Abstract
In this paper we derive some generalizations of certain Gronwall- Bellman-Bihari-Gamidov type integral inequalities and their weakly singular analogues, which provide explicit bounds on unknown functions. To show the feasibility of the obtained inequalities, two illustrative examples are also introduced.
Highlights
The integral inequalities which provide explicit bounds on unknown functions have proved to be very useful in the study of qualitative properties of the solutions of differential and integral equations
During the past few years, many such new inequalities have been discovered, which are motivated by certain applications
Throughout the paper, R denotes the set of real numbers, R0 = (0, ∞), R+ = [0, +∞) and I = [0, T ] (T ≥ 0 is a constant), C(X, Y ) denotes the collection of continuous functions from the set X to the set Y, p, q, r are real constants such that p = 0, 0 ≤ q, r ≤ p
Summary
The integral inequalities which provide explicit bounds on unknown functions have proved to be very useful in the study of qualitative properties of the solutions of differential and integral equations. Gamidov [6], while studying the boundary value problem for higher order differential equations, initiated the study of obtaining explicit upper bounds on the integral inequalities of the forms (1.1). Zheng [16] established a weakly singular version of the Gronwall-Bellman-Gamidov inequality as follows:. [5], we discuss more general form of nonlinear weakly singular integral inequalities of Gronwall-Bellman-Bihari-Gamidov t up(t) ≤a(t) + b(t) (tα1 − sα1)β1−1sγ1−1f (s)uq(s)ds.
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