory of certain classes of retarded dynamic equations on time scales. Some applications are also presented to illustrate the usefulness of our results.
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1 Introduction
In 1988, Hilger introduced the theory of time scales in order to unify and extend the difference and differential calculus in a consistent way (see [1]). Since then, more and more researchers are getting involved in this fast-growing field, for example, [2–10] and the references therein. Among various aspects of the theory, we notice that dynamic inequalities on time scales is an object of long standing interest [11–29]. However, to the best of our knowledge, there are few results dealing with Volterra-Fredholm type integral inequalities on time scales. Recent results in this direction include the works of Gu and Meng [16] and Meng and Shao [20].
The purpose of this paper is to investigate some new retarded Volterra-Fredholm type integral inequalities on time scales, which not only generalize and extend the results of [16, 20] and some known integral inequalities but also provide a handy and effective tool for the study of qualitative properties of solutions of some complicated Volterra-Fredholm type dynamic equations.
The paper is organized as follows. In Section 2, some necessary definition and lemmas are presented. In Section 3, some new retarded Volterra-Fredholm type integral inequalities on time scales are investigated. Finally, Section 4 is devoted to applying our results to a retarded Volterra-Fredholm type dynamic integral equation on time scales.
2 Preliminaries
Throughout this paper, knowledge and understanding of time scales and time scale notation is assumed. For an excellent introduction to the calculus on time scales, we refer the reader to [30] and [31].
List of abbreviations. In what follows, we always assume that \(\mathbb{R}\) denotes the set of real numbers, \(\mathbb{R}_{+}=[0,\infty)\), \(\mathbb{Z}\) denotes the set of integers, \(\mathbb{T}\) is an arbitrary time scale (nonempty closed subset of \(\mathbb{R}\)), \(\mathcal{R}\) denotes the set of all regressive and rd-continuous functions, \(\mathcal{R}^{+}=\{p\in\mathcal{R}:1+\mu (t)p(t)>0\mbox{ for all }t\in\mathbb{T}\}\) and \(I=[t_{0},T]\cap\mathbb{T}^{\kappa}\), where \(t_{0}\in\mathbb{T}^{\kappa}\), \(T\in\mathbb{T}^{\kappa}\), \(T>t_{0}\). The set \(\mathbb{T}^{\kappa}\) is defined as follows: If \(\mathbb{T}\) has a maximum m and m is left-scattered, then \(\mathbb{T}^{\kappa}=\mathbb{T}-\{m\}\). Otherwise \(\mathbb{T}^{\kappa}=\mathbb{T}\). The graininess function \(\mu: \mathbb {T}\rightarrow[0,\infty)\) is defined by \(\mu(t):=\sigma(t)-t\), the forward jump operator \(\sigma: \mathbb{T}\rightarrow\mathbb{T}\) by \(\sigma(t):=\inf\{s\in\mathbb{T}: s>t\}\), and the “circle plus” addition ⊕ is defined by \((p\oplus q)(t):=p(t)+q(t)+\mu(t)p(t)q(t)\) for all \(t\in\mathbb {T}^{\kappa}\).
The following lemmas and definition are useful in the proof of the main results of this paper.
Lemma 2.1
([30, Theorem 1.16])
Assume that \(f:\mathbb {T}\rightarrow\mathbb{R}\) is a function and let \(t\in\mathbb{T}\). If f is differentiable at t, then
Lemma 2.2
([30, Theorem 1.98])
Assume that \(\nu :\mathbb{T}\rightarrow\mathbb{R}\) is a strictly increasing function and \(\widetilde{\mathbb{T}}:=\nu(\mathbb{T})\) is a time scale. If \(f: \mathbb{T}\rightarrow\mathbb{R}\) is an rd-continuous function and ν is differentiable with rd-continuous derivative, then for \(a, b\in \mathbb{T}\),
Lemma 2.3
Let \(\alpha:I\rightarrow I\) be a continuous and strictly increasing function such that \(\alpha(t)\leq t\), and \(\alpha^{\Delta}\) is rd-continuous. Assume that \(f: I\rightarrow\mathbb{R}\) is an rd-continuous function, then
implies
Proof
From (2.1), we get, for any \(t\in I\),
By Lemma 2.2, we obtain
so we get
□
Lemma 2.4
([30, Theorem 1.117])
Suppose that, for each \(\varepsilon>0\), there exists a neighborhood U of t, independent of \(\tau\in[t_{0},\sigma(t)]\), such that
where \(w: \mathbb{T}\times\mathbb{T}^{\kappa}\rightarrow\mathbb{R}_{+}\) is continuous at \((t,t)\), \(t\in\mathbb{T}^{\kappa}\) with \(t>t_{0}\), and \(w^{\Delta}_{t}(t,\cdot)\) are rd-continuous on \([t_{0},\sigma(t)]\). Then
implies
Lemma 2.5
([30, Theorem 6.1])
Suppose that y and f are rd-continuous functions and \(p\in\mathcal {R}^{+}\). Then
implies
Definition 2.1
A function \(x:I\rightarrow I\) is said to belong to the class ϒ if
-
(1)
x is continuous and strictly increasing, and
-
(2)
\(x(t)\leq t\) and \(x^{\Delta}\) is rd-continuous.
