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In this paper, we study the extremal solutions of nonlinear fractional p‐Laplacian differential system with the fractional conformable derivative by applying monotone iterative method and a half‐pair of upper and lower solutions. For the smooth running of our work, we develop a comparison principle about linear system, which play a very crucial role in this article. At last, an illustrative example is given for the main result. [ABSTRACT FROM AUTHOR]

In this work, we consider a class of fuzzy fractional delay integro-differential equations with the generalized Caputo-type Atangana-Baleanu (ABC) fractional derivative. By using the monotone iterative method, we not only obtain the existence and uniqueness of the solution for the given problem with the initial condition but also give the monotone iteration sequence converging to the unique solution of the problem. Furthermore, we also give the continuous dependence of the unique solution on initial value. Finally, an example is presented to illustrate the main results obtained. The results presented in this study are new and open a new avenue of research for fuzzy fractional delay integro-differential equations with the generalized ABC fractional derivative.

The purpose of this paper is to consider the fractional delayed evolution equation of order γ ∈ (1 , 2) in ordered Banach space. In the absence of assumptions about the compactness of cosine families or related sine families, the existence results of positive solutions are studied by using some fixed point theorems and monotone iterative method under the conditions that nonlinear function satisfies the non-compactness measure conditions and some appropriate growth conditions or order conditions. [ABSTRACT FROM AUTHOR]

In this paper, we study the following second order differential equation: − (Φ(u ' (t))) ' + φp(u(t)) = εf(t, u(t)) a.e. on Ω = [0, T] under nonlinear multivalued boundary value conditions which incorporate as special cases the classicals boundary value conditions of type Dirichlet, Neumann, and Sturm-Liouville. Using monotone iterative method coupled with lower and upper solutions method, multifunction analysis, theory of monotone operators, and theory of topological degree, we show existence of solution and extremal solutions when the lower and upper solutions are well ordered or not. Since the boundary value conditions do not include the periodic one, we show that our method stay true for the periodic problem. [ABSTRACT FROM AUTHOR]

In this paper we discuss the boundary value problem for a first order delay differential equation of the type, y(t) + y(t) = f(t, y(t - r)). We prove the existence of solution between weakly coupled lower and up- per solution by assuming f to be a non-decreasing function in the second coordinate. Further, we use this existence result to establish monotone ite- rative method, where we obtain increasing as well as decreasing sequence of functions whose limits are a solution of the boundary value problem. The sequence of functions obtained are solutions of some defined boundary value problem with linear condition of linear delay differential equation. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of extremal mild solutions for a class of Hilfer fractional evolution equation with non-instantaneous impulses and nonlocal conditions in ordered Banach spaces. The definition of mild solutions for our concerned problem was given based on a C 0 -semigroup W (·) generated by the operator - A and probability density function. By means of monotone iterative technique and the method of lower and upper solutions, the existence of extremal mild solutions between lower and upper mild solutions for nonlinear Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions is obtained under the situation that the corresponding C 0 -semigroup W (·) and non-instantaneous impulsive function γ k are compact, W (·) is not compact and γ k is compact, W (·) and γ k are not compact, respectively. At the end, in order to illustrate our results, three concrete examples are given. [ABSTRACT FROM AUTHOR]

In this article, we study the existence of mild solutions for a class of Hilfer fractional evolution equations with non-instantaneous impulses in ordered Banach spaces. The definition of mild solutions for our problem was given based on a C 0 -semigroup W (·) generated by the operator - A and probability density function. By means of monotone iterative technique and the method of lower and upper, the existence of extremal mild solutions between lower and upper mild solutions for nonlinear evolution equation with non-instantaneous impulses is obtained under the situation that the corresponding C 0 -semigroup W (·) and non-instantaneous impulsive function γ k are compact, W (·) is not compact and γ k is compact, W (·) and γ k are not compact, respectively. At last, two examples are given to illustrate the abstract results. [ABSTRACT FROM AUTHOR]

This paper introduces a new concept of Caputo type interval-valued fractional conformable calculus. Based on this, some theorems and properties related to fractional conformable calculus are presented. As an application, this paper attempts to investigate an initial value problem of functional integro-differential equations with Caputo type interval-valued fractional conformable derivatives. We not only prove the existence of at least one solution, the extremal solutions and the unique solution of the above problem, but also establish some monotone iterative sequences converging to the extremal solutions. More meaningful and important, some comparison principles and generalized Gronwall inequality related to the proposed problem are also proved. Two examples are presented to illustrate the main results.We firmly believe that the concept of interval-valued fractional conformable calculus, new comparison principles and generalized Gronwall inequality introduced here can be conveniently applied to qualitative analysis of a variety of new interval-valued fractional conformable biological models, economic models, etc., which will better describe and explain various complex phenomena derived from the real world.

