Abstract
The Dirichlet problem and the problem with functional boundary condition for ϕ-Laplacian on the semi-infinite interval are studied as well as solutions between the lower and upper functions.
Highlights
The Dirichlet problem on the bounded interval for (φ(t, x, x )) = f (t, x, x ), t ∈ I = [a, b], x(a) = A, x(b) = B, is well studied [1,2,5,10]
Under additional conditions of the Nagumo or Schrader type one can prove the existence of a solution x of the Dirichlet problem satisfying the estimates α ≤ x ≤ β
In [11] it is proved that there exists a generalized solution of the Dirichlet problem provided that there only exist lower and upper functions
Summary
Let us assume that for all (t, x, x ) ∈ J × R × R and for each compact interval J the function φJ : J × R2 → R satisfies the conditions: φJ ∈ C(J × R2, R) and for fixed t and x is strictly increasing on x . The class BB−(I, R) consists of functions β : I → R satisfying the following conditions: for each t ∈ (a, b] there exists a left derivative βl(t) and a limit lim τ →t−
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