### Continuous solutions of the functional equation $\phi \left(f\right(x\left)\right)=G(x,\phi (x\left)\right)$ for vector-valued functions φ

Z. Krzeszowiak (1969)

Annales Polonici Mathematici

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Z. Krzeszowiak (1969)

Annales Polonici Mathematici

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Teresa Janiak, Elżbieta Łuczak-Kumorek (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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The basic idea of this paper is to give the existence theorem and the method of averaging for the system of functional-differential inclusions of the form ⎧$\u1e8b\left(t\right)\in F(t,{x}_{t},{y}_{t})$ (0) ⎨ ⎩$\u1e8b\left(t\right)\in G(t,{x}_{t},{y}_{t})$ (1)

James C. Lillo (1967)

Annales Polonici Mathematici

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László Simon (2015)

Mathematica Bohemica

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We consider second order quasilinear evolution equations where also the main part contains functional dependence on the unknown function. First, existence of solutions in $(0,T)$ is proved and examples satisfying the assumptions of the existence theorem are formulated. Then a uniqueness theorem is proved. Finally, existence and some qualitative properties of the solutions in $(0,\infty )$ (boundedness and stabilization as $t\to \infty $) are shown.

Min Zhang, Jianguo Si (2014)

Annales Polonici Mathematici

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This work deals with Feigenbaum’s functional equation ⎧ $h\left(g\left(x\right)\right)={g}^{p}\left(h\left(x\right)\right)$, ⎨ ⎩ g(0) = 1, -1 ≤ g(x) ≤ 1, x∈[-1,1] where p ≥ 2 is an integer, ${g}^{p}$ is the p-fold iteration of g, and h is a strictly monotone odd continuous function on [-1,1] with h(0) = 0 and |h(x)| < |x| (x ∈ [-1,1], x ≠ 0). Using a constructive method, we discuss the existence of continuous unimodal even solutions of the above equation.

H. Swiatak (1968)

Matematički Vesnik

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H. G. Dales, M. Daws, H. L. Pham, P. Ramsden

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The theory of multi-norms was developed by H. G. Dales and M. E. Polyakov in a memoir that was published in Dissertationes Mathematicae. In that memoir, the notion of ’equivalence’ of multi-norms was defined. In the present memoir, we make a systematic study of when various pairs of multi-norms are mutually equivalent. In particular, we study when (p,q)-multi-norms defined on spaces ${L}^{r}\left(\Omega \right)$ are equivalent, resolving most cases; we have stronger results in the case where r = 2. We also show...

Giovanni Dore, Alberto Venni (2005)

Studia Mathematica

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We give a concise exposition of the basic theory of ${H}^{\infty}$ functional calculus for N-tuples of sectorial or bisectorial operators, with respect to operator-valued functions; moreover we restate and prove in our setting a result of N. Kalton and L. Weis about the boundedness of the operator $f(T\u2081,...,{T}_{N})$ when f is an R-bounded operator-valued holomorphic function.

Z. Kominek (1974)

Annales Polonici Mathematici

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C. T. Ng (1973)

Annales Polonici Mathematici

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M. Malenica (1982)

Matematički Vesnik

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H. Światak (1967)

Annales Polonici Mathematici

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Maciej Sablik (1998)

Annales Polonici Mathematici

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We deal with the linear functional equation (E) $g\left(x\right)={\sum}_{i=1}^{r}{p}_{i}g\left({c}_{i}x\right)$, where g:(0,∞) → (0,∞) is unknown, $(p\u2081,...,{p}_{r})$ is a probability distribution, and ${c}_{i}$’s are positive numbers. The equation (or some equivalent forms) was considered earlier under different assumptions (cf. [1], [2], [4], [5] and [6]). Using Bernoulli’s Law of Large Numbers we prove that g has to be constant provided it has a limit at one end of the domain and is bounded at the other end.

Dorota Krassowska, Janusz Matkowski (2005)

Annales Polonici Mathematici

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It is shown that, under some general algebraic conditions on fixed real numbers a,b,α,β, every solution f:ℝ → ℝ of the system of functional inequalities f(x+a) ≤ f(x)+α, f(x+b) ≤ f(x)+β that is continuous at some point must be a linear function (up to an additive constant). Analogous results for three other similar simultaneous systems are presented. An application to a characterization of ${L}^{p}$-norm is given.

J. Eweret (1989)

Matematički Vesnik

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Malgorzata Wójcicka (1986)

Colloquium Mathematicae

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Hôǹg Thái Nguyêñ, Maciej Juniewicz, Jolanta Ziemińska (2001)

Studia Mathematica

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We present a new continuous selection theorem, which unifies in some sense two well known selection theorems; namely we prove that if F is an H-upper semicontinuous multivalued map on a separable metric space X, G is a lower semicontinuous multivalued map on X, both F and G take nonconvex ${L}_{p}(T,E)$-decomposable closed values, the measure space T with a σ-finite measure μ is nonatomic, 1 ≤ p < ∞, ${L}_{p}(T,E)$ is the Bochner-Lebesgue space of functions defined on T with values in a Banach space E, F(x)...

Kaori Yamazaki (2010)

Studia Mathematica

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Let c be the Banach space consisting of all convergent sequences of reals with the sup-norm, ${C}_{\infty}(A,c)$ the set of all bounded continuous functions f: A → c, and ${C}_{A}(X,c)$ the set of all functions f: X → c which are continuous at each point of A ⊂ X. We show that a Tikhonov subspace A of a topological space X is strong Choquet in X if there exists a monotone extender $u:{C}_{\infty}(A,c)\to {C}_{A}(X,c)$. This shows that the monotone extension property for bounded c-valued functions can fail in GO-spaces, which provides a negative answer...

Masami Sakai (2009)

Colloquium Mathematicae

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In connection with a conjecture of Scheepers, Bukovský introduced properties wQN* and SSP* and asked whether wQN* implies SSP*. We prove it in this paper. We also give characterizations of properties S₁(Γ,Ω) and ${S}_{fin}(\Gamma ,\Omega )$ in terms of upper semicontinuous functions

D. Przeworska-Rolewicz, S. Rolewicz (1967)

Annales Polonici Mathematici

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Yongkun Li, Changzhao Li, Juan Zhang (2010)

Annales Polonici Mathematici

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By using the well-known Leggett–Williams multiple fixed point theorem for cones, some new criteria are established for the existence of three positive periodic solutions for a class of n-dimensional functional differential equations with impulses of the form ⎧y’(t) = A(t)y(t) + g(t,yt), $t\ne {t}_{j}$, j ∈ ℤ, ⎨ ⎩$y\left(t{\u207a}_{j}\right)=y\left(t{\xaf}_{j}\right)+{I}_{j}\left(y\left({t}_{j}\right)\right)$, where $A\left(t\right)={\left({a}_{ij}\left(t\right)\right)}_{n\times n}$ is a nonsingular matrix with continuous real-valued entries.

S. Rolewicz (2006)

Studia Mathematica

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Let X be an arbitrary set, and γ: X × X → ℝ any function. Let Φ be a family of real-valued functions defined on X. Let $\Gamma :X\to {2}^{\Phi}$ be a cyclic ${\Phi}^{\gamma (\xb7,\xb7)}$-monotone multifunction with non-empty values. It is shown that the following generalization of the Rockafellar theorem holds. There is a function f: X → ℝ such that Γ is contained in the ${\Phi}^{\gamma (\xb7,\xb7)}$-subdifferential of f, $\Gamma \left(x\right)\subset {\partial}_{\Phi}^{\gamma (\xb7,\xb7)}{f|}_{x}$.