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Some new integral inequalities with mixed nonlinearities for discontinuous functions
Advances in Difference Equations volume 2018, Article number: 22 (2018)
Abstract
In this paper, we establish some new integral inequalities with mixed nonlinearities for discontinuous functions, which provide a handy tool in deriving the explicit bounds for the solutions of impulsive differential equations and differential-integral equations with impulsive conditions.
1 Introduction
In recent years, the theory of impulsive differential systems has been attracting the attention of many mathematicians, and the interest in the subject is still growing. This is partly due to broad applications of it in many areas including threshold theory in biology, ecosystems management and orbital transfer of satellite, see [1]. One effective method for investigating the properties of solutions to impulsive differential systems is related to the integral inequalities for discontinuous functions (integro-sum inequalities). Up to now, a lot of integro-sum inequalities (for example, [2–18] and the references therein) have been discovered. For example, in 2003, Borysenko [3] considered the following integro-sum inequality:
In 2009, Gallo and Piccirillo [8] further discussed the following nonlinear integro-sum inequality:
In 2012, Wang et al. [17] considered the nonlinear integro-sum inequality as follows:
Very recently, in 2016, Zheng et al. [18] considered the following nonlinear integro-sum inequality under the condition \(p>q>0\):
Motivated by [3, 8, 17, 18], in this paper, we investigate some new integro-sum inequality with mixed nonlinearities under the condition \(p>0\), \(q>0\) (\(p\neq q\)):
and the more general form
We also discuss some nonlinear integro-sum inequality with positive and negative coefficients under the condition \(0< q< p< r\):
and the more general form under the condition \(0< q_{j}< p< r_{j}\) (\(j=1,2,\ldots, L\)):
Based on these inequalities, we provide explicit bounds for unknown functions concerned and then apply the results to research the qualitative properties of solutions of certain impulsive differential equations.
2 Preliminaries
Throughout the present paper, R denotes the set of real numbers; \(\mathrm {R}_{+}=[0,+\infty)\) is the subset of R; \(C(D, E)\) denotes the class of all continuous functions defined on the set D with range in the set E.
Lemma 2.1
([19])
Assume that the following conditions for \(t\geq t_{0}\) hold:
-
(i)
\(x_{0}\) is a nonnegative constant,
-
(ii)
$$x(t)\leq x_{0}+ \int_{t_{0}}^{t} \bigl[e(s) x(s) +l(s) x^{\alpha}(s) \bigr]\,\mathrm {d}s, $$
where x, e and l are nonnegative continuous functions and \(\alpha\neq1\) is a positive constant.
If
holds, then
Lemma 2.2
([20])
Let x be a nonnegative function, \(0< q< p< r\), \(c_{1}\geq0\), \(k_{2}\geq0\), \(c_{2}>0\) and \(k_{1}>0\). Then
where
3 Main results
Theorem 3.1
Suppose that x is a nonnegative piecewise continuous function defined on \([t_{0},\infty)\) with discontinuities of the first kind in the points \(t_{i}\) (\(i=1,2,\ldots\)) and satisfies the integro-sum inequality
where \(0\leq t_{0}< t_{1} < t_{2}<\cdots\) , \(\lim_{i\rightarrow\infty}t _{i}=\infty\), functions \(a(t)\geq0\) and \(c(t)\geq0\) are defined on \([t_{0},\infty)\), \(f_{1},f_{2},f_{3},g_{1},g_{2}\in C(\mathrm {R}_{+},\mathrm {R}_{+})\), \(\beta_{i}\geq0\) (\(i=1,2,\ldots\)), \(p>0\), \(q>0\), \(p\neq q\) and \(m>0\) are constants. If
then, for \(t\geq t_{0}\), the following estimates hold:
where
Proof
From (3.1) and (3.5), we have, for \(t\in I_{0}=[t_{0},t _{1}]\),
and \(r_{1}(t)\) is non-decreasing on \([t_{0},\infty)\). Take any fixed \(T\in[t_{0},t_{1}]\), and for arbitrary \(t\in[t_{0},T]\), we have
Let \(u(t)=x^{p}(t)\). Inequality (3.7) is equivalent to
Let
It follows from (3.8) and (3.9) that
\(V(t)\) is non-decreasing and
Since \(V(t)\) is non-decreasing, from (3.11) we have
where \(e(t)\) and \(l(t)\) are defined as in (3.4). Integrating (3.12) from \(t_{0}\) to t yields
From the above and Lemma 2.1, we get
and then from (3.10) and the assumption \(u(t)=x^{p}(t)\), we have
Since the above inequality is true for any \(t\in[t_{0},T]\), we obtain
Replacing T by t yields
This means that (3.1) is true.