In this work, we propose the following fourth‐order non‐self‐adjoint SBVPs to investigate 1rαrα1rα(rαϕ′)′′′=12rαμ′ϕ′2+2μϕ′ϕ″+λG(r),for00$$ \mu (r)=p{r}^{2\alpha -2},p\in {\mathbb{R}}^{+},G(r)\in {L}^1\left[0,1\right]\kern0.4em \mathrm{such}\ \mathrm{that}\kern0.50em {M}_1\ge G(r)\ge M>0 $$ and α>1$$ \alpha >1 $$ with respect to three different homogeneous boundary conditions. By using monotone iterative technique we show the existence of at least one solution, which depends on the parameter λ$$ \lambda $$. We report presence of two solutions computationally. We show the sign of the both solutions and derive the estimations of the parameter λ$$ \lambda $$ which indicates the region of nonexistence. To validate the estimations of λ$$ \lambda $$ we propose an iterative technique with the help of Greens function. We place the numerical results as an evidence of our derived theoretical results. [ABSTRACT FROM AUTHOR]

This paper introduces a new concept of Caputo type interval-valued fractional conformable calculus. Based on this, some theorems and properties related to fractional conformable calculus are presented. As an application, this paper attempts to investigate an initial value problem of functional integro-differential equations with Caputo type interval-valued fractional conformable derivatives. We not only prove the existence of at least one solution, the extremal solutions and the unique solution of the above problem, but also establish some monotone iterative sequences converging to the extremal solutions. More meaningful and important, some comparison principles and generalized Gronwall inequality related to the proposed problem are also proved. Two examples are presented to illustrate the main results. We firmly believe that the concept of interval-valued fractional conformable calculus, new comparison principles and generalized Gronwall inequality introduced here can be conveniently applied to qualitative analysis of a variety of new interval-valued fractional conformable biological models, economic models, etc., which will better describe and explain various complex phenomena derived from the real world. [ABSTRACT FROM AUTHOR]

This article is devoted to a numerical analysis of two types of fractional nonlinear reaction-diffusion problems with periodic conditions or with initial conditions. We also consider two types of derivative in time, the first one is that of Caputo and the second is the conformable derivative; we discretize the problem via a finite difference method. We construct two iterative schemes by the upper and lower solutions method which converges monotonically towards a maximal solution or a minimal solution of the considered problem when the mesh decreases to zero, depending on whether the initial iteration is an upper solution or a lower solution. A comparison lemma for the different monotone sequences is also proved. The presented iterative scheme is used to show that the finite difference system converges to continuous solutions of the fractional reaction-diffusion problem. Finally to validate the theoretical results some examples with numerical simulations of reaction-diffusion problem are also presented and discussed in detail. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a mixed fractional integral boundary value problem with p(t)-Laplacian operator. Firstly, we derive the Green function through the direct computation and obtain the properties of Green function. For p (t) ≠ constant, under the appropriate conditions of the nonlinear term, we establish the existence result of at least one positive solution of the above problem by means of the Leray–Schauder fixed point theorem. Meanwhile, we also obtain the positive extremal solutions and iterative schemes in view of applying a monotone iterative method. For p (t) = constant, by using Guo–Krasnoselskii fixed point theorem, we study the existence of positive solutions of the above problem. These results enrich the ones in the existing literatures. Finally, some examples are included to demonstrate our main results in this paper and we give out an open problem. [ABSTRACT FROM AUTHOR]

In the paper, nonlinear systems of mixed-type functional differential equations are analyzed and the existence of semi-global and global solutions is proved. In proofs, the monotone iterative technique and Schauder-Tychonov fixed-point theorem are used. In addition to proving the existence of global solutions, estimates of their co-ordinates are derived as well. Linear variants of results are considered and the results are illustrated by selected examples.

We investigate the existence of solutions for a Caputo fractional differential equation with periodic boundary condition. Using the positivity of Green's function of the corresponding linear equation, we show the existence of positive solutions by using Krasnosel'skii fixed point theorem. Meanwhile, by using monotone iterative method and lower and upper solutions method, we also discuss the existence of extremal solutions for a special case.

