Nonlinear Integral Inequalities Research Articles

Some new generalized nonlinear integral inequalities for functions of two independent variables

The purpose of this paper is to establish some nonlinear GronwalBellman integral inequalities in the case of functions of two independent variables which can be used as handy tools in the theory of partial differential and integral equations. The results extend and improve the earlier publication [8], an application is given to illustrate the efficiency of the obtained results.

Nonlinear Gronwall-Bellman type integral inequalities with maxima

Integral inequalities with maxima of the unknown function are useful in the study of control theory. Known results were given for Gronwall-Bellman type integral inequalities with the maxima in the form of linear dependence on the unknown function with a single delay term. In this paper we consider a general form of nonlinear integral inequalities with the maxima and more than one delay terms. Requiring neither monotonicity nor separability of given functions, we apply monotonization to estimate the unknown function. Our result can be used to weaken conditions for some known results. We apply our result to prove boundedness of solutions for a differential equation with the maxima and an integral equation with maxima separately. Mathematics subject classification (2010): 26D10, 26D15.

On nonlinear integral inequalities of Gronwall type in two independent variables

The aim of the present paper is to establish some nonlinear 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.

A Generalized Nonlinear Gronwall-Bellman Inequality with Maxima in Two Variables

This paper deals with a generalized form of nonlinear retarded Gronwall-Bellman type integral inequality in which the maximum of the unknown function of two variables is involved. This form includes both a nonconstant term outside the integrals and more than one distinct nonlinear integrals. Requiring neither monotonicity nor separability of given functions, we apply a technique of monotonization to estimate the unknown function. Our result can be used to weaken conditions for some known results. We apply our result to a boundary value problem of a partial differential equation with maxima for uniqueness.

Erratum to “Note on Some Nonlinear Integral Inequalities and Applications to Differential Equations”

First, we apologize for the misprints in the original article. Now we correct those misprints. Page 1, line 2: Between equations and The Gronwall-Bellman insert “Among various types of integral inequalities.” Page 1, line 5: Replace diferential by differential. Page 2, line 1: Replace of the situation by to the situation. Page 2, Lemma 2.1: between For x ∈ R , y ∈ R , 1/p 1/q 1, and one has insert “with p > 1.” Page 2, Theorem 2.3: Replace and there by “If there.” Page 2, Theorem 2.3: Replace and u t satisfy by and the function u t satisfies. Page 3, Proof of Theorem 2.3, line 9: Replace it yields by we obtain. Page 4, Remark 2.4, line 2: Replace become by becomes. Page 4, Theorem 2.5, line 1: Replace holds by hold. Page 4, Theorem 2.5, 2.15 : Replace ∫ t 0 b s ∑i n i 1 hi t a t p /p p /p −1ds by ∫ t 0 b s ∑i n i 1 hi s a s p /p p /p −1ds. Page 4, Theorem 2.5, 2.16 : Replace ∫ t 0 b s ∑i n i 1 hi t a t p /p p /p −1ds by ∫ t 0 b s ∑i n i 1 hi s a s p /p p /p −1ds. Page 5, 2.28 : Replace ∫ t 0 p /p − 1 b t ∑i n i 1 hi t a t p /p p /p −1ds by ∫ t 0 p /p − 1 b s ∑i n i 1 hi s a s p /p p /p −1ds.

Some new generalized retarded nonlinear integral inequalities with iterated integrals and their applications

Some new generalized retarded nonlinear integral inequalities with iterated integrals are discussed and upper bound estimations of unknown functions are given by analysis technique. These estimations can be used as tools in the study of differential-integral equations with the initial conditions.MSC:26D10, 26D15, 26D20, 34A12, 34A40.

Note on Some Nonlinear Integral Inequalities in Two Independent Variables on Time Scales and Applications

Our aim in this note is to investigate some nonlinear integral inequalities in two independent variables on time scales.The inequalities given here can be used as handy tools to study the properties of certain partial dynamic equations on time scales.

A class of retarded nonlinear integral inequalities and its application in nonlinear differential-integral equation

In this paper, we discuss a class of retarded nonlinear integral inequalities and give an upper bound estimation of an unknown function by the integral inequality technique. This estimation can be used as a tool in the study of differential-integral equations with the initial conditions.MSC:26D10, 26D15, 26D20, 34A12, 34A40.

Some new retarded nonlinear integral inequalities and their applications in differential–integral equations

In this paper, we discuss some new retarded nonlinear Gronwall–Bellman–Pachpatte integral inequalities and give upper bound estimation of the unknown function by integral inequality technique. Our results generalize those of Abdeldaim and Yakout [A. Abdeldaim, M. Yakout, On some new integral inequalities of Gronwall–Bellman–Pachpatte type, Appl. Math. Comput. 217 (2011) 7887–7899] to some inequalities with both retarded argument and nonlinear integrand. An application to estimate the solutions of differential–integral equations is also given.

