Abstract
In 2020, El-Deeb et al. proved several dynamic inequalities. It is our aim in this paper to give the retarded time scales case of these inequalities. We also give a new proof and formula of Leibniz integral rule on time scales. Beside that, we also apply our inequalities to discrete and continuous calculus to obtain some new inequalities as special cases. Furthermore, we study boundedness of some delay initial value problems by applying our results as application.
Highlights
Theorem 1.2 (Chain rule on time scales [3]) Let f : R → R be continuously differentiable and suppose g : T → R is -differentiable
In 2020, El-Deeb et al [1] have proved the following inequalities: ̃ u(ς, ˆ ) ς ˆ ≤ a(ς, ˆ ) +f (ξ1, ξ2) ̃ u(ξ1, ξ2) + p(ξ1, ξ2) ξ2 ξ1 ξ1 +b(ξ1, ξ2) h(ξ1, ξ2) ̃ u(ξ1, ξ2) + g(ζ, ξ2) ̃ u(ζ, ξ2) ζξˆ2 ξ1 andu(ς, ˆ )̃ u(ξ1, ξ2) f (ξ1, ξ2) ̃ u(ξ1, ξ2) + p(ξ1, ξ2) ξ2 ξ1f (ξ1, ξ2) ̃ u(ξ1, ξ2)ξ1 g(ζ, ξ2) ̃ u(ζ, ξ2) ζξˆ2 ξ1
We will discuss the retarded time scale case of the inequalities obtained in [1] using new techniques by replacing the upper limit ςandof the integral by the delay function α (ς) ≤ ςand β( ˆ ) ≤ ˆ
Summary
Theorem 1.2 (Chain rule on time scales [3]) Let f : R → R be continuously differentiable and suppose g : T → R is -differentiable. We will discuss the retarded time scale case of the inequalities obtained in [1] using new techniques by replacing the upper limit ςandof the integral by the delay function α (ς) ≤ ςand β( ˆ ) ≤ ˆ . Theorem 2.1 (Leibniz integral rule on time scales) In the following by f (t, s) we mean the delta derivative of f (t, s) with respect to t.
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