ArticleOriginal scientific text
Title
On a class of functional boundary value problems for the equation x'' = f(t,x,x',x'',λ)
Authors 1
Affiliations
- Department of Mathematical Analysis, Faculty of Science, Palacký University, Tomkova 38, 779 06 Olomouc, Czech Republic
Abstract
The Leray-Schauder degree theory is used to obtain sufficient conditions for the existence and uniqueness of solutions for the boundary value problem x'' = f(t,x,x',x'',λ), α(x) = 0, β(x̅) = 0, γ(x̿)=0, depending on the parameter λ. Here α, β, γ are linear bounded functionals defined on the Banach space of C⁰-functions on [0,1] and x̅(t) = x(0) - x(t), x̿(t)=x(1)-x(t).
Keywords
Leray-Schauder degree theory, functional boundary conditions, boundary value problem depending on the parameter
Bibliography
- C. Fabry and P. Habets, The Picard boundary value problem for second order vector differential equations, J. Differential Equations 42 (1981), 186-198.
- P. Hartman, Ordinary Differential Equations, Wiley-Interscience, New York, 1964.
- W. V. Petryshyn, Solvability of various boundary value problems for the equation x'' = F(t,x,x',x'') - y, Pacific J. Math. 122 (1986), 169-195.
- S. Staněk, Three-point boundary value problem for nonlinear third-order differential equations with parameter, Acta Univ. Palack. Olomuc. Fac. Rerum Natur., 100, Math. 30 (1991), 61-74.
- S. Staněk, Multi-point boundary value problem for a class of functional differential equations with parameter, Math. Slovaca 42 (1992), 85-96.
- S. Staněk, Three-point boundary value problem for nonlinear second-order differential equation with parameter, Czechoslovak Math. J. 42 (117) (1992), 241-256.
- S. Staněk, On a class of five-point boundary value problems in second-order functional differential equations with parameter, Acta Math. Hungar. 62 (1993), 253-262.
- S. Staněk, On a class of functional boundary value problems for second-order functional differential equations with parameter, Czechoslovak Math. J. 43 (118) (1993), 339-348.
- S. Staněk, Leray-Schauder degree method in functional boundary problems depending on the parameter, Math. Nachr. 164 (1993), 333-344.
- S. Staněk, On certain three-point regular boundary value problems for nonlinear second-order differential equations depending on the parameter, 1992, submitted for publication.
- A. Tineo, Existence of solutions for a class of boundary value problems for the equation x'' = F(t,x,x',x''), Comment. Math. Univ. Carolin. 29 (1988), 285-291.
Pages:
225-237
Main language of publication
English
Received
1993-04-15
Accepted
1993-10-28
Published
1994
Exact and natural sciences