Pub Date : 2024-03-29DOI: 10.1186/s13661-024-01856-0
Meiyu Liu, Minghe Pei, Libo Wang
We study the existence of solutions of the nonlinear second order m-point boundary value problem with p-Laplacian at resonance $$ textstylebegin{cases} (phi _{p}(x'))'=f(t,x,x'),quad tin [0,1], x'(0)=0, qquad x(1)=sum_{i=1}^{m-2}a_{i}x(xi _{i}), end{cases} $$ where $phi _{p}(s)=|s|^{p-2}s$ , $p>1$ , $f:[0,1]times mathbb{R}^{2}to mathbb{R}$ is a continuous function, $a_{i}>0$ ( $i=1,2,ldots ,m-2$ ) with $sum_{i=1}^{m-2}a_{i}=1$ , $0
我们研究了非线性二阶 m 点边界值问题的共振时 p-Laplacian 的解的存在性 $$ textstylebegin{cases} (phi _{p}(x'))'=f(t. x,x'),quad tin [0,1],qquad x'(0)=0、x,x'),quad tin [0,1],x'(0)=0, qquad x(1)=sum_{i=1}^{m-2}a_{i}x(xi _{i}), end{cases} $$ 其中 $phi _{p}(s)=|s|^{p-2}s$ , $p>1$ , $f：$f: [0,1]times mathbb{R}^{2}to mathbb{R}$ 是一个连续函数, $a_{i}>0$ ( $i=1,2,ldots ,m-2$ ) with $sum_{i=1}^{m-2}a_{i}=1$ , $0
{"title":"Solvability of a nonlinear second order m-point boundary value problem with p-Laplacian at resonance","authors":"Meiyu Liu, Minghe Pei, Libo Wang","doi":"10.1186/s13661-024-01856-0","DOIUrl":"https://doi.org/10.1186/s13661-024-01856-0","url":null,"abstract":"We study the existence of solutions of the nonlinear second order m-point boundary value problem with p-Laplacian at resonance $$ textstylebegin{cases} (phi _{p}(x'))'=f(t,x,x'),quad tin [0,1], x'(0)=0, qquad x(1)=sum_{i=1}^{m-2}a_{i}x(xi _{i}), end{cases} $$ where $phi _{p}(s)=|s|^{p-2}s$ , $p>1$ , $f:[0,1]times mathbb{R}^{2}to mathbb{R}$ is a continuous function, $a_{i}>0$ ( $i=1,2,ldots ,m-2$ ) with $sum_{i=1}^{m-2}a_{i}=1$ , $0<xi _{1}<xi _{2}<cdots <xi _{m-2}<1$ . Based on the topological transversality method together with the barrier strip technique and the cut-off technique, we obtain new existence results of solutions of the above problem. Meanwhile some examples are also given to illustrate our main results.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"34 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140323590","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-29DOI: 10.1186/s13661-024-01851-5
Hassan Eltayeb
The current paper concentrates on discovering the exact solutions of the time-fractional regular and singular coupled Burger’s equations by involving a new technique known as the double Sumudu-generalized Laplace and Adomian decomposition method. Furthermore, some theorems of the double Sumudu-generalized Laplace properties are proved. Further, the offered method is a powerful tool for solving an enormous number of problems. The precision of the technique is evaluated with the aid of some examples, this method offers a solution precisely and successfully in a series form with smoothly calculated coefficients. The relation between both the approximate and exact solution is represented by a graph to display the high speed of this method’s convergence.
