Analysis of a Nonlinear Integral Inequality on Time Scales

During the past decades, many integral inequalities have been established since then, for example [1-10], which have played an important role in the research of qualitative properties of solutions of dynamic equations. Our aim in this paper is to establish a new Volterra-Fredholm type delay integral inequality on time scales, which provides new bound for unknown functions. In the rest of the paper, R denotes the set of real numbers and [0, ) R+ = ∞ . T denotes an arbitrary time scale and 0 0 0 0 [ , ) , [ , ) T x T T y T = ∞ = ∞ ∩ ∩ , where 0 0 , x y T ∈ . The set T κ is defined to be T if T does not have a left-scattered maximum, otherwise it is T without the left-scattered maximum. On T we define the forward and backward jump operators ( , ) T T σ ∈ and ( , ) T T ρ∈ such that ( ) inf{ , } t s T s t σ = ∈ > , ( ) sup{ , } t s T s t ρ = ∈ < . Definition 1: A point t T ∈ with t infT > is said to be left-dense if ( ) t t ρ = and right-dense if ( ) t t σ = , left-scattered if ( ) t t ρ < and right-scattered if ( ) t t σ > . Definition 2: A function ( , ) f T R ∈ is called rd-continuous if it is continuous in right-dense points and if the left-sided limits exist in left-dense points, while f is called regressive if 1 ( ) ( ) 0 t f t μ + ≠ , where ( ) ( ) t t t μ σ = − .