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In this article we consider the inverse problem of simultaneously determining three coefficients of a FitzHugh–Nagumo type system defined in a bounded domain. We use a Carleman estimate to establish Hölder estimates for these coefficients by a finite number of measurements of only one component of the system. [ABSTRACT FROM AUTHOR]

Abstract The logical progression from the constant order differential equations is the field of variable-order differential equations. Such equations can frequently give a more succinct description of problems in the real world. In light of this, we therefore take into account a class of coupled boundary value problems under variable-order differentiation. By utilizing the fixed-point techniques of Banach and Schauder, we investigate the existence and uniqueness of solutions to the proposed problem. Also, sufficient results are documented for the necessary needs. Furthermore, some stability results based on the ideas of Ulam, Hyers, and Rassias are elaborated upon. Ultimately, appropriate examples and in-depth analysis are presented to support our results.

In this paper, we study an (A , Q) -functional equation f (x + y , z + w) + f (x + y , z − w) + f (x − y , z + w) + f (x − y , z − w) = 4 f (x , z) + 4 f (x , w) in a Banach space and determine its general solutions (in certain groups). Moreover, we verify some stability results of the (A , Q) -functional equation as a consequence of a fixed point theorem in a complete metric space investigated by Brzdȩk et al. [A fixed point approach to stability of functional equations, Nonlinear Anal. 74 (2011) 6728–6732]. [ABSTRACT FROM AUTHOR]

The objective of the manuscript is to build coupled singular fractional-order differential equations with time delay. To study the underline problem, an integral representation is initially discussed and the operator form of the solution is investigated using various supplementary hypotheses. Also, the existence and uniqueness of the considered problem are investigated by using the Lebesgue-dominated convergence theorem and some analysis results. Moreover, the stability analysis to determine the nature of the proposed model's solution is examined. Finally, two supportive examples are provided to demonstrate our analysis as applications.

The logical progression from the constant order differential equations is the field of variable-order differential equations. Such equations can frequently give a more succinct description of problems in the real world. In light of this, we therefore take into account a class of coupled boundary value problems under variable-order differentiation. By utilizing the fixed-point techniques of Banach and Schauder, we investigate the existence and uniqueness of solutions to the proposed problem. Also, sufficient results are documented for the necessary needs. Furthermore, some stability results based on the ideas of Ulam, Hyers, and Rassias are elaborated upon. Ultimately, appropriate examples and in-depth analysis are presented to support our results. [ABSTRACT FROM AUTHOR]

The objective of the manuscript is to build coupled singular fractional-order differential equations with time delay. To study the underline problem, an integral representation is initially discussed and the operator form of the solution is investigated using various supplementary hypotheses. Also, the existence and uniqueness of the considered problem are investigated by using the Lebesgue-dominated convergence theorem and some analysis results. Moreover, the stability analysis to determine the nature of the proposed model's solution is examined. Finally, two supportive examples are provided to demonstrate our analysis as applications. [ABSTRACT FROM AUTHOR]

Abstract In this article, we introduce a new type of non-expansive mapping, namely weakly K-nonexpansive mapping, which is weaker than non-expansiveness and stronger than quasi-nonexpansiveness. We prove some weak and strong convergence results using weakly K-nonexpansive mappings. Also, we define weakly ( α , K ) $(\alpha ,K)$ -nonexpansive mapping and using it prove one stability result for JF-iterative process. Some prominent examples are presented illustrating the facts. A numerical example is given to compare the convergence behavior of some known iterative algorithms for weakly K-nonexpansive mappings. Moreover, we show by example that the class of α-nonexpansive mappings due to Aoyama and Kohsaka and the class of generalized α-nonexpansive mappings due to Pant and Shukla are independent. Finally, our fixed point theorem is applied to obtain a solution of a nonlinear fractional differential equation.

The paper deals with an inverse problem of identifying parameters in a nonlinear subdiffusion model from a final observation. The nonlinear subdiffusion model involves a Caputo fractional derivative of order α ∈ (0 , 1) in time. Such problem has important application in a large field of applied science. To treat our model, we first study the regularity of the solution for the direct problem by means of Mittag–Leffler functions. Second, to study our inverse parameter problem, we reformulate it into an optimal control one with a Least Squares cost function and we establish the existence of the optimal solution. Third, we show the uniqueness and stability with respect to the data of our inverse problem based on the optimality conditions of the considered functional and the regularity of the solution for the direct problem. Finally, we present some numerical experiments using descent gradient algorithm. [ABSTRACT FROM AUTHOR]

In this research work, we establish some new results about piecewise equation involving Caputo Fabrizio derivative (CFD). The concerned class has been recently introduced and these results are fundamental for investigation of qualitative theory and numerical interpretation. We derive some necessary results for the existence, uniqueness and various form of Hyers-Ulam (H-U) type stability for the considered problem. For the required results, we need to utilize usual classical fixed point theorems due to Banach and Krasnoselskii's. Moreover, results devoted to H-U stability are derived by using classical tools of nonlinear functional analysis. Some pertinent test problems are given to demonstrate our results.

