## Abstract and Applied Analysis

### New Existence Results for Fractional Integrodifferential Equations with Nonlocal Integral Boundary Conditions

#### Abstract

We consider a boundary value problem of fractional integrodifferential equations with new nonlocal integral boundary conditions of the form: $x(0)=\beta x(\theta ), x(\xi )=\alpha {\int }_{\eta }^{1}\mathrm{‍}x(s)ds$, and $0<\theta <\xi <\eta <1$. According to these conditions, the value of the unknown function at the left end point $t=0$ is proportional to its value at a nonlocal point $\theta$ while the value at an arbitrary (local) point $\xi$ is proportional to the contribution due to a substrip of arbitrary length $(1-\eta )$. These conditions appear in the mathematical modelling of physical problems when different parts (nonlocal points and substrips of arbitrary length) of the domain are involved in the input data for the process under consideration. We discuss the existence of solutions for the given problem by means of the Sadovski fixed point theorem for condensing maps and a fixed point theorem due to O’Regan. Some illustrative examples are also presented.

#### Article information

Source
Abstr. Appl. Anal., Volume 2015, Special Issue (2014), Article ID 205452, 10 pages.

Dates
First available in Project Euclid: 15 April 2015

https://projecteuclid.org/euclid.aaa/1429104834

Digital Object Identifier
doi:10.1155/2015/205452

Mathematical Reviews number (MathSciNet)
MR3332056

Zentralblatt MATH identifier
06663001

#### Citation

Alsaedi, Ahmed; Ntouyas, Sotiris K.; Ahmad, Bashir. New Existence Results for Fractional Integrodifferential Equations with Nonlocal Integral Boundary Conditions. Abstr. Appl. Anal. 2015, Special Issue (2014), Article ID 205452, 10 pages. doi:10.1155/2015/205452. https://projecteuclid.org/euclid.aaa/1429104834

#### References

• G. S. Wang and A. F. Blom, “A strip model for fatigue crack growth predictions under general load conditions,” Engineering Fracture Mechanics, vol. 40, no. 3, pp. 507–533, 1991.
• B. Ahmad, T. Hayat, and S. Asghar, “Diffraction of a plane wave by an elastic knife-edge adjacent to a strip,” The Canadian Applied Mathematics Quarterly, vol. 9, pp. 303–316, 2001.
• I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.
• A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science, Amsterdam, The Netherlands, 2006.
• 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.
• D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012.
• M. Benchohra, S. Hamani, and S. K. Ntouyas, “Boundary value problems for differential equations with fractional order and nonlocal conditions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2391–2396, 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.
• 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.
• B. Ahmad, “On nonlocal boundary value problems for nonlinear integro-differential equations of arbitrary fractional order,” Results in Mathematics, vol. 63, no. 1-2, pp. 183–194, 2013.
• B. Ahmad and J. J. Nieto, “Sequential fractional differential equations with three-point boundary conditions,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3046–3052, 2012.
• J. R. Graef, L. Kong, and Q. Kong, “Application of the mixed monotone operator method to fractional boundary value problems,” Fractional Differential Calculus, vol. 2, no. 1, pp. 87–98, 2012.
• C. Hu, B. Liu, and S. Xie, “Monotone iterative solutions for nonlinear boundary value problems of fractional differential equation with deviating arguments,” Applied Mathematics and Computation, vol. 222, pp. 72–81, 2013.
• S. Staněk, “Limit properties of positive solutions of fractional boundary value problems,” Applied Mathematics and Computation, vol. 219, no. 5, pp. 2361–2370, 2012.
• B. Ahmad and S. K. Ntouyas, “Existence results for nonlocal boundary value problems of fractional differential equations and inclusions with strip conditions,” Boundary Value Problems, vol. 2012, 2012.
• P. Zhang and Y. Gong, “Existence and multiplicity results for a class of fractional differential inclusions with boundary conditions,” Boundary Value Problems, vol. 2012, article 82, 2012.
• A. Alsaedi, S. K. Ntouyas, and B. Ahmad, “Existence results for Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions,” Abstract and Applied Analysis, vol. 2013, Article ID 869837, 17 pages, 2013.
• R. Kamocki and C. Obczynski, “On fractional differential inclusions with the Jumarie derivative,” Journal of Mathematical Physics, vol. 55, no. 2, Article ID 2902, 10 pages, 2014.
• J. Tariboon, T. Sitthiwirattham, and S. K. Ntouyas, “Existence results for fractional differential inclusions with multi-point and fractional integral boundary conditions,” Journal of Computational Analysis and Applications, vol. 17, no. 2, pp. 343–360, 2014.
• B. Ahmad and S. K. Ntouyas, “An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions,” Abstract and Applied Analysis, vol. 2014, Article ID 705809, 7 pages, 2014.
• D. O'Regan, “Fixed-point theory for the sum of two operators,” Applied Mathematics Letters, vol. 9, no. 1, pp. 1–8, 1996.
• A. Granas and J. Dugundji, Fixed Point Theory, Springer, New York, NY, USA, 2005.
• E. Zeidler, Nonlinear Functional Analysis and Its Application: Fixed Point-Theorems, vol. 1, Springer, New York, NY, USA, 1986.
• B. N. Sadovskii, “On a fixed point principle,” Functional Analysis and Its Applications, vol. 1, pp. 74–76, 1967. \endinput