## Abstract and Applied Analysis

- Abstr. Appl. Anal.
- Volume 2012, Special Issue (2012), Article ID 172963, 19 pages.

### Existence of Solutions for Fractional Integro-Differential Equation with Multipoint Boundary Value Problem in Banach Spaces

Yulin Zhao, Li Huang, Xuebin Wang, and Xianyang Zhu

**Full-text: Open access**

#### Abstract

By means of the fixed-point theorem in the cone of strict-set-contraction operators, we consider the existence of a nonlinear multi-point boundary value problem of fractional integro-differential equation in a Banach space. In addition, an example to illustrate the main results is given.

#### Article information

**Source**

Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 172963, 19 pages.

**Dates**

First available in Project Euclid: 7 May 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.aaa/1399486655

**Digital Object Identifier**

doi:10.1155/2012/172963

**Mathematical Reviews number (MathSciNet)**

MR2999916

**Zentralblatt MATH identifier**

1296.47110

#### Citation

Zhao, Yulin; Huang, Li; Wang, Xuebin; Zhu, Xianyang. Existence of Solutions for Fractional Integro-Differential Equation with Multipoint Boundary Value Problem in Banach Spaces. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 172963, 19 pages. doi:10.1155/2012/172963. https://projecteuclid.org/euclid.aaa/1399486655

#### References

- K. S. Miller and B. Ross,
*An Introduction to the Fractional Calculus and Fractional Differential Equations*, John Wiley & Sons, New York, NY, USA, 1993.Zentralblatt MATH: 0943.82582 - I. Podlubny,
*Fractional Differential Equations*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, London, UK, 1999. - A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo,
*Theory and Applications of Fractional Differential Equations*, vol. 204, Elsevier Science, Amsterdam, The Netherlands, 2006. - V. Lakshmikantham and A. S. Vatsala, “Basic theory of fractional differential equations,”
*Nonlinear Analysis*, vol. 69, no. 8, pp. 2677–2682, 2008. - V. Lakshmikantham, “Theory of fractional functional differential equations,”
*Nonlinear Analysis*, vol. 69, no. 10, pp. 3337–3343, 2008.Zentralblatt MATH: 1162.34344 - V. Lakshmikantham and A. S. Vatsala, “General uniqueness and monotone iterative technique for fractional differential equations,”
*Applied Mathematics Letters*, vol. 21, no. 8, pp. 828–834, 2008.Zentralblatt MATH: 1161.34031 - Z. Bai and H. Lü, “Positive solutions for boundary value problem of nonlinear fractional differential equation,”
*Journal of Mathematical Analysis and Applications*, vol. 311, no. 2, pp. 495–505, 2005.Zentralblatt MATH: 1079.34048

Mathematical Reviews (MathSciNet): MR2168413

Digital Object Identifier: doi:10.1016/j.jmaa.2005.02.052 - S. Zhang, “Positive solutions to singular boundary value problem for nonlinear fractional differential equation,”
*Computers & Mathematics with Applications*, vol. 59, no. 3, pp. 1300–1309, 2010. - Z. Bai, “On positive solutions of a nonlocal fractional boundary value problem,”
*Nonlinear Analysis*, vol. 72, no. 2, pp. 916–924, 2010. - C. F. Li, X. N. Luo, and Y. Zhou, “Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations,”
*Computers & Mathematics with Applications*, vol. 59, no. 3, pp. 1363–1375, 2010. - M. El-Shahed and J. J. Nieto, “Nontrivial solutions for a nonlinear multi-point boundary value problem of fractional order,”
*Computers & Mathematics with Applications*, vol. 59, no. 11, pp. 3438–3443, 2010. - Y. Zhao, H. Chen, and L. Huang, “Existence of positive solutions for nonlinear fractional functional differential equation,”
*Computers & Mathematics with Applications*, vol. 64, no. 10, pp. 3456–3467, 2012.Mathematical Reviews (MathSciNet): MR2989373 - W.-X. Zhou and Y.-D. Chu, “Existence of solutions for fractional differential equations with multi-point boundary conditions,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 3, pp. 1142–1148, 2012.Zentralblatt MATH: 1245.35153

