## Abstract and Applied Analysis

### A Note on Fractional Equations of Volterra Type with Nonlocal Boundary Condition

#### Abstract

We deal with nonlocal boundary value problems of fractional equations of Volterra type involving Riemann-Liouville derivative. Firstly, by defining a weighted norm and using the Banach fixed point theorem, we show the existence and uniqueness of solutions. Then, we obtain the existence of extremal solutions by use of the monotone iterative technique. Finally, an example illustrates the results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 432941, 8 pages.

Dates
First available in Project Euclid: 27 February 2014

https://projecteuclid.org/euclid.aaa/1393511967

Digital Object Identifier
doi:10.1155/2013/432941

Mathematical Reviews number (MathSciNet)
MR3081579

Zentralblatt MATH identifier
1294.45004

#### Citation

Liu, Zhenhai; Wang, Rui. A Note on Fractional Equations of Volterra Type with Nonlocal Boundary Condition. Abstr. Appl. Anal. 2013 (2013), Article ID 432941, 8 pages. doi:10.1155/2013/432941. https://projecteuclid.org/euclid.aaa/1393511967

#### References

• B. Ahmad and S. Sivasundaram, “On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 480–487, 2010.
• Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 2, pp. 916–924, 2010.
• T. Jankowski, “Fractional equations of Volterra type involving a Riemann-Liouville derivative,” Applied Mathematics Letters, vol. 26, no. 3, pp. 344–350, 2013.
• S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993.
• A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
• V. Lakshmikanthan, S. Leela, and J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Academic Publishers, Cambridge, Mass, USA, 2009.
• X. Liu and Z. Liu, “Existence results for fractional differential inclusions with multivalued term depending on lower-order derivative,” Abstract and Applied Analysis, vol. 2012, Article ID 423796, 24 pages, 2012.
• Z. Liu and X. W. Li, “Existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 6, pp. 1362–1373, 2013.
• Z. Liu, J. Sun, and I. Szántó, “Monotone iterative technique for Riemann-Liouville fractional integro-differential equations with advanced arguments,” Results in Mathematics, vol. 63, no. 3-4, pp. 1277–1287, 2013.
• Z. Liu and J. Sun, “Nonlinear boundary value problems of fractional functional integro-differential equations,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3228–3234, 2012.
• I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
• M. ur Rehman and R. A. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1038–1044, 2010.
• J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Eds., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, Dordrecht, The Netherlands, 2007.
• H. A. H. Salem, “On the fractional order $m$-point boundary value problem in reflexive Banach spaces and weak topologies,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 565–572, 2009.
• F. Wang, Z. Liu, and J. Li, “Complete controllability of fractional neutral differential systems in abstract space,” Abstract and Applied Analysis, vol. 2013, Article ID 529025, 11 pages, 2013.
• G. Wang, R. P. Agarwal, and A. Cabada, “Existence results and the monotone iterative technique for systems of nonlinear fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 6, pp. 1019–1024, 2012.
• S. Zhang, “Monotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, pp. 2087–2093, 2009.
• W. Zhong and W. Lin, “Nonlocal and multiple-point boundary value problem for fractional differential equations,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1345–1351, 2010.