Abstract: Abstract Let \(T > 2\) be an integer, \(\mathbb {T}=\{1, 2,\ldots ,T\}\) . We are considered with the discrete nonlinear two-point boundary value problem at resonance: $$\begin{aligned} \begin{aligned}&\mathcal {L} u(j)=\nu _1 u(j)+g(u(j))-e(j),\ \ j\in \mathbb {T}, \\&u(0)=u(T+1)=0,&\quad \quad (P)\\ \end{aligned} \end{aligned}$$ where $$\begin{aligned} \mathcal {L}u(j)=\left\{ \begin{array}{ll} -\triangle ^2 u(j-1)+b(j)\Delta u(j)+a_0(j) u(j), &{}\quad j\in \mathbb {T},\\ \quad 0,&{}\quad j\in \{0, T+1\},\\ \end{array} \right. \end{aligned}$$ \(b, e: \mathbb {T}\rightarrow \mathbb {R}\) , \(a_0:\mathbb {T}\rightarrow [0, \infty )\) , \(\nu _1\) is the principal eigenvalue of \(\mathcal {L}\) . We show that there exists a constant \(d_0 > \nu _1\) , depending only on b and \(a_0\) , such that if $$\begin{aligned} \underset{ \xi \rightarrow \infty }{\lim \sup } \; \frac{g(\xi )}{\xi }<d_0-\nu _1, \end{aligned}$$ and $$\begin{aligned} {\overline{g}}(-\infty )\sum ^T_{j=1} \Theta ^*(j)<\sum ^T_{j=1} \Theta ^*(j)h(j)<{\underline{g}}(\infty )\sum ^T_{j=1} \Theta ^*(j), \end{aligned}$$ then (P) has at least one solution. Here, \(\underline{g}(\infty )=\liminf \nolimits _{\xi \rightarrow \infty } g(\xi )\) , \(\overline{g}(-\infty )=\limsup \nolimits _{\xi \rightarrow -\infty } g(\xi )\) , and \(\Theta ^*\) is an eigenfunction of \(\mathcal {L}^*\) corresponding to the principle eigenvalue \(\nu _1\) . PubDate: 2019-03-14 DOI: 10.1007/s00009-019-1329-7
Abstract: Abstract In this work, we have proved a Hardy–Littlewood–Sobolev inequality for variable exponents. After that, we use this inequality together with the variational method to establish the existence of solution for a class of Choquard equations involving the p(x)-Laplacian operator. PubDate: 2019-03-14 DOI: 10.1007/s00009-019-1316-z
Abstract: Abstract The Waldschmidt constant \({{\,\mathrm{{\widehat{\alpha }}}\,}}(I)\) of a radical ideal I in the coordinate ring of \({\mathbb {P}}^N\) measures (asymptotically) the degree of a hypersurface passing through the set defined by I in \({\mathbb {P}}^N\) . Nagata’s approach to the 14th Hilbert Problem was based on computing such constant for the set of points in \({\mathbb {P}}^2\) . Since then, these constants drew much attention, but still there are no methods to compute them (except for trivial cases). Therefore, the research focuses on looking for accurate bounds for \({{\,\mathrm{{\widehat{\alpha }}}\,}}(I)\) . In the paper, we deal with \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s)\) , the Waldschmidt constant for s very general lines in \({\mathbb {P}}^3\) . We prove that \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s) \ge \lfloor \sqrt{2s-1}\rfloor \) holds for all s, whereas the much stronger bound \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s) \ge \lfloor \sqrt{2.5 s}\rfloor \) holds for all s but \(s=4\) , 7 and 10. We also provide an algorithm which gives even better bounds for \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s)\) , very close to the known upper bounds, which are conjecturally equal to \({{\,\mathrm{{\widehat{\alpha }}}\,}}(s)\) for s large enough. PubDate: 2019-03-11 DOI: 10.1007/s00009-019-1328-8
Abstract: Abstract In this paper, we are mainly concerned with a one-dimensional wave control system. We assert the existence of non-orthogonal fusion frames by extending this problem to a theoretical one introduced by Sz. Nagy (Acta Sci Math Szeged 14, 1951). The key idea of this work is based on the estimate inspired from Sz. Nagy (1951) using the spectral analysis method. More precisely, we prove that if the eigenvalues of the unperturbed operator are isolated and with finite multiplicity, we can construct non-orthogonal fusion frames. PubDate: 2019-03-11 DOI: 10.1007/s00009-019-1318-x
Abstract: Abstract We set up the basic theory of existence and uniqueness of solutions for systems of differential equations with usual derivatives replaced by Stieltjes derivatives. In this occasion, we are concerned about those systems where each component has a different derivator. We also show the interest of these systems of equations with a population model. PubDate: 2019-03-08 DOI: 10.1007/s00009-019-1321-2
