Existence and multiplicity of solutions for nonlinear fractional differential equations

Volume 9, Issue 6, pp 4639--4646 http://dx.doi.org/10.22436/jnsa.009.06.102

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Authors

Hamidreza Marasi - Department of Mathematical sciences, Basic Science Faculty, Bonab University, Bonab, Iran. Hossein Piri - Department of Mathematical sciences, Basic Science Faculty, Bonab University, Bonab, Iran. Hassen Aydi - Department of Mathematics, College of Education of Jubail, University of Dammam, P. O: 12020, Industrial Jubail 31961, Saudi Arabia. - Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan.

Abstract

In this paper, we consider the following fractional initial value problems: \[D^\alpha u(t) = f(t; u(t);D^\beta u(t)); t \in (0; 1];\] \[u^{(k)}(0) = \eta_k; k = 0; 1; ...; n - 1;\] where \(n - 1 < \beta < \alpha < n; (n \in N)\) are real numbers, \(D^\alpha\) and \(D^\beta\) are the Caputo fractional derivatives and \(f \in C([0; 1] \times R)\). Using the fixed point index theory, we study the existence and multiplicity of positive solutions and obtain some new results.

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Keywords

• Fractional differential equation
• positive solution
• index fixed point theorem.

MSC

•  37C25
•  34A08
•  74H20

References

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