Explicit bounds of unknown function of some new weakly singular retarded integral inequalities for discontinuous functions and their applications

The purpose of the present paper is to establish some new retarded weakly singular integral inequalities of Gronwall-Bellman type for discontinuous functions, which generalize some known weakly singular and impulsive integral inequalities. The inequalities given here can be used in the analysis of the qualitative properties of certain classes of singular differential equations and singular impulsive equations.


Introduction
Being an important tool in the study of qualitative properties of solutions of differential equations and integral equations, various generalizations of Gronwall-Bellman integral inequality and their applications have attracted great interest of many mathematicians (such as [-] and the references therein). Gronwall  Usually, this type integral inequalities have regular or continuous integral kernels, but some problems of theory and practicality require us to solve integral inequalities with singular kernels. For example, to prove a global existence and an exponential decay result for a parabolic Cauchy problem. Henry [] investigated the following linear singular integral inequality:

Sano and Kunimatsu[] generalized Henry's type inequality to
and gave a sufficient condition for stabilization of semilinear parabolic distributed systems. Ye et al. [] discussed the linear singular integral inequality and they used it to study the dependence of the solution and the initial condition to a certain fractional differential equation with Riemann-Liouville fractional derivatives. All inequalities of this type are proved by an iteration argument and the estimation formulas are expressed by a complicated power series which is sometimes not very convenient for applications. To avoid the weakness, Medveď [] presented a new method to solve integral inequalities of Henry-Gronwall type, then he got the explicit bounds with a quite simple formula, similar to the classic Gronwall-Bellman inequalities. Furthermore, he also obtained global solutions of the semilinear evolutions in []. In , Ma and Pečarić [] used the modification of Medveď's method to study a new weakly singular integral inequality, Besides the results mentioned above, various investigators have discovered many useful and new weakly singular integral inequalities, mainly inspired by their applications in various branches of fractional differential equations (see [, -] and the references therein).
In analyzing the impulsive phenomenon of a physical system governed by certain differential and integral equations, by estimating the unknown function in the integral inequality of the discontinuous functions, Some properties of the solution of some impulsive differential equations can be studied. These inequalities and their various linear and nonlinear generalizations are crucial in the discussion of the existence, uniqueness, boundedness, stability, and other qualitative properties of solutions of differential and integral equations (see [, , -] and the references therein). Tatar [] discussed the following class of integral inequalities: Iovane [] studied the following discontinuous function integral inequality: Gllo et al. [] studied the impulsive integral inequality Mi et al. [] studied the integral inequality of complex functions with unknown function where w(u) is monotone decreasing continuous function defined on [, ∞), and w(u) >  when u > . Liu et al. [] investigated the impulsive integral inequality with delay Zheng et al. [] studied the following integral inequality for discontinuous function: where u(t), a(t) and g i (t), b j (t), c j (t) ( ≤ i ≤ N ,  ≤ j ≤ L) are positive and continuous functions on [t  , ∞), and c j (t) are nondecreasing functions on [t  , ∞), and φ i (t), w j (t) are continuous functions on [t  , ∞) and However, in certain situations, such as some classes of delay impulsive differential equations and delay impulsive integral equations, it is desirable to find some new delay impulsive inequalities, in order to achieve a diversity of desired goals. In this paper, we discuss a class of retarded integral inequalities with weak singularity for discontinuous functions,

Main results
Throughout this paper, R denotes the set of real numbers and R + = [, ∞) is the given subset of R, and C(M, S) denotes the class of all continuous functions defined on set M with range in the set S.
The following lemmas are very useful in the procedures of our proof in our main results.

Lemma  Suppose that f (x) and g(x) are nonnegative and continuous functions on
Let α(t) be a continuous, differentiable and increasing function on Proof We prove the inequality (). Using the inequality (), we obtain . . , a n be nonnegative real numbers, m >  is a real number, and n is a natural number. Then

Let α(t) be a continuous, differentiable and increasing function on
is the well-known beta-function and θ = p[β(γ -) + ξ -] + . Suppose that the positive constants β, γ , ξ , p  and p  satisfy conditions: , If u(t) satisfies the following inequality: where Differentiating v(t) with respect to t and using () and (), we have Integrating both sides of the inequality () from  to t, From () and (), we obtain

h(s)a(s)e(s) ds.
() Substituting the inequality () into () we get the required estimation (). The proof is completed.

Theorem  Let u(t) is a nonnegative piecewise continuous function with discontinuous
of the first kind in the points t i (t  < t  < t  < · · · , lim i→∞ t i = ∞), a(t), f (t) ∈ C(R + , R + ), a(t) ≥ , and let α(t) be a continuous, differentiable and increasing function on [t  , +∞) with α(t) ≤ t, α(t i ) = t i , i = , , , . . . . Let p, β, γ be positive constants, β i ∈ [, ∞). If u(t) satisfies the inequality (), then we have whereã Proof Firstly, we consider the case t ∈ [t  , t  ), denoting then v(t) is a nonnegative and nondecreasing continuous function, and Differentiating () with respect to t, we have then (t) is a nonnegative and nondecreasing function, and (t  ) = a  (t  ), since a(t) ≥ , we can conclude that v(t) ≤ (t), differentiating (), from (), we obtain Let η(t) = -(p+) (t), then η (t) = -(p + ) -(p+) (t), () can be restated as η (t) + (p + )η(t) α (t)a (t) + α (t)g α(t) Multiplying by exp((p + ) t  (a (α - (s)) + g(s)) ds) on both sides of (), we have integrating both sides of () from t  to t, we obtain by η(t) = -(p+) (t), from (), we have Integrating both side of () from t  to t, we get Equation () has the same form as Lemma , and the functions of () satisfy the conditions of Theorem . Consequently, by using a similar procedure to Lemma  and Theorem , we can get the desired estimations () for t ∈ [t  , t  ). Next, let us consider the interval [t  , t  ), when t ∈ [t  , t  ), () can be restated as setting then (t) is a nonnegative and nondecreasing function, and Differentiating with respect to t both sides of (), we obtain () has the same form of (), and using a similar procedure for t ∈ [t  , t  ), we can get the desired estimations () for t ∈ [t  , t  ). Consequently, by using a similar procedure for t ∈ [t i , t i+ ), we can get the desired estimations () for t ∈ [t i , t i+ ). Thus we complete the proof of Theorem .

Conclusion
In this paper, we generalized the weakly singular integral inequality. The first inequality was a generally weak singular type, the second inequality was a like-weakly singular type with discontinuous functions, the third inequality was a type of weakly singular integral inequality with impulsive. We used analytical methods, reducing the inequality with the known results in the lemma, and the estimations of the upper bound of the unknown functions were given. The results were applied to the weakly singular integral equation and the impulsive differential system.