Some retarded nonlinear integral inequalities and their applications in retarded differential equations

In this article, we discuss some generalized retarded nonlinear integral inequalities, which not only include nonlinear compound function of unknown function but also include retard items, and give upper bound estimation of the unknown function by integral inequality technique. This estimation can be used as tool in the study of differential equations with the initial conditions. 2000 MSC: 26D10; 26D15; 26D20; 34A12; 34A40.

A non-local inequality and global existence

In this article we prove a collection of new non-linear and non-local integral inequalities. As an example for u⩾0 and p∈(0,∞) we obtain∫R3dxup+1(x)⩽(p+1p)2∫R3dx{(−Δ)−1u(x)}|∇up2(x)|2. We use these inequalities to deduce global existence of solutions to a non-local heat equation with a quadratic non-linearity for large radial monotonic positive initial conditions. Specifically, we improve Krieger and Strain (in press) [4] to include all α∈(0,7475).

Generalized Gronwall–Bellman-type delay dynamic inequalities on time scales and their applications

In this paper, we establish some new Gronwall–Bellman-type delay dynamic inequalities on time scales, which can be used as a handy tool in the investigation of qualitative as well as quantitative analysis of solutions of delay dynamic equations on time scales. Our results generalize the main results of Ou-Iang [L. Ou-Iang, The boundedness of solutions of linear differential equations y″+A(t)y=0, Shuxue Jinzhan 3 (1957) 409-415], Jiang [F.C. Jiang, F.W. Meng, Explicit bounds on some new nonlinear integral inequality with delay, J. Comput. Appl. Math. 205 (2007) 479-486.] and Yuan [Z.L. Yuan, X.W. Yuan, F.W. Meng, Some new delay integral inequalities and their applications, Appl. Math. Comput. 208 (2009) 231-237].

Some generalized nonlinear retarded integral inequalities with applications

In this article we discuss some new generalized nonlinear Gronwall-Bellman-Type integral inequalities with two variables, which include a non-constant term outside the integrals. We use our result to deal with the estimate on the solutions of partial differential equations with the initial and boundary conditions. Mathematics Subject Classification 2000: 26D10; 26D15; 26D20; 34A40.

On the viability of nonlinear Hadamard fractional integral equations

In this paper, viability properties of nonlinear Hadamard fractional integral equations are investigated. We give a sufficient condition that guarantees fractional viability of a locally closed set with respect to the vector-valued continuous function. In particular, we derive the uniqueness of viable solutions by using a new nonlinear integral inequality with Hadamard type singular kernel.

Some new integral inequalities on time scales

By introducing two adjusting parameters, we investigate some new nonlinear integral inequalities on time scales, which provide explicit bounds on unknown functions, and can be used as tools in the qualitative theory of certain classes of dynamic equations on time scales. Mathematics subject classification (2010): 26D15, 26E70.

A new nonlinear integral inequality of Wendroff type with continuous and weakly singular kernel and its application

The main objective of this paper is to establish some new explicit bounds for nonlinear integral inequalities of Wendroff type with continuous and weakly singular kernel, which gener- alized some known inequalities for functions in two variables and can be furnished a handy tool for the study of qualitative as well as quantitative properties of solutions of nonlinear differential equations. Some applications are also given to illustrate the usefulness of our results.

Nonlinear integral inequalities involving maxima of the unknown scalar functions

Nonlinear integral inequalities involving maxima of the unknown scalar functions

On the nonexistence of blowing-up solutions to a fractional functional-differential equation

A sufficient condition for the nonexistence of blowing-up mild solutions of a nonlinear evolution fractional functional-differential equation associated with a strongly continuous semigroup and with a nonlinearity containing the Riemann–Liouville fractional integral is established. We prove a result on a new type of nonlinear integral inequalities with weakly singular kernels and delay and apply it in the proof of the result on the nonexistence of blowing-up solutions. This result is applied to a fractionally damped pendulum equation with a time delay forcing term (a feedback control).

Some nonlinear delay integral inequalities on time scales arising in the theory of dynamics equations

In this paper, some new nonlinear delay integral inequalities on time scales are established, which provide a handy tool in the research of boundedness of unknown functions in delay dynamic equations on time scales. The established results generalize some of the results in Lipovan [J. Math. Anal. Appl. 322, 349-358 (2006)], Pachpatte [J. Math. Anal. Appl. 251, 736-751 (2000)], Li [Comput. Math. Appl. 59, 1929-1936 (2010)], and Sun [J. Math. Anal. Appl. 301, 265-275 (2005)].MSC 2010: 26E70; 26D15; 26D10.

Some new nonlinear integral inequalities and their applications in the qualitative analysis of differential equations

In this paper, some new nonlinear integral inequalities are established, which provide a handy tool for analyzing the global existence and boundedness of solutions of differential and integral equations. The established results generalize the main results in Sun (J. Math. Anal. Appl. 301, 265-275, 2005), Ferreira and Torres (Appl. Math. Lett. 22, 876-881, 2009), Xu and Sun (Appl. Math. Comput. 182, 1260-1266, 2006) and Li et al. (J. Math. Anal. Appl. 372, 339-349 2010). MSC 2010: 26D15; 26D10