Nonlinear Integral Inequalities Research Articles

A New Nonlinear Integral Inequality with a Tempered Ψ–Hilfer Fractional Integral and Its Application to a Class of Tempered Ψ–Caputo Fractional Differential Equations

In this paper, the tempered Ψ–Riemann–Liouville fractional derivative and the tempered Ψ–Caputo fractional derivative of order n−1<α<n∈N are introduced for Cn−1–functions. A nonlinear version of the second Henry–Gronwall inequality for integral inequalities with the tempered Ψ–Hilfer fractional integral is derived. By using this inequality, an existence and uniqueness result and a sufficient condition for the non-existence of blow-up solutions of nonlinear tempered Ψ–Caputo fractional differential equations are proved. Illustrative examples are given.

New retarded nonlinear integral inequalities of Gronwall–Bellman–Pachpatte type and their applications

Abstract The goal of the present article is to offer several new retarded nonlinear inequalities of Gronwall, Bellman, and Pachpatte kind for a class of integral and integro-differential equations. These inequalities generalize and provide new formulations of some well-known results in the mathematical framework of integral and differential inequalities that have been derived currently as well as in earlier times. These results can be utilized to investigate diverse aspects, both qualitative and quantitative, of a class of aforementioned equations. We propose a few applications to ensure the effectiveness of these inequalities.

Remarks on a controlled degenerate diffusion process

We establish explicit estimates for a two-dimensional LQG homing problem with a zero terminal cost under the assumption that the controlled diffusion process is degenerate. The present results extend and complete those in Lefebvre (2013) which are given explicitly for a particular interesting case. Our main tool is a nonlinear integral inequality of the Gronwall type. As an application, we consider a simple example to illustrate the present results.

Nonlinear Integral Inequalities Involving Tempered Ψ-Hilfer Fractional Integral and Fractional Equations with Tempered Ψ-Caputo Fractional Derivative

In this paper, the nonlinear version of the Henry–Gronwall integral inequality with the tempered Ψ-Hilfer fractional integral is proved. The particular cases, including the linear one and the nonlinear integral inequality of this type with multiple tempered Ψ-Hilfer fractional integrals, are presented as corollaries. To illustrate the results, the problem of the nonexistence of blowing-up solutions of initial value problems for fractional differential equations with tempered Ψ-Caputo fractional derivative of order 0<α<1, where the right side may depend on time, the solution, or its tempered Ψ-Caputo fractional derivative of lower order, is investigated. As another application of the integral inequalities, sufficient conditions for the Ψ-exponential stability of trivial solutions are proven for these kinds of differential equations.

Results for Retarded Nonlinear Integral Inequalities with Mixed Powers and Their Applications to Delay Integrodifferential Equations

Results for Retarded Nonlinear Integral Inequalities with Mixed Powers and Their Applications to Delay Integrodifferential Equations

On the Analysis of Regularized Fuzzy Systems of Uncertain Differential Equations.

This article analyzes a regularized set of fuzzy differential equations with respect to an uncertain parameter. We provide sufficient conditions for the correctness of a new regularization scheme. For the resulting family of regularized fuzzy differential equations, the following properties are analyzed, and efficient criteria are proposed: successive approximations, continuity, global existence of solutions, existence of approximate solutions, existence of solutions in the autonomous case. In addition, we develop stability criteria for the regularized family of fuzzy differential equations on the basis of the comparison technique and the method of nonlinear integral inequalities. We expect that the derived results will inspire future research work in this direction.

A class of nonlinear delay integral inequalities for two-variable functions and their applications in Volterra integral equations

A class of nonlinear delay integral inequalities for two-variable functions and their applications in Volterra integral equations

Results for retarded nonlinear integral inequalities with mixed powers and their applications to delay integro-differential equations

UDC 517.9 We present new retarded nonlinear integral inequalities with mixеd powers. The obtained inequalities can be used to study the boundedness and global existence of the solutions of integro-differential equation with delay and Volterra-type integral equation with delay. These inequalities extend some results available in the literature. Finally, we present two examples to demonstrate the usefulness of our main results.

On New Generalized Viscosity Implicit Double Midpoint Rule for Hierarchical Problem

The implicit midpoint rules are employed as a powerful numerical technique, and in this article we attend a class of viscosity iteration approximations on hierarchical problems for the implicit double midpoint rules. We prove the strong convergence theorem to the unique solution on hierarchical problem of this technique is established under some favorable conditions imposed on the control parameters in Hilbert spaces. Furthermore, we propose some applications to the constrained convex minimization problem, nonlinear Fredholm integral equation and variational inequality on fixed point problem. Moreover, some numerical examples are also presented to illustrate the different proposed methods and convergence results. Our results modified the implicit double midpoint rules with the hierarchical problem.

Some new weakly singular nonlinear integral inequalities and their application

We establish some new weakly singular nonlinear integral inequalities, which generalize some known integral inequalities. The inequalities given here can be used in the analysis of the qualitative properties of fractional differential equations and integral equations. Applications are also provided to illustrate the usefulness of our results.

