In the present manuscript we incorporate fractional order Caputo derivative
to study a class of non-integer order differential equation. For existence
and uniqueness of solution some results from fixed point theory is on our
disposal. The method used for exploring these existence results is
topological degree method and some auxiliary conditions are developed for
stability analysis. For further elaboration an illustrative example is
provided in the last part of the research article.
In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results.
Abstract
We present an approach that facilitates the generation of explicit solutions to atmospheric Ekman flows with a height-dependent eddy viscosity. The approach relies on applying to the governing equations, of Sturm–Liouville type, a suitable Liouville substitution and then reducing the outcome to a nonlinear first-order differential equation of Riccati type.
In this paper, we study the existence of periodic solutions to nonlinear fully third-order differential equation x‴+ft,x,x′,x″=0,t∈ℝ≔−∞,∞, where f:ℝ4⟶ℝ is continuous and T-periodic in t. By using the topological transversality method together with the barrier strip technique, we obtain new existence results of periodic solutions to the above equation without growth restrictions on the nonlinearity. Meanwhile, as applications, an example is given to demonstrate our results.
We consider an eigenvalue problem for a certain type of quasi-linear second-order differential equation on the interval (0, ∞). Using an appropriate version of the mountain pass theorem, we establish the existence of a positive solution in [Formula: see text] for a range of values of the eigenvalue. It is shown that these solutions generate solutions of Maxwell's equations having the form of guided travelling waves propagating through a self-focusing dielectric. Motivated by models of optical fibres, the refractive index of the dielectric has an axial symmetry but may vary with distance for the axis. Previous existence results for this problem deal only with the homogeneous case.
We prove existence results for solutions of periodic boundary value problems concerning thenth-order differential equation withp-Laplacian[φ(x(n−1)(t))]'=f(t,x(t),x'(t),...,x(n−1)(t))and the boundary value conditionsx(i)(0)=x(i)(T),i=0,...,n−1. Our method is based upon the coincidence degree theory of Mawhin. It is interesting thatfmay be a polynomial and the degree of some variables amongx0,x1,...,xn−1in the functionf(t,x0,x1,...,xn−1)is allowed to be greater than1.
Application of topological degree method in quantitative behavior of fractional differential equations
