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12 pages, 282 KiB

Open AccessArticle

On an Integral Equation with the Riemann Function Kernel

by Sergei Sitnik and Abdul Ahad Arian

Axioms 2022, 11(4), 166; https://doi.org/10.3390/axioms11040166 - 07 Apr 2022

Viewed by 1489

This paper is concerned with a study of a special integral equation. This integral equation arises in many applied problems, including transmutation theory, inverse scattering problems, the solution of singular Sturm–Liouville and Shrödinger equations, and the representation of solutions of singular Sturm–Liouville and [...] Read more.

This paper is concerned with a study of a special integral equation. This integral equation arises in many applied problems, including transmutation theory, inverse scattering problems, the solution of singular Sturm–Liouville and Shrödinger equations, and the representation of solutions of singular Sturm–Liouville and Shrödinger equations. A special integral equation is derived and formulated using the Riemann function of a singular hyperbolic equation. In the paper, the existence of a unique solution to this equation is proven by the method of successive approximations. The results can be applied, for example, to representations of solutions to Sturm–Liouville equations with singular potentials, such as Bargmann and Miura potentials, and similiar. The treatment of problems with such potentials are very important in mathematical physics, and inverse, scattering and related problems. The estimates received do not contain any undefined constants, and for transmutation kernels all estimates are explicitly written. Full article

(This article belongs to the Special Issue Advances in Nonlinear Boundary Value Problems: Theory and Applications)

22 pages, 623 KiB

Open AccessFeature PaperArticle

Constant-Sign Green’s Function of a Second-Order Perturbed Periodic Problem

by Alberto Cabada, Lucía López-Somoza and Mouhcine Yousfi

Axioms 2022, 11(3), 139; https://doi.org/10.3390/axioms11030139 - 17 Mar 2022

Cited by 1 | Viewed by 1742

Abstract

In this paper, we were interested in obtaining the exact expression and studying the regions of constant sign of Green’s function related to a second-order perturbed periodic problem coupled with integral boundary conditions at the extremes of the interval of the definition. To [...] Read more.

In this paper, we were interested in obtaining the exact expression and studying the regions of constant sign of Green’s function related to a second-order perturbed periodic problem coupled with integral boundary conditions at the extremes of the interval of the definition. To obtain the expression of Green’s function related to this problem, we used the theory presented in a previous paper of the authors for general non-local perturbed boundary-value problems. Moreover, we characterized the parameter set where such a Green’s function has a constant sign. To this end, we needed to consider first a related second-order problem without integral boundary conditions, obtaining the properties of its Green’s function and then using them to compute the sign of the one related to the main problem. Full article

(This article belongs to the Special Issue Advances in Nonlinear Boundary Value Problems: Theory and Applications)

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14 pages, 288 KiB

Open AccessArticle

Existence Results for Coupled Implicit \({\psi}\)-Riemann–Liouville Fractional Differential Equations with Nonlocal Conditions

by Dinghong Jiang and Chuanzhi Bai

Axioms 2022, 11(3), 103; https://doi.org/10.3390/axioms11030103 - 25 Feb 2022

Cited by 4 | Viewed by 1877

Abstract

In this paper, we study the existence and uniqueness of solutions for a coupled implicit system involving

ψ

-Riemann–Liouville fractional derivative with nonlocal conditions. We first transformed the coupled implicit problem into an integral system and then analyzed the uniqueness and existence of [...] Read more.

In this paper, we study the existence and uniqueness of solutions for a coupled implicit system involving

ψ

-Riemann–Liouville fractional derivative with nonlocal conditions. We first transformed the coupled implicit problem into an integral system and then analyzed the uniqueness and existence of this integral system by means of Banach fixed-point theorem and Krasnoselskiis fixed-point theorem. Some known results in the literature are extended. Finally, an example is given to illustrate our theoretical result. Full article

(This article belongs to the Special Issue Advances in Nonlinear Boundary Value Problems: Theory and Applications)

14 pages, 325 KiB

Open AccessArticle

Mild Solutions for Impulsive Integro-Differential Equations Involving Hilfer Fractional Derivative with almost Sectorial Operators

by Kulandhaivel Karthikeyan, Panjaiyan Karthikeyan, Nichaphat Patanarapeelert and Thanin Sitthiwirattham

Axioms 2021, 10(4), 313; https://doi.org/10.3390/axioms10040313 - 22 Nov 2021

Cited by 2 | Viewed by 1269

Abstract

In this manuscript, we establish the mild solutions for Hilfer fractional derivative integro-differential equations involving jump conditions and almost sectorial operator. For this purpose, we identify the suitable definition of a mild solution for this evolution equations and obtain the existence results. In [...] Read more.

In this manuscript, we establish the mild solutions for Hilfer fractional derivative integro-differential equations involving jump conditions and almost sectorial operator. For this purpose, we identify the suitable definition of a mild solution for this evolution equations and obtain the existence results. In addition, an application is also considered. Full article

(This article belongs to the Special Issue Advances in Nonlinear Boundary Value Problems: Theory and Applications)

16 pages, 310 KiB

Open AccessArticle

Sequential Riemann–Liouville and Hadamard–Caputo Fractional Differential Equation with Iterated Fractional Integrals Conditions

by Sotiris K. Ntouyas, Surang Sitho, Teerasak Khoployklang and Jessada Tariboon

Axioms 2021, 10(4), 277; https://doi.org/10.3390/axioms10040277 - 27 Oct 2021

Cited by 3 | Viewed by 1168

Abstract

In the present research, we initiate the study of boundary value problems for sequential Riemann–Liouville and Hadamard–Caputo fractional derivatives, supplemented with iterated fractional integral boundary conditions. Firstly, we convert the given nonlinear problem into a fixed point problem by considering a linear variant [...] Read more.

