Abstract
We will consider the nonhomogeneous ϕ-Laplacian differential equation { ( ϕ ( u ′ ( t ) ) ) ′ = − h ( t ) f ( u ( t ) ) , t ∈ ( 0 , T ) , u ( 0 ) = ∑ i = 1 k α i u ( η i ) , ϕ ( u ′ ( T ) ) = β , where ϕ:R→(−b,b) (0<b≤+∞) is an increasing homeomorphism such that ϕ(0)=0, h:[0,T]→ R + and f: R + → R + are continuous, β≥0 and η i ∈(0,T) and α i ∈R, i=1,2,…,k. Based on the Krasnosel’skii fixed point theorem, the existence of a positive solution is obtained, even if some of the α i coefficients are negative. Two examples are also given to illustrate our main results.
Highlights
1 Introduction We are concerned with the φ-Laplacian differential equation with the nonhomogeneous Dirichlet-Neumann boundary conditions
The existence of positive solutions for homogeneous and nonhomogeneous boundary value problems have been studied by several authors and many interesting results have been obtained
If the coefficient takes a negative value, it is sometimes difficult to find an appropriate cone to guarantee the existence of a positive solution of the corresponding differential equation
Summary
Abstract We will consider the nonhomogeneous φ-Laplacian differential equation (φ(u (t))) = –h(t)f (u(t)), t ∈ (0, T), u(0) = Based on the Krasnosel’skii fixed point theorem, the existence of a positive solution is obtained, even if some of the αi coefficients are negative. 1 Introduction We are concerned with the φ-Laplacian differential equation with the nonhomogeneous Dirichlet-Neumann boundary conditions Boundary value problems, including the φ-Laplacian operator, have received a lot of attention with respect to the existence and multiplicity of solutions.
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