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2. On four-point boundary value problem without growth conditions
- Creator:
- Rachůnková, Irena
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- math, four-point boundary value problem, and Carathéodory conditions
- Language:
- English
- Description:
- We prove the existence of solutions of four-point boundary value problems under the assumption that $f$ fulfils various combinations of sign conditions and no growth restrictions are imposed on $f$. In contrast to earlier works all our results are proved for the Carathéodory case.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
3. Periodic singular problem with quasilinear differential operator
- Creator:
- Rachůnková, Irena and Tvrdý, Milan
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- singular periodic boundary value problem, positive solution, φ-Laplacian, pLaplacian, attractive singularity, repulsive singularity, strong singularity, lower function, and upper function
- Language:
- English
- Description:
- We study the singular periodic boundary value problem of the form (φ(u ' ))' + h(u)u ' = g(u) + e(t), u(0) = u(T), u ' (0) = u ' (T), where φ: R→R is an increasing and odd homeomorphism such that φ(R ) = R, h ∈ C[0, ∞), e ∈ L1[0, T] and g ∈ C(0, ∞) can have a space singularity at x = 0, i.e. lim sup x→0+ |g(x)| = ∞ may hold. We prove new existence results both for the case of an attractive singularity, when lim inf x→0+ g(x) = −∞, and for the case of a strong repulsive singularity, when lim x→0+ R 1 x g(ξ)dξ = ∞. In the latter case we assume that φ(y) = φp(y) = |y| p−2 y, p > 1, is the well-known p-Laplacian. Our results extend and complete those obtained recently by Jebelean and Mawhin and by Liu Bing.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
4. Resonance and multiplicity in periodic boundary value problems with singularity
- Creator:
- Rachůnková, Irena, Tvrdý, Milan, and Vrkoč, Ivo
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- second order nonlinear ordinary differential equation, periodic problem, and lower and upper functions
- Language:
- English
- Description:
- The paper deals with the boundary value problem u '' + k u = g(u) + e(t), u(0) = u(2π), u ' (0) = u ' (2π), where k ∈ R, g : (0, ∞) → R is continuous, e ∈ L[0, 2π] and lim x→0+ ∫1 x g(s) ds = ∞. In particular, the existence and multiplicity results are obtained by using the method of lower and upper functions which are constructed as solutions of related auxiliary linear problems.
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
5. Singular Dirichlet problem for ordinary differential equations with φ-Laplacian
- Creator:
- Polášek, Vladimír and Rachůnková, Irena
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- singular Dirichlet problem, φ-Laplacian, existence of smooth solution, and lower and upper function
- Language:
- English
- Description:
- We provide sufficient conditions for solvability of a singular Dirichlet boundary value problem with φ-Laplacian (φ(u ' ))' = f(t, u, u ' ), u(0) = A, u(T) = B, where φ is an increasing homeomorphism, φ(R) = R, φ(0) = 0, f satisfies the Carathéodory conditions on each set [a, b] × R 2 with [a, b] ⊂ (0, T) and f is not integrable on [0, T] for some fixed values of its phase variables. We prove the existence of a solution which has continuous first derivative on [0, T].
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public
6. Strong singularities in mixed boundary value problems
- Creator:
- Rachůnková, Irena
- Format:
- bez média and svazek
- Type:
- model:article and TEXT
- Subject:
- singular mixed boundary value problem, positive solution, lower function, upper function, and convergence of approximate regular problems
- Language:
- English
- Description:
- We study singular boundary value problems with mixed boundary conditions of the form (p(t)u ' ) ' + p(t)f(t, u, p(t)u ' ) = 0, lim t→0+ p(t)u ' (t) = 0, u(T) = 0, where [0, T] ⊂ . We assume that D ⊂ R 2 , f satisfies the Carathéodory conditions on (0, T) × D, p ∈ C[0, T] and 1/p need not be integrable on [0, T]. Here f can have time singularities at t = 0 and/or t = T and a space singularity at x = 0. Moreover, f can change its sign. Provided f is nonnegative it can have even a space singularity at y = 0. We present conditions for the existence of solutions positive on [0, T).
- Rights:
- http://creativecommons.org/publicdomain/mark/1.0/ and policy:public