In the present research, we initiate the study of boundary value problems for sequential Riemann–Liouville and Hadamard–Caputo fractional derivatives, supplemented with iterated fractional integral boundary conditions. Firstly, we convert the given nonlinear problem into a fixed point problem by considering a linear variant of the given problem. Once the fixed point operator is available, we use a variety of fixed point theorems to establish results regarding existence and uniqueness. Some properties of iteration that will be used in our study are also discussed. Examples illustrating our main results are also constructed. At the end, a brief conclusion is given. Our results are new in the given configuration and enrich the literature on boundary value problems for fractional differential equations. Full article

(This article belongs to the Special Issue Advances in Nonlinear Boundary Value Problems: Theory and Applications)

12 pages, 315 KiB

Open AccessArticle

Periodic Third-Order Problems with a Parameter

by Feliz Minhós and Nuno Oliveira

Axioms 2021, 10(3), 222; https://doi.org/10.3390/axioms10030222 - 11 Sep 2021

Cited by 3 | Viewed by 1672

Abstract

This work concerns with the solvability of third-order periodic fully problems with a weighted parameter, where the nonlinearity must verify only a local monotone condition and no periodic, coercivity or super or sublinearity restrictions are assumed, as usual in the literature. The arguments [...] Read more.

This work concerns with the solvability of third-order periodic fully problems with a weighted parameter, where the nonlinearity must verify only a local monotone condition and no periodic, coercivity or super or sublinearity restrictions are assumed, as usual in the literature. The arguments are based on a new type of lower and upper solutions method, not necessarily well ordered. A Nagumo growth condition and Leray–Schauder’s topological degree theory are the existence tools. Only the existence of solution is studied here and it will remain open the discussion on the non-existence and the multiplicity of solutions. Last section contains a nonlinear third-order differential model for periodic catatonic phenomena, depending on biological and/or chemical parameters. Full article

(This article belongs to the Special Issue Advances in Nonlinear Boundary Value Problems: Theory and Applications)

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16 pages, 407 KiB

Open AccessArticle

Numerical Solution for Singular Boundary Value Problems Using a Pair of Hybrid Nyström Techniques

by Mufutau Ajani Rufai and Higinio Ramos

Axioms 2021, 10(3), 202; https://doi.org/10.3390/axioms10030202 - 25 Aug 2021

Cited by 10 | Viewed by 1712

Abstract

This manuscript presents an efficient pair of hybrid Nyström techniques to solve second-order Lane–Emden singular boundary value problems directly. One of the proposed strategies uses three off-step points. The obtained formulas are paired with an appropriate set of formulas implemented for the first [...] Read more.

This manuscript presents an efficient pair of hybrid Nyström techniques to solve second-order Lane–Emden singular boundary value problems directly. One of the proposed strategies uses three off-step points. The obtained formulas are paired with an appropriate set of formulas implemented for the first step to avoid singularity at the left end of the integration interval. The fundamental properties of the proposed scheme are analyzed. Some test problems, including chemical kinetics and physical model problems, are solved numerically to determine the efficiency and validity of the proposed approach. Full article

(This article belongs to the Special Issue Advances in Nonlinear Boundary Value Problems: Theory and Applications)

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19 pages, 340 KiB

Open AccessArticle

On a Nonlinear Mixed Problem for a Parabolic Equation with a Nonlocal Condition

by Abdelkader Djerad, Ameur Memou and Ali Hameida

Axioms 2021, 10(3), 181; https://doi.org/10.3390/axioms10030181 - 06 Aug 2021

Cited by 1 | Viewed by 1465

Abstract

The aim of this work is to prove the well-posedness of some linear and nonlinear mixed problems with integral conditions defined only on two parts of the considered boundary. First, we establish for the associated linear problem a priori estimate and prove that [...] Read more.

The aim of this work is to prove the well-posedness of some linear and nonlinear mixed problems with integral conditions defined only on two parts of the considered boundary. First, we establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense using a functional analysis method. Then by applying an iterative process based on the obtained results for the linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the nonlinear problem. Full article

(This article belongs to the Special Issue Advances in Nonlinear Boundary Value Problems: Theory and Applications)

13 pages, 254 KiB

Open AccessArticle

On the Existence of Coupled Fractional Jerk Equations with Multi-Point Boundary Conditions

by Lei Hu, Yaozhen Han and Shuqin Zhang

Axioms 2021, 10(2), 103; https://doi.org/10.3390/axioms10020103 - 24 May 2021

Cited by 2 | Viewed by 1619

Abstract

By coincidence degree theory due to Mawhin, some sufficient conditions for the existence of solution for a class of coupled jerk equations with multi-point conditions are established. The new existence results have not yet been reported before. Novel coupled fractional jerk equations with [...] Read more.

By coincidence degree theory due to Mawhin, some sufficient conditions for the existence of solution for a class of coupled jerk equations with multi-point conditions are established. The new existence results have not yet been reported before. Novel coupled fractional jerk equations with resonant boundary value conditions are discussed in detail for the first time. Our work is interesting and complements known results. Full article

(This article belongs to the Special Issue Advances in Nonlinear Boundary Value Problems: Theory and Applications)

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