3 Main results
Theorem 3.1
Assume that \(\alpha\in\Upsilon\) and \(u, a, b, f_{1}, f_{2}, g: I\rightarrow\mathbb{R}_{+}\) are rd-continuous functions, a is nondecreasing, \(\lambda\geq0\) is a constant, \(\alpha^{\Delta}(t)\geq0\), \(b^{\Delta}(t)\geq0\), \(\mu(t)A(t)<1\). Suppose that u satisfies
If
then
where we use the convention that \(\frac{1}{0}=+\infty\),
Proof
Denote
Then z is nondecreasing on I,
and
From Lemma 2.3 and (3.4)-(3.7), we have
Note that from (3.4) we get
and from (3.9) we have
which yields
i.e.,
Note that z is rd-continuous and \(B\oplus C\in\mathcal{R}^{+}\), from Lemma 2.5 and (3.10), we obtain
i.e.,
From (3.2), (3.11) and (3.12), we have
In view of (3.2) and (3.13), we get
Substituting (3.14) into (3.11), we obtain
Noting \(u(t)\leq z(t)\), we get the desired inequality (3.3). This completes the proof. □
If we let \(\lambda=\frac{1}{b(T)}\) in Theorem 3.1, then we obtain the following corollary.
Corollary 3.1
Assume that α, u, a, b, \(f_{1}\), \(f_{2}\), g, A, B, C are the same as in Theorem 3.1 and \(b(T)\neq0\). Suppose that u satisfies
If
then
Remark 3.1
If we take \(a(t)\equiv u_{0}\), \(b(t)\equiv1\), \(\alpha(t)=t\) and \(\lambda =1\), then Theorem 3.1 reduces to [16, Theorem 3]. If we take \(a(t)\equiv u_{0}\), \(b(t)\equiv1\), \(\alpha(t)=t\), \(f_{1}(t)=f_{2}(t)\) and \(\lambda=1\), then Theorem 3.1 reduces to [20, Theorem 2.3].
Theorem 3.2
Assume that α, u, a, b, λ are the same as in Theorem 3.1 and \(\mu(t)\widetilde{A}(t)<1\). Let \(v(t,s)\) and \(w(t,s)\) be defined as in Lemma 2.3 such that \(v^{\Delta}_{t}(t,s)\geq0\), \(w^{\Delta}_{t}(t,s)\geq0\) for \(t\geq s\) and (2.3) holds. Suppose that u satisfies
If
then
where
Proof
Denote
Then z is nondecreasing on I,
and
From Lemma 2.3 and (3.15)-(3.21), we have
Similar to the proof of Theorem 3.1, we get (3.17). This completes the proof. □
If we let \(\lambda=\frac{1}{b(T)}\) in Theorem 3.2, then we obtain the following corollary.
Corollary 3.2
Assume that u, a, b, v, w, Ã, B̃, C̃ are the same as in Theorem 3.2 and \(b(T)\neq0\). Suppose that u satisfies
If
then
Theorem 3.3
Assume that u, a, f, λ are the same as in Theorem 3.1. Let \(g(t,s)\) be defined as in Lemma 2.4 such that \(g^{\Delta}_{t}(t,s)\geq0\) for \(t\geq s\) and (2.3) holds. Suppose that u satisfies
If
then
where
Proof
Denote
Then z is nondecreasing on I,
and
From Lemma 2.4 and (3.26)-(3.29), we get
Let
Obviously,
From Lemma 2.4, (3.27) and (3.32), we obtain
It is easy to see that \(A\in\mathcal{R}^{+}\). Therefore, from Lemma 2.5 and the above inequality, we have
Combining (3.33) and (3.34), we get
Setting \(t=\tau\) in (3.35), integrating it from \(t_{0}\) to t, we easily obtain
From (3.28) and (3.30), we have
i.e.,
From (3.25), (3.36) and (3.37), we have
In view of (3.25), we get
Substituting (3.38) into (3.36), we have
Noting \(u(t)\leq z(t)\), we get the desired inequality (3.26). This completes the proof. □
If we let \(\lambda=1\) in Theorem 3.3, then we obtain the following corollary.
Corollary 3.3
Assume that u, a, f, g and A are the same as in Theorem 3.3. Suppose that u satisfies
If
then
Remark 3.2
If we take \(a(t)\equiv u_{0}\), \(g(s,t)=g(t)\) and \(\lambda=0\), then Theorem 3.3 reduces to [27, Theorem 1]. If we take \(\mathbb{T}=\mathbb{R}\), \(a(t)\equiv u_{0}\) and \(\lambda=0\), then Theorem 3.3 reduces to [32, Theorem 2.1 \((a_{1})\)]. If we take \(\mathbb{T}=\mathbb{Z}\), \(a(t)\equiv u_{0}\) and \(\lambda=0\), then Theorem 3.3 reduces to [32, Theorem 2.3 \((c_{1})\)].
4 Applications
In this section, we will present some simple applications for our results.