Mathematical Reviews (MathSciNet): MR2843780

Digital Object Identifier: doi:10.1016/j.cnsns.2011.07.019 - G. Wang, B. Ahmad, and L. Zhang, “Some existence results for impulsive nonlinear fractional differential equations with mixed boundary conditions,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1389–1397, 2011. - C. Yuan, “Two positive solutions for $(n-1,1)$-type semipositone integral boundary value problems for coupled systems of nonlinear fractional differential equations,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 2, pp. 930–942, 2012.Mathematical Reviews (MathSciNet): MR2834462

Digital Object Identifier: doi:10.1016/j.cnsns.2011.06.008 - B. Ahmad and S. Sivasundaram, “Existence of solutions for impulsive integral boundary value problems of fractional order,”
*Nonlinear Analysis*, vol. 4, no. 1, pp. 134–141, 2010.Zentralblatt MATH: 1187.34038 - 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.Zentralblatt MATH: 1207.45014

Mathematical Reviews (MathSciNet): MR2678559

Digital Object Identifier: doi:10.1016/j.amc.2010.05.080 - S. Hamani, M. Benchohra, and J. R. Graef, “Existence results for boundary-value problems with nonlinear fractional differential inclusions and integral conditions,”
*Electronic Journal of Differential Equations*, vol. 2010, pp. 1–16, 2010.Mathematical Reviews (MathSciNet): MR2592005 - A. Arikoglu and I. Ozkol, “Solution of fractional integro-differential equations by using fractional differential transform method,”
*Chaos, Solitons and Fractals*, vol. 40, no. 2, pp. 521–529, 2009.Zentralblatt MATH: 1197.45001

Mathematical Reviews (MathSciNet): MR2527812

Digital Object Identifier: doi:10.1016/j.chaos.2007.08.001 - S. Staněk, “The existence of positive solutions of singular fractional boundary value problems,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1379–1388, 2011. - B. Ahmad and J. J. Nieto, “Existence of solutions for nonlocal boundary value problems of higher-order nonlinear fractional differential equations,”
*Abstract and Applied Analysis*, vol. 2009, Article ID 494720, 9 pages, 2009.Zentralblatt MATH: 1186.34009

Mathematical Reviews (MathSciNet): MR2516016

Digital Object Identifier: doi:10.1155/2009/494720 - M. A. Darwish, “On a perturbed functional integral equation of Urysohn type,”
*Applied Mathematics and Computation*, vol. 218, no. 17, pp. 8800–8805, 2012.Zentralblatt MATH: 1245.45004 - M. A. Darwish and S. K. Ntouyas, “Boundary value problems for fractional functional differential equations of mixed type,”
*Communications in Applied Analysis*, vol. 13, no. 1, pp. 31–38, 2009. - C. S. Goodrich, “Existence of a positive solution to a class of fractional differential equations,”
*Applied Mathematics Letters*, vol. 23, no. 9, pp. 1050–1055, 2010.Zentralblatt MATH: 1204.34007 - 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.Mathematical Reviews (MathSciNet): MR2492889

Digital Object Identifier: doi:10.1016/j.cam.2008.05.033 - Z.-W. Lv, J. Liang, and T.-J. Xiao, “Solutions to the Cauchy problem for differential equations in Banach spaces with fractional order,”
*Computers & Mathematics with Applications*, vol. 62, no. 3, pp. 1303–1311, 2011.Zentralblatt MATH: 1228.65136 - X. Su, “Solutions to boundary value problem of fractional order on unbounded domains in a banach space,”
*Nonlinear Analysis*, vol. 74, no. 8, pp. 2844–2852, 2011.Mathematical Reviews (MathSciNet): MR2776532 - Y.-L. Zhao and H.-B. Chen, “Existence of multiple positive solutions for $m$-point boundary value problems in banach spaces,”
*Journal of Computational and Applied Mathematics*, vol. 215, no. 1, pp. 79–90, 2008.Mathematical Reviews (MathSciNet): MR2400619

