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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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[1] Babakhani, A.; Gejji, V.D., Existence of positive solutions of nonlinear fractional differential equations, J. math. anal. appl., 278, 434-442, (2003) · Zbl 1027.34003
[2] Delbosco, D., Fractional calculus and function spaces, J. fract. calc., 6, 45-53, (1996) · Zbl 0829.46018
[3] Delbosco, D.; Rodino, L., Existence and uniqueness for a nonlinear fractional differential equation, J. math. anal. appl., 204, 609-625, (1996) · Zbl 0881.34005
[4] El-Sayed, A.M.A., Nonlinear functional differential equations of arbitrary orders, Nonlinear anal., 33, 181-186, (1998) · Zbl 0934.34055
[5] Gejji, V.D.; Babakhani, A., Analysis of a system of fractional differential equations, J. math. anal. appl., 293, 511-522, (2004) · Zbl 1058.34002
[6] Kilbas, A.A.; Marichev, O.I.; Samko, S.G., Fractional integral and derivatives (theory and applications), (1993), Gordon and Breach Switzerland · Zbl 0818.26003
[7] Kilbas, A.A.; Trujillo, J.J., Differential equations of fractional order: methods, results and problems I, Appl. anal., 78, 153-192, (2001) · Zbl 1031.34002
[8] Kilbas, A.A.; Trujillo, J.J., Differential equations of fractional order: methods, results and problems II, Appl. anal., 81, 435-493, (2002) · Zbl 1033.34007
[9] Krasnosel’skii, M.A., Positive solutions of operator equations, (1964), Noordhoff Groningen · Zbl 0121.10604
[10] Leggett, R.W.; Williams, L.R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana univ. math. J., 28, 673-688, (1979) · Zbl 0421.47033
[11] Miller, K.S., Fractional differential equations, J. fract. calc., 3, 49-57, (1993) · Zbl 0781.34006
[12] Miller, K.S.; Ross, B., An introduction to the fractional calculus and fractional differential equations, (1993), Wiley New York · Zbl 0789.26002
[13] Nakhushev, A.M., The sturm – liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms, Dokl. akad. nauk SSSR, 234, 308-311, (1977) · Zbl 0376.34015
[14] Podlubny, I., Fractional differential equations, mathematics in science and engineering, (1999), Academic Press New York
[15] I. Podlubny, The Laplace transform method for linear differential equations of the fractional order, Inst. Expe. Phys., Slov. Acad. Sci., UEF-02-94, Kosice, 1994
[16] Zhang, S.Q., The existence of a positive solution for a nonlinear fractional differential equation, J. math. anal. appl., 252, 804-812, (2000) · Zbl 0972.34004
[17] Zhang, S.Q., Existence of positive solution for some class of nonlinear fractional differential equations, J. math. anal. appl., 278, 136-148, (2003) · Zbl 1026.34008
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