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Solutions to boundary-value problems for nonlinear differential equations of fractional order. (English) Zbl 1173.34011

Summary: We discuss the existence, uniqueness and continuous dependence of solutions for a boundary value problem of nonlinear fractional differential equation.
We present the analysis of the boundary value problem
\[ \begin{gathered} ^CD^\alpha_{0^+}u(t)=f(t,u(t),{}^CD^\beta_{0^+}u(t)), \quad 0<t<1,\\ a_1u(0)-a_2u'(0)=A,\;b_1u(1)+b_2u'(1)=B,\tag{1}\end{gathered} \]
where \(1<\alpha\leq 2\), \(0<\beta\leq 1\), \(a_ib_i\geq 0\), \(i=1,2\), \(a_1b_1+a_1b_2+a_2b_1>0\), \(^CD^\alpha_{0+}\) and \(^CD^\beta_{0^+}\) are the Caputo’s fractional derivatives and \(f:[0,1]\times\mathbb R\times \mathbb R\to\mathbb R\) is continuous. We impose a growth condition on the function \(f\) to prove an existence result for (1).

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
26A33 Fractional derivatives and integrals
47N20 Applications of operator theory to differential and integral equations
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