Some new integral inequalities of Wendorff type for discontinuous functions with integral jump conditions

In this paper, we investigate some new integral inequalities of Wendorff type for discontinuous functions with two independent variables and integral jump conditions. These integral inequalities with discontinuities are of non-Lipschitz type. New lower bounds are obtained, integral inequalities with retardation are also involved.

In this paper, in a similar way to [8][9][10][11][12] for the inequalities of the functions with one independent variable, we investigate a new Wendorff type inequality for discontinuous functions with two independent variables and give some integro-sum functional inequalities with delay.

Integral inequalities for discontinuous functions with discontinuities of non-Lipschitz type
For a given function a defined in a domain Ω with two variables, we say a is a nondecreasing function if, for all (p, q), (P, Q) ∈ Ω with p ≤ P, q ≤ Q, one always has a(p, q) ≤ a(P, Q).

Theorem 2.1 Let a nonnegative function ϕ(t, x), determined in the domain
be continuous in Ω, with the exception of the points (t i , x i ) where there are finite jumps and satisfy a certain integro-sum inequality in Ω then the function ϕ(t, x) satisfies the following estimates: Proof Due to a(t, x) > 0, we can obtain that Set with W (t 0 , We give the proof by induction. Firstly, we consider the domain set thus Then Differentiating K(t, x) with respect to t, the following equation holds: because b(t, x) and V (t, x) are continuous in Ω 11 . Besides, V (t, x) > 0, it means that V (t, x) maintains the sign in Ω 11 . So, on account of generalized mean value theorem of integrals, we can get that Integrating this inequality from t 0 to t implies so we can get that This shows that the estimates are true in Ω 11 . Secondly, suppose that (9) and (10) are true in the domain Ω kk . If 0 < m ≤ 1, then for (t, x) ∈ Ω k+1,k+1 the following inequality holds: so when 0 < m ≤ 1, (9) stands.

Theorem 2.2 Suppose that there exists a nonnegative piecewise continuous function
, and it satisfies the inequality . . , the following estimates hold: Proof Because of a(t, x) > 0, we get Set then By mathematical induction, we consider the function in the domain set then ϕ(t, x) ≤ a(t, x)K(t, x), K(t 0 , x) = 1, K(t, x 0 ) = 1.
Differentiating K(t, x) with respect to t, the following equation holds: Since b(t, x) and W (t, x) are continuous in Ω 11 , besides W (t, x) > 0, it means that W (t, x) maintains the same sign in Ω 11 . So, on account of the generalized first mean value theorem of integrals, we can get that Integrating this inequality from t 0 to t, we get For the case of m > 1, Integrating this inequality from t 0 to t, we get Now, we firstly consider the case of 0 < m < 1. Suppose that (19) is justified in the domain Ω kk , then for (t, x) ∈ Ω k+1,k+1 the following inequality holds: The right-hand side of this inequality is defined as x) with respect to t, and on account of the generalized first mean value theorem of integrals, we can get that Integrating the above inequality from t k to t, we get then we can get that so when 0 < m < 1, (19) stands.
Next, we prove the case of m > 1. Assume that (20) is fulfilled in the domain Ω kk , then for (t, x) ∈ Ω k+1,k+1 the following inequality holds: The right-hand side of the last inequality is defined as and W (t, x) ≤ U(t, x). Differentiating U(t, x) with respect to t, and on account of the generalized first mean value theorem of integrals, we can get that Integrating this inequality from t k to t, we have Then we can get that ∀(t, x) ∈ Ω k+1,k+1 satisfying that So when m > 1, (20) stands. By mathematical induction, this completes the proof.

Theorem 2.3 Suppose that there exists a nonnegative piecewise continuous function ϕ(t, x)
determined in the domain Ω, with discontinuity of the first kind in the points (t k , x k ) (t 0 < t 1 < t 2 < · · · , x 0 < x 1 < x 2 < · · · , lim i→∞ t i = ∞, lim i→∞ x i = ∞), and it satisfies the inequality m > 0, m = 1, where a, b, γ i , β i satisfy the conditions of Theorem 2.1. Then, for (t, x) ∈ Ω, k = 1, 2, . . . , the following estimates hold: if m = 1; and Using the procedure for W (t, x) in Theorem 2.2, it is possible to obtain for W (t, x) the following estimates: if m = 1;