. In this paper, we study the multiplicity of solutions for a class of concave-convex p - Laplacian equations with the combined effect of coef fi cient functions of concave-convex terms. By the Nehari method and some analysis techniques, we obtain an exact constant for the effect of coef fi cient functions of concave-convex terms to ensure this problem has two nonzero and nonnegative solutions and give the relation of size of the two solutions. Moreover, under some stronger conditions, we prove that the two solutions are positive. Our results generalize and improve some known results in the literature.
{"title":"The Nehari Manifold for a p-Laplacian equation with concave-convex nonlinearities and sign-changing potential","authors":"Hong-Ying Li","doi":"10.7153/DEA-2019-11-16","DOIUrl":"https://doi.org/10.7153/DEA-2019-11-16","url":null,"abstract":". In this paper, we study the multiplicity of solutions for a class of concave-convex p - Laplacian equations with the combined effect of coef fi cient functions of concave-convex terms. By the Nehari method and some analysis techniques, we obtain an exact constant for the effect of coef fi cient functions of concave-convex terms to ensure this problem has two nonzero and nonnegative solutions and give the relation of size of the two solutions. Moreover, under some stronger conditions, we prove that the two solutions are positive. Our results generalize and improve some known results in the literature.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we are concern with the existence of positive solutions for a singular system of nonlinear fractional q -difference equations with coupled integral boundary conditions and two parameters. By using the properties of the Green’s function and Guo-Krasnosel’skii fixed point theorem, some existence results of at least one positive solution are obtained. As applications, two examples are presented to illustrate the main results. Mathematics subject classification (2010): 39A13, 34B18, 34A08.
{"title":"Positive solutions for a singular coupled system of nonlinear higher-order fractional q-difference boundary value problems with two parameters","authors":"Wengui Yang","doi":"10.7153/dea-2019-11-25","DOIUrl":"https://doi.org/10.7153/dea-2019-11-25","url":null,"abstract":"In this paper, we are concern with the existence of positive solutions for a singular system of nonlinear fractional q -difference equations with coupled integral boundary conditions and two parameters. By using the properties of the Green’s function and Guo-Krasnosel’skii fixed point theorem, some existence results of at least one positive solution are obtained. As applications, two examples are presented to illustrate the main results. Mathematics subject classification (2010): 39A13, 34B18, 34A08.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a sixth order two point boundary value problem. Upper and lower estimates for positive solutions of the problem are proved. Sufficient conditions for the existence and nonexistence of positive solutions for the problem are obtained. An example is included to illustrate the results.
{"title":"Positive solutions to a nonlinear sixth order boundary value problem","authors":"Bo Yang","doi":"10.7153/DEA-2019-11-13","DOIUrl":"https://doi.org/10.7153/DEA-2019-11-13","url":null,"abstract":"We consider a sixth order two point boundary value problem. Upper and lower estimates for positive solutions of the problem are proved. Sufficient conditions for the existence and nonexistence of positive solutions for the problem are obtained. An example is included to illustrate the results.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"38 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this manuscript, we study a system of fractional integro boundary value problem on unbounded domain. The solution of the system is de fi ned in terms of the Green’s function. We have established the existence and uniqueness results by utilizing the fi xed point theorems. The main outcomes and assumptions are veri fi ed via some examples.
{"title":"Positive solutions for fractional integro-boundary value problem of order (1,2) on an unbounded domain","authors":"V. Gupta, J. Dabas","doi":"10.7153/DEA-2019-11-14","DOIUrl":"https://doi.org/10.7153/DEA-2019-11-14","url":null,"abstract":". In this manuscript, we study a system of fractional integro boundary value problem on unbounded domain. The solution of the system is de fi ned in terms of the Green’s function. We have established the existence and uniqueness results by utilizing the fi xed point theorems. The main outcomes and assumptions are veri fi ed via some examples.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Weconsider the asymptotic behavior of solutions to the nonlocal nonlinear Schr¨odinger equation with dissipative nonlinearity. We prove that there exists a solution which has different behavior from that of the typical cubic nonlinear Schr¨odinger equation.
{"title":"Final state problem for the nonlocal nonlinear Schrödinger equation with dissipative nonlinearity","authors":"Mamoru Okamoto, Kota Uriya","doi":"10.7153/dea-2019-11-23","DOIUrl":"https://doi.org/10.7153/dea-2019-11-23","url":null,"abstract":". Weconsider the asymptotic behavior of solutions to the nonlocal nonlinear Schr¨odinger equation with dissipative nonlinearity. We prove that there exists a solution which has different behavior from that of the typical cubic nonlinear Schr¨odinger equation.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study second order two-parametric quantum boundary value problems. The main aims of this paper are presented in two steps. In the first step, we consider second order two-parametric quantum boundary value problems with general nonlinearities and by the use of Krasnoselskii fixed point theorem on positive cones we provide some sufficient conditions to reach the existence, multiplicity and nonexistence of positive solutions. At the and of this step, some illustrative examples are given to show practical implementability of the obtained theoretical results. In the second step, we consider the corresponding two-parametric quantum eigenvalue problems and in the light of Lyapunov inequalities, we present a lower bound estimation for positive eigenvalues. We complete this step with a numerical evaluation to identify validity of the obtained lower bound.
