Existence of positive solution for BVP of nonlinear fractional differential equation with integral boundary conditions

Abstract

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 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.

Keywords

• Fractional differential equation
• Integral boundary condition
• Boundary value problem
• Positive solution
• Existence