A family of measures of noncompactness in the Hölder space Cn,γ(R+) and its application to some fractional differential equations and numerical methods

In this paper, we prove the existence of solutions for the following fractional boundary value problem c D α u ( t ) = f ( t , u ( t ) ) , α ∈ ( n , n + 1 ) , 0 ≤ t + ∞ , u ( 0 ) = 0 , u ′ ′ ( 0 ) = 0 , … , u ( n ) ( 0 ) = 0 , lim t → + ∞ c D α − 1 u ( t ) = β u ( ξ ) . The considerations of this paper are based on the concept of a new family of measures of noncompactness in the space of functions C n , γ ( R + ) satisfying the Holder condition and a fixed point theorem of Darbo type. We also provide an illustrative example in support of our existence theorems. Finally, to credibility, we apply successive approximation and homotopy perturbation method to find solution of the above problem with high accuracy.