{"title":"Second order two-parametric quantum boundary value problems","authors":"Yousef Gholami","doi":"10.7153/dea-2019-11-10","DOIUrl":"https://doi.org/10.7153/dea-2019-11-10","url":null,"abstract":"In this paper we study second order two-parametric quantum boundary value problems. The main aims of this paper are presented in two steps. In the first step, we consider second order two-parametric quantum boundary value problems with general nonlinearities and by the use of Krasnoselskii fixed point theorem on positive cones we provide some sufficient conditions to reach the existence, multiplicity and nonexistence of positive solutions. At the and of this step, some illustrative examples are given to show practical implementability of the obtained theoretical results. In the second step, we consider the corresponding two-parametric quantum eigenvalue problems and in the light of Lyapunov inequalities, we present a lower bound estimation for positive eigenvalues. We complete this step with a numerical evaluation to identify validity of the obtained lower bound.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"158 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130054","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Classical fixed point theorems often begin with the assumption that we have a mapping P of a non-empty, closed, bounded, convex set G in a Banach space into itself. Then a number of conditions are added which will ensure that there is at least one fixed point in the set G . These fixed point theorems have been very effective with many problems in applied mathematics, particularly for integral equations containing a term ∫ t 0 A(t− s)v(t,s,x(s))ds, because such terms frequently map sets of bounded continuous functions into compact sets. But there is a large and important class of integral equations from applied mathematics containing such a term with a coefficient function f (t,x) which destroys all compactness. Investigators have then turned to Darbo’s fixed point theorem and measures of non-compactness to get a (possibly non-unique) fixed point. In this paper: a) We offer an elementary alternative to measures of non-compactness and Darbo’s theorem by using progressive contractions. This method yields a unique fixed point (unlike Darbo’s theorem) which, in turn, by default yields asymptotic stability as introduced in [1]. b) We lift the growth requirements in both x and t seen using Darbo’s theorem. c) We offer a technique for finding the mapping set G .
经典不动点定理通常以一个假设开始:我们有一个映射P，它是Banach空间中一个非空的、封闭的、有界的凸集G到它自身的映射。然后添加一些条件，以确保集合G中至少有一个不动点。这些不动点定理对于应用数学中的许多问题都是非常有效的，特别是对于包含一项∫t0 a (t - s)v(t,s,x(s))ds的积分方程，因为这些项经常将有界连续函数的集合映射成紧集。但在应用数学中，有一类重要的积分方程包含这样一个项，它的系数函数是f (t,x)，它破坏了所有的紧性。研究人员随后转向Darbo的不动点定理和非紧性的度量来获得一个(可能非唯一的)不动点。在这篇论文中:a)我们提供了一种基本的替代非紧性度量和用渐进收缩的Darbo定理。这种方法产生一个唯一的不动点(与Darbo定理不同)，而这个不动点反过来又默认产生[1]中介绍的渐近稳定性。b)我们用Darbo定理提高了x和t的增长要求。c)我们提供了一种寻找映射集G的技术。
{"title":"Progressive contractions, measures of non-compactness and quadratic integral equations","authors":"T. Burton, I. Purnaras","doi":"10.7153/dea-2019-11-12","DOIUrl":"https://doi.org/10.7153/dea-2019-11-12","url":null,"abstract":"Classical fixed point theorems often begin with the assumption that we have a mapping P of a non-empty, closed, bounded, convex set G in a Banach space into itself. Then a number of conditions are added which will ensure that there is at least one fixed point in the set G . These fixed point theorems have been very effective with many problems in applied mathematics, particularly for integral equations containing a term ∫ t 0 A(t− s)v(t,s,x(s))ds, because such terms frequently map sets of bounded continuous functions into compact sets. But there is a large and important class of integral equations from applied mathematics containing such a term with a coefficient function f (t,x) which destroys all compactness. Investigators have then turned to Darbo’s fixed point theorem and measures of non-compactness to get a (possibly non-unique) fixed point. In this paper: a) We offer an elementary alternative to measures of non-compactness and Darbo’s theorem by using progressive contractions. This method yields a unique fixed point (unlike Darbo’s theorem) which, in turn, by default yields asymptotic stability as introduced in [1]. b) We lift the growth requirements in both x and t seen using Darbo’s theorem. c) We offer a technique for finding the mapping set G .","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130107","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We derive Lyapunov-type inequalities for certain fractional differential equations of order α , where 1 < α (cid:2) 2 or 2 < α (cid:2) 3. The methods used within rely on considering the maximum value of a nontrivial solution in a given interval as opposed to traditional methods which utilize the Green’s function. This particular method provides versatility and can be applied to other fractional boundary value problems where the Green’s function is inaccessible. Furthermore, we demonstrate how the inequalities may be extended to fractional multivariate equations in both the left and right-fractional cases.