Abstract: Abstract In this paper, we propose some new coupled coincidence point and fixed point theorems for the mappings F and g. Our results are obtained by exploring the corresponding initial value problems for the mapping F and weakening the involved contractive conditions. As an application, we study the existence and uniqueness of solution to integro-differential equations. PubDate: 2019-03-08 DOI: 10.1007/s00009-019-1326-x
Abstract: Abstract In this paper, we present new fixed point theorems for multivalued nonexpansive mappings. Since Banach space can have any geometric structure, we consider mappings such that their perturbation by the identity operator is expansive. Then we derive some fixed point results including existence theorems for the sum and product of some classes of nonlinear operators. Three illustrating examples for functional and differential inclusions are supplied. PubDate: 2019-03-06 DOI: 10.1007/s00009-019-1327-9
Abstract: Abstract We study the semigroups \(({\mathcal {C}}_p)_+\) , \(({\mathcal {C}}_p^{\text {dual}})_-\) , \({\mathcal {G}}_-\) and \({\mathcal {H}}_-\) associated with the operator ideals \({\mathcal {C}}_p\) of p-converging operators ( \(1<p<\infty \) ), \({\mathcal {G}}\) of Grothendieck operators, and \({\mathcal {H}}\) of \(c_0\) -cosingular operators. We show that these semigroups admit a perturbative characterization and satisfy the 3-operator property. PubDate: 2019-03-05 DOI: 10.1007/s00009-019-1314-1
Abstract: Abstract Let \({{\mathfrak {F}}}_0\) the class of fully zero-simple semihypergroups. In this paper, we study the main properties of residual semihypergroup \((H_+, \star )\) of a semihypergroup \((H, \circ )\) in \({{\mathfrak {F}}}_0\) . We prove that the quotient semigroup \(H_+/\beta ^*_{H_+}\) is a completely simple and periodic semigroup. Moreover, we find the necessary and sufficient conditions for \((H_+, \star )\) to be a torsion group and, in particular, an abelian 2-group. PubDate: 2019-03-05 DOI: 10.1007/s00009-019-1324-z
Abstract: Abstract In this paper, we study the p(x)-curl systems: $$\begin{aligned} \left\{ \begin{array}{ll} \nabla \times \big ( \nabla \times \mathbf {u} ^{p(x)-2}\nabla \times \mathbf {u}\big )+a(x) \mathbf {u} ^{p(x)-2}\mathbf {u} =\mathbf {f}(x,\mathbf {u}),&{} \mathrm{in}\; \Omega ,\\ \nabla \cdot \mathbf {u}=0, &{}\mathrm{in}\; \Omega ,\\ \nabla \times \mathbf {u} ^{p(x)-2}\nabla \times \mathbf {u} \times \mathbf {n}=0, \mathbf {u}\cdot \mathbf {n}=0,&{} \mathrm{on} \; \partial \Omega ,\\ \end{array} \right. \end{aligned}$$ where \(\Omega \subset \mathbb {R}^{3}\) is a bounded simply connected domain with a \(C^{1,1}\) boundary denoted by \(\partial \Omega \) , \(p:\overline{\Omega }\rightarrow (1,+\infty )\) is a continuous function, \(a\in L^{\infty }(\Omega )\) , and \(\mathbf {f}:\overline{\Omega }\times \mathbb {R}^{3}\rightarrow \mathbb {R}^{3}\) is a Carath \(\mathrm{{\acute{e}}}\) odory function. We use mountain pass theorem and symmetric mountain pass theorem to obtain the existence and multiplicity of solutions for a class of p(x)-curl systems in the absence of Ambrosetti–Rabinowitz condition. PubDate: 2019-03-02 DOI: 10.1007/s00009-019-1312-3
Abstract: Abstract In this paper, mutually sn-permutable subgroups of groups belonging to a class of generalised supersoluble groups are studied. Some analogs of known theorems on mutually sn-permutable products are established. PubDate: 2019-03-02 DOI: 10.1007/s00009-019-1323-0
Abstract: We consider initial value problems for abstract evolution equations with fractional time derivative. Concerning the Caputo derivative \(\mathbb {D}^\alpha u\) , we show that certain assumptions, which are known to be sufficient to get a unique solution with a prescribed regularity, are also necessary. So we establish a maximal regularity result. We consider similar problems with the Riemann–Liouville derivative \(\partial ^\alpha u\) . Here, we give a complete proof (necessity and sufficiency of the assumptions) of the corresponding maximal regularity results. PubDate: 2019-03-01 DOI: 10.1007/s00009-019-1309-y
Abstract: Abstract We study the convergence in variation for the sampling Kantorovich operators in both the cases of averaged-type kernels and classical band-limited kernels. In the first case, a characterization of the space of the absolutely continuous functions in terms of the convergence in variation is obtained. PubDate: 2019-03-01 DOI: 10.1007/s00009-019-1315-0
