Abstract
In this paper, we consider the existence of solutions to the p(r)-Laplacian equation with multi-point boundary conditions. Under some new criteria and by utilizing degree methods and also the Leray–Schauder fixed point theorem, the new existence results of the solutions have been established. Some results in the literature can be generalized and improved. And as an application, two examples are provided to demonstrate the effectiveness of our theoretical results.
Highlights
In recent years, there has been extensive interest in boundary value problems (BVPs) with variable exponent in a Banach space, see [1–9]
This paper focuses on the following p(r)-Laplacian differential equations with multipoint boundary conditions:
5 Conclusions In this paper, we are concerned with a class of differential equations involving a p(r)Laplacian operator
Summary
There has been extensive interest in boundary value problems (BVPs) with variable exponent in a Banach space, see [1–9]. Aiming to obtain the existence of solutions to problem (1.1), we need the following lemmas. From the continuity of f , φ–1 and the definition of a, it is easy to see that u is a solution of problem (1.1) if and only if u is a fixed point of the integral operator T when λ = 1. Applying Lemma 2.2, we can obtain that T(u, 1) has a fixed point in U, that is to say, problem (1.1) has at least one solution. Example 4.1 Consider the following p(r)-Laplacian differential equations with six-point boundary conditions:
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