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The paper is about a sub-supersolution method for the prescribed mean curvature problem. We formulate the problem as a variational inequality and propose appropriate concepts of sub- and supersolutions for such inequality. Existence and enclosure results for solutions and extremal solutions between sub- and supersolutions are established.

On a variational property of integral functionals and related conjectures

Biagio Ricceri (1996)

Banach Center Publications

The aim of this paper is to give the proofs of those results that in [4] were only announced, and, at the same time, to propose some possible developments, indicating some of the most significant open problems.

On automatic boundedness of Nemytskiĭ set-valued operators

S. Rolewicz, Wen Song (1995)

Studia Mathematica

Let X, Y be two separable F-spaces. Let (Ω,Σ,μ) be a measure space with μ complete, non-atomic and σ-finite. Let $N_{F}$ be the Nemytskiĭ set-valued operator induced by a sup-measurable set-valued function $F : Ω \times X \to 2^{Y}$ . It is shown that if $N_{F}$ maps a modular space $(N (L (Ω, Σ, μ; X)), ϱ_{N, μ})$ into subsets of a modular space $(M (L (Ω, Σ, μ; Y)), ϱ_{M, μ})$ , then $N_{F}$ is automatically modular bounded, i.e. for each set K ⊂ N(L(Ω,Σ,μ;X)) such that $r_{K} = s u p ϱ_{N, μ} (x) : x \in K < \infty$ we have $s u p ϱ_{M, μ} (y) : y \in N_{F} (K) < \infty$ .

On characterization of the Lipschitzian composition operator between spaces of functions of bounded $p$ -variation

Nelson Merentes, Sergio Rivas (1995)

Czechoslovak Mathematical Journal

On continuous composition operators

Wilhelmina Smajdor (2010)

Annales Polonici Mathematici

Let I ⊂ ℝ be an interval, Y be a normed linear space and Z be a Banach space. We investigate the Banach space Lip₂(I,Z) of all functions ψ: I → Z such that $M_{ψ} : = s u p | | [r, s, t; ψ] | | : r < s < t, r, s, t \in I < \infty$ , where [r,s,t;ψ]:= ((s-r)ψ(t)+(t-s)ψ(r)-(t-r)ψ(s))/((t-r)(t-s)(s-r)). We show that ψ ∈ Lip₂(I,Z) if and only if ψ is differentiable and its derivative ψ’ is Lipschitzian. Suppose the composition operator N generated by h: I × Y → Z, (Nφ)(t):= h(t,φ(t)), maps the set (I,Y) of all affine functions φ: I → Y into Lip₂(I,Z). We prove that if N is...

On Hammerstein equations with natural growth conditions.

Moroz, V., Zabreiko, P. (1999)

Zeitschrift für Analysis und ihre Anwendungen

On quadratic integral equations of Urysohn type in Fréchet spaces.

Benchohra, M., Darwish, M.A. (2010)

Acta Mathematica Universitatis Comenianae. New Series

On some nonlinear potential problems.

Efendiev, M.A., Schmitz, Hermann, Wendland, Wolfgang L. (1999)

Electronic Journal of Differential Equations (EJDE) [electronic only]

On spaces $L^{p (x)}$ and $W^{k, p (x)}$

Ondrej Kováčik, Jiří Rákosník (1991)

Czechoslovak Mathematical Journal

On the autonomous Nemytskij operator in Hölder spaces.

Goebel, M., Sachweh, F. (1999)

Zeitschrift für Analysis und ihre Anwendungen

On the compactness and condensing of nonlinear superposition operators

F. Dedagić, N. Okičić (2005)

Kragujevac Journal of Mathematics

On the Conley index in Hilbert spaces in the absence of uniqueness

Marek Izydorek, Krzysztof P. Rybakowski (2002)

Fundamenta Mathematicae

Consider the ordinary differential equation (1) ẋ = Lx + K(x) on an infinite-dimensional Hilbert space E, where L is a bounded linear operator on E which is assumed to be strongly indefinite and K: E → E is a completely continuous but not necessarily locally Lipschitzian map. Given any isolating neighborhood N relative to equation (1) we define a Conley-type index of N. This index is based on Galerkin approximation of equation (1) by finite-dimensional ODEs and extends...

On the existence of a fuzzy integral equation of Urysohn-Volterra type

Mohamed Abdalla Darwish (2008)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

We present an existence theorem for integral equations of Urysohn-Volterra type involving fuzzy set valued mappings. A fixed point theorem due to Schauder is the main tool in our analysis.

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On a certain Volterra-Fredholm type integral equation.

On a characterization of the Preisach model for hysteresis

On a class of parabolic integro-differential equations.

On a nonlinear compactness lemma in $L^{p} (0, T; B)$ .

On a sub-supersolution method for the prescribed mean curvature problem