Digital Object Identifier: doi:10.1016/j.cam.2007.03.025 - Y. Zhao, H. Chen, and C. Xu, “Existence of multiple solutions for three-point boundary-value problems on infinite intervals in banach spaces,”
*Electronic Journal of Differential Equations*, vol. 44, pp. 1–11, 2012.Zentralblatt MATH: 1244.34023 - X. Zhang, M. Feng, and W. Ge, “Existence and nonexistence of positive solutions for a class of $n$th-order three-point boundary value problems in Banach spaces,”
*Nonlinear Analysis*, vol. 70, no. 2, pp. 584–597, 2009. - D. J. Guo, V. Lakshmikantham, and X. Z. Liu,
*Nonlinear Integral Equations in Abstract Spaces*, vol. 373, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.

### More like this

- Fractional Differential Equations with Fractional Impulsive and Nonseparated Boundary Conditions

Fu, Xi and Liu, Xiaoyou, Abstract and Applied Analysis, 2014 - Existence Results for Fractional Differential Equations with Separated
Boundary Conditions and Fractional Impulsive Conditions

Fu, Xi and Liu, Xiaoyou, Abstract and Applied Analysis, 2013 - A New Impulsive Multi-Orders Fractional Differential Equation Involving Multipoint Fractional Integral Boundary Conditions

Wang, Guotao, Liu, Sanyang, Baleanu, Dumitru, and Zhang, Lihong, Abstract and Applied Analysis, 2014

- Fractional Differential Equations with Fractional Impulsive and Nonseparated Boundary Conditions

Fu, Xi and Liu, Xiaoyou, Abstract and Applied Analysis, 2014 - Existence Results for Fractional Differential Equations with Separated
Boundary Conditions and Fractional Impulsive Conditions

Fu, Xi and Liu, Xiaoyou, Abstract and Applied Analysis, 2013 - A New Impulsive Multi-Orders Fractional Differential Equation Involving Multipoint Fractional Integral Boundary Conditions

Wang, Guotao, Liu, Sanyang, Baleanu, Dumitru, and Zhang, Lihong, Abstract and Applied Analysis, 2014 - Positive Solutions for Multipoint Boundary Value Problems for Singular Fractional Differential Equations

Jleli, Mohamed, Karapinar, Erdal, and Samet, Bessem, Journal of Applied Mathematics, 2014 - Positive Solutions for Boundary Value Problem of Nonlinear
Fractional Differential Equation

El-Shahed, Moustafa, Abstract and Applied Analysis, 2007 - Existence Results for a Coupled System of Nonlinear Fractional Differential Equations in Banach Spaces

Cao, Yuping and Bai, Chuanzhi, Journal of Applied Mathematics, 2014 - A Class of Fractional
p
-Laplacian Integrodifferential Equations in Banach Spaces

Liu, Yiliang and Lu, Liang, Abstract and Applied Analysis, 2013 - The Existence of Positive Solutions for Fractional Differential Equations with Sign Changing Nonlinearities

Jiang, Weihua, Qiu, Jiqing, and Guo, Weiwei, Abstract and Applied Analysis, 2012 - Existence and Positivity of Solutions for a Second-Order Boundary Value Problem with Integral Condition

Guezane-Lakoud, Assia, Nacira, Hamidane, and Rabah, Khaldi, International Journal of Differential Equations, 2012 - Positive Solutions for Nonlinear Integro-Differential Equations of Mixed
Type in Banach Spaces

Sun, Yan, Abstract and Applied Analysis, 2013