On the global uniform stability analysis of non-autonomous dynamical systems: A survey
Mathematica Moravica, Vol. 26, No. 2 (2022), 1–48.
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Abstract. In this survey, we introduce the notion of stability of time varying nonlinear systems. In particular we investigate the notion of global practical exponential stability for non-autonomous systems. The proposed approach for stability analysis is based on the determination of the bounds of perturbations that characterize the asymptotic convergence of the solutions to a closed ball centered at the origin.
Keywords. Lyapunov theory, Perturbed systems, Practical stability.
Anti-periodic boundary value problems for Caputo-Fabrizio fractional impulsive differential equations
Mathematica Moravica, Vol. 26, No. 2 (2022), 49–62.
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Abstract. In this paper, we shall discuss the existence and uniqueness of solutions for a nonlinear anti-periodic boundary value problem for fractional impulsive differential equations involving a Caputo-Fabrizio fractional derivative of order $r\in(0,1)$. Our results are based on some fixed point theorem, nonlinear alternative of Leray-Schauder type and coupled lower and upper solutions.
Keywords. Fractional differential equation, Caputo-Fabrizio integral of fractional order, Caputo-Fabrizio fractional derivatives, Anti-periodic boundary value problem, Fixed point, Lower and upper solutions, coupled lower and upper solutions.
Weak preopen sets and weak bicontinuity in texture spaces
Mathematica Moravica, Vol. 26, No. 2 (2022), 63–71.
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Abstract. The aim of this paper is introduce and study the notion of weak preopen sets and weak prebicontinuity on weak structures in texture spaces. It is presented some characterizations of weak prebicontinuity, and a link is given between weak spaces and weak structure on discrete texture spaces.
Keywords. Texture, Fuzzy set, Weak structure, Weak preopen set, Weak prebicontinuity.
Effects of the ARA transform method for time fractional problems
Mathematica Moravica, Vol. 26, No. 2 (2022), 73–84.
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Abstract. The aim of this study is to establish the solutions of time fractional mathematical problems with the aid of new integral transforms called the ARA transform. The fractional derivative is taken in the sense of Liouville-Caputo derivative. The fractional partial differential equations are reduced into ordinary differential equations. Later solving this fractional equation and applying inverse the ARA transform, the solution is acquired. The implementation of this transform for fractional differential equations is very similar to the implementation of the Laplace transform. However, the ARA transform allows us to take the integral transform of some functions for which we can not take the Laplace transform. The illustrated examples justify that the implementation and efficiency of this method are better than any other integral transforms to tackle time fractional differential equations (TFDEs).
Keywords. Liouville-Caputo fractional derivative, time fractional partial differential equation, ARA integral transform.
Generalized fixed point theorems on metric spaces
Mathematica Moravica, Vol. 26, No. 2 (2022), 85–101.
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Abstract. In this paper, we establish some fixed point theorems for single valued and multi-valued mappings on a complete metric space. Suzuki's and some other fixed point theorems are generalized by taking a more general contractive condition for single valued mappings. It is also proved that our result characterizes the completeness of the metric space. Further, taking generalized contractive condition, a fixed point theorem is also established for multi-valued mappings.
Keywords. Single valued and multi-valued mappings, Suzuki type contraction, fixed points and metric completeness.
Differential subordination and superordination results for generalized “Srivastava–Attiya” fractional integral operator
Mathematica Moravica, Vol. 26, No. 2 (2022), 103–112.
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Abstract. In this paper, we derive some subordination and superordination results for the generalized “Srivastava- Attiya” fractional integral operator. Some interesting corollaries for this operator is also obtained.
Keywords. $p$-valent analytic function, Differential subordination, Differential superordination, “Srivastava- Attiya” fractional integral operator.
Delay-dependent input-output stability conditions for non-autonomous neutral type differential equations in a Banach space
Mathematica Moravica, Vol. 26, No. 2 (2022), 113–121.
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Abstract. In a Banach space we consider a class of linear non-autonomous neutral type differential equations with several delays. For the considered equations we derive explicit delay-dependent input-output stability conditions. Applications to neutral type integro-differential equations are also discussed.
Keywords. Banach space, non-autonomous neutral type differential equation, input-output stability, integro-differential equations.
Existence of beam-equation solutions with strong damping and $p(x)$-biharmonic operator
Mathematica Moravica, Vol. 26, No. 2 (2022), 123–145.
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Abstract. In this paper, we consider a nonlinear beam equation with a strong damping and the $p(x)$-biharmonic operator. The exponent $p(\cdot)$ of nonlinearity is a given function satisfying some condition to be specified. Using Faedo-Galerkin method, the local and global existence of weak solutions is established with mild assumptions on the variable exponent $p(\cdot)$. This work improves and extends many other results in the literature.
Keywords. $p(x)$-biharmonic operator, weak solutions, beam equation, variable exponent.