Solvability of nonlinear fractional integro-differential equation with nonlocal condition

Abstract

Purpose – This study describes the applicability of the a priori estimate method on a nonlocal nonlinear fractional differential equation for which the weak solution's existence and uniqueness are proved. The authors divide the proof into two sections for the linear associated problem; the authors derive the a priori bound and demonstrate the operator range density that is generated. The authors solve the nonlinear problem by introducing an iterative process depending on the preceding results. Design/methodology/approach – The functional analysis method is the a priori estimate method or energy inequality method. Findings – The results show the efficiency of a priori estimate method in the case of time-fractional order differential equations with nonlocal conditions. Our results also illustrate the existence and uniqueness of the continuous dependence of solutions on fractional order differential equations with nonlocal conditions. Research limitations/implications – The authors’ work can be considered a contribution to the development of the functional analysis method that is used to prove well-positioned problems with fractional order. Originality/value – The authors confirm that this work is original and has not been published elsewhere, nor is it currently under consideration for publication elsewhere.

Keywords

• Existence and uniqueness
• A priori estimate
• Fractional derivatives and integrals
• Integral condition