FRACTIONAL-ORDER BOUNDARY VALUE PROBLEM WITH STURM-LIOUVILLE BOUNDARY CONDITIONS

Using the new conformable fractional derivative, which diers from the Riemann-Liouville and Caputo fractional derivatives, we reformu- late the second-order conjugate boundary value problem in this new setting. Utilizing the corresponding positive fractional Green's function, we apply a functional compression-expansion xed point theorem to prove the existence of a positive solution. We then compare our results favorably to those based on the Riemann-Liouville fractional derivative.

[1]  Yongxiang Li,et al.  The Existence of Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equations , 2014 .

[2]  Existence of positive solutions for eigenvalue problem of nonlinear fractional differential equations , 2012 .

[3]  Xinguang Zhang,et al.  Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations , 2012 .

[4]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[5]  A. Cabada,et al.  Positive solutions of nonlinear fractional differential equations with integral boundary value conditions , 2012 .

[6]  Yuji Liu,et al.  Solvability of boundary value problems for fractional order elastic beam equations , 2014 .

[7]  Wu Wenquan.  UNIQUENESS AND ASYMPTOTIC BEHAVIOR OF POSITIVE SOLUTIONS FOR A FRACTIONAL-ORDER INTEGRAL BOUNDARY-VALUE PROBLEM , 2013 .

[8]  N. Mahmudov,et al.  Existence of Solutions of Order Fractional Three-Point Boundary Value Problems with Integral Conditions , 2014 .

[9]  Udita N. Katugampola.  A New Fractional Derivative with Classical Properties , 2014, 1410.6535.

[10]  Xiaoyou Liu,et al.  FRACTIONAL DIFFERENTIAL EQUATIONS WITH FRACTIONAL NON-SEPARATED BOUNDARY CONDITIONS , 2013 .

[11]  Johnny Henderson,et al.  Multiple symmetric positive solutions for a second order boundary value problem , 2000 .

[12]  Yongping Sun,et al.  Existence and nonexistence of positive solutions for fractional-order two-point boundary value problems , 2014, Advances in Difference Equations.

[13]  Bashir Ahmad,et al.  Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions , 2009, Comput. Math. Appl..

[14]  Xinguang Zhang,et al.  Positive Solutions of Eigenvalue Problems for a Class of Fractional Differential Equations with Derivatives , 2012 .

[15]  E. Kaufmann,et al.  Positive solutions of a boundary value problem for a nonlinear fractional differential equation. , 2008 .

[16]  Jingxian Sun,et al.  A generalization of the cone expansion and compression fixed point theorem and applications , 2007 .

[17]  M. A. Hammad,et al.  ABEL'S FORMULA AND WRONSKIAN FOR CONFORMABLE FRACTIONAL DIFFERENTIAL EQUATIONS , 2014 .

[18]  Jeffrey T. Neugebauer,et al.  Existence and comparison of smallest eigenvalues for a fractional boundary-value problem , 2014 .

[19]  M. Sababheh,et al.  A new definition of fractional derivative , 2014, J. Comput. Appl. Math..

[20]  Xuemai Gu,et al.  Uniqueness and Asymptotic Behavior of Positive Solutions for a Fractional-Order Integral Boundary Value Problem , 2012 .

[21]  J. Henderson,et al.  FUNCTIONAL COMPRESSION-EXPANSION FIXED POINT THEOREM , 2008 .

[22]  Douglas R. Anderson,et al.  Fixed Point Theorem of Cone Expansion and Compression of Functional Type , 2002 .

[23]  Songlin Hu,et al.  Existence of positive solutions for a fractional high-order three-point boundary value problem , 2014 .