Cohomological tautness of singular Riemannian foliations The original version of this article was revised due to a retrospective Open Access order.

RETRACTED ARTICLE: Existence of solutions of nonlocal initial value problems for differential equations with Hilfer–Katugampola fractional derivative

Boundary value problems for second order linear difference equations: application to the computation of the inverse of generalized Jacobi matrices Abstract We have named generalized Jacobi matrices to those that are practically tridiagonal, except for the two final entries and the two first entries of its first and its last row respectively. This class of matrices encompasses both standard Jacobi and periodic Jacobi matrices that appear in many contexts in pure and applied mathematics. Therefore, the study of the inverse of these matrices becomes of specific interest. However, explicit formulas for inverses are known only in a few cases, in particular when the coefficients of the diagonal entries are subjected to some restrictions. We will show that the inverse of generalized Jacobi matrices can be raised in terms of the resolution of a boundary value problem associated with a second order linear difference equation. In fact, recent advances in the study of linear difference equations, allow us to compute the solution of this kind of boundary value problems. So, the conditions that ensure the uniqueness of the solution of the boundary value problem leads to the invertibility conditions for the matrix, whereas that solutions for suitable problems provide explicitly the entries of the inverse matrix.

On the solutions of Caputo–Hadamard Pettis-type fractional differential equations Abstract Let E be a Banach space with the topological dual \(E^*\) . The aim of this paper is two-fold. On the one hand, we prove some basic properties of Hadamard-type fractional integral operators. These results are related to earlier results about integral operators acting on different function spaces, but for the vector-valued case they are of independent interest. Note that we discuss it in a rather general setting. We study Hadamard–Pettis integral operators in both single and multivalued case. On the other hand, we apply these results to obtain the existence of solutions of the fractional-type problem $$\begin{aligned} \frac{d^\alpha x(t)}{d t^\alpha }= \lambda f(t,x(t)), \quad \alpha \in (0,1),~~t \in [1,e], \quad x(1)+ b x(e)=h \end{aligned}$$ with certain constants \(\lambda , b\) , where \(h \in E\) and \(f: [1,e]\times E \rightarrow E\) is Pettis integrable function such that, for every \(\varphi \in E^*\) , \(\varphi f\) lies in an appropriate Orlicz spaces. Here \(\frac{d^\alpha }{d t^\alpha }\) stands the Caputo–Hadamard fractional differential operator.

Reckoning solution of split common fixed point problems by using inertial self-adaptive algorithms Abstract In this paper, we construct a novel algorithm for the split common fixed point problem for two demicontractive operators in Hilbert spaces. By using inertial self-adaptive algorithms, we obtain strong convergence results for finding a solution of the split common fixed point problems. Applications to solving the split minimization problem and the split feasibility problem are included. Our results extend and generalize many previously known results in this research area. Moreover, numerical experiments are supplied to demonstrate the convergence behavior and efficiency of the proposed algorithm.

A remark on the topology of complex polynomial functions Abstract We begin with a survey of the notion of atypical values of R. Thom. We give a theorem on the atypical values of complex polynomials which generalizes a theorem of Broughton (On the topology of polynomial hypersurfaces in singularities, Part 1, 167–178, proceedings of symposia in pure mathematics, vol 40. American Mathematicaol Society, Providence, 1983) and Broughton (Milnor numbers and the topology of polynomial hypersurfaces. Invent Math 92:217–241, 1988). For this purpose we introduce the notion of atypical value from infinity. Our proofs are geometrical. We end with some open problems to understand the difference between tame polynomials at infinity and polynomials without atypical values from infinity.

Completability and optimal factorization norms in tensor products of Banach function spaces Abstract Given \(\sigma \) -finite measure spaces \((\Omega _1,\Sigma _1, \mu _1)\) and \((\Omega _2,\Sigma _2,\mu _2)\) , we consider Banach spaces \(X_1(\mu _1)\) and \(X_2(\mu _2)\) , consisting of \(L^0 (\mu _1)\) and \(L^0 (\mu _2)\) measurable functions respectively, and study when the completion of the simple tensors in the projective tensor product \(X_1(\mu _1) \otimes _\pi X_2(\mu _2)\) is continuously included in the metric space of measurable functions \(L^0(\mu _1 \otimes \mu _2)\) . In particular, we prove that the elements of the completion of the projective tensor product of \(L^p\) -spaces are measurable functions with respect to the product measure. Assuming certain conditions, we finally show that given a bounded linear operator \(T:X_1(\mu _1) \otimes _\pi X_2(\mu _2) \rightarrow E\) (where E is a Banach space), a norm can be found for T to be bounded, which is ‘minimal’ with respect to a given property (2-rectangularity). The same technique may work for the case of n-spaces.

Bivariate Baskakov type operators Abstract The goal of this paper is to propose a modern and comprehensive exposition of the main aspects of functions for bivariate Baskakov type operators. Primarily, we prove that operators preserve Lipschitzs constant of a Lipschitz continuous function. Then, we demonstrate that given operators can maintain some properties of the function f. Ultimately, we deal with the monotony of the bivariate Baskakov type operators for which the approximating function is convex. That is to say, we discuss that investigated operators are monotonically nonincreasing for n while f is \(\varsigma \) -convex.

Existence and multiplicity of solutions for Kirchhoff type equations involving fractional p -Laplacian without compact condition Abstract The purpose of this paper is mainly to investigate the following fractional Kirchhoff equation in \({{\mathbb {R}}}^N\) $$\begin{aligned} \left( a+b\iint _{{{\mathbb {R}}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\mathrm{d}x\mathrm{d}y\right) ^{p-1} (-\Delta )^s_p u+\lambda V(x)|u|^{p-2}u=f(x,u), \end{aligned}$$ where \(0<s<1\) , \(2\le p<\infty \) , \(a,b>0\) are constants, \(\lambda \) is a parameter, V is sign-changing potential function satisfying some assumptions which may not guarantee the compactness of the corresponding Sobolev embedding. Under some suitable conditions, we prove the existence and multiplicity of nontrivial solutions by applying some new tricks for the above equation.

Tridiagonal M-matrices whose inverse is tridiagonal and related pentadiagonal matrices Abstract A necessary and sufficient condition in order to guarantee that the inverse of a tridiagonal M-matrix is tridiagonal is provided. Pentadiagonal M-matrices whose inverse is pentadiagonal are also analyzed.