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In [6], a sufficient condition for a space to be δ-stratifiable was presented and it was asked whether the condition is necessary. Jin et al [4] gave a negative answer to the question by showing that a space with the condition is zero-dimensional. In this paper, we show that a space with the condition is precisely an almost discrete space. Moreover, we introduce the notions of strongly lower (upper) semi-continuous functions, with which the characterizations of δ-stratifiable spaces are presented.

This paper focuses on existence and uniqueness of common fixed points for a pair of self‐mapping satisfying generalized rational type ( ϑ , ψ , φ )‐weak contractive condition in which one of the mapping is α ‐admissible with respect to the other and weakly compatible mappings in the framework of b ‐metric spaces. The results presented herein generalize and improve some well‐known results in the existing literature. Furthermore, we draw some corollaries from our results and provide an example for illustrating the validity of our findings. As an application of our result, we discuss the existence of a solution to a fractional order differential equation.

According to a 2002 theorem by Cardaliaguet and Tahraoui, an isotropic, compact and connected subset of the group $\\textrm {GL}^{\\!+}(2)$ of invertible $2\\times 2$ - - matrices is rank-one convex if and only if it is polyconvex. In a 2005 Journal of Convex Analysis article by Alexander Mielke, it has been conjectured that the equivalence of rank-one convexity and polyconvexity holds for isotropic functions on $\\textrm {GL}^{\\!+}(2)$ as well, provided their sublevel sets satisfy the corresponding requirements. We negatively answer this conjecture by giving an explicit example of a function $W\\colon \\textrm {GL}^{\\!+}(2)\\to \\mathbb {R}$ which is not polyconvex, but rank-one convex as well as isotropic with compact and connected sublevel sets.

In this paper, two kinds of parametric generalized vector equilibrium problems in normed spaces are studied. The sufficient conditions for the continuity of the solution mappings to the two kinds of parametric generalized vector equilibrium problems are established under suitable conditions. The results presented in this paper extend and improve some main results in Chen and Gong (Pac J Optim 3:511–520, 2010 ), Chen and Li (Pac J Optim 6:141–152, 2010 ), Chen et al. (J Glob Optim 45:309–318, 2009 ), Cheng and Zhu (J Glob Optim 32:543–550, 2005 ), Gong (J Optim Theory Appl 139:35–46, 2008 ), Li and Fang (J Optim Theory Appl 147:507–515, 2010 ), Li et al. (Bull Aust Math Soc 81:85–95, 2010 ) and Peng et al. (J Optim Theory Appl 152(1):256–264, 2011 ).

This note establishes an extension of the acute angle lemma (also known as the Hairy Ball Theorem or the Hedgehog Theorem in the case of single-valued mappings) to multifunctions with noncompact image sets. The main result establishes the existence of solutions for operator inclusions involving upper semi-continuous multifunctions with convex values. By relaxing the coercivity assumptions typically required in such analyses, we extend the applicability to scenarios where standard dissipation conditions do not hold. The introduced framework leverages the concept of K-inf-compact support to ensure the existence of zeroes for multifunctions under less restrictive conditions. Applications to hemivariational inequalities and related variational problems are discussed. The examples and counterexamples that demonstrate the obtained generalizations are provided.

In this work, we reformulate and investigate the well-known pantograph differential equation by applying newly-defined conformable operators in both Caputo and Riemann–Liouville settings simultaneously for the first time. In fact, we derive the required existence criteria of solutions corresponding to the inclusion version of the three-point Caputo conformable pantograph BVP subject to Riemann–Liouville conformable integral conditions. To achieve this aim, we establish our main results in some cases including the lower semi-continuous, the upper semi-continuous and the Lipschitz set-valued maps. Eventually, the last part of the present research is devoted to proposing two numerical simulative examples to confirm the consistency of our findings.

This paper is devoted to the asymptotic behavior of solutions to a class of non-autonomous random dispersive-dissipative wave equations driven by nonlinear colored noise defined on unbounded domains. We first prove the existence of pullback random attractors of the random wave equation driven by nonlinear colored noise, and then establish the upper semi-continuity of the random attractors to a class of random wave equations driven by linear operator-type colored noise as the correlation time tends to zero. The operator-type noise is unbounded which is multiplied by a Laplace operator. Both methods of spectral decomposition and uniform tail-estimates are combined to achieve the pullback asymptotic compactness of the solutions in order to overcome the difficulty arising from the lack of compact Sobolev embeddings on unbounded domains.

We prove some fixed point theorems for asymptotically regular self-mappings, not necessarily orbitally continuous or k -continuous, satisfying weaker Proinov contraction. Our results extend several recent results in the literature and provide more answers to an open question raised by Rhoades. Some nontrivial examples are also given to illustrate our results.

For differentiable dynamical systems with dominated splittings, we give upper estimates on the measure-theoretic tail entropy in terms of Lyapunov exponents. As our primary application, we verify the upper semi-continuity of metric entropy in various settings with domination.

As an extension of quasi-continuity in general topology, we define soft quasi-continuity. We show that this notion is equivalent to the known notion of soft semi-continuity. Next, we define soft weak quasi-continuity. With the help of examples, we prove that soft weak quasi-continuity is strictly weaker than both soft semi-continuity and soft weak continuity. We introduce many characterizations of soft weak quasi-continuity. Moreover, we study the relationship between soft quasi-continuity and weak quasi-continuity with their analogous notions in general topology. Furthermore, we show that soft regularity of the co-domain of a soft function is a sufficient condition for equivalence between soft semi-continuity and soft weakly quasi-continuity. Furthermore, we provide several results of soft composition, restrictions, preservation, and soft graph theorems in terms of soft weak quasi-continuity.