The unique solvability issues of the Cauchy problem with a fractional derivative is considered in the case when the coefficient at the desired function is a continuous function. The solution of the problem is found in an explicit form. The uniqueness theorem is proved. The existence theorem for a solution to the problem is proved by reducing it to a Volterra equation of the second kind with a singularity in the kernel, and the necessary and sufficient conditions for the existence of a solution to the problem are obtained.
{"title":"On the unique solvability of a Cauchy problem with a fractional derivative","authors":"M. Kosmakova, A. Akhmetshin","doi":"10.31197/atnaa.1216018","DOIUrl":"https://doi.org/10.31197/atnaa.1216018","url":null,"abstract":"The unique solvability issues of the Cauchy problem with a fractional derivative is considered in the case when the coefficient at the desired function is a continuous function. The solution of the problem is found in an explicit form. The uniqueness theorem is proved. The existence theorem for a solution to the problem is proved by reducing it to a Volterra equation of the second kind with a singularity in the kernel, and the necessary and sufficient conditions for the existence of a solution to the problem are obtained.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75201265","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}
This paper is first study for considering nonlocal elliptic equation with Caputo derivative. We obtain the upper bound of the mild solution. The second contribution is to provide the lower bound of the solution at terminal time. We prove the non-correction of the problem in the sense of Hadamard. The main tool is the use of upper and lower bounds of the Mittag-Lefler function, combined with analysis in Hilbert scales space.
{"title":"On Caputo fractional elliptic equation with nonlocal condition","authors":"Tien Nguyen","doi":"10.31197/atnaa.1197560","DOIUrl":"https://doi.org/10.31197/atnaa.1197560","url":null,"abstract":"This paper is first study for considering nonlocal elliptic equation with Caputo derivative. We obtain the upper bound of the mild solution. The second contribution is to provide the lower bound of the solution at terminal time. We prove the non-correction of the problem in the sense of Hadamard. The main tool is the use of upper and lower bounds of the Mittag-Lefler function, combined with analysis in Hilbert scales space.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83526401","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 work four uncertain delay differential equations of Volterra-Levin type will be considered. Applying suitable contraction mapping and fixed point method, the stability of the equations will be studied. It will be shown that the solutions are bounded and, with additional condition, the solutions tend to zero. Also, a necessary and sufficient condition for the asymptotic stability of the solutions of an uncertain differential equation will be presented.
{"title":"Stability of Uncertain Equations of Volterra-Levin type and an Uncertain Delay Differential Equation Via Fixed Point Method","authors":"V. Roomi, Hamid Reza Ahmadi̇","doi":"10.31197/atnaa.1212287","DOIUrl":"https://doi.org/10.31197/atnaa.1212287","url":null,"abstract":"In this work four uncertain delay differential equations of Volterra-Levin type will be considered. Applying suitable contraction mapping and fixed point method, the stability of the equations will be studied. It will be shown that the solutions are bounded and, with additional condition, the solutions tend to zero. Also, a necessary and sufficient condition for the asymptotic stability of the solutions of an uncertain differential equation will be presented.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75552247","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 propose the triangular orthogonal functions as a basis functions for solution of weakly singular Volterra integral equations of the second kind. Powerful properties of these functions and some operational matrices are utilized in a direct method to reduce singular integral equation to some algebraic equations. The presented method does not need any integration for obtaining the constant coefficients. The method is computationally attractive, and applications are demonstrated through illustrative examples.
{"title":"Triangular functions in solving Weakly Singular Volterra integral equations","authors":"Monireh Nosrati̇, H. Afshari","doi":"10.31197/atnaa.1236577","DOIUrl":"https://doi.org/10.31197/atnaa.1236577","url":null,"abstract":"In this paper, we propose the triangular orthogonal functions as a basis functions \u0000for solution of weakly singular Volterra integral equations of the second \u0000kind. Powerful properties of these functions and some operational matrices \u0000are utilized in a direct method to reduce singular integral equation to \u0000some algebraic equations. The presented method does not need any integration \u0000for obtaining the constant coefficients. The method is computationally \u0000attractive, and applications are demonstrated through illustrative examples.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91021455","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}
This paper is mainly devoted to study a class of first-order differential inclusions governed by time-dependent subdifferential operators involving an integral perturbation. Employing then the constructive method used there, we also handle the associated second-order differential inclusion. Our final topic, accomplished in infinite-dimensional Hilbert spaces, is to develop some variants related to coupled systems by such differential inclusions and fractional differential equations.
{"title":"Coupled systems of subdifferential type with integral perturbation and fractional differential equations","authors":"Aya Bouabsa, S. Saidi","doi":"10.31197/atnaa.1149751","DOIUrl":"https://doi.org/10.31197/atnaa.1149751","url":null,"abstract":"This paper is mainly devoted to study a class of first-order differential inclusions governed by time-dependent subdifferential operators involving an integral perturbation. Employing then the constructive method used there, we also handle the associated second-order differential inclusion. \u0000Our final topic, accomplished in infinite-dimensional Hilbert spaces, is to develop some variants related to coupled systems by such differential inclusions and fractional differential equations.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86666070","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 propose and study convergence of neutrosophic random variables. Besides, some relations among these convergences are proved. Besides, we define the notion of neutrosophic weak law of large number and neutrosophic central limit theorem, also some applications examples are shown.
{"title":"Convergence of Neutrosophic Random Variables","authors":"Carlos Granados","doi":"10.31197/atnaa.1145837","DOIUrl":"https://doi.org/10.31197/atnaa.1145837","url":null,"abstract":"In this paper, we propose and study convergence of neutrosophic random variables. Besides, some relations among these convergences are proved. Besides, we define the notion of neutrosophic weak law of large number and neutrosophic central limit theorem, also some applications examples are shown.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74192769","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}
Fixed point theory is very useful in nonlinear analysis, diferential equations, differential and random differen- tial inclusions. It is well known that different types of fixed points implies the existence of specific solutions of the respective problem concerning differential equations or inclusions. There are several classifications of fixed points for single valued mappings