Abstract: We study the existence and uniqueness of mild and classical solutions for abstract impulsive differential equations with state-dependent time impulses and an example is presented. PubDate: 2019-03-01 DOI: 10.1007/s00009-019-1308-z
Abstract: Abstract We describe a unifying approach for studying the power series of the positive linear operators from a certain class. For the same operators, we give simpler proofs of some known ergodic theorems. PubDate: 2019-03-01 DOI: 10.1007/s00009-019-1313-2
Abstract: Abstract The aim of this study is to investigate the existence and other properties of solution of nonlinear fractional integro–differential equations with constant coefficient. Also with the help of Pachpatte’s inequality, we prove the continuous dependence of the solutions. PubDate: 2019-03-01 DOI: 10.1007/s00009-019-1325-y
Abstract: Abstract Let \(\varphi \) be a complex-valued function in the plane \({\mathbb {C}}.\) The superposition operator is defined by \( S_\varphi (f)=\varphi \circ f\) . In this paper, we characterize the nonlinear superposition operators \(S_\varphi \) acting between the Zygmund-type and Bloch-type spaces in terms of the order and type or the degree of \(\varphi \) . PubDate: 2019-03-01 DOI: 10.1007/s00009-019-1304-3
Abstract: Abstract We present a new integration method for evaluating infinite series involving alternating harmonic numbers. Using this technique, we provide new evaluations for series containing factors of the form \(\left( {\begin{array}{c}2n\\ n\end{array}}\right) ^{2} H_{2n}'\) that cannot be evaluated using known generating functions involving harmonic-type numbers. A closed-form evaluation is given for the series: $$\begin{aligned} \sum _{n = 1}^{\infty } \left( - \frac{1}{16} \right) ^{n} \frac{ \left( {\begin{array}{c}2 n\\ n\end{array}}\right) ^2 H_{2n}' }{ n + 1}, \end{aligned}$$ and we show how the integration method given in our article may be applied to evaluate natural generalizations and variants of the above series, such as the binomial-harmonic series: $$\begin{aligned} \sum _{n = 1}^{\infty } \frac{\left( -\frac{1}{16}\right) ^n \left( {\begin{array}{c}2 n\\ n\end{array}}\right) ^2 H_{2n}' }{2 n - 1} = \frac{(\pi -4 \ln (2)) \Gamma ^2\left( \frac{1}{4}\right) }{8 \sqrt{2} \pi ^{3/2}} - \frac{\sqrt{\frac{\pi }{2}} (\pi +4 \ln (2) - 4 )}{\Gamma ^2\left( \frac{1}{4}\right) } \end{aligned}$$ introduced in our article. PubDate: 2019-02-28 DOI: 10.1007/s00009-019-1311-4
Abstract: Abstract Given a vector function \(\mathbf F =(F_1,\ldots ,F_d),\) analytic on a neighborhood of some compact subset E of the complex plane with simply connected complement, we define a sequence of vector rational functions with common denominator in terms of the expansions of the components \(F_k, k=1,\ldots ,d,\) with respect to the sequence of Faber polynomials associated with E. Such sequences of vector rational functions are analogous to row sequences of type II Hermite–Padé approximation. We give necessary and sufficient conditions for the convergence with geometric rate of the common denominators of the sequence of vector rational functions so constructed. The exact rate of convergence of these denominators is provided and the rate of convergence of the approximants is estimated. It is shown that the common denominators of the approximants detect the poles of the system of functions “closest” to E and their order. PubDate: 2019-02-28 DOI: 10.1007/s00009-019-1307-0
Abstract: For a commutative semi-simple Banach algebra A which is an ideal in its second dual, we give a necessary and sufficient condition for an essential abstract Segal algebra in A to be a BSE-algebra. We show that a large class of abstract Segal algebras in the Fourier algebra A(G) of a locally compact group G are BSE-algebra if and only if they have bounded weak approximate identities. In addition, in the case that G is discrete, we show that \(A_{\mathrm{cb}}(G)\) is a BSE-algebra if and only if G is weakly amenable. We study the BSE-property of some certain Segal algebras implemented by local functions that were recently introduced by J. Inoue and S.-E. Takahasi. Finally, we give a similar construction for the group algebra implemented by a measurable and sub-multiplicative function. PubDate: 2019-02-28 DOI: 10.1007/s00009-019-1305-2
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