Nonlocal differential equations with convolution coefficients and applications to fractional calculus. (English) Zbl 1504.34051
Author’s abstract: The existence of at least one positive solution to a large class of both integer- and fractional-order nonlocal differential equations, of which one model case is
\[
-A((b*u^q)(1))u^{\prime\prime}(t)=\lambda f(t,u(t)),\quad t\in(0,1),\,q\geq 1,
\]
is considered. Due to the coefficient \(A((b*u^q)(1))\) appearing in the differential equation, the equation has a coefficient containing a convolution term. By choosing the kernel \(b\) in various ways, specific nonlocal coefficients can be recovered such as nonlocal coefficients equivalent to a fractional integral of Riemann-Liouville type. The results rely on the use of a nonstandard order cone together with topological fixed point theory. Applications to fractional differential equations are given, including a problem related to the \((n-1,1)\)-conjugate problem.
Reviewer: Gennaro Infante (Arcavata di Rende)
MSC:
| 34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
| 34A09 | Implicit ordinary differential equations, differential-algebraic equations |
| 34B08 | Parameter dependent boundary value problems for ordinary differential equations |
| 26A33 | Fractional derivatives and integrals |
| 47H07 | Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces |
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
Keywords:
finite convolution; nonlocal differential equation; fractional calculus; positive solution; coercivityReferences:
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