The Alexandrov–Fenchel inequality bounds from below the square of the mixed volume V (K 1 , K 2 , K 3 , ... , K n) of convex bodies K 1 , ... , K n in ℝ n by the product of the mixed volumes V (K 1 , K 1 , K 3 , ... , K n) and V (K 2 , K 2 , K 3 , ... , K n). As a consequence, for integers α 1 , ... , α m ∈ ℕ with α 1 + ⋯ + α m = n the product V n (K 1) α 1 n ⋯ V n (K m) α m n of suitable powers of the volumes V n (K i) of the convex bodies K i , i = 1 , ... , m , is a lower bound for the mixed volume V (K 1 [ α 1 ] , ... , K m [ α m ]) , where α i is the multiplicity with which K i appears in the mixed volume. It has been conjectured by Betke and Weil that there is a reverse inequality, that is, a sharp upper bound for the mixed volume V (K 1 [ α 1 ] , ... , K m [ α m ]) in terms of the product of the intrinsic volumes V α i (K i) , for i = 1 , ... , m. The case where m = 2 , α 1 = 1 , α 2 = n − 1 has recently been settled by the present authors (2020). The case where m = 3 , α 1 = α 2 = 1 , α 3 = n − 2 has been treated by Artstein-Avidan et al. under the assumption that K 2 is a zonoid and K 3 is the Euclidean unit ball. The case where α 2 = ⋯ = α m = 1 , K 1 is the unit ball and K 2 , ... , K m are zonoids has been considered by Hug and Schneider. Here, we substantially generalize these previous contributions, in cases where most of the bodies are zonoids, and thus we provide further evidence supporting the conjectured reverse Alexandrov–Fenchel inequality. The equality cases in all considered inequalities are characterized. More generally, stronger stability results are established as well. [ABSTRACT FROM AUTHOR]

In this paper we consider $r(x)-$Kirchhoff type equation with variable-exponent nonlinearity of the form $$ u_{tt}-\Delta u-\big(a+b\int_{\Omega}\frac{1}{r(x)}|\nabla u|^{r(x)}dx\big)\Delta_{r(x)}u+\beta u_{t}=|u|^{p(x)-2}u, $$ associated with initial and Dirichlet boundary conditions. Under appropriate conditions on $r(.)$ and $p(.)$, stability result along the solution energy is proved. It is also shown that regarding arbitrary positive initial energy and suitable range of variable exponents, solutions blow-up in a finite time.

In this research work, we establish some new results about piecewise equation involving Caputo Fabrizio derivative (CFD). The concerned class has been recently introduced and these results are fundamental for investigation of qualitative theory and numerical interpretation. We derive some necessary results for the existence, uniqueness and various form of Hyers-Ulam (H-U) type stability for the considered problem. For the required results, we need to utilize usual classical fixed point theorems due to Banach and Krasnoselskii's. Moreover, results devoted to H-U stability are derived by using classical tools of nonlinear functional analysis. Some pertinent test problems are given to demonstrate our results. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new type of non-expansive mapping, namely weakly K-nonexpansive mapping, which is weaker than non-expansiveness and stronger than quasi-nonexpansiveness. We prove some weak and strong convergence results using weakly K-nonexpansive mappings. Also, we define weakly (α , K) -nonexpansive mapping and using it prove one stability result for JF-iterative process. Some prominent examples are presented illustrating the facts. A numerical example is given to compare the convergence behavior of some known iterative algorithms for weakly K-nonexpansive mappings. Moreover, we show by example that the class of α-nonexpansive mappings due to Aoyama and Kohsaka and the class of generalized α-nonexpansive mappings due to Pant and Shukla are independent. Finally, our fixed point theorem is applied to obtain a solution of a nonlinear fractional differential equation. [ABSTRACT FROM AUTHOR]

