scholarly journals Positive Solutions for a System of Fractional Integral Boundary Value Problems Involving Hadamard-Type Fractional Derivatives

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Haiyan Zhang ◽  
Yaohong Li ◽  
Jiafa Xu
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fractional Derivatives ◽  
Fixed Point Index ◽  
Fractional Integral ◽  
Boundary Value ◽  
Integral Boundary ◽  
Existence Of Positive Solutions ◽  
Integral Boundary Value Problems ◽  
Hadamard Fractional Integral

In this paper, we use fixed-point index to study the existence of positive solutions for a system of Hadamard fractional integral boundary value problems involving nonnegative nonlinearities. By virtue of integral-type Jensen inequalities, some appropriate concave and convex functions are used to depict the coupling behaviors for our nonlinearities fii=1, 2.

scholarly journals Positive Solutions for a System of Fractional Integral Boundary Value Problems of Riemann–Liouville Type Involving Semipositone Nonlinearities

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 970
Author(s):  
Youzheng Ding ◽  
Jiafa Xu ◽  
Zhengqing Fu
Keyword(s):  
Fixed Point ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fractional Integral ◽  
Matrix Theory ◽  
Boundary Value ◽  
Liouville Type ◽  
Integral Boundary ◽  
Integral Boundary Value Problems ◽  
Fractional Boundary Value Problems

In this work by the index of fixed point and matrix theory, we discuss the positive solutions for the system of Riemann–Liouville type fractional boundary value problems D 0 + α u ( t ) + f 1 ( t , u ( t ) , v ( t ) , w ( t ) ) = 0 , t ∈ ( 0 , 1 ) , D 0 + α v ( t ) + f 2 ( t , u ( t ) , v ( t ) , w ( t ) ) = 0 , t ∈ ( 0 , 1 ) , D 0 + α w ( t ) + f 3 ( t , u ( t ) , v ( t ) , w ( t ) ) = 0 , t ∈ ( 0 , 1 ) , u ( 0 ) = u ′ ( 0 ) = ⋯ = u ( n − 2 ) ( 0 ) = 0 , D 0 + p u ( t ) | t = 1 = ∫ 0 1 h ( t ) D 0 + q u ( t ) d t , v ( 0 ) = v ′ ( 0 ) = ⋯ = v ( n − 2 ) ( 0 ) = 0 , D 0 + p v ( t ) | t = 1 = ∫ 0 1 h ( t ) D 0 + q v ( t ) d t , w ( 0 ) = w ′ ( 0 ) = ⋯ = w ( n − 2 ) ( 0 ) = 0 , D 0 + p w ( t ) | t = 1 = ∫ 0 1 h ( t ) D 0 + q w ( t ) d t , where α ∈ ( n − 1 , n ] with n ∈ N , n ≥ 3 , p , q ∈ R with p ∈ [ 1 , n − 2 ] , q ∈ [ 0 , p ] , D 0 + α is the α order Riemann–Liouville type fractional derivative, and f i ( i = 1 , 2 , 3 ) ∈ C ( [ 0 , 1 ] × R + × R + × R + , R ) are semipositone nonlinearities.

scholarly journals Positive Solutions for a System of Nonlinear Semipositone Boundary Value Problems with Riemann-Liouville Fractional Derivatives

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Xiaowei Qiu ◽  
Jiafa Xu ◽  
Donal O’Regan ◽  
Yujun Cui
Keyword(s):  
Boundary Value Problems ◽  
Positive Solutions ◽  
Fractional Derivatives ◽  
Fixed Point Index ◽  
Existence Theorems ◽  
Boundary Value ◽  
Point Index ◽  
Coupling Behavior ◽  
Existence Of Positive Solutions ◽  
Liouville Fractional Derivative

We study the existence of positive solutions for the system of nonlinear semipositone boundary value problems with Riemann-Liouville fractional derivatives D0+αD0+αu=f1t,u,u′,v,v′,  0<t<1, D0+αD0+αv=f2(t,u,u′,v,v′),  0<t<1, u0=u′0=u′(1)=D0+αu(0)=D0+α+1u(0)=D0+α+1u(1)=0, and v(0)=v′(0)=v′(1)=D0+αv(0)=D0+α+1v(0)=D0+α+1v(1)=0, where α∈(2,3] is a real number and D0+α is the standard Riemann-Liouville fractional derivative of order α. Under some appropriate conditions for semipositone nonlinearities, we use the fixed point index to establish two existence theorems. Moreover, nonnegative concave and convex functions are used to depict the coupling behavior of our nonlinearities.

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