On Oleck-Opial-Beesack-Troy integro-differential inequalities

It is proved that if a linear operator l : C((a;b);R)!L((a;b);R) is nonpositive and for the Cauchy problem u00(t) = l(u)(t) + q(t), u(a) = c the theorem on dierential inequalities is valid, then l is a Volterra operator.

Applied Mathematics and Computation, 2004

The aim of the present paper is to establish some new integral inequalities in two independent variables which provide explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain partial differential equations.

Applied Mathematics and Computation, 2013

a r t i c l e i n f o Keywords: Bihari inequality Fractional differential equations Riemann-Liouville integral Cauchy-type problem singular differential equations a b s t r a c t

Advances in inequalities and applications, 2013

In this paper, some basic results concerning the strict and nonstrict integro-differential inequalities and existence of the maximal and minimal solutions are proved for a first order hybrid integro-differential equation with a linear perturbations of second type.

2015

In this paper, an integral identity for twice differentiable functions is generalized. Then, by using convexity of |f ′′ | or |f ′′ | q and with the aid of power mean and Hölder's inequalities we achieved some new results. We also gave some applications to quadrature formulas and some special means. Therewithal, by choosing α = 1 2 in our main results, we obtained some findings in [13].

International Journal of Nonlinear Analysis and Applications, 2021

In this paper, several new integral inequalities are presented, which are effective in dealing with the integro-differential inequalities whose variable exponents are greater than one. Compared with existing integral inequalities, those proposed here can be applied to more complicated differential equations. The notions of uniform Lipschitz stability are generalized and the relations between these notions are analyzed. Several sufficient conditions about uniform Lipschitz asymptotic stability of nonlinear systems are established by the proposed integral inequalities. These sufficiently conditions can be similarly generalized to linearly perturbed differential systems that appear in the literature. Finally, an example of uniform Lipschitz asymptotic stability of nonlinear differential systems is shown.

Applied Mathematics and Computation, 1999

In this paper, we establish some new integral inequalities of the Gronwall±Bellman type that have a wide range of the applications in the theory of ordinary dierential equations. The purpose of this paper is to extend the results proved in G.B. Pachpatte [J.

International Scholarly Research Notices, 2011

The object of this paper is to establish some nonlinear integrodifferential integral inequalities in n independent variables. These new inequalities represent a generalization of the results obtained by Pachpatte in the case of a function with one and two variables. Our results can be used as tools in the qualitative theory of a certain class of partial integrodifferential equation.

Applied Mathematics and Computation, 2011

In this article, some new explicit bounds on solutions to a class of new nonlinear integral inequalities of Gronwall-Bellman-Bihari type with delay for discontinuous functions are established. These inequalities generalize and improve some former famous results about inequalities, and which provide an excellent tool to discuss the qualitative and quantitative properties for solutions to some nonlinear differential and integral equations. To illustrate our results, we present an example to show estimated solutions for an impulsive differential system.