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The aim of this research work is to derive some appropriate results for extremal solutions to a class of generalized Caputo-type nonlinear fractional differential equations (FDEs) under nonlinear boundary conditions (NBCs). The aforesaid results are derived by using the monotone iterative method, which exercises the procedure of upper and lower solutions. Two sequences of extremal solutions are generated in which one converges to the upper and the other to the corresponding lower solution. The method does not need any prior discretization or collocation for generating the aforesaid two sequences for upper and lower solutions. Further, the aforesaid techniques produce a fruitful combination of upper and lower solutions. To demonstrate our results, we provide some pertinent examples. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the non-cooperative elliptic Schrödinger systems arising in Bose–Einstein condensation phenomena and some nonlinear optical materials. The more delicate case of systems of negative potentials is considered. We prove the existence and multiplicity of nontrivial solutions for the above system in space dimensions N ≥ 3 . Our proofs are based on symmetric mountain pass method, the monotone iterative method, as well as suitable Schrödinger test-function arguments. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of at least one positive solution to the fourth-order two-point boundary value problem (BVP) which models a cantilever beam equation, where one end is kept free. Here f ∈ C [ 0 , 1 ] × R + , R + , g ∈ C [ 0 , 1 ] , R + and λ is a positive parameter. The sufficient conditions are interesting, new and easy to verify. We have used some inequalities on the nonlinear function f and eigenvalues of a linear integral operator as bounds for the parameter λ to obtain our results. Our approach is based on a revised version of a fixed point theorem due to Gustafson and Schmitt. [ABSTRACT FROM AUTHOR]

In this paper, we consider the following generalized nonlinear k-Hessian system { G (S k 1 k (λ (D 2 z 1))) S k 1 k (λ (D 2 z 1)) = φ (| x | , z 1 , z 2) , x ∈ ℝ N , G (S k 1 k (λ (D 2 z 2))) S k 1 k (λ (D 2 z 2)) = φ (| x | , z 1 , z 2) , x ∈ ℝ N , where G is a nonlinear operator and Sk (λ(D2z)) stands for the k-Hessian operator. We first are interested in the classification of positive entire k-convex radial solutions for the k-Hessian system if φ(∣x∣, z1, z2) = b(∣x∣)φ(z1, z2) and ψ(∣x∣, z1, z2) = h(∣x∣)ψ(z1). Moreover, with the help of the monotone iterative method, some new existence results on the positive entire k-convex radial solutions of the k-Hessian system with the special non-linearities ψ,φ are given, which improve and extend many previous works. [ABSTRACT FROM AUTHOR]

In this paper, existence and uniqueness of weak solutions for a linear parabolic problem with conformable derivative are proved, the existence of weak periodic solutions for conformable fractional parabolic nonlinear dierential equation is proved by using a more generalized monotone iterative method combined with the method of upper and lower solutions. We prove the convergence of monotone sequence to weak periodic minimal and maximal solutions. Moreover, the conformable version of the Lions-Magness and AubinLions lemmas are also proved.

In this paper, we discuss the asymptotically periodic problem for the abstract fractional evolution equation under order conditions and growth conditions. Without assuming the existence of upper and lower solutions, some new results on the existence of the positive S-asymptotically σ-periodic mild solutions are obtained by using monotone iterative method and fixed point theorem. It is worth noting that Lipschitz condition is no longer needed, which makes our results more widely applicable. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to present an existence-comparison and uniqueness theorem as well as an iterative method for finite difference system, which correspond to a class of semilinear weakly coupled system of time degenerate parabolic initial boundary value problems. The basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and construction of monotone sequences from a suitable linear discrete system. Using upper and lower solutions as two distinct initial iterations, two monotone sequences are constructed from a suitable linear system under mixed quasimonotonicity condition. It is shown that these sequences converge monotonically to a solution of the discrete system. [ABSTRACT FROM AUTHOR]

Abstract By using monotone iterative method, the extremal solutions and the unique solution are obtained for a nonlinear fractional p-Laplacian boundary value problem involving fractional conformable derivatives and nonlocal integral boundary conditions. Comparison theorems related to the proposed study are also proved. The paper concludes with an illustrative example for the main result.

In this paper, we investigate a class of nonlinear Schrödinger systems containing a nonlinear operator under Osgood-type conditions. By employing the iterative technique, the existence conditions for entire positive radial solutions of the above problem are given under the cases where components μ and ν are bounded, μ and ν are blow-up, and one of the components is bounded, while the other is blow-up. Finally, we present two examples to verify our results.

