Non-existence of solutions for two-point fractional and third-order boundary-value problems

Abstract

In this article, we provide sufficient conditions for the non-existence of solutions of the boundary-value problems with fractional derivative of order $alphain(2,3)$ in the Riemann-Liouville sense $$displaylines{ D_{0+}^{alpha}x(t)+lambda a(t)f(x(t))=0,quad tin(0,1),cr x(0)=x'(0)=x'(1)=0, }$$ and in the Caputo sense $$displaylines{ ^CD^{alpha}x(t)+f(t,x(t))=0,quad tin(0,1),cr x(0)=x'(0)=0, quad x(1)=lambdaint_0^1x(s)ds; }$$ and for the third-order differential equation $$ x'''(t)+(Fx)(t)=0, quad hbox{a.e. }tin [0,1], $$ associated with three among the following six conditions $$ x(0)=0,quad x(1)=0,quad x'(0)=0, quad x'(1)=0, quad x''(0)=0, quad x''(1)=0. $$ Thus, fourteen boundary-value problems at resonance and six boundary-value problems at non-resonanse are studied. Applications of the results are, also, given.

Keywords

• Third order differential equation
• two-point boundary-value problem
• fractional boundary condition
• nonexistence of solutions