Example 4.1
Consider the following retarded Volterra-Fredholm type dynamic integral equation on time scales:
where \(u, a, b: I\rightarrow\mathbb{R}\) are rd-continuous functions, \(\vert a \vert \) is nondecreasing, \(b(T)\neq0\), \(\alpha\in \Upsilon\), \(F, \widetilde{F}:I\times\mathbb{R}\times\mathbb{R}\rightarrow\mathbb{R}\) and \(H, \widetilde{H}:I\times\mathbb{R}\rightarrow\mathbb{R}\) are continuous functions.
The following theorem gives an estimate for the solutions of Eq. (4.1).
Theorem 4.1
Suppose that the functions \(F, H, \widetilde{F}\) and H̃ in (4.1) satisfy the conditions
where \(f_{1}, f_{2}, g: I\rightarrow\mathbb{R}_{+}\) are rd-continuous functions. If
then all solutions of Eq. (4.1) satisfy
where \(B(t)\) and \(C(t)\) are defined as in Theorem 3.1.
Proof
Using Corollary 3.1, we obtain the desired inequality (4.4). □
The next result deals with the uniqueness of solutions of Eq. (4.1).
Theorem 4.2
Suppose that the functions F, H, F̃ and H̃ in (4.1) satisfy the conditions
where \(f_{1}, f_{2}, g: I\rightarrow\mathbb{R}_{+}\) are rd-continuous functions. If
then Eq. (4.1) has at most one solution on I, where \(B(t)\) and \(C(t)\) are defined as in Theorem 3.1.
Proof
Let \(u(t)\) and \(v(t)\) be two solutions of Eq. (4.1) on I. From (4.1) and (4.7)-(4.10), we have
Applying Corollary 3.1 to (4.11), we get \(\vert u(t)-v(t) \vert \leq0\), \(t\in I\). Therefore \(u(t)\equiv v(t)\), that is, there is at most one solution to Eq. (4.1). □
Example 4.2
Assume \(\mathbb{T}=q^{\mathbb{N}_{0}}=\{ 1,q,q^{2},\ldots\}\) with \(q>1\) and consider the following Volterra-Fredholm type q-difference equation:
where \(u, a, b, f_{1}, f_{2}, g: I\rightarrow\mathbb{R}_{+}\) are functions, \(t_{0}, T\in q^{\mathbb{N}_{0}}\), \(T>t_{0}\) and \(I=[t_{0},T]\cap q^{\mathbb{N}_{0}}\).
Theorem 4.3
Assume that a is nondecreasing and \((q-1)tA(t)<1\). If
then the solution \(u(t)\) of Eq. (4.12) satisfies the following inequality:
where
Proof
Note that \(\sigma(t)=qt\), \(\mu(t)=(q-1)t\) for any \(t\in q^{\mathbb{N}_{0}}\) and
for \(t>t_{0}\), where \(t, t_{0}, \tau\in q^{\mathbb{N}_{0}}\). Let \(u(t)\) be a solution of Eq. (4.12), we get
Then a suitable application of Corollary 3.1 to (4.14) yields the desired result (4.13). □
Example 4.3
Assume \(\mathbb{T}=\mathbb{R}\) and consider the following retarded Volterra-Fredholm type integral equation:
where \(u, a, b,\alpha, f_{1}, f_{2}, g: I\rightarrow\mathbb{R}_{+}\) are continuous functions, \(t_{0}, T\in\mathbb{R}\), \(T>t_{0}\) and \(I=[t_{0},T]\).
Theorem 4.4
Assume that a is nondecreasing, \(\lambda\geq0\) is a constant, \(\alpha'(t)>0\), \(\alpha(t)\leq t\) and \(b'(t)\geq0\). If
then the solution \(u(t)\) of Eq. (4.15) satisfies the following inequality:
where we use the convention that \(\frac{1}{0}=+\infty\),
Proof
Note that \(\sigma(t)=t\), \(\mu(t)=0\) for any \(t\in\mathbb{R}\), and
for \(t>t_{0}\). Let \(u(t)\) be a solution of Eq. (4.15), we have
Then a suitable application of Theorem 3.1 to (4.17) yields the desired result (4.16). □
5 Conclusions
In this paper we have established some new retarded Volterra-Fredholm type integral inequalities on time scales. Unlike some existing results in the literature (e.g., [16, 20]), the integral inequalities considered in this paper involve the retarded term, which results in difficulties in the estimation on the explicit bounds of unknown functions \(u(t)\). These inequalities generalize and extend some known inequalities and can be used as tools in the qualitative theory of certain classes of retarded dynamic equations on time scales.
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Acknowledgements
The author thanks the reviewers for their helpful and valuable suggestions and comments on this paper. This research was supported by the National Natural Science Foundation of China (No. 11671227) and the Project of Shandong Province Higher Educational Science and Technology Program (China) (No. J14LI09).
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Liu, H. A class of retarded Volterra-Fredholm type integral inequalities on time scales and their applications. J Inequal Appl 2017, 293 (2017). https://doi.org/10.1186/s13660-017-1573-y
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DOI: https://doi.org/10.1186/s13660-017-1573-y