Nonlinear Inequalities with Double Riesz Potentials

We investigate the nonnegative solutions to the nonlinear integral inequality u ≥ Iα ∗((Iβ ∗ up)uq) a.e. in {mathbb R}^{N}, where α, β ∈ (0, N), p, q > 0 and Iα, Iβ denote the Riesz potentials of order α and β respectively. Our approach relies on a nonlocal positivity principle which allows us to derive optimal ranges for the parameters α, β, p and q to describe the existence and the nonexistence of a solution. The optimal decay at infinity for such solutions is also discussed.

New Principles of Non-Linear Integral Inequalities on Time Scales

Abstract The concept of inequalities in time scales has attracted the attention of mathematicians for a quarter century. And these studies have inspired the solution of many problems in the branches of physics, biology, mechanics and economics etc. In this article, new principles of non-linear integral inequalities are presented in time scales via diamond-α dynamic integral and the nabla integral.

Some Nonlinear Integral Inequalities Connected with Retarded Terms on Time Scales

The objective of this research is to formulate a specific class of integral inequalities of Gronwall kind concerning retarded term and nonlinear integrals with time scales theory. Our results generate several new inequalities that reflect continuous and discrete form, as well as giving the unknown function an upper bound estimate. The effectiveness of such inequalities arises from the belief that it is widely relevant in unique circumstances where there is no valid utilization of various available inequalities. Applications are additionally represented to display the legitimacy of built-up hypotheses.

Computational approach of dynamic integral inequalities with applications to timescale calculus

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.

Reconstruction of nonlinear integral inequalities associated with time scales calculus

In this paper, we build up some generalizations of nonlinear integral inequalities and recreate the results of some Pachpatte’s inequalities on time scales. We not just settle new estimated bounds of a particular class of nonlinear retarded dynamic inequalities, but additionally determine and unify continuous analogs alongside a subjective time scale mathbb{T}. We demonstrate applications of the treated inequalities to reflect the benefits of our work. The key effects will be proven by using the analysis procedure and the standard time-scale comparison theorem technique.

A note on a conjecture for the critical curve of a weakly coupled system of semilinear wave equations with scale‐invariant lower order terms

In this note, two blow‐up results are proved for a weakly coupled system of semilinear wave equations with distinct scale‐invariant lower order terms both in the subcritical case and in the critical case when the damping and the mass terms make both equations in some sense “wave‐like.” In the proof of the subcritical case, an iteration argument is used. This approach is based on a coupled system of nonlinear ordinary integral inequalities and lower bound estimates for the spatial integral of the nonlinearities. In the critical case, we employ a test function‐type method that has been developed recently by Ikeda‐Sobajima‐Wakasa and relies strongly on a family of certain self‐similar solutions of the adjoint linear equation. Therefore, as critical curve in the p−q plane of the exponents of the power nonlinearities for this weakly coupled system, we conjecture a shift of the critical curve for the corresponding weakly coupled system of semilinear wave equations.

Nonlinear integral inequality with power and its application in delay integro-differential equations

New nonlinear integral inequalities (NII) are presented in this paper. Based on mathematical analysis technique, several estimation results are obtained, which not only complement the aforementioned results, but also generalize the inequalities to the more general nonlinearities. As an application, they can be employed to estimate the bound on the solutions of power integro-differential equations (IDE).

Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source

<p style="text-indent:20px;">In this paper, we consider the initial boundary value problem for the fourth order wave equation with nonlinear boundary velocity feedbacks <inline-formula><tex-math id="M1">$ f_{1}(u_{\nu{t}}) $</tex-math></inline-formula>, <inline-formula><tex-math id="M2">$ f_{2}(u_{t}) $</tex-math></inline-formula> and internal source <inline-formula><tex-math id="M3">$ |u|^{\rho}u $</tex-math></inline-formula>. Under some geometrical conditions, the existence and uniform decay rates of the solutions are proved even if the nonlinear boundary velocity feedbacks <inline-formula><tex-math id="M4">$ f_{1}(u_{\nu{t}}) $</tex-math></inline-formula>, <inline-formula><tex-math id="M5">$ f_{2}(u_{t}) $</tex-math></inline-formula> have not polynomial growth near the origin respectively. By the combination of the Galerkin approximation, potential well method and a special basis constructed, we first obtain the global existence and uniqueness of regular solutions and weak solutions. In addition, we also investigate the explicit decay rate estimates of the energy, the ideas of which are based on the construction of a special weight function <inline-formula><tex-math id="M6">$ \phi(t) $</tex-math></inline-formula> (that depends on the behaviors of the functions <inline-formula><tex-math id="M7">$ f_{1}(u_{\nu{t}}) $</tex-math></inline-formula>, <inline-formula><tex-math id="M8">$ f_{2}(u_{t}) $</tex-math></inline-formula> near the origin), nonlinear integral inequality and the Multiplier method.</p>

Refinements of some $2D$ nonlinear integral inequalities and applications in fractional integral equations

Refinements of some $2D$ nonlinear integral inequalities and applications in fractional integral equations

Some generalizations of retarded nonlinear integral inequalities and its applications

Some generalizations of retarded nonlinear integral inequalities and its applications