Integral Inequalities of Gronwall-Bellman Type

Author(s)
Zareen A. Khan

ABSTRACT

The goal of the present paper is to establish some new approach on the basic integral inequality of Gronwall-Bellman type and its generalizations involving function of one independent variable which provides explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain partial differential and integral equations.

The goal of the present paper is to establish some new approach on the basic integral inequality of Gronwall-Bellman type and its generalizations involving function of one independent variable which provides explicit bounds on unknown functions. The inequalities given here can be used as tools in the qualitative theory of certain partial differential and integral equations.

KEYWORDS

Integral Inequalities, One Independent Variable, Partial Differential Equations, Nondecreasing, Nonincreasing

Integral Inequalities, One Independent Variable, Partial Differential Equations, Nondecreasing, Nonincreasing

Cite this paper

Khan, Z. (2014) Integral Inequalities of Gronwall-Bellman Type.*Applied Mathematics*, **5**, 3484-3488. doi: 10.4236/am.2014.521326.

Khan, Z. (2014) Integral Inequalities of Gronwall-Bellman Type.

References

[1] Abdeldaim, A. and Yakout, M. (2011) On Some New Integral Inequalities of Gronwall-Bellman-Pachpatte Type. Applied Mathematics and Computation, 217, 7887-7899.

http://dx.doi.org/10.1016/j.amc.2011.02.093

[2] Pachpatte, B.G. (2001) On Some Fundamental Integral Inequalities and Their Discrete Analogues. Journal of Inequalities in Pure and Applied Mathematics, 2, Article 15.

[3] Pachpatte, B.G. (1994) On Some Fundamental Integral Inequalities Arising in the Theory of Differential Equations. Chinese Journal of Mathematics, 22, 261-273.

[4] Pachpatte, B.G. (1996) Comparison Theorems Related to a Certain Inequality Used in the Theory of Differential Equations. Soochow Journal of Mathematics, 22, 383-394.

[5] Langenhop, C.E. (1960) Bounds on the Norm of a Solution of a General Differential Equation. Proceedings of the American Mathematical Society, 11, 795-799.

[6] Bainov, D. and Simeonov, P. (1992) Integral Inequalities and Applications. Kluwer Academic Publishers, Dordrecht.

[7] Mitrinovíc, D.S., Pecaríc, J.E. and Fink, A.M. (1991) Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic Publishers, Dordrecht.

[8] Beckenbach, E.F. and Bellman, R. (1961) Inequalities. Springer-Verlag, New York.

http://dx.doi.org/10.1007/978-3-642-64971-4

[9] Bihari, I. (1956) A Generalization of a Lemma of Bellman and Its Application to Uniqueness Problem of Differential Equations. Acta Mathematica Academiae Scientiarum Hungarica, 7, 71-94.

http://dx.doi.org/10.1007/BF02022967

[10] Guiliano, L. (1946) Generalazzioni di un lemma di Gronwall. Rend. Accad., Lincei, 1264-1271.

[11] Bellman, R. (1943) The Stability of Solutions of Linear Di?erential Equations. Duke Mathematical Journal, 10, 643- 647.

http://dx.doi.org/10.1215/S0012-7094-43-01059-2

[12] Dragomir, S.S. and Kim, Y.H. (2002) On Certain New Integral Inequalities and Their Applications. JIPAM, 3, Article 65.

[13] Dragomir, S.S. and Kim, Y.H. (2003) Some Integral Inequalities for Functions of Two Variables. Electronic Journal of Differential Equations, 10, 1-13.

[14] Gronwall, T.H. (1919) Note on the Derivatives with Respect to a Parameter of Solutions of a System of Differential Equations. Annals of Mathematics, 20, 292-296.

http://dx.doi.org/10.2307/1967124

[15] Nemyckii, V.V. and Stepanov, V.V. (1947) Qualitative Theory of Differential Equations. OGIZ, Moscow.

[1] Abdeldaim, A. and Yakout, M. (2011) On Some New Integral Inequalities of Gronwall-Bellman-Pachpatte Type. Applied Mathematics and Computation, 217, 7887-7899.

http://dx.doi.org/10.1016/j.amc.2011.02.093

[2] Pachpatte, B.G. (2001) On Some Fundamental Integral Inequalities and Their Discrete Analogues. Journal of Inequalities in Pure and Applied Mathematics, 2, Article 15.

[3] Pachpatte, B.G. (1994) On Some Fundamental Integral Inequalities Arising in the Theory of Differential Equations. Chinese Journal of Mathematics, 22, 261-273.

[4] Pachpatte, B.G. (1996) Comparison Theorems Related to a Certain Inequality Used in the Theory of Differential Equations. Soochow Journal of Mathematics, 22, 383-394.

[5] Langenhop, C.E. (1960) Bounds on the Norm of a Solution of a General Differential Equation. Proceedings of the American Mathematical Society, 11, 795-799.

[6] Bainov, D. and Simeonov, P. (1992) Integral Inequalities and Applications. Kluwer Academic Publishers, Dordrecht.

[7] Mitrinovíc, D.S., Pecaríc, J.E. and Fink, A.M. (1991) Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic Publishers, Dordrecht.

[8] Beckenbach, E.F. and Bellman, R. (1961) Inequalities. Springer-Verlag, New York.

http://dx.doi.org/10.1007/978-3-642-64971-4

[9] Bihari, I. (1956) A Generalization of a Lemma of Bellman and Its Application to Uniqueness Problem of Differential Equations. Acta Mathematica Academiae Scientiarum Hungarica, 7, 71-94.

http://dx.doi.org/10.1007/BF02022967

[10] Guiliano, L. (1946) Generalazzioni di un lemma di Gronwall. Rend. Accad., Lincei, 1264-1271.

[11] Bellman, R. (1943) The Stability of Solutions of Linear Di?erential Equations. Duke Mathematical Journal, 10, 643- 647.

http://dx.doi.org/10.1215/S0012-7094-43-01059-2

[12] Dragomir, S.S. and Kim, Y.H. (2002) On Certain New Integral Inequalities and Their Applications. JIPAM, 3, Article 65.

[13] Dragomir, S.S. and Kim, Y.H. (2003) Some Integral Inequalities for Functions of Two Variables. Electronic Journal of Differential Equations, 10, 1-13.

[14] Gronwall, T.H. (1919) Note on the Derivatives with Respect to a Parameter of Solutions of a System of Differential Equations. Annals of Mathematics, 20, 292-296.

http://dx.doi.org/10.2307/1967124

[15] Nemyckii, V.V. and Stepanov, V.V. (1947) Qualitative Theory of Differential Equations. OGIZ, Moscow.