Existence and uniqueness of positive solutions for fractional relaxation equation in terms of ψ-Caputo fractional derivative

This manuscript is committed to deal with the existence and uniqueness of positive solutions for fractional relaxation equation involving ψ-Caputo fractional derivative. The existence of solution is carried out with the help of Schauder’s fixed point theorem, while the uniqueness of the solution is obtained by applying the Banach contraction principle, along with Bielecki type norm. Moreover, two explicit monotone iterative sequences are constructed for the approximation of the extreme positive solutions to the proposed problem. Lastly, two examples are presented to support the obtained results.

Monotone Iterative and Upper–Lower Solution Techniques for Solving the Nonlinear ψ-Caputo Fractional Boundary Value Problem

The objective of this paper is to study the existence of extremal solutions for nonlinear boundary value problems of fractional differential equations involving the ψ−Caputo derivative CDa+σ;ψϱ(t)=V(t,ϱ(t)) under integral boundary conditions ϱ(a)=λIν;ψϱ(η)+δ. Our main results are obtained by applying the monotone iterative technique combined with the method of upper and lower solutions. Further, we consider three cases for ψ*(t) as t, Caputo, 2t, t, and Katugampola (for ρ=0.5) derivatives and examine the validity of the acquired outcomes with the help of two different particular examples.

Extremal Solutions for Caputo Conformable Differential Equations with p-Laplacian Operator and Integral Boundary Condition

The Caputo conformable derivative is a new Caputo-type fractional differential operator generated by conformable derivatives. In this paper, using Banach fixed point theorem, we obtain the uniqueness of the solution of nonlinear and linear Cauchy problem with the conformable derivatives in the Caputo setting, respectively. We also establish two comparison principles and prove the extremal solutions for nonlinear fractional p -Laplacian differential system with Caputo conformable derivatives by utilizing the monotone iterative technique. An example is given to verify the validity of the results.

Existence and nonexistence of entire k-convex radial solutions to Hessian type system

In this paper, a Hessian type system is studied. After converting the existence of an entire solution to the existence of a fixed point of a continuous mapping, the existence of entire k-convex radial solutions is established by the monotone iterative method. Moreover, a nonexistence result is also obtained.

Existence of Solutions for a Class of Nonlinear Neumann BVPs in the Presence of Upper and Lower Solutions

In this article, we explore the monotone iterative technique (MI-technique) to study the existence of solutions for a class of nonlinear Neumann 4-point, boundary value problems (BVPs) defined as, \begin{eqnarray*} \begin{split} -\z^{(2)}(\y)=\x(\y,\z,\z^{(1)}),\quad 0<\y<1,\\ \z^{(1)}(0)=\lambda \z^{(1)}(\beta_1 ),\quad \z^{(1)}(1)=\delta \z^{(1)}(\beta_2), \end{split} \end{eqnarray*} where $ 0<\beta_1 \leq \beta_2 <1$ and $\lambda$, $\delta\in (0,1)$. The nonlinear term $ \x(\y,\z,\z^{(1)}): \Omega\rightarrow \mathbb{R} $, where $\Omega =[0,1]\times \mathbb{R}^2 $, is Lipschitz in $ \z^{(1)}(\y)$ and one sided Lipschitz in $ \z(\y)$. Using lower solution $l(\y)$ and upper solutions $u(\y)$, we develop MI-technique, which is based on quasilinearization. To construct the sequences of upper and lower solutions which are monotone, we prove maximum principle as well as anti maximum principle. Then under certain assumptions, we prove that these sequence converges uniformly to the solution $ \z(\y)$ in the specific region, where $ \frac{\partial\x}{\partial\z}<0 $ or $ \frac{\partial\x}{\partial\z}>0 $. To demonstrate that the proposed technique is effective, we compute the solution of the nonlinear multi-point BVPs. We don’t require sign restriction which is very common and very strict condition.

Local existence–uniqueness and monotone iterative approximation of positive solutions for p-Laplacian differential equations involving tempered fractional derivatives

In this paper, we are concerned with a kind of tempered fractional differential equation Riemann–Stieltjes integral boundary value problems with p-Laplacian operators. By means of the sum-type mixed monotone operators fixed point theorem based on the cone $P_{h}$ P h , we obtain not only the local existence with a unique positive solution, but also construct two successively monotone iterative sequences for approximating the unique positive solution. Finally, we present an example to illustrate our main results.

Convergence of approximate solutions for singular difference systems with maxima

In this paper, by introducing a new singular fractional difference comparison theorem, the existence of maximal and minimal quasi-solutions are proved for the singular fractional difference system with maxima combined with the method of upper and lower solutions and the monotone iterative technique. Finally, we give an example to show the validity of the established results.

Transversality conditions for the existence of solutions of first-order discontinuous functional differential equations

We are concerned with the existence of extremal solutions to a large class of first-order functional differential problems under weak regularity assumptions. Our technique involves multivalued analysis and the method of lower and upper solutions in order to obtain a new existence result to a scalar Cauchy problem. As a consequence of this result and a monotone iterative method for discontinuous operators, we derive our main existence result which is illustrated by several examples concerning well-known models: a generalized logistic equation or second-order problems in the presence of dry friction.