Volume 28, Issue 1

In this paper, we study the existence of integrable solutions for initial value problems for fractional order implicit differential equations with Hadamard fractional derivative. Our results are based on Schauder’s fixed point theorem and the Banach contraction principle fixed point theorem.

In the present paper, the problem of estimating the contingent cone to the solution set associated with certain set-valued inclusions is addressed by variational analysis methods and tools. As a main result, inner (resp. outer) approximations, which are expressed in terms of outer (resp. inner) prederivatives of the set-valued term appearing in the inclusion problem, are provided. For the analysis of inner approximations, the evidence arises that the metric increase property for set-valued mappings turns out to play a crucial role. Some of the results obtained in this context are then exploited for formulating necessary optimality conditions for constrained problems, whose feasible region is defined by a set-valued inclusion.

A wave equation in a bounded and smooth domain of ℝ n {\mathbb{R}^{n}} with a delay term in the nonlinear boundary feedback is considered. Under suitable assumptions, global existence and uniform decay rates for the solutions are established. The proof of existence of solutions relies on a construction of suitable approximating problems for which the existence of the unique solution will be established using nonlinear semigroup theory and then passage to the limit gives the existence of solutions to the original problem. The uniform decay rates for the solutions are obtained by proving certain integral inequalities for the energy function and by establishing a comparison theorem which relates the asymptotic behavior of the energy and of the solutions to an appropriate dissipative ordinary differential equation.

We familiarize a notion of a fixed circle in a partial metric space, which is not necessarily the same as a circle in a Euclidean space. Next, we establish novel fixed circle theorems and verify these by illustrative examples with geometric interpretation to demonstrate the authenticity of the postulates. Also, we study the geometric properties of the set of non-unique fixed points of a discontinuous self-map in reference to fixed circle problems and responded to an open problem regarding the existence of a maximum number of points for which there exist circles. This paper is concluded by giving an application to activation function to exhibit the feasibility of results, thereby providing a better insight into the analogous explorations.

Time-harmonic electromagnetic waves in vacuum are described by the Helmholtz equation Δ ⁢ u + ω 2 ⁢ u = 0   for  ⁢ ( x , y , z ) ∈ ℝ 3 . \Delta u+\omega^{2}u=0\quad\text{for }(x,y,z)\in{\mathbb{R}}^{3}. For the evolution of such waves along the z -axis, a Schrödinger equation can be derived through a multiple scaling ansatz. It is the purpose of this paper to justify this formal approximation by proving bounds between this formal approximation and true solutions of the original system. The challenge of the presented validity analysis is the fact that the Helmholtz equation is ill-posed as an evolutionary system along the z -axis.

In this work, we introduce and investigate a new class k - U ⁢ S ~ s ⁢ ( b , μ , γ , t ) {k-\widetilde{US}_{s}(b,\mu,\gamma,t)} of analytic functions in the open unit disk U with negative coefficients. The object of the present paper is to determine coefficient estimates, neighborhoods and partial sums for functions f belonging to this class.

In this paper, we have a new matrix generalization with absolute matrix summability factor of an infinite series by using quasi-β-power increasing sequences. That theorem also includes some new and known results dealing with some basic summability methods

We offer a solution to a functional equation using properties of the Mellin transform. A new criteria for the Riemann Hypothesis is offered as an application of our main result, through a functional relationship with the Riemann xi function.

The stagnation point flow of a non-Newtonian Reiner–Rivlin fluid has been studied in the presence of a uniform magnetic field. The technique of similarity transformation has been used to obtain the self-similar ordinary differential equations. In this paper, an attempt has been made to prove the existence and uniqueness of the solution of the resulting free boundary value problem. Monotonic behavior of the solution is discussed. The numerical results, shown through a table and graphs, elucidate that the flow is significantly affected by the non-Newtonian cross-viscous parameter L and the magnetic parameter M .

In this work, we argue about the Lesche stability of some systems that are motivated by the use of fractional derivatives.

In this work, we discuss the long time behavior of solutions of the Whitham–Broer–Kaup system with Lipschitz nonlinearity and negative dispersion term. We prove the global well-posedness when α + β 2 < 0 {\alpha+\beta^{2}<0} as well as the convergence to 0 of small solutions at rate 𝒪 ⁢ ( t - 1 / 2 ) {\mathcal{O}(t^{-1/2})} .

A time dependent singularly perturbed differential-difference equation is considered. The problem involves time delay and general small space shift terms. Taylor series approximation is used to expand the space shift term. A robust numerical scheme based on the backward Euler scheme for the time and classical upwind scheme for space is proposed. The convergence analysis is carried out. It is observed that the proposed scheme converges almost first order up to a logarithm term and optimal first order in space on the Shishkin and Bakhvalov–Shishkin mesh, respectively. Numerical results confirm the efficiency of the proposed scheme, which are in agreement with the theoretical bounds.

In this paper, we introduce the g -Navier–Stokes equations with time-fractional derivative of order α ∈ ( 0 , 1 ) {\alpha\in(0,1)} in domains of ℝ 2 {\mathbb{R}^{2}} . We then study the existence and uniqueness of weak solutions by means of the Galerkin approximation. Finally, an optimal control problem is considered and solved.

In this paper, we provide a rigorous derivation of an asymptotic formula for the perturbation of eigenvalues associated to the Stokes eigenvalue problem with Dirichlet conditions and in the presence of small deformable inclusions. Taking advantage of the small sizes of the inclusions immersed in an incompressible Newtonian fluid having kinematic viscosity different from the background one, we show that our asymptotic formula can be expressed in terms of the eigenvalue in the absence of the inclusions and in terms of the viscous moment tensor (VMT).

The sub-equation method is a kind of straightforward algebraic method to construct exact solutions of nonlinear evolution equations. In this paper, the sub-equation method is improved by proposing some extended homogeneous balance conditions. By applying them to several examples, it can be seen that new solutions could indeed be obtained.