{"title":"Application of double Sumudu-generalized Laplace decomposition method and two-dimensional time-fractional coupled Burger’s equation","authors":"Hassan Eltayeb","doi":"10.1186/s13661-024-01851-5","DOIUrl":"https://doi.org/10.1186/s13661-024-01851-5","url":null,"abstract":"The current paper concentrates on discovering the exact solutions of the time-fractional regular and singular coupled Burger’s equations by involving a new technique known as the double Sumudu-generalized Laplace and Adomian decomposition method. Furthermore, some theorems of the double Sumudu-generalized Laplace properties are proved. Further, the offered method is a powerful tool for solving an enormous number of problems. The precision of the technique is evaluated with the aid of some examples, this method offers a solution precisely and successfully in a series form with smoothly calculated coefficients. The relation between both the approximate and exact solution is represented by a graph to display the high speed of this method’s convergence.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"2012 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140323570","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-28DOI: 10.1186/s13661-024-01852-4
Tuba Tunç, İzzettin Demir
In mathematics and the applied sciences, as a very useful tool, fractional calculus is a basic concept. Furthermore, in many areas of mathematics, it is better to use a new hybrid fractional operator, which combines the proportional and Caputo operators. So we concentrate on the proportional Caputo-hybrid operator because of its numerous applications. In this research, we introduce a novel extension of the Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator and establish an identity. Then, taking into account this novel generalized identity, we develop some integral inequalities associated with the left-side of Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator. Moreover, to illustrate the newly established inequalities, we give some examples with the help of graphs.
{"title":"On a new version of Hermite–Hadamard-type inequality based on proportional Caputo-hybrid operator","authors":"Tuba Tunç, İzzettin Demir","doi":"10.1186/s13661-024-01852-4","DOIUrl":"https://doi.org/10.1186/s13661-024-01852-4","url":null,"abstract":"In mathematics and the applied sciences, as a very useful tool, fractional calculus is a basic concept. Furthermore, in many areas of mathematics, it is better to use a new hybrid fractional operator, which combines the proportional and Caputo operators. So we concentrate on the proportional Caputo-hybrid operator because of its numerous applications. In this research, we introduce a novel extension of the Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator and establish an identity. Then, taking into account this novel generalized identity, we develop some integral inequalities associated with the left-side of Hermite–Hadamard-type inequalities for proportional Caputo-hybrid operator. Moreover, to illustrate the newly established inequalities, we give some examples with the help of graphs.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"74 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140323472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to the study of a mixed boundary value problem for a complete Sturm–Liouville equation, where the coefficients can also be negative. In particular, the existence of infinitely many distinct positive solutions to the given problem is obtained by using critical point theory.
{"title":"Mixed boundary value problems involving Sturm–Liouville differential equations with possibly negative coefficients","authors":"Gabriele Bonanno, Giuseppina D’Aguì, Valeria Morabito","doi":"10.1186/s13661-024-01848-0","DOIUrl":"https://doi.org/10.1186/s13661-024-01848-0","url":null,"abstract":"This paper is devoted to the study of a mixed boundary value problem for a complete Sturm–Liouville equation, where the coefficients can also be negative. In particular, the existence of infinitely many distinct positive solutions to the given problem is obtained by using critical point theory.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"13 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140312689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-28DOI: 10.1186/s13661-024-01853-3
Xinyu Wang, Na Wang
In the study of singularly perturbed boundary problems with turning points, the solution undergoes sharp changes near these points and exhibits various interior phenomena. We employ the matching asymptotic expansion method to analyze and solve a singularly perturbed boundary and interior layers problem with multiple turning points, resulting in a composite expansion that fits well with the numerical solution. The solution demonstrates a strong association with special functions, which is verified by the theory of differential inequalities.