For \(t\in I_{1}=(t_{1},t_{2}]\), from (3.1), (3.2), (3.5) and (3.13), we get
Inequality (3.14) is the same as (3.6) if we replace \(r_{1}(t)\) and \(t_{0}\) with \(r_{2}(t)\) and \(t_{1}\) in (3.6), respectively. Thus, by (3.14), we have, for \(t\in I_{1}=(t_{1},t_{2}]\),
Suppose that
holds for \(t\in I_{i-1}=(t_{i-1}, t_{i}]\), \(i=2,3,\ldots\) . Then, for \(t\in I_{i}=(t_{i}, t_{i+1}]\), from (3.1), (3.2), (3.5) and (3.15) we obtain
Inequality (3.16) is the same as (3.6) if we replace \(r_{1}(t)\) and \(t_{0}\) with \(r_{i+1}(t)\) and \(t_{i}\) in (3.6), respectively. Thus, by (3.16), we have, for \(t\in I_{i}=(t_{i}, t_{i+1}]\),
By induction, we know that (3.3) holds for \(t\in(t_{i},t_{i+1}]\), for any nonnegative integer i. This completes the proof of Theorem 3.1. □
Theorem 3.2
Suppose that x is a nonnegative piecewise continuous function defined on \([t_{0},\infty)\) with discontinuities of the first kind in the points \(t_{i}\) (\(i=1,2,\ldots\)) and satisfies the integro-sum inequality
where \(0\leq t_{0}< t_{1} < t_{2}<\cdots\) , \(\lim_{i\rightarrow\infty}t _{i}=\infty\), \(a(t)\geq0\) is defined on \([t_{0},\infty)\), \(f\in C(\mathrm {R}_{+},\mathrm {R}_{+})\), \(b_{j},g_{j}\in C( \mathrm {R}_{+},\mathrm {R}_{+})\) (\(j=1,2,\ldots, L\)), \(c_{j},\theta _{j}\in C(\mathrm {R}_{+},\mathrm {R}_{+})\) (\(j=1,2,\ldots, M\)), \(\beta_{i}\geq0\) (\(i=1,2,\ldots\)), \(p>0\), \(q>0\), \(p\neq q\), and \(m>0\) are constants. If
then, for \(t\geq t_{0}\), the following estimates hold:
where
The proof is similar to that of Theorem 3.1, and we omit these details.
Theorem 3.3
Suppose that x is a nonnegative piecewise continuous function defined on \([t_{0},\infty)\) with discontinuities of the first kind in the points \(t_{i}\) (\(i=1,2,\ldots\)) and satisfies the integro-sum inequality:
where \(0\leq t_{0}< t_{1} < t_{2}<\cdots\) , \(\lim_{i\rightarrow\infty}t _{i}=\infty\), \(a(t)\) is defined on \([t_{0},\infty)\) and \(a(t_{0}) \neq0\), \(b(t)\geq0\) and \(c(t)\geq0\) are defined on \([t_{0},\infty )\), \(f,g\in C(\mathrm {R}_{+},\mathrm {R}_{+})\), \(h\in C(\mathrm {R}_{+},(0,+\infty))\), \(0< q< p< r\), \(\beta_{i}\geq0\) (\(i=1,2,\ldots\)) and \(m>0\) are constants.
Then, for any continuous functions \(k_{1}(t)>0\) and \(k_{2}(t)\geq0\) on \([t_{0},\infty)\) satisfying \(k(t)= k_{1}(t)-k_{2}(t)\geq0\), the following estimates hold:
where
Proof
From (3.20) and (3.24), we obtain, for \(t\in I_{0}=[t _{0},t_{1}]\),
From Lemma 2.1, (3.24)-(3.27) and (3.29), we have
\(r_{1}(t)\) and \(e(t)\) are non-decreasing on \([t_{0},\infty)\). Take any fixed \(T\in[t_{0},t_{1}]\), and for arbitrary \(t\in[t_{0},T]\), we have
Let \(u(t)=x^{p}(t)\). Inequality (3.31) is equivalent to
Define a function \(V(t)\) by the right-hand side of (3.32). Then \(V(t)\) is positive and
We have
and then, from (3.33), (3.34) and the assumption \(u(t)=x^{p}(t)\), we get
Since the above inequality is true for any \(t\in[t_{0},T]\), we obtain
Replacing T by t yields
This means that (3.21) is true for \(t\in[t_{0},t_{1}]\).
