Boundary value problems for fractional differential equations with integral and ordinary-fractional flux boundary conditions

Volume 9, Issue 6, pp 3622--3637 http://dx.doi.org/10.22436/jnsa.009.06.15

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Authors

Bashir Ahmad - Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. Sotiris K. Ntouyas - Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. - Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece.

Abstract

In this paper, we consider a new class of boundary value problems of Caputo type fractional differential equations supplemented with classical/nonlocal Riemann-Liouville integral and flux boundary conditions and obtain some existence results for the given problems. The flux boundary condition \(x'(0) = b ^cD^\beta x(1)\) states that the ordinary flux \(x'(0)\) at the left-end point of the interval [0; 1] is proportional to a flux \(^cD^\beta x(1)\) of fractional order \(\beta \in (0; 1]\) at the right-end point of the given interval. The coupling of integral and flux boundary conditions introduced in this paper owes to the novelty of the work. We illustrate our results with the aid of examples. Our work not only generalizes some known results but also produces new results for specific values of the parameters involved in the problems at hand.

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Keywords

• Differential equations
• ractional order
• integral boundary conditions
• flux
• existence
• fixed point.

MSC

•  34A08
•  34B10

References

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