{"title":"Singular perturbation boundary and interior layers problems with multiple turning points","authors":"Xinyu Wang, Na Wang","doi":"10.1186/s13661-024-01853-3","DOIUrl":"https://doi.org/10.1186/s13661-024-01853-3","url":null,"abstract":"In the study of singularly perturbed boundary problems with turning points, the solution undergoes sharp changes near these points and exhibits various interior phenomena. We employ the matching asymptotic expansion method to analyze and solve a singularly perturbed boundary and interior layers problem with multiple turning points, resulting in a composite expansion that fits well with the numerical solution. The solution demonstrates a strong association with special functions, which is verified by the theory of differential inequalities.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"13 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140312696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-28DOI: 10.1186/s13661-024-01846-2
Yan Pang, Junping Xie, Xingyong Zhang
We investigate a generalized poly-Laplacian system with a parameter on weighted finite graphs, a generalized poly-Laplacian system with a parameter and Dirichlet boundary value on weighted locally finite graphs, and a $(p,q)$ -Laplacian system with a parameter on weighted locally finite graphs. We utilize a critical points theorem built by Bonanno and Bisci [Bonanno, Bisci, and Regan, Math. Comput. Model. 52(1-2):152–160, 2010], which is an abstract critical points theorem without compactness condition, to obtain that these systems have infinitely many nontrivial solutions with unbounded norm when the parameters locate some well-determined range.
{"title":"Infinitely many solutions for three quasilinear Laplacian systems on weighted graphs","authors":"Yan Pang, Junping Xie, Xingyong Zhang","doi":"10.1186/s13661-024-01846-2","DOIUrl":"https://doi.org/10.1186/s13661-024-01846-2","url":null,"abstract":"We investigate a generalized poly-Laplacian system with a parameter on weighted finite graphs, a generalized poly-Laplacian system with a parameter and Dirichlet boundary value on weighted locally finite graphs, and a $(p,q)$ -Laplacian system with a parameter on weighted locally finite graphs. We utilize a critical points theorem built by Bonanno and Bisci [Bonanno, Bisci, and Regan, Math. Comput. Model. 52(1-2):152–160, 2010], which is an abstract critical points theorem without compactness condition, to obtain that these systems have infinitely many nontrivial solutions with unbounded norm when the parameters locate some well-determined range.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"112 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140323509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-27DOI: 10.1186/s13661-024-01850-6
Yang Yu, Qi Ge
In this paper, we study the existence and uniqueness of positive solutions for a class of a fractional differential equation system of Riemann–Liouville type on infinite intervals with infinite-point boundary conditions. First, the higher-order equation is reduced to the lower-order equation, and then it is transformed into the equivalent integral equation. Secondly, we obtain the existence and uniqueness of positive solutions for each fixed parameter $lambda >0$ by using the mixed monotone operators fixed-point theorem. The results obtained in this paper show that the unique positive solution has good properties: continuity, monotonicity, iteration, and approximation. Finally, an example is given to demonstrate the application of our main results.
{"title":"Positive solutions for the Riemann–Liouville-type fractional differential equation system with infinite-point boundary conditions on infinite intervals","authors":"Yang Yu, Qi Ge","doi":"10.1186/s13661-024-01850-6","DOIUrl":"https://doi.org/10.1186/s13661-024-01850-6","url":null,"abstract":"In this paper, we study the existence and uniqueness of positive solutions for a class of a fractional differential equation system of Riemann–Liouville type on infinite intervals with infinite-point boundary conditions. First, the higher-order equation is reduced to the lower-order equation, and then it is transformed into the equivalent integral equation. Secondly, we obtain the existence and uniqueness of positive solutions for each fixed parameter $lambda >0$ by using the mixed monotone operators fixed-point theorem. The results obtained in this paper show that the unique positive solution has good properties: continuity, monotonicity, iteration, and approximation. Finally, an example is given to demonstrate the application of our main results.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"20 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140312621","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-25DOI: 10.1186/s13661-024-01845-3
Ze Gao, Qiming Guo
We obtain the gradient estimates of the positive solutions to a nonlinear elliptic equation on an n-dimensional complete Riemannian manifold $(M, g)$ $$ Delta u +au(ln{u})^{p}+buln{u}=0, $$ where $ane 0$ , b are two constants and $p=frac{k_{1}}{2k_{2}+1}ge 2$ , here $k_{1}$ and $k_{2}$ are two positive integers. The gradient bound is independent of the bounds of the solution and the Laplacian of the distance function. As the applications of the estimates, we show the Harnack inequality and the upper bound of the solution.
