Computational approach of dynamic integral inequalities with applications to timescale calculus
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Computational approach of dynamic integral inequalities with applications to timescale calculus Zareen A. Khan1 · Pooja Arora1 Received: 20 January 2020 / Revised: 19 August 2020 / Accepted: 4 September 2020 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2020
Abstract Based on some known results and simple technique, we emphasize in this article, certain nonlinear dynamic integral inequalities in one variable on timescales. Part of the novelty herein not only unifies and extends some integral inequalities related to different cases of positive constants but also explores the explicit bounds for discontinuous functions on timescales. We contribute to the ongoing research by providing mathematical results that can be used as necessary tools in the theory of certain classes of differential, integral, finite difference and sum–difference equations on timescales. The consequences of the computational experiments show that the proposed strategy can produce adequate and reliable results. Examples are also discussed to demonstrate the importance of the tests. Keywords Timescales · Integral inequality · Dynamic equation · Discontinuous functions Mathematics Subject Classification 26D15 · 26D20 · 39A12
1 Introduction Throughout the most recent decades, with the advancement of the hypothesis of differential and integral equations, a few essential and distinction inequalities have been examined, which assume a great job in the study of boundedness, global existence, stability of solutions of differential and integral equations as well as difference equations. Among the ones, Gronwall– Bellman inequalities are of particular status which provide explicit bounds for unknown functions. Bellman (1943) demonstrated the fundamental integral inequality
Communicated by José Tenreiro Machado.
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Zareen A. Khan [email protected] Pooja Arora [email protected]
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Department of Mathematics, College of Science, Princess Nourah Bint Abdulrahman University, Riyadh, Saudi Arabia 0123456789().: V,-vol
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Z. A. Khan, P. Arora
x(v ∗ ) ≤ b +
v∗
l(s ∗ )x(s ∗ )ds ∗ , v ∗ ∈ [a, b],
a
for some b ≥ 0, which commits significantly to the study of equilibrium and asymptotic behavior to identify solutions to integral equations. Such inequalities have been greatly enhanced in the ongoing years by affirming their significance and natural incentive in various parts of the applied sciences. Later, Pachpatte (1975) replaced the constant b from the above integral inequality by a non-decreasing function b(t ∗ ) and studied v∗ x(v ∗ ) ≤ b(v ∗ ) + g(v ∗ ) l(s ∗ )x(s ∗ )ds ∗ , v ∗ ∈ [0, ∞). (1) 0
Inequality (1) which was proposed first by Gollwitzer, then by Pachpatte and Beesack boosts new speculations and can be tried as astounding instruments in the examination of explicit classes of differential and fundamental conditions. El-Owaidy et al. (1999) presented another basic inequality of the type v∗ s∗ ∗ ∗ ∗ ∗ ∗ ∗ x(v ) ≤ x0 + l(s ) x (s ) + j(ς )x(ς )dς ds ∗ , v ∗ ∈ [0, ∞), 0
0
in which th
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