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Positive solutions for boundary value problem of nonlinear fractional differential equation. (English) Zbl 1079.34048

Summary: We investigate the existence and multiplicity of positive solutions to the boundary value problem \[ D^\alpha_{0+}u(t)+f \bigl(t,u(t) \bigr)=0,\;0<t<1, \quad u(0)=u(1)=0, \] where \(1<\alpha\leq 2\) is a real number, \(D_{0+}^\alpha\) is the standard Riemann-Liouville differentiation, and \(f:[0,1] \times[0,\infty) \to [0,\infty)\) is continuous. By means of some fixed-point theorems in a cone, existence and multiplicity results positive solutions are obtained. The proofs are based upon the reduction of the problem considered to the equivalent Fredholm integral equation of second kind.

MSC:

34K05 General theory of functional-differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
26A33 Fractional derivatives and integrals
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References:

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