Existence of solutions for integral boundary value problems of mixed fractional differential equations under resonance

Abstract

In this paper, we concerned the existence of solutions of the following nonlinear mixed fractional differential equation with the integral boundary value problem:{D1−αCD0+βu(t)=f(t,u(t),D0+β+1u(t),D0+βu(t)),0<t<1,u(0)=u′(0)=0,u(1)=∫01u(t)dA(t),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\left \\{ \\textstyle\\begin{array}{l} {}^{C}D^{\\alpha}_{1-} D^{\\beta}_{0+}u(t)=f(t,u(t),D^{\\beta +1}_{0+}u(t),D^{\\beta}_{0+}u(t)),\\quad 0< t< 1,\\\\ u(0)=u'(0)=0,\\qquad u(1)=\\int^{1}_{0}u(t)\\,dA(t), \\end{array}\\displaystyle \\right . $$\\end{document} where {}^{C}D^{alpha}_{1-} is the left Caputo fractional derivative of order alphain(1,2], and D^{beta}_{0+} is the right Riemann–Liouville fractional derivative of order betain(0,1]. The coincidence degree theory is the main theoretical basis to prove the existence of solutions of such problems.