Convolution differential equations with Riemann–Liouville nonlocal elements

Existence of at least one positive solution to the nonlocal differential equation − M ( ( a ∗ ( g ∘ u ) ) ′ ( 1 ) ) u ( t ) = λ f ( t , u ( t ) ) , 0 > t > 1 , {equation} -M ( (a*(g∘ u) )’(1) )u(t)=λ f (t,u(t) ),\;0>t>1, {equation} subject to the right-focal boundary data u ( 0 ) = 0 = u ′ ( 1 ) u(0)=0=u’(1) , is considered. Depending upon the choice of the kernel a a in the finite convolution, ( a ∗ ( g ∘ u ) ) ′ ( 1 ) (a*(g∘ u) )’(1) , this formulation includes as a special case the Riemann–Liouville fractional derivative. Topological fixed point theory, carried out via a specialized order cone, is employed to deduce the main result. Due to the presence of the derivative in the nonlocal coefficient, we use a nonstandard index theory applicable to unbounded sets.