Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions
Abstract This paper is concerned with the following boundary value problem of nonlinear fractional differential equation with integral boundary conditions: { ( C D 0 + q u ) ( t ) + f ( t , u ( t ) ) = 0 , t ∈ [ 0 , 1 ] , u ″ ( 0 ) = 0 , α u ( 0 ) − β u ′ ( 0 ) = ∫ 0 1 h 1 ( s ) u ( s ) d s , γ u (...
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| Format: | Article |
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SpringerOpen
2020-04-01
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| Series: | Advances in Difference Equations |
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| Online Access: | http://link.springer.com/article/10.1186/s13662-020-02618-9 |
| _version_ | 1818957698484404224 |
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| author | Min Li Jian-Ping Sun Ya-Hong Zhao |
| author_facet | Min Li Jian-Ping Sun Ya-Hong Zhao |
| author_sort | Min Li |
| collection | DOAJ |
| description | Abstract This paper is concerned with the following boundary value problem of nonlinear fractional differential equation with integral boundary conditions: { ( C D 0 + q u ) ( t ) + f ( t , u ( t ) ) = 0 , t ∈ [ 0 , 1 ] , u ″ ( 0 ) = 0 , α u ( 0 ) − β u ′ ( 0 ) = ∫ 0 1 h 1 ( s ) u ( s ) d s , γ u ( 1 ) + δ ( C D 0 + σ u ) ( 1 ) = ∫ 0 1 h 2 ( s ) u ( s ) d s , $$ \textstyle\begin{cases} ({}^{C}D_{0+}^{q}u)(t)+f(t,u(t))=0,\quad t\in [0,1], \\ u^{\prime \prime }(0)=0, \\ \alpha u(0)-\beta u^{\prime }(0)=\int _{0}^{1}h_{1}(s)u(s)\,ds, \\ \gamma u(1)+\delta ({}^{C}D_{0+}^{\sigma }u)(1) =\int _{0}^{1}h_{2}(s)u(s)\,ds, \end{cases} $$ where 2 < q ≤ 3 $2< q\leq 3$ , 0 < σ ≤ 1 $0<\sigma \leq 1$ , α , γ , δ ≥ 0 $\alpha , \gamma , \delta \geq 0$ , and β > 0 $\beta >0$ satisfying 0 < ρ : = ( α + β ) γ + α δ Γ ( 2 − σ ) < β [ γ + δ Γ ( q ) Γ ( q − σ ) ] $0<\rho :=(\alpha +\beta )\gamma + \frac{\alpha \delta }{\varGamma (2-\sigma )}<\beta [\gamma + \frac{\delta \varGamma (q)}{\varGamma (q-\sigma )} ]$ . D 0 + q C ${}^{C}D_{0+}^{q}$ denotes the standard Caputo fractional derivative. First, Green’s function of the corresponding linear boundary value problem is constructed. Next, some useful properties of the Green’s function are obtained. Finally, existence results of at least one positive solution for the above problem are established by imposing some suitable conditions on f and h i $h_{i}$ ( i = 1 , 2 $i=1,2$ ). The method employed is Guo–Krasnoselskii’s fixed point theorem. An example is also included to illustrate the main results of this paper. |
| first_indexed | 2024-12-20T11:13:59Z |
| format | Article |
| id | doaj.art-5f607bc61a8b4db89b00bf55c9028c04 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-20T11:13:59Z |
| publishDate | 2020-04-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-5f607bc61a8b4db89b00bf55c9028c042022-12-21T19:42:41ZengSpringerOpenAdvances in Difference Equations1687-18472020-04-012020111310.1186/s13662-020-02618-9Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditionsMin Li0Jian-Ping Sun1Ya-Hong Zhao2Department of Applied Mathematics, Lanzhou University of TechnologyDepartment of Applied Mathematics, Lanzhou University of TechnologyDepartment of Applied Mathematics, Lanzhou University of TechnologyAbstract This paper is concerned with the following boundary value problem of nonlinear fractional differential equation with integral boundary conditions: { ( C D 0 + q u ) ( t ) + f ( t , u ( t ) ) = 0 , t ∈ [ 0 , 1 ] , u ″ ( 0 ) = 0 , α u ( 0 ) − β u ′ ( 0 ) = ∫ 0 1 h 1 ( s ) u ( s ) d s , γ u ( 1 ) + δ ( C D 0 + σ u ) ( 1 ) = ∫ 0 1 h 2 ( s ) u ( s ) d s , $$ \textstyle\begin{cases} ({}^{C}D_{0+}^{q}u)(t)+f(t,u(t))=0,\quad t\in [0,1], \\ u^{\prime \prime }(0)=0, \\ \alpha u(0)-\beta u^{\prime }(0)=\int _{0}^{1}h_{1}(s)u(s)\,ds, \\ \gamma u(1)+\delta ({}^{C}D_{0+}^{\sigma }u)(1) =\int _{0}^{1}h_{2}(s)u(s)\,ds, \end{cases} $$ where 2 < q ≤ 3 $2< q\leq 3$ , 0 < σ ≤ 1 $0<\sigma \leq 1$ , α , γ , δ ≥ 0 $\alpha , \gamma , \delta \geq 0$ , and β > 0 $\beta >0$ satisfying 0 < ρ : = ( α + β ) γ + α δ Γ ( 2 − σ ) < β [ γ + δ Γ ( q ) Γ ( q − σ ) ] $0<\rho :=(\alpha +\beta )\gamma + \frac{\alpha \delta }{\varGamma (2-\sigma )}<\beta [\gamma + \frac{\delta \varGamma (q)}{\varGamma (q-\sigma )} ]$ . D 0 + q C ${}^{C}D_{0+}^{q}$ denotes the standard Caputo fractional derivative. First, Green’s function of the corresponding linear boundary value problem is constructed. Next, some useful properties of the Green’s function are obtained. Finally, existence results of at least one positive solution for the above problem are established by imposing some suitable conditions on f and h i $h_{i}$ ( i = 1 , 2 $i=1,2$ ). The method employed is Guo–Krasnoselskii’s fixed point theorem. An example is also included to illustrate the main results of this paper.http://link.springer.com/article/10.1186/s13662-020-02618-9Fractional differential equationIntegral boundary conditionBoundary value problemPositive solutionExistence |
| spellingShingle | Min Li Jian-Ping Sun Ya-Hong Zhao Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions Advances in Difference Equations Fractional differential equation Integral boundary condition Boundary value problem Positive solution Existence |
| title | Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions |
| title_full | Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions |
| title_fullStr | Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions |
| title_full_unstemmed | Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions |
| title_short | Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions |
| title_sort | existence of positive solution for bvp of nonlinear fractional differential equation with integral boundary conditions |
| topic | Fractional differential equation Integral boundary condition Boundary value problem Positive solution Existence |
| url | http://link.springer.com/article/10.1186/s13662-020-02618-9 |
| work_keys_str_mv | AT minli existenceofpositivesolutionforbvpofnonlinearfractionaldifferentialequationwithintegralboundaryconditions AT jianpingsun existenceofpositivesolutionforbvpofnonlinearfractionaldifferentialequationwithintegralboundaryconditions AT yahongzhao existenceofpositivesolutionforbvpofnonlinearfractionaldifferentialequationwithintegralboundaryconditions |