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Journal: Annales Mathematicae Silesianae

On the existence of two solutions of functional boundary value problems

Svatoslav Staněk

Language: EN | Published: 23-07-2022 | Abstract | pp. 93-109

The functional differential equation (g(x'(t)))' = (Fx)(t) is considered. Here g is an increasing homomorphism on ℝ, g(0) = 0 and F: C¹(J)→L₁(J) is a continuous operator satisfying a growth condition with respect to x. A class of nonlinear functional boundary conditions is considered and sufficient conditions for the existence at least one positive and one negative solutions of the boundary value problems are given. Results are proved by the homotopy theory, the Leray-Schauder degree and the Borsuk theorem.

2022-07-23



Journal: Annales Mathematicae Silesianae

Boundary value problems for one-parameter second-order differential equations

Svatoslav Staněk

Language: EN | Published: 30-09-1993 | Abstract | pp. 89-98

The paper establishes sufficient conditions for the existence of solutions of a one-parameter differential equation x" = f(t,x,x',λ) satisfying some of the following boundary conditions:
γ(x) = 0,  x'(a) = x'(b) = 0,
x'(a) = x'(b) = 0,  x(c)-x(d) = 0
and
x'(a) = x'(t₀) = x'(b) = 0.
Here γ is a functional. The application is given for a class of one-parameter functional boundary value problems.

1993-09-30



Journal: Annales Mathematicae Silesianae

Leray-Schauder degree method in one-parameter functional boundary value problem

Svatoslav Staněk

Language: EN | Published: 30-09-1996 | Abstract | pp. 111-125

Sufficient conditions for the existence of solutions of one-parameter functional boundary value problems of the type
x" = f(t,x,x_t,x',x'_t,λ),
(x₀,x'₀) ∈ {(ϕ,χ+c); c∈R}, α(x|_J) = A, β(x(T)-x|_J) = B
are given. Here f: J×R×C_r×R×C_r×R→R is continuous, ϕ,χ∈C_r, α,β are continuous increasing functionals, A,B∈R and x|_J is the restriction of x to J=[0,T]. Results are proved by the Leray-Schauder degree method.

1996-09-30