Fractional boundary value problem in Orlicz spaces

Abstract

The aim of this paper is devoted to investigate the existence of positive continuous solutions for boundary value problem of fractional type    dα,gx(t) dtα = λu(t, x(t))[ζ(x(t)) + η(x(t))], t ∈ [a, b], 0 <α< 1, λ ∈ R+, x(a) − px(b) = h, (1) under the monotonicity conditions imposed on η and ζ. Here h ∈ R+, p ∈ [0, 1), and u is ”possibly singular” function from an appropriate Orlicz space. By the singularity of the above problem, we mean that the possibility of η(0) being undefined is allowed. To encompass the full scope of this paper, we present some examples illustrating the main results.