In this paper we employ the method of maximal and minimal solutions coupled with comparison principles and the monotone iterative technique to obtain results of existence and approximation of solutions for differential equations with piecewise constant delay of generalized type (DEPCAG). [ABSTRACT FROM AUTHOR]

Based on the theory of lower and upper solutions, we study the monotone iterative method for the nonlinear integral boundary value problems of fractional p-Laplacian equations with delay, which involves both Riemann–Liouville derivative and Caputo derivative. Some new results on the existence of positive solutions are established and the iterative methods for finding approximate solutions of the boundary value problem are obtained. Finally, two examples are given out to illustrate the numerical solution and the related graphic simulations are also provided. [ABSTRACT FROM AUTHOR]

The aim of this research work is to derive some appropriate results for extremal solutions to a class of generalized Caputo-type nonlinear fractional differential equations (FDEs) under nonlinear boundary conditions (NBCs). The aforesaid results are derived by using the monotone iterative method, which exercises the procedure of upper and lower solutions. Two sequences of extremal solutions are generated in which one converges to the upper and the other to the corresponding lower solution. The method does not need any prior discretization or collocation for generating the aforesaid two sequences for upper and lower solutions. Further, the aforesaid techniques produce a fruitful combination of upper and lower solutions. To demonstrate our results, we provide some pertinent examples.

Abstract In this paper, we get a new form equivalent integral equation for a class of evolution equations of fractional order with nonlocal conditions on the half-line. With the aid of it, the uniqueness of the mild solution is obtained by the Banach contraction theorem. Also, we present the existence and uniqueness theorem of positive mild solutions by the monotone iterative method without assumption of lower and upper solutions.

Abstract The equivalent integral equation of a new form for a class of fractional evolution equations is obtained by the method of Laplace transform, which is different from those given in the existing literature. By the monotone iterative method without the assumption of lower and upper solutions, we present some new results on the existence of positive mild solutions for the abstract fractional evolution equations on the half-line.

Abstract Based on an equivalent integral equation of a new type for a class of fractional evolution equations, which is different from those obtained in the existing literature, the paper investigates a class of fractional evolution equations with nonlocal conditions on infinite interval. Without the assumption of lower and upper solutions, we present a new result on the existence and uniqueness of positive mild solutions for the abstract fractional evolution equations by using the monotone iterative method.

Synthetic aperture radar (SAR) monopulse angle measurement is an adequate method of target angle measurement in low signal-to-clutter and noise ratio. However, there is a contradiction between the instantaneous angle measurement and the SAR coherent accumulation. In this study, the authors use the total variation method to solve the above problem, and propose the monotone iterative method to enhance the efficiency of the total variation method, which is verified in the simulation.

Abstract This paper deals with a parameter robust numerical method for solving a nonlinear singularly perturbed elliptic reaction–diffusion problem. Existence and uniqueness of a solution to a nonlinear difference scheme are established. The uniform convergence of the nonlinear difference scheme on piecewise uniform and log-meshes is established. Numerical experiments are presented. [ABSTRACT FROM AUTHOR]

In this paper, we discuss two point boundary value problems of second-order fuzzy boundary value problem. Using monotone iterative method, we investigate the existence of solutions to boundary value problem of fuzzy integro-differential equation firstly, then the results are applied to discuss second-order fuzzy boundary value problem. [ABSTRACT FROM AUTHOR]

In this paper, we discuss two point boundary value problems of second-order fuzzy boundary value problem. Using monotone iterative method, we investigate the existence of solutions to boundary value problem of fuzzy integro-differential equation firstly, then the results are applied to discuss second-order fuzzy boundary value problem. [ABSTRACT FROM AUTHOR]

Synthetic aperture radar (SAR) monopulse angle measurement is an adequate method of target angle measurement in low signal-to-clutter and noise ratio. However, there is a contradiction between the instantaneous angle measurement and the SAR coherent accumulation. In this study, the authors use the total variation method to solve the above problem, and propose the monotone iterative method to enhance the efficiency of the total variation method, which is verified in the simulation. [ABSTRACT FROM AUTHOR]

Positive and negative definite comparison results for nonlinear q-th fractional differential equations of Riemann-Liouville type are derived without requiring Hölder continuity assumption. Monotone iterative method is then developed to a class of nonlinear boundary value problems for fractional differential equations, using coupled upper and lower solutions. Existence of the multiplicity solutions for the nonlinear fractional differential equations is presented. [ABSTRACT FROM AUTHOR]

In this paper, we get a new form equivalent integral equation for a class of evolution equations of fractional order with nonlocal conditions on the half-line. With the aid of it, the uniqueness of the mild solution is obtained by the Banach contraction theorem. Also, we present the existence and uniqueness theorem of positive mild solutions by the monotone iterative method without assumption of lower and upper solutions. [ABSTRACT FROM AUTHOR]

In this article, we discuss a class of boundary value problems for nonlinear singular fractional order differential equations. By applying monotone iterative techniques, we not only obtain the existence of positive solutions but also establish an iterative scheme for approximating the solutions. An example is also included to illustrate the main result. [ABSTRACT FROM AUTHOR]