×
Alsaedi, Ahmed; Ntouyas, Sotiris K.; Ahmad, Bashir

New existence results for fractional integrodifferential equations with nonlocal integral boundary conditions. (English) Zbl 1448.45007

Abstr. Appl. Anal. 2015, Article ID 205452, 10 p. (2015).
Summary: We consider a boundary value problem of fractional integrodifferential equations with new nonlocal integral boundary conditions of the form: \(x(0) = \beta x(\theta)\), \(x(\xi) = \alpha \int_\eta^1 x(s) d s\), 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.

Cited in 5 Documents

MSC:

45J05 Integro-ordinary differential equations
Cite Review PDF
Full Text: DOI
OA License

References:

[1] Wang, G. S.; Blom, A. F., A strip model for fatigue crack growth predictions under general load conditions, Engineering Fracture Mechanics, 40, 3, 507-533 (1991) · doi:10.1016/0013-7944(91)90148-T
[2] Ahmad, B.; Hayat, T.; Asghar, S., Diffraction of a plane wave by an elastic knife-edge adjacent to a strip, The Canadian Applied Mathematics Quarterly, 9, 303-316 (2001) · Zbl 1049.76059
[3] Podlubny, I., Fractional Differential Equations (1999), San Diego, Calif, USA: Academic Press, San Diego, Calif, USA · Zbl 0918.34010
[4] Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J., Theory and Applications of Fractional Differential Equations. Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies (2006), Amsterdam, The Netherlands: Elsevier Science, Amsterdam, The Netherlands · Zbl 1092.45003
[5] Sabatier, J.; Agrawal, O. P.; Machado, J. A. T., Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering (2007), Dordrecht, The Netherlands: Springer, Dordrecht, The Netherlands · Zbl 1116.00014
[6] Baleanu, D.; Diethelm, K.; Scalas, E.; Trujillo, J. J., Fractional Calculus Models and Numerical Methods. Fractional Calculus Models and Numerical Methods, Series on Complexity, Nonlinearity and Chaos (2012), Boston, Mass, USA: World Scientific, Boston, Mass, USA · Zbl 1248.26011
[7] Benchohra, M.; Hamani, S.; Ntouyas, S. K., Boundary value problems for differential equations with fractional order and nonlocal conditions, Nonlinear Analysis: Theory, Methods & Applications, 71, 7-8, 2391-2396 (2009) · Zbl 1198.26007 · doi:10.1016/j.na.2009.01.073
[8] Zhong, W.; Lin, W., Nonlocal and multiple-point boundary value problem for fractional differential equations, Computers & Mathematics with Applications, 59, 3, 1345-1351 (2010) · Zbl 1189.34036 · doi:10.1016/j.camwa.2009.06.032
[9] Ahmad, B.; Sivasundaram, S., On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order, Applied Mathematics and Computation, 217, 2, 480-487 (2010) · Zbl 1207.45014 · doi:10.1016/j.amc.2010.05.080
[10] Ahmad, B., On nonlocal boundary value problems for nonlinear integro-differential equations of arbitrary fractional order, Results in Mathematics, 63, 1-2, 183-194 (2013) · Zbl 1270.45004 · doi:10.1007/s00025-011-0187-9
[11] Ahmad, B.; Nieto, J. J., Sequential fractional differential equations with three-point boundary conditions, Computers & Mathematics with Applications, 64, 10, 3046-3052 (2012) · Zbl 1268.34006 · doi:10.1016/j.camwa.2012.02.036
[12] Graef, J. R.; Kong, L.; Kong, Q., Application of the mixed monotone operator method to fractional boundary value problems, Fractional Differential Calculus, 2, 1, 87-98 (2012) · Zbl 1412.34086 · doi:10.7153/fdc-02-06
[13] Hu, C.; Liu, B.; Xie, S., Monotone iterative solutions for nonlinear boundary value problems of fractional differential equation with deviating arguments, Applied Mathematics and Computation, 222, 72-81 (2013) · Zbl 1333.34117 · doi:10.1016/j.amc.2013.07.048
[14] Staněk, S., Limit properties of positive solutions of fractional boundary value problems, Applied Mathematics and Computation, 219, 5, 2361-2370 (2012) · Zbl 1308.34104 · doi:10.1016/j.amc.2012.09.008
[15] Ahmad, B.; Ntouyas, S. K., Existence results for nonlocal boundary value problems of fractional differential equations and inclusions with strip conditions, Boundary Value Problems, 2012 (2012) · Zbl 1279.26010
[16] Zhang, P.; Gong, Y., Existence and multiplicity results for a class of fractional differential inclusions with boundary conditions, Boundary Value Problems, 2012, article 82 (2012) · Zbl 1281.26007 · doi:10.1186/1687-2770-2012-82
[17] Alsaedi, A.; Ntouyas, S. K.; Ahmad, B., Existence results for Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions, Abstract and Applied Analysis, 2013 (2013) · Zbl 1276.26008 · doi:10.1155/2013/869837
[18] Kamocki, R.; Obczynski, C., On fractional differential inclusions with the Jumarie derivative, Journal of Mathematical Physics, 55, 2 (2014) · Zbl 1293.26010 · doi:10.1063/1.4865997
[19] Tariboon, J.; Sitthiwirattham, T.; Ntouyas, S. K., Existence results for fractional differential inclusions with multi-point and fractional integral boundary conditions, Journal of Computational Analysis and Applications, 17, 2, 343-360 (2014) · Zbl 1295.34011
[20] Ahmad, B.; Ntouyas, S. K., An existence theorem for fractional hybrid differential inclusions of Hadamard type with Dirichlet boundary conditions, Abstract and Applied Analysis, 2014 (2014) · Zbl 1474.34094 · doi:10.1155/2014/705809
[21] O’Regan, D., Fixed-point theory for the sum of two operators, Applied Mathematics Letters, 9, 1, 1-8 (1996) · Zbl 0858.34049 · doi:10.1016/0893-9659(95)00093-3
[22] Granas, A.; Dugundji, J., Fixed Point Theory (2005), New York, NY, USA: Springer, New York, NY, USA
[23] Zeidler, E., Nonlinear Functional Analysis and Its Application: Fixed Point-Theorems, 1 (1986), New York, NY, USA: Springer, New York, NY, USA · Zbl 0583.47050
[24] Sadovskii, B. N., On a fixed point principle, Functional Analysis and Its Applications, 1, 74-76 (1967) · Zbl 0165.49102
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.