Nonlocal differential equations with \({(p,q)}\) growth. (English) Zbl 1531.45012
Summary: The author considers a class of nonlocal convolution-type ordinary differential equations of the form
\[
-A{({(a*(g\,\circ\, u))}(1))}u^{\prime \prime}(t)=\lambda f{(t,u(t))}, \quad 0 < t < 1,
\]
where \(g\) is a continuous function satisfying \((p,q)\) growth – that is, there exist constants \(c_1\), \(c_2\in (0,\infty)\) and \(c_3\in [0,\infty)\) such that
\[
c_1u^p\leqslant g(u)\leqslant c_2{(c_3+u^q)}, \quad u\geqslant 0,
\]
where \(1\leqslant p\leqslant q {<} +\infty\). Existence of at least one positive solution to this equation when subjected to given boundary data is studied by means of topological fixed point theory.
{© 2023 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society.}
{© 2023 The Authors. Bulletin of the London Mathematical Society is copyright © London Mathematical Society.}
MSC:
| 45J05 | Integro-ordinary differential equations |
| 45P05 | Integral operators |
| 42A85 | Convolution, factorization for one variable harmonic analysis |
| 44A35 | Convolution as an integral transform |
| 26A51 | Convexity of real functions in one variable, generalizations |
| 47H30 | Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.) |
| 47G10 | Integral operators |
| 47N20 | Applications of operator theory to differential and integral equations |
| 47H10 | Fixed-point theorems |
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