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A Class of Iterative Nonlinear Difference Inequality with Weakly Singularity

A Class of Iterative Nonlinear Difference Inequality with Weakly Singularity A Class of Iterative Nonlinear Difference Inequality with Weakly Singularity div.banner_title_bkg div.trangle { border-color: #000000 transparent transparent transparent; opacity:0.45; /*new styles start*/ -ms-filter:"progid:DXImageTransform.Microsoft.Alpha(Opacity=45)" ;filter: alpha(opacity=45); /*new styles end*/ } div.banner_title_bkg_if div.trangle { border-color: transparent transparent #000000 transparent ; opacity:0.45; /*new styles start*/ -ms-filter:"progid:DXImageTransform.Microsoft.Alpha(Opacity=45)" ;filter: alpha(opacity=45); /*new styles end*/ } div.banner_title_bkg div.trangle { width: 297px; }div.banner_title_bkg_if div.trangle { width: 202px; } #banner { background-image: url('http://images.hindawi.com/journals/jam/jam.banner.jpg'); background-position: 50% 0;} Hindawi Publishing Corporation Home Journals About Us Journal of Applied Mathematics Impact Factor 0.834 About this Journal Submit a Manuscript Table of Contents Journal Menu About this Journal · Abstracting and Indexing · Aims and Scope · Annual Issues · Article Processing Charges · Articles in Press · Author Guidelines · Bibliographic Information · Citations to this Journal · Contact Information · Editorial Board · Editorial Workflow · Free eTOC Alerts · Publication Ethics · Reviewers Acknowledgment · Submit a Manuscript · Subscription Information · Table of Contents Open Special Issues · Published Special Issues · Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Full-Text ePUB Linked References How to Cite this Article Journal of Applied Mathematics Volume 2014 (2014), Article ID 236965, 9 pages http://dx.doi.org/10.1155/2014/236965 Research Article A Class of Iterative Nonlinear Difference Inequality with Weakly Singularity Chunmiao Huang , 1 Wu-Sheng Wang , 1 and Xiaoliang Zhou 2 1 School of Mathematics and Statistics, Hechi University, Yizhou, Guangxi 546300, China 2 Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524088, China Received 27 March 2014; Accepted 28 April 2014; Published 8 May 2014 Academic Editor: Junjie Wei Copyright © 2014 Chunmiao Huang et al. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We discuss a class of new nonlinear weakly singular difference inequality, which is solved by change of variable, discrete Hölder inequality, discrete Jensen inequality, the mean-value theorem for integrals and amplification method, and Gamma function. Explicit bound for the unknown function is given clearly. Moreover, an example is presented to show the usefulness of our results. 1. Introduction Being an important tool in the study of qualitative properties of solutions of differential equations and integral equations, various generalizations of Gronwall inequalities and their applications have attracted great interests of many mathematicians (such as [ 1 – 21 ]). Gronwall-Bellman inequality [ 22 , 23 ] can be stated as follows: if and are nonnegative and continuous functions on an interval satisfying for some constant , then In 1981, Henry [ 2 ] discussed the following linear singular integral inequality: In 2007, Ye et al. [ 20 ] discussed linear singular integral inequality: In 2011, Abdeldaim and Yakout [ 21 ] studied a new integral inequality of Gronwall-Bellman-Pachpatte type On the other hand, many physical problems arising in a wide variety of applications are governed by finite difference equations. The theory of difference equations has been developed as a natural discrete analogue of corresponding theory of differential equations. Difference inequalities which give explicit bounds on unknown functions provide a very useful and important tool in the study of many qualitative as well as quantitative properties of solutions of nonlinear difference equations (such as [ 24 – 32 ]). Sugiyama [ 26 ] established the most precise and complete discrete analogue of the Gronwall inequality in the following form: For instance, Pachpatte [ 27 ] considered the following discrete inequality: In 2006, Cheung and Ren [ 29 ] studied Later, Zheng et al. [ 31 ] discussed the following discrete inequality: Motivated by the results given in [ 2 , 20 , 21 , 32 ], in this paper, we discuss a new linear singular integral inequality where , , , and . For the reader’s convenience, we present some necessary lemmas. Lemma 1 (discrete Jensen inequality [ 28 ]). Let be nonnegative real numbers, is a real number, and is a natural number. Then Lemma 2 (discrete Hölder inequality [ 30 ]). Let be nonnegative real numbers, and let be positive numbers such that ; then Lemma 3. Let , , , and . If , then where , is the well-known -function. Proof. By the definition of integration and the conditions in Lemma 3 , we have Using a change of variables and , we have the estimation Since , , , and , from ( 14 ) and ( 15 ), we have the relation ( 13 ). 