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Erschienen in:
01.10.2014 | Original Research
Existence results of fractional differential equations with irregular boundary conditions and p-Laplacian operator
Erschienen in: Journal of Applied Mathematics and Computing | Ausgabe 1-2/2014
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Abstract
In this paper, we discuss the existence of solutions for irregular boundary value problems of nonlinear fractional differential equations with p-Laplacian operator
where \(^{c}D_{0+}^{\alpha}\) is the Caputo fractional derivative, ϕ
p
(s)=|s|
p−2
s, p>1, \({\phi}_{p}^{-1}={\phi}_{q}\), \(\frac {1}{p}+\frac{1}{q}=1\) and \(f: [0,1] \times\mathbb{R} \times\mathbb {R} \longrightarrow\mathbb{R}\). Our results are based on the Schauder and Banach fixed point theorems. Furthermore, two examples are also given to illustrate the results.
$$\left \{ \begin{array}{l} {\phi}_p(^cD_{0+}^{\alpha}u(t))=f(t,u(t),u'(t)), \quad 0< t<1, \ 1< \alpha \leq2, \\ u(0)+(-1)^{\theta}u'(0)+bu(1)=\lambda, \qquad u(1)+(-1)^{\theta}u'(1)=\int_0^1g(s,u(s))ds,\\ \quad \theta=0,1, \ b \neq \pm1, \end{array} \right . $$