For \(t\in I_{1}=(t_{1},t_{2}]\), from Lemma 2.1 and (3.20), (2.24)-(2.27) and (3.35), we obtain
Inequality (3.36) is the same as (3.30) if we replace \(r_{1}(t)\) and \(t_{0}\) with \(r_{2}(t)\) and \(t_{1}\) in (3.36), respectively. Thus, by (3.35) and (3.36), we get, for \(t\in I_{1}=(t_{1},t_{2}]\),
Suppose that
Then, for \(t\in I_{i}=(t_{i}, t_{i+1}]\), from Lemma 2.1 and (3.20), (3.24)-(3.27) and (3.37), we have
Inequality (3.38) is the same as (3.30) if we replace \(r_{1}(t)\) and \(t_{0}\) with \(r_{i+1}(t)\) and \(t_{i}\) in (3.38), respectively. Thus, by (3.35) and (3.38), we have, for \(t\in I_{i}=(t_{i}, t_{i+1}]\),
By induction, we know that (3.30) holds for \(t\in(t_{i},t_{i+1}]\), for any nonnegative integer i. This completes the proof of Theorem 3.3. □
Theorem 3.4
Suppose that x is a nonnegative piecewise continuous function defined on \([t_{0},\infty)\) with discontinuities of the first kind in the points \(t_{i}\) (\(i=1,2,\ldots\)) and satisfies the integro-sum inequality
where \(0\leq t_{0}< t_{1} < t_{2}<\cdots\) , \(\lim_{i\rightarrow\infty}t _{i}=\infty\), \(a(t)\) is defined on \([t_{0},\infty)\) and \(a(t_{0}) \neq0\), \(b(t)\geq0\) and \(c(t)\geq0\) are defined on \([t_{0},\infty )\), \(f,g\in C(\mathrm {R}_{+},\mathrm {R}_{+})\), \(h\in C(\mathrm {R}_{+},(0,+\infty))\), \(0< q_{j}< p< r_{j}\) (\(j=1,2,\ldots, L\)), \(\beta_{i}\geq0\), \(i=1,2,\ldots\) , and \(m>0\) are constants.
Then, for any continuous functions \(k_{1}(t)>0\) and \(k_{2}(t)\geq0\) on \([t_{0},\infty)\) satisfying \(k(t)= k_{1}(t)-k_{2}(t)\geq0\), the following estimates hold:
where
The proof is similar to that of Theorem 3.3, and we omit these details.
4 Application
In this section, we will apply the results which we have established above to the estimates of solutions of certain impulsive differential equations.
Example 4.1
Consider the following impulsive differential equation:
where \(p>0\), \(m>0\) are constants, the functions \(d(t)\geq0\), \(t\in[t _{0},\infty)\), \(F\in C(\mathrm {R}\times \mathrm {R}\times \mathrm {R}, \mathrm {R}_{+})\) and \(G\in C(\mathrm {R}\times \mathrm {R}\times \mathrm {R},\mathrm {R}_{+})\) satisfy the following conditions:
where \(q>0\) (\(q\neq p\)) is a constant, and \(f(t)\), \(g_{j}(t)\), \(b_{j}(t)\) (\(j=1,2,\ldots, L\)), \(c_{j}(t)\), \(\theta_{j}(t)\) (\(j=1,2,\ldots, M\)) are defined as in Theorem 3.2. If
then for \(t\geq t_{0}\), every solution \(x(t)\) of Eq. (4.1) satisfies the following estimates:
where \(l(t)\), \(e(t)\), \(r_{i}(t)\) and \(v_{i}(t)\) (\(i=1,2,\ldots\)) are defined as in Theorem 3.2.