{"title":"A non Green's function approach to fractional Lyapunov-type inequalities with applications to multivariate domains","authors":"Sougata Dhar, J. Kelly","doi":"10.7153/DEA-2019-11-19","DOIUrl":"https://doi.org/10.7153/DEA-2019-11-19","url":null,"abstract":". We derive Lyapunov-type inequalities for certain fractional differential equations of order α , where 1 < α (cid:2) 2 or 2 < α (cid:2) 3. The methods used within rely on considering the maximum value of a nontrivial solution in a given interval as opposed to traditional methods which utilize the Green’s function. This particular method provides versatility and can be applied to other fractional boundary value problems where the Green’s function is inaccessible. Furthermore, we demonstrate how the inequalities may be extended to fractional multivariate equations in both the left and right-fractional cases.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"4 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130329","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a bounded open set Ω ⊂ Rn , n 3 , we consider the nonlinear fourth-order partial differential equation ∑|α|=1,2 (−1)|α|Dα Aα(x,u,Du,Du)+B(x,u,Du,Du) = 0. It is assumed that the principal coefficients {Aα}|α|=1,2 satisfy the growth and coercivity conditions suitable for the energy space W̊ 1,q 2,p (Ω) = W̊ 1,q(Ω)∩W̊ 2,p(Ω) , 1 < p< n/2 , 2p < q < n . The lower-order term B(x,u,Du,D2u) behaves as b(u) {|Du|q + |D2u|p}+g(x) where g ∈ Lτ (Ω) , τ > n/q . We establish the Hölder continuity up to the boundary of any solution u∈ W̊ 1,q 2,p (Ω)∩L∞(Ω) by using the measure density condition on ∂Ω , an interior local result and a modified Moser method with special test function.
{"title":"Hölder continuity up to the boundary of solutions to nonlinear fourth-order elliptic equations with natural growth terms","authors":"S. Bonafede, M. Voitovych","doi":"10.7153/DEA-2019-11-03","DOIUrl":"https://doi.org/10.7153/DEA-2019-11-03","url":null,"abstract":"In a bounded open set Ω ⊂ Rn , n 3 , we consider the nonlinear fourth-order partial differential equation ∑|α|=1,2 (−1)|α|Dα Aα(x,u,Du,Du)+B(x,u,Du,Du) = 0. It is assumed that the principal coefficients {Aα}|α|=1,2 satisfy the growth and coercivity conditions suitable for the energy space W̊ 1,q 2,p (Ω) = W̊ 1,q(Ω)∩W̊ 2,p(Ω) , 1 < p< n/2 , 2p < q < n . The lower-order term B(x,u,Du,D2u) behaves as b(u) {|Du|q + |D2u|p}+g(x) where g ∈ Lτ (Ω) , τ > n/q . We establish the Hölder continuity up to the boundary of any solution u∈ W̊ 1,q 2,p (Ω)∩L∞(Ω) by using the measure density condition on ∂Ω , an interior local result and a modified Moser method with special test function.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
C. Martinez, A. Martinez, G. Bressan, E. V. Castelani, Roberto Molina de Souza
. We consider in this work the fourth order equation with nonlinear boundary condi- tions. We present the result for the existence of multiple solutions based on the Avery-Peterson fi xed-point theorem. This work is also a study for numerical solutions based on the Levenberg- Maquardt method with a heuristic strategy for initial points that proposes to numerically deter-mine multiple solutions to the problem addressed.
{"title":"Multiple solutions for a fourth order equation with nonlinear boundary conditions: theoretical and numerical aspects","authors":"C. Martinez, A. Martinez, G. Bressan, E. V. Castelani, Roberto Molina de Souza","doi":"10.7153/DEA-2019-11-15","DOIUrl":"https://doi.org/10.7153/DEA-2019-11-15","url":null,"abstract":". We consider in this work the fourth order equation with nonlinear boundary condi- tions. We present the result for the existence of multiple solutions based on the Avery-Peterson fi xed-point theorem. This work is also a study for numerical solutions based on the Levenberg- Maquardt method with a heuristic strategy for initial points that proposes to numerically deter-mine multiple solutions to the problem addressed.","PeriodicalId":51863,"journal":{"name":"Differential Equations & Applications","volume":"1 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71130181","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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