In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fra...In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel'skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.展开更多
In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form （Pa,bα）{D^au（x）＋f（x,u（x））=0,x∈（0,1）,u（0）=u（1）=0,D^a-3u（0）=a,u^（1）=-6w...In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form （Pa,bα）{D^au（x）＋f（x,u（x））=0,x∈（0,1）,u（0）=u（1）=0,D^a-3u（0）=a,u^（1）=-6where 3 ≤ a 〈 4, D^ is the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f（x, t） = -p（x）t^σ, with cr ~ （-1, 1） and p being a nonnegative continuous function that may be singular at x - 0 or x - 1 and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch^uder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem （P0,0）. Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a ＋ b 〉 0 and f（x, t） -= tφ（x, t）, with φ（x, t） being a nonnegative continuous function in （0, 1） × [0, ∞） that is required to satisfy some suitable integrability condition. Using estimates on the Green＇s function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem （Pa,b）, which behaves like the unique solution of the homogeneous problem corresponding the existence results. to （Pa,b）. Some examples are given to illustrate the existence results.,展开更多
基金Supported by the Research Fund for the Doctoral Program of High Education of China(20094407110001)Supported by the NSF of Guangdong Province(10151063101000003)
文摘In this paper,we study a Dirichlet-type boundary value problem(BVP) of nonlinear fractional differential equation with an order α∈(3,4],where the fractional derivative D~α_(o^+)is the standard Riemann-Liouville fractional derivative.By constructing the Green function and investigating its properties,we obtain some criteria for the existence of one positive solution and two positive solutions for the above BVP.The Krasnosel'skii fixedpoint theorem in cones is used here.We also give an example to illustrate the applicability of our results.
基金funded by the National Plan for Science,Technology and Innovation(MAARIFAH),King Abdulaziz City for Science and Technology,Kingdom of Saudi Arabia,Award Number(No.13-MAT2137-02)
文摘In this paper, the authors aim at proving two existence results of fractional differential boundary value problems of the form （Pa,bα）{D^au（x）＋f（x,u（x））=0,x∈（0,1）,u（0）=u（1）=0,D^a-3u（0）=a,u^（1）=-6where 3 ≤ a 〈 4, D^ is the standard Riemann-Liouville fractional derivative and a, b are nonnegative constants. First the authors suppose that f（x, t） = -p（x）t^σ, with cr ~ （-1, 1） and p being a nonnegative continuous function that may be singular at x - 0 or x - 1 and satisfies some conditions related to the Karamata regular variation theory. Combining sharp estimates on some potential functions and the Sch^uder fixed point theorem, the authors prove the existence of a unique positive continuous solution to problem （P0,0）. Global estimates on such a solution are also obtained. To state the second existence result, the authors assume that a, b are nonnegative constants such that a ＋ b 〉 0 and f（x, t） -= tφ（x, t）, with φ（x, t） being a nonnegative continuous function in （0, 1） × [0, ∞） that is required to satisfy some suitable integrability condition. Using estimates on the Green＇s function and a perturbation argument, the authors prove the existence and uniqueness of a positive continuous solution u to problem （Pa,b）, which behaves like the unique solution of the homogeneous problem corresponding the existence results. to （Pa,b）. Some examples are given to illustrate the existence results.,