我们得到了 n 维完整黎曼流形 $(M. g)$$ 上非线性椭圆方程正解的梯度估计值、g)$ $$ Delta u +au(ln{u})^{p}+buln{u}=0, $$ 其中 $ane 0$ , b 是两个常数，$p=frac{k_{1}}{2k_{2}+1}ge 2$ , 这里 $k_{1}$ 和 $k_{2}$ 是两个正整数。梯度边界与解的边界和距离函数的拉普拉奇无关。作为估计值的应用，我们展示了哈纳克不等式和解的上界。
{"title":"Gradient estimates for a class of elliptic equations with logarithmic terms","authors":"Ze Gao, Qiming Guo","doi":"10.1186/s13661-024-01845-3","DOIUrl":"https://doi.org/10.1186/s13661-024-01845-3","url":null,"abstract":"We obtain the gradient estimates of the positive solutions to a nonlinear elliptic equation on an n-dimensional complete Riemannian manifold $(M, g)$ $$ Delta u +au(ln{u})^{p}+buln{u}=0, $$ where $ane 0$ , b are two constants and $p=frac{k_{1}}{2k_{2}+1}ge 2$ , here $k_{1}$ and $k_{2}$ are two positive integers. The gradient bound is independent of the bounds of the solution and the Laplacian of the distance function. As the applications of the estimates, we show the Harnack inequality and the upper bound of the solution.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"24 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140301095","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-25DOI: 10.1186/s13661-024-01847-1
Dumitru Motreanu, Abdolrahman Razani
The aim of this paper is to prove the existence of generalized variational solutions for nonlinear Dirichlet problems driven by anisotropic and Finsler Laplacian competing operators. The main difficulty consists in the lack of ellipticity and monotonicity in the principal part of the equations. This difficulty is overcome by developing a Galerkin-type procedure.
{"title":"Competing anisotropic and Finsler ((p,q))-Laplacian problems","authors":"Dumitru Motreanu, Abdolrahman Razani","doi":"10.1186/s13661-024-01847-1","DOIUrl":"https://doi.org/10.1186/s13661-024-01847-1","url":null,"abstract":"The aim of this paper is to prove the existence of generalized variational solutions for nonlinear Dirichlet problems driven by anisotropic and Finsler Laplacian competing operators. The main difficulty consists in the lack of ellipticity and monotonicity in the principal part of the equations. This difficulty is overcome by developing a Galerkin-type procedure.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"32 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140301096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-18DOI: 10.1186/s13661-023-01795-2
Umut Bektaş, Halil Anaç
This paper solves fractional differential equations using the Shehu transform in combination with the q-homotopy analysis transform method (q-HATM). As the Shehu transform is only applicable to linear equations, q-HATM is an efficient technique for approximating solutions to nonlinear differential equations. In nonlinear systems that explain the emergence of stripes in 2D systems, the Newell–Whitehead–Segel equation plays a significant role. The findings indicate that the outcomes derived from the tables yield superior results compared to the existing LTDM in the literature. Maple is utilized to depict three-dimensional surfaces and find numerical values that are displayed in a table.
{"title":"A hybrid method to solve a fractional-order Newell–Whitehead–Segel equation","authors":"Umut Bektaş, Halil Anaç","doi":"10.1186/s13661-023-01795-2","DOIUrl":"https://doi.org/10.1186/s13661-023-01795-2","url":null,"abstract":"This paper solves fractional differential equations using the Shehu transform in combination with the q-homotopy analysis transform method (q-HATM). As the Shehu transform is only applicable to linear equations, q-HATM is an efficient technique for approximating solutions to nonlinear differential equations. In nonlinear systems that explain the emergence of stripes in 2D systems, the Newell–Whitehead–Segel equation plays a significant role. The findings indicate that the outcomes derived from the tables yield superior results compared to the existing LTDM in the literature. Maple is utilized to depict three-dimensional surfaces and find numerical values that are displayed in a table.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"98 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140153027","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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