Abstract
By using the Leggett-Williams norm-type theorem due to O’Regan and Zima and constructing suitable Banach spaces and operators, we investigate the existence of positive solutions for fractional p-Laplacian boundary value problems at resonance. An example is given to illustrate the main results.
Highlights
Boundary value problems at resonance have attracted more and more attention
To the best of our knowledge, there is no paper to show the existence of a positive solution for boundary value problems with a nonlinear derivative operator at resonance by using Leggett-Williams norm-type theorems
Motivated by the excellent results mentioned above, we will discuss the existence of positive solutions for the p-Laplacian boundary value problem
Summary
Boundary value problems at resonance have attracted more and more attention. Many authors studied the existence of solutions for these problems by using Mawhin’s continuous theorem [ ] and its extension obtained by Ge and Ren [ ]; see [ – ] and the references cited therein. By using Leggett-Williams norm-type theorems due to O’Regan and Zima [ ], the existence of positive solutions for the boundary value problems at resonance with a linear derivative operator has been investigated (see [ – ]). To the best of our knowledge, there is no paper to show the existence of a positive solution for boundary value problems with a nonlinear derivative operator (for instance, p-Laplacian operator) at resonance by using Leggett-Williams norm-type theorems. Motivated by the excellent results mentioned above, we will discuss the existence of positive solutions for the p-Laplacian boundary value problem. Since CDβ + [φp(CDα + ·)] is a nonlinear operator, we cannot solve the problem U(t) is a solution of the following problem:. Proof Assume that u(t) is a solution of the problem
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