Proof
The solution \(x(t)\) of Eq. (4.1) satisfies the following equivalent equation:
From conditions (4.2) and (4.3), it is easy to have
By using Theorem 3.2, we easily obtain estimates (4.4) and (4.5) of solutions of Eq. (4.1). □
Example 4.2
Consider the following impulsive differential equation:
where \(0\leq t_{0}< t_{1} < t_{2}<\cdots\) , \(\lim_{i\rightarrow\infty}t _{i}=\infty\), \(f,g\in C(\mathrm {R}_{+},\mathrm {R}_{+})\), \(h\in C(\mathrm {R}_{+},(0,+\infty))\), \(a(t)\geq0\) is defined on \([t_{0},\infty)\) and \(\beta_{i}\geq0\) (\(i=1,2,\ldots\)) are constants. Then, for any continuous functions \(k_{1}(t)>0\) and \(k_{2}(t)\geq0\) on \([t_{0},\infty)\) satisfying \(k(t)= k_{1}(t)-k_{2}(t)\geq0\), the following estimates hold:
where
Proof
The solution \(x(t)\) of Eq. (4.6) satisfies the following equivalent equation:
From the assumptions of f, g and h, it follows
By using Theorem 3.3, we easily obtain estimates (4.7) and (4.8) of solutions of Eq. (4.6). □
References
Bainov, DD, Simeonov, PS: Impulsive Differential Equations: Periodic Solution and Applications. Longman, Harlow (1993)
Mitropolskiy, YuA, Iovane, G, Borysenko, SD: About a generalization of Bellman-Bihari type inequalities for discontinuous functions and their applications. Nonlinear Anal. 66, 2140-2165 (2007)
Borysenko, SD: About one integral inequality for piece-wise continuous functions. In: Proc. X Int. Kravchuk Conf., Kyiv, p. 323 (2004)
Iovane, G: Some new integral inequalities of Bellman-Bihari type with delay for discontinuous functions. Nonlinear Anal. 66, 498-508 (2007)
Gallo, A, Piccirillo, AM: About new analogies of Gronwall-Bellman-Bihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems. Nonlinear Anal. 67, 1550-1559 (2007)
Borysenko, SD, Ciarletta, M, Iovane, G: Integro-sum inequalities and motion stability of systems with impulse perturbations. Nonlinear Anal. 62, 417-428 (2005)
Borysenko, S, Iovane, G: About some new integral inequalities of Wendroff type for discontinuous functions. Nonlinear Anal. 66, 2190-2203 (2007)
Gallo, A, Piccirillo, AM: On some generalizations Bellman-Bihari result for integro-functional inequalities for discontinuous functions and their applications. Bound. Value Probl. 2009, Article ID 808124 (2009)
Iovane, G: On Gronwall-Bellman-Bihari type integral inequalities in several variables with retardation for discontinuous functions. Math. Inequal. Appl. 11, 599-606 (2008)
Gallo, A, Piccirillo, AM: About some new generalizations of Bellman-Bihari results for integro-functional inequalities with discontinuous functions and applications. Nonlinear Anal. 71, 2276-2287 (2009)
Deng, SF, Prather, C: Generalization of an impulsive nonlinear singular Gronwall-Bihari inequality with delay. J. Inequal. Pure Appl. Math. 9, Article ID 34 (2008)
Wang, WS, Zhou, XL: A generalized Gronwall-Bellman-Ou-Iang type inequality for discontinuous functions and applications to BVP. Appl. Math. Comput. 216, 3335-3342 (2010)
Shao, J, Meng, FW: Nonlinear impulsive differential and integral inequalities with integral jump conditions. Adv. Differ. Equ. 2016, 112 (2016)
Zheng, B: Some generalized Gronwall-Bellman type nonlinear delay integral inequalities for discontinuous functions. J. Inequal. Appl. 2013, 297 (2013)
Mi, YZ, Deng, SF, Li, XP: Nonlinear integral inequalities with delay for discontinuous functions and their applications. J. Inequal. Appl. 2013, 430 (2013)
Mi, YZ: Generalized integral inequalities for discontinuous functions with two independent variables and their applications. J. Inequal. Appl. 2014, 524 (2014)
Wang, WS, Li, ZZ, Tang, AM: Nonlinear retarded integral inequalities for discontinuous functions and their applications. In: Computer, Informatics, Cybernetics and Applications, vol. 107, pp. 149-159 (2012)
Zheng, ZW, Gao, X, Shao, J: Some new generalized retarded inequalities for discontinuous functions and their applications. J. Inequal. Appl. 2016, 7 (2016)
Li, YS: The bound, stability and error estimates of the solution of nonlinear different equations. Acta Math. Sin. 12(1), 28-36 (1962) (in Chinese)
Tian, YZ, Cai, YL, Li, LZ, Li, TX: Some dynamic integral inequalities with mixed nonlinearities on time scales. J. Inequal. Appl. 2015, 12 (2015)
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) A Project of Shandong Province Higher Educational Science and Technology Program (China) (No. J14LI09).
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Liu, H. Some new integral inequalities with mixed nonlinearities for discontinuous functions. Adv Differ Equ 2018, 22 (2018). https://doi.org/10.1186/s13662-017-1450-5
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DOI: https://doi.org/10.1186/s13662-017-1450-5
Keywords
- integral inequalities
- discontinuous functions
- mixed nonlinearities
- impulsive differential equations