In the present paper we prove existence results for solutions to nonlinear elliptic Neumann problems whose prototype is λ | u | p − 2 u − Δ p u − div (c (x) | u | p − 2 u) + b (x) | ∇ u | p − 2 ∇ u = f in Ω , | ∇ u | p − 2 ∇ u + c (x) | u | p − 2 u ⋅ n ̲ = 0 on ∂ Ω where Ω is a bounded domain of R N , N ≥ 2 , with Lipschitz boundary, 1 < p < N , n ̲ is the outer unit normal to ∂ Ω , λ > 0 , the datum f belongs to the dual space of W 1 , p (Ω) or to Lebesgue space L 1 (Ω). Finally the coefficients b , c belong to appropriate Lebesgue spaces or Lorentz spaces. Existence results for weak solutions or renormalized solutions are proved under smallness assumptions on the coefficients b and c. [ABSTRACT FROM AUTHOR]

In this research, the three-dimensional (3D) steady and incompressible laminar Homann stagnation point nanofluid flow over a porous moving surface is addressed. The disturbance in the porous medium has been characterized by the Darcy-Forchheimer relation. The slip for viscous fluid is considered. The energy equation is organized in view of radiative heat flux which plays an important role in the heat transfer rate. The governing flow expressions are first altered into first-order ordinary ones and then solved numerically by the shooting method. Dual solutions are obtained for the velocity, skin friction coefficient, temperature, and Nusselt number subject to sundry flow parameters, magnetic parameter, Darcy-Forchheimer number, thermal radiation parameter, suction parameter, and dimensionless slip parameter. In this research, the main consideration is given to the engineering interest like skin friction coefficient (velocity gradient or surface drag force) and Nusselt number (temperature gradient or heat transfer rate) and discussed numerically through tables. In conclusion, it is noticed from the stability results that the upper branch solution (UBS) is more reliable and physically stable than the lower branch solution (LBS). [ABSTRACT FROM AUTHOR]

In this study, the stability problem of continuous‐time switched systems composed fully of unstable subsystems is considered. Unlike the hybrid conditions derived in previous literature, this study used a novel time‐dependent quadratic Lyapunov function approach and a convex sufficient condition for switched linear systems is proposed in the framework of bounded maximum average dwell time (BMADT). The resulting condition in this study is enforced using the sum of squares programming. Then, the condition for switched linear systems is extended to uncertain systems and the robust stability condition is derived. Moreover, a new stability result of continuous‐time switched systems with all subsystems unstable is investigated based on the BMADT conditions. The simulation shows that it is better than the results in previous literature. Two numerical examples are proposed to illustrate the authors' approach. [ABSTRACT FROM AUTHOR]

In this study, the stability problem of discrete‐time switched systems without stable subsystems is considered. Using the k ‐sample‐like method, the authors construct a Lyapunov function whose value at the switching instant is less than the value at the last switching instant when the corresponding dwell time belongs to a special time span. According to whether the dwell time belongs to the time span, switchings are divided into two categories: the switching without divergence time and the switching with divergence time. Based on that, a new less conservative exponential stability theorem is established with the bounded maximum average dwell time. Furthermore, by jointly considering the dynamic characteristics of the subsystems before and after switching instants, they also give the stability result via the dwell time with the floating lower and upper bounds. Finally, some numerical examples are given to illustrate the effectiveness of the theoretical results. [ABSTRACT FROM AUTHOR]

We strengthen the volume inequalities for Lp zonoids of even isotropic measures and for their duals, which are originally due to Ball, Barthe, and Lutwak, Yang, and Zhang. The special case p = ∞ yields a stability version of the reverse isoperimetric inequality for centrally symmetric convex bodies. Adding to known inequalities and stability results for the reverse isoperimetric inequality of arbitrary convex bodies, we state a conjecture on volume inequalities for Lp zonoids of general centered (non-symmetric) isotropic measures. We achieve our main results by strengthening Barthe's measure transportation proofs of the rank one case of the geometric Brascamp--Lieb and reverse Brascamp--Lieb inequalities by estimating the derivatives of the transportation maps for a special class of probability density functions. Based on our argument, we phrase a conjecture about a possible stability version of the Brascamp--Lieb and reverse Brascamp--Lieb inequalities. We also establish some geometric properties of the distribution of general isotropic measures that are essential to our argument. In particular, we prove a measure theoretic analog of the Dvoretzky--Rogers lemma. [ABSTRACT FROM AUTHOR]

We study the problem of whether the curvature-dimension condition with negative values of the generalized dimension parameter is stable under a suitable notion of convergence. To this purpose, first of all we propose an appropriate setting to introduce the CD (K , N) condition for N < 0 , allowing metric measure structures in which the reference measure is quasi-Radon. Then in this class of spaces we define the distance d iKRW , which extends the already existing notions of distance between metric measure spaces. Finally, we prove that if a sequence of metric measure spaces satisfying the CD (K , N) condition with N < 0 is converging with respect to the distance d iKRW to some metric measure space, then this limit structure is still a CD (K , N) space. [ABSTRACT FROM AUTHOR]