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In this paper, we investigate the existence and uniqueness of minimizers of a fractional variational problem generalized from the energy functional associated with a cantilever beam under a uniformly distributed load. We apply the fractional Euler–Lagrange condition to formulate the minimization problem as a boundary value problem and obtain existence and uniqueness results in both L 2 and L ∞ settings. Additionally, we characterize the continuous dependence of the minimizers on varying loads in the energy functional. Moreover, an approximate solution is derived via the homotopy perturbation method, which is numerically demonstrated in various examples. The results show that the deformations are larger for smaller orders of the fractional derivative. [ABSTRACT FROM AUTHOR]

This paper studies the dynamics of an opinion formation model with a leader associated with a system of fractional differential equations. We applied the concept of α -exponential stability and the uniqueness of equilibrium to show the consensus of the followers with the leader. A sufficient condition for the consensus is obtained for both fractional formation models with and without time-dependent external inputs. Moreover, numerical results are provided to illustrate the dynamical behavior. [ABSTRACT FROM AUTHOR]

We study the optimal feedback control for fractional evolution equations with a nonlinear perturbation of the time-fractional derivative term involving Caputo fractional derivatives with arbitrary kernels. Firstly, we derive a mild solution in terms of the semigroup operator generated by resolvents and a kernel from the general Caputo fractional operators and establish the existence and uniqueness of mild solutions for the feedback control systems. Then, the existence of feasible pairs by applying Filippov's theorem is obtained. In addition, the existence of optimal control pairs for the Lagrange problem has been investigated. [ABSTRACT FROM AUTHOR]

We consider an optimal control problem for a time-dependent obstacle variational inequality involving fractional Liouville–Caputo derivative. The obstacle is considered as the control, and the corresponding solution to the obstacle problem is regarded as the state. Our aim is to find the optimal control with the properties that the state is closed to a given target profile and the obstacle is not excessively large in terms of its norm. We prove existence results and establish necessary conditions of obstacle problems via the approximated time fractional-order partial differential equations and their adjoint problems. The result in this paper is a generalization of the obstacle problem for a parabolic variational inequalities as the Liouville–Caputo fractional derivatives were used instead of the classical derivatives. [ABSTRACT FROM AUTHOR]

In this paper, we study the local and global existence, and uniqueness of mild solution to initial value problems for fractional semilinear evolution equations with compact and noncompact semigroup in Banach spaces. In particular, we derive the form of fundamental solution in terms of semigroup induced by resolvent and ψ-function from Caputo fractional derivatives. These results generalize previous work where the classical Caputo fractional derivative is considered. Moreover, we prove the Mittag-Leffler–Ulam–Hyers stability result. Finally, we give examples of time-fractional heat equation to illustrate the result. [ABSTRACT FROM AUTHOR]

We consider fractional hybrid differential equations involving the Caputo fractional derivative of order $0<\alpha<1$ . Using fixed point theorems developed by Dhage et al. in Applied Mathematics Letters 34, 76-80 (2014), we prove the existence and approximation of mild solutions. In addition, we provide a numerical example to illustrate the results obtained. [ABSTRACT FROM AUTHOR]

Using Krasnoselskii's fixed point theorem and dichotomy theory, we prove the existence of periodic solutions for differential equations with multiple delays of the form $x'(t)+ cx'(t-\tau) = A(t)x(t) +f(t, x(t-\alpha_{1}(t)),\ldots, x(t-\alpha_{m}(t)))$, where the parameter $c \ll1$ is a small perturbation for a delayed forced term. Moreover, we discuss the convergence of these solutions to a solution of the unperturbed problem as $c \rightarrow0$. [ABSTRACT FROM AUTHOR]

Abstract: We study the effect of domain perturbation on invariant manifolds for semilinear parabolic equations subject to the Dirichlet boundary condition. Under the Mosco convergence assumption on the domains, we prove the upper and lower semicontinuity of both the local unstable invariant manifold and the local stable invariant manifold near a hyperbolic equilibrium. The continuity results are obtained by keeping track of the construction of invariant manifolds in [P.W. Bates, C.K.R.T. Jones, Invariant manifolds for semilinear partial differential equations, in: Dynamics Reported, Vol. 2, in: Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 1–38]. [Copyright &y& Elsevier]

The article offers information on the parabolic equations of domain perturbation. It says that domain perturbation is one vital topic in the partial differential equations analysis. It mentions that the said equation is utilized in finding solutions for complicated nonlinear problems. Moreover, methods for finding domain perturbation are also discussed.

The purpose of the present paper is to construct a common solution of the split null point problem associated with the maximal monotone operators and the fixed point problem associated with a finite family of k‐demicontractive operators in Hilbert spaces. We compute the optimal common solution via inertial parallel hybrid algorithm under a suitable set of control conditions. The viability of parallel implementation of the algorithm is demonstrated for various theoretical as well as numerical results. The results presented in this paper improve various existing results in the current literature. [ABSTRACT FROM AUTHOR]

In this paper, the authors present a strategy based on fixed point iterative methods to solve a nonlinear dynamical problem in a form of Green's function with boundary value problems. First, the authors construct the sequence named Green's normal-S iteration to show that the sequence converges strongly to a fixed point, this sequence was constructed based on the kinetics of the amperometric enzyme problem. Finally, the authors show numerical examples to analyze the solution of that problem. [ABSTRACT FROM AUTHOR]

This paper provides iterative construction of a common solution associated with the fixed point problem of an infinite family of |-demicontractive mappings, pseudomonotone equilibrium problem satisfying Lipschitz-type continuity and the split common null point problem. We propose an iterative algorithm that combines the classical extragradient method with the inertial extrapolation technique. The analysis of the proposed algorithm is two-fold: firstly, we establish strong convergence results under suitable set of constraints and secondly we verify the viability of the proposed algorithm via numerical experiment with applications. [ABSTRACT FROM AUTHOR]

In this article, we introduce a new multi-step iteration for approximating a common fixed point of a finite class of multi-valued Bregman relatively nonexpansive mappings in the setting of reflexive Banach spaces. We prove a strong convergence theorem for the proposed iterative algorithm under certain hypotheses. Additionally, we also use our results for the solution of variational inequality problems and to find the zero points of maximal monotone operators. The theorems furnished in this work are new and well-established and generalize many well-known recent research works in this field. [ABSTRACT FROM AUTHOR]