2. Main Result In this section, we give the estimation of unknown function in ( 10 ). Let . For function , its difference is defined by . Obviously, the linear difference equation with the initial condition has the solution . For convenience, in the sequel we complementarily define that . Theorem 4. Suppose that is a constant, , are nonnegative and nondecreasing functions defined on , are nonnegative, nondecreasing, and continuous functions defined on , , , , and . If satisfies ( 10 ), then where and , , , is the largest integer number such that Proof. From ( 10 ), we have Applying Lemma 2 with , to ( 24 ), we obtain that where is used. Applying Lemma 3 , we have By discrete Jensen inequality ( 11 ) with , from ( 26 ) we obtain that Again using discrete Jensen inequality ( 11 ) with , , from ( 27 ) we obtain that For in ( 28 ), applying Lemma 2 with , , we obtain that here Lemma 3 is used. Substituting ( 29 ) into ( 28 ), we have where and are defined by ( 21 ) and ( 22 ), respectively. Let ; from ( 30 ) we have Since , are nondecreasing functions, from ( 31 ) we have where is chosen arbitrarily. Let denote the function on the right-hand side of ( 32 ), which is a positive and nondecreasing function on . From ( 32 ), we have Using and ( 33 ), we obtain for all . Let Then From ( 35 ), we have It implies that, for all , On the other hand, by the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that for all , where is defined by ( 18 ). From ( 38 ) and ( 39 ), we have Taking in ( 40 ) and summing up over from to , from ( 40 ) we obtain Let denote the function on the right-hand side of ( 41 ), which is a positive and nondecreasing function on . From ( 41 ), we have Using and ( 42 ), we obtain From ( 43 ), we have for all . Again by the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that where is defined by ( 19 ). From ( 44 ) and ( 45 ), we have Taking in ( 46 ) and summing up over from to , from ( 46 ) we obtain for all . Let Then From ( 48 ) and ( 49 ), we have Using the mean-value theorem for integrals, from ( 50 ) we have where is defined by ( 20 ). From ( 36 ), ( 42 ), ( 49 ), and ( 51 ), we have Using and ( 33 ), from ( 52 ) we obtain that Since is chosen arbitrarily, from ( 53 ) we have This is our required estimation ( 16 ) of unknown function in ( 10 ). 3. Application In this section, we apply our results to discuss the boundedness of solutions of an iterative difference equation with a weakly singular kernel. Example 5. Suppose that satisfies the difference equation where , , , and . Then we have for all . Let , . From ( 18 ) to ( 20 ) we obtain that Using Theorem 4 , we get which is an upper bound of in ( 55 ). Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This research was supported by National Natural Science Foundation of China (Project no. 11161018), the NSF of Guangxi Zhuang Autonomous Region (no. 2012GXNSFAA053009), the high school specialty and curriculum integration project of Guangxi Zhuang Autonomous Region (no. GXTSZY2220), the Science Innovation Project of Department of Education of Guangdong province (Grant 2013KJCX0125), and the NSF of Guangdong Province (no. s2013010013385). The authors are very grateful to the editor and the referees for their careful comments and valuable suggestions on this paper. References I. A. Bihari, “A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations,” Acta Mathematica Academiae Scientiarum Hungaricae , vol. 7, pp. 81–94, 1956. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet D. Henry, Geometric Theory of Semilinear Parabolic Equations , vol. 840, Springer, Berlin, Germany, 1981. View at MathSciNet D. Bainov and P. Simeonov, Integral Inequalities and Applications , vol. 57, Kluwer Academic, Dordrecht, The Netherlands, 1992. View at Publisher · View at Google Scholar · View at MathSciNet M. Medveď, “A new approach to an analysis of Henry type integral inequalities and their Bihari type versions,” Journal of Mathematical Analysis and Applications , vol. 214, no. 2, pp. 349–366, 1997. View at Publisher · View at Google Scholar · View at MathSciNet B. G. Pachpatte, Inequalities for Differential and Integral Equations , vol. 197, Academic Press, London, UK, 1998. View at MathSciNet I. Podlubny, Fractional Differential Equations , vol. 198, Academic Press, New York, NY, USA, 1999. View at MathSciNet M. Medveď, “Nonlinear singular integral inequalities for functions in two and n independent variables,” Journal of Inequalities and Applications , vol. 5, no. 3, pp. 287–308, 2000. View at Publisher · View at Google Scholar · View at MathSciNet E. H. Yang, “A new nonlinear discrete inequality and its application,” Annals of Differential Equations. Weifen Fangcheng Niankan , vol. 17, no. 3, pp. 261–267, 2001. View at Zentralblatt MATH · View at MathSciNet O. Lipovan, “A retarded Gronwall-like inequality and its applications,” Journal of Mathematical Analysis and Applications , vol. 252, no. 1, pp. 389–401, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet W. Zhang and S. Deng, “Projected Gronwall-Bellman's inequality for integrable functions,” Mathematical and Computer Modelling , vol. 34, no. 3-4, pp. 393–402, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet Q. H. Ma and E. H. Yang, “Estimates on solutions of some weakly singular Volterra integral inequalities,” Acta Mathematicae Applicatae Sinica. Yingyong Shuxue Xuebao , vol. 25, no. 3, pp. 505–515, 2002 (Chinese). View at Zentralblatt MATH · View at MathSciNet F. W. Meng and W. N. Li, “On some new nonlinear discrete inequalities and their applications,” Journal of Computational and Applied Mathematics , vol. 158, no. 2, pp. 407–417, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet R. P. Agarwal, S. Deng, and W. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,” Applied Mathematics and Computation , vol. 165, no. 3, pp. 599–612, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet B.-I. Kim, “On some Gronwall type inequalities for a system integral equation,” Bulletin of the Korean Mathematical Society , vol. 42, no. 4, pp. 789–805, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet B. G. Pachpatte, “On certain nonlinear integral inequalities involving iterated integrals,” Tamkang Journal of Mathematics , vol. 37, no. 3, pp. 261–271, 2006. View at Zentralblatt MATH · View at MathSciNet W. S. Cheung, “Some new nonlinear inequalities and applications to boundary value problems,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods , vol. 64, no. 9, pp. 2112–2128, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet W. S. Wang, “A generalized retarded Gronwall-like inequality in two variables and applications to BVP,” Applied Mathematics and Computation , vol. 191, no. 1, pp. 144–154, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet Q. H. Ma and W. S. Cheung, “Some new nonlinear difference inequalities and their applications,” Journal of Computational and Applied Mathematics , vol. 202, no. 2, pp. 339–351, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet M. Medveď, “On singular versions of Bihari and Wendroff-Pachpatte type integral inequalities and their application,” Tatra Mountains Mathematical Publications , vol. 38, pp. 163–174, 2007. View at MathSciNet H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications , vol. 328, no. 2, pp. 1075–1081, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet A. Abdeldaim and M. Yakout, “On some new integral inequalities of Gronwall-Bellman-Pachpatte type,” Applied Mathematics and Computation , vol. 217, no. 20, pp. 7887–7899, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,” Annals of Mathematics , vol. 20, no. 4, pp. 292–296, 1919. View at Publisher · View at Google Scholar · View at MathSciNet R. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal , vol. 10, pp. 643–647, 1943. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet T. E. Hull and W. A. J. Luxemburg, “Numerical methods and existence theorems for ordinary differential equations,” Numerische Mathematik , vol. 2, pp. 30–41, 1960. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet D. Willett and J. S. W. Wong, “On the discrete analogues of some generalizations of Gronwall's inequality,” vol. 69, pp. 362–367, 1965. View at Zentralblatt MATH · View at MathSciNet S. Sugiyama, “On the stability problems of difference equations,” Bulletin of Science and Engineering Research Laboratory. Waseda University , vol. 45, pp. 140–144, 1969. View at MathSciNet B. G. Pachpatte, “Finite difference inequalities and discrete time control systems,” Indian Journal of Pure and Applied Mathematics , vol. 9, no. 12, pp. 1282–1290, 1978. View at Zentralblatt MATH · View at MathSciNet M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities: Cauchys Equation and Jensens Inequality , vol. 489, University of Katowice, Katowice, Poland, 1985. View at MathSciNet W.-S. Cheung and J. Ren, “Discrete non-linear inequalities and applications to boundary value problems,” Journal of Mathematical Analysis and Applications , vol. 319, no. 2, pp. 708–724, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet E. H. Yang, Q. H. Ma, and M. C. Tan, “Discrete analogues of a new class of nonlinear Volterra singular integral inequalities,” Journal of Jinan University , vol. 28, no. 1, pp. 1–6, 2007. K. Zheng, S. Zhong, and M. Ye, “Discrete nonlinear inequalities in time control systems,” in Proceedings of the International Conference on Apperceiving Computing and Intelligence Analysis (ICACIA '09) , pp. 403–406, October 2009. View at Publisher · View at Google Scholar · View at Scopus K. Zheng, H. Wang, and C. Guo, “On nonlinear discrete weakly singular inequalities and applications to Volterra-type difference equations,” Advances in Difference Equations , vol. 2013, p. 239, 2013. View at Publisher · View at Google Scholar · View at MathSciNet var _gaq = _gaq || []; _gaq.push(['_setAccount', 'UA-8578054-2']); _gaq.push(['_trackPageview']); (function () { var ga = document.createElement('script'); ga.type = 'text/javascript'; ga.async = true; ga.src = ('https:' == document.location.protocol ? 'https://ssl' : 'http://www') + '.google-analytics.com/ga.js'; var s = document.getElementsByTagName('script')[0]; s.parentNode.insertBefore(ga, s); })(); http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Applied Mathematics Hindawi Publishing Corporation

A Class of Iterative Nonlinear Difference Inequality with Weakly Singularity

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A Class of Iterative Nonlinear Difference Inequality with Weakly Singularity div.banner_title_bkg div.trangle { border-color: #000000 transparent transparent transparent; opacity:0.45; /*new styles start*/ -ms-filter:"progid:DXImageTransform.Microsoft.Alpha(Opacity=45)" ;filter: alpha(opacity=45); /*new styles end*/ } div.banner_title_bkg_if div.trangle { border-color: transparent transparent #000000 transparent ; opacity:0.45; /*new styles start*/ -ms-filter:"progid:DXImageTransform.Microsoft.Alpha(Opacity=45)" ;filter: alpha(opacity=45); /*new styles end*/ } div.banner_title_bkg div.trangle { width: 297px; }div.banner_title_bkg_if div.trangle { width: 202px; } #banner { background-image: url('http://images.hindawi.com/journals/jam/jam.banner.jpg'); background-position: 50% 0;} Hindawi Publishing Corporation Home Journals About Us Journal of Applied Mathematics Impact Factor 0.834 About this Journal Submit a Manuscript Table of Contents Journal Menu About this Journal · Abstracting and Indexing · Aims and Scope · Annual Issues · Article Processing Charges · Articles in Press · Author Guidelines · Bibliographic Information · Citations to this Journal · Contact Information · Editorial Board · Editorial Workflow · Free eTOC Alerts · Publication Ethics · Reviewers Acknowledgment · Submit a Manuscript · Subscription Information · Table of Contents Open Special Issues · Published Special Issues · Special Issue Guidelines Abstract Full-Text PDF Full-Text HTML Full-Text ePUB Linked References How to Cite this Article Journal of Applied Mathematics Volume 2014 (2014), Article ID 236965, 9 pages http://dx.doi.org/10.1155/2014/236965 Research Article A Class of Iterative Nonlinear Difference Inequality with Weakly Singularity Chunmiao Huang , 1 Wu-Sheng Wang , 1 and Xiaoliang Zhou 2 1 School of Mathematics and Statistics, Hechi University, Yizhou, Guangxi 546300, China 2 Department of Mathematics, Zhanjiang Normal University, Zhanjiang, Guangdong 524088, China Received 27 March 2014; Accepted 28 April 2014; Published 8 May 2014 Academic Editor: Junjie Wei Copyright © 2014 Chunmiao Huang et al. This is an open access article distributed under the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We discuss a class of new nonlinear weakly singular difference inequality, which is solved by change of variable, discrete Hölder inequality, discrete Jensen inequality, the mean-value theorem for integrals and amplification method, and Gamma function. Explicit bound for the unknown function is given clearly. Moreover, an example is presented to show the usefulness of our results. 1. Introduction Being an important tool in the study of qualitative properties of solutions of differential equations and integral equations, various generalizations of Gronwall inequalities and their applications have attracted great interests of many mathematicians (such as [ 1 – 21 ]). Gronwall-Bellman inequality [ 22 , 23 ] can be stated as follows: if and are nonnegative and continuous functions on an interval satisfying for some constant , then In 1981, Henry [ 2 ] discussed the following linear singular integral inequality: In 2007, Ye et al. [ 20 ] discussed linear singular integral inequality: In 2011, Abdeldaim and Yakout [ 21 ] studied a new integral inequality of Gronwall-Bellman-Pachpatte type On the other hand, many physical problems arising in a wide variety of applications are governed by finite difference equations. The theory of difference equations has been developed as a natural discrete analogue of corresponding theory of differential equations. Difference inequalities which give explicit bounds on unknown functions provide a very useful and important tool in the study of many qualitative as well as quantitative properties of solutions of nonlinear difference equations (such as [ 24 – 32 ]). Sugiyama [ 26 ] established the most precise and complete discrete analogue of the Gronwall inequality in the following form: For instance, Pachpatte [ 27 ] considered the following discrete inequality: In 2006, Cheung and Ren [ 29 ] studied Later, Zheng et al. [ 31 ] discussed the following discrete inequality: Motivated by the results given in [ 2 , 20 , 21 , 32 ], in this paper, we discuss a new linear singular integral inequality where , , , and . For the reader’s convenience, we present some necessary lemmas. Lemma 1 (discrete Jensen inequality [ 28 ]). Let be nonnegative real numbers, is a real number, and is a natural number. Then Lemma 2 (discrete Hölder inequality [ 30 ]). Let be nonnegative real numbers, and let be positive numbers such that ; then Lemma 3. Let , , , and . If , then where , is the well-known -function. Proof. By the definition of integration and the conditions in Lemma 3 , we have Using a change of variables and , we have the estimation Since , , , and , from ( 14 ) and ( 15 ), we have the relation ( 13 ). 2. Main Result In this section, we give the estimation of unknown function in ( 10 ). Let . For function , its difference is defined by . Obviously, the linear difference equation with the initial condition has the solution . For convenience, in the sequel we complementarily define that . Theorem 4. Suppose that is a constant, , are nonnegative and nondecreasing functions defined on , are nonnegative, nondecreasing, and continuous functions defined on , , , , and . If satisfies ( 10 ), then where and , , , is the largest integer number such that Proof. From ( 10 ), we have Applying Lemma 2 with , to ( 24 ), we obtain that where is used. Applying Lemma 3 , we have By discrete Jensen inequality ( 11 ) with , from ( 26 ) we obtain that Again using discrete Jensen inequality ( 11 ) with , , from ( 27 ) we obtain that For in ( 28 ), applying Lemma 2 with , , we obtain that here Lemma 3 is used. Substituting ( 29 ) into ( 28 ), we have where and are defined by ( 21 ) and ( 22 ), respectively. Let ; from ( 30 ) we have Since , are nondecreasing functions, from ( 31 ) we have where is chosen arbitrarily. Let denote the function on the right-hand side of ( 32 ), which is a positive and nondecreasing function on . From ( 32 ), we have Using and ( 33 ), we obtain for all . Let Then From ( 35 ), we have It implies that, for all , On the other hand, by the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that for all , where is defined by ( 18 ). From ( 38 ) and ( 39 ), we have Taking in ( 40 ) and summing up over from to , from ( 40 ) we obtain Let denote the function on the right-hand side of ( 41 ), which is a positive and nondecreasing function on . From ( 41 ), we have Using and ( 42 ), we obtain From ( 43 ), we have for all . Again by the mean-value theorem for integrals, for arbitrarily given integers , there exists in the open interval such that where is defined by ( 19 ). From ( 44 ) and ( 45 ), we have Taking in ( 46 ) and summing up over from to , from ( 46 ) we obtain for all . Let Then From ( 48 ) and ( 49 ), we have Using the mean-value theorem for integrals, from ( 50 ) we have where is defined by ( 20 ). From ( 36 ), ( 42 ), ( 49 ), and ( 51 ), we have Using and ( 33 ), from ( 52 ) we obtain that Since is chosen arbitrarily, from ( 53 ) we have This is our required estimation ( 16 ) of unknown function in ( 10 ). 3. Application In this section, we apply our results to discuss the boundedness of solutions of an iterative difference equation with a weakly singular kernel. Example 5. Suppose that satisfies the difference equation where , , , and . Then we have for all . Let , . From ( 18 ) to ( 20 ) we obtain that Using Theorem 4 , we get which is an upper bound of in ( 55 ). Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This research was supported by National Natural Science Foundation of China (Project no. 11161018), the NSF of Guangxi Zhuang Autonomous Region (no. 2012GXNSFAA053009), the high school specialty and curriculum integration project of Guangxi Zhuang Autonomous Region (no. GXTSZY2220), the Science Innovation Project of Department of Education of Guangdong province (Grant 2013KJCX0125), and the NSF of Guangdong Province (no. s2013010013385). The authors are very grateful to the editor and the referees for their careful comments and valuable suggestions on this paper. References I. A. Bihari, “A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations,” Acta Mathematica Academiae Scientiarum Hungaricae , vol. 7, pp. 81–94, 1956. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet D. Henry, Geometric Theory of Semilinear Parabolic Equations , vol. 840, Springer, Berlin, Germany, 1981. View at MathSciNet D. Bainov and P. Simeonov, Integral Inequalities and Applications , vol. 57, Kluwer Academic, Dordrecht, The Netherlands, 1992. View at Publisher · View at Google Scholar · View at MathSciNet M. Medveď, “A new approach to an analysis of Henry type integral inequalities and their Bihari type versions,” Journal of Mathematical Analysis and Applications , vol. 214, no. 2, pp. 349–366, 1997. View at Publisher · View at Google Scholar · View at MathSciNet B. G. Pachpatte, Inequalities for Differential and Integral Equations , vol. 197, Academic Press, London, UK, 1998. View at MathSciNet I. Podlubny, Fractional Differential Equations , vol. 198, Academic Press, New York, NY, USA, 1999. View at MathSciNet M. Medveď, “Nonlinear singular integral inequalities for functions in two and n independent variables,” Journal of Inequalities and Applications , vol. 5, no. 3, pp. 287–308, 2000. View at Publisher · View at Google Scholar · View at MathSciNet E. H. Yang, “A new nonlinear discrete inequality and its application,” Annals of Differential Equations. Weifen Fangcheng Niankan , vol. 17, no. 3, pp. 261–267, 2001. View at Zentralblatt MATH · View at MathSciNet O. Lipovan, “A retarded Gronwall-like inequality and its applications,” Journal of Mathematical Analysis and Applications , vol. 252, no. 1, pp. 389–401, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet W. Zhang and S. Deng, “Projected Gronwall-Bellman's inequality for integrable functions,” Mathematical and Computer Modelling , vol. 34, no. 3-4, pp. 393–402, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet Q. H. Ma and E. H. Yang, “Estimates on solutions of some weakly singular Volterra integral inequalities,” Acta Mathematicae Applicatae Sinica. Yingyong Shuxue Xuebao , vol. 25, no. 3, pp. 505–515, 2002 (Chinese). View at Zentralblatt MATH · View at MathSciNet F. W. Meng and W. N. Li, “On some new nonlinear discrete inequalities and their applications,” Journal of Computational and Applied Mathematics , vol. 158, no. 2, pp. 407–417, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet R. P. Agarwal, S. Deng, and W. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,” Applied Mathematics and Computation , vol. 165, no. 3, pp. 599–612, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet B.-I. Kim, “On some Gronwall type inequalities for a system integral equation,” Bulletin of the Korean Mathematical Society , vol. 42, no. 4, pp. 789–805, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet B. G. Pachpatte, “On certain nonlinear integral inequalities involving iterated integrals,” Tamkang Journal of Mathematics , vol. 37, no. 3, pp. 261–271, 2006. View at Zentralblatt MATH · View at MathSciNet W. S. Cheung, “Some new nonlinear inequalities and applications to boundary value problems,” Nonlinear Analysis. Theory, Methods & Applications A: Theory and Methods , vol. 64, no. 9, pp. 2112–2128, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet W. S. Wang, “A generalized retarded Gronwall-like inequality in two variables and applications to BVP,” Applied Mathematics and Computation , vol. 191, no. 1, pp. 144–154, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet Q. H. Ma and W. S. Cheung, “Some new nonlinear difference inequalities and their applications,” Journal of Computational and Applied Mathematics , vol. 202, no. 2, pp. 339–351, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet M. Medveď, “On singular versions of Bihari and Wendroff-Pachpatte type integral inequalities and their application,” Tatra Mountains Mathematical Publications , vol. 38, pp. 163–174, 2007. View at MathSciNet H. Ye, J. Gao, and Y. 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Journal of Applied MathematicsHindawi Publishing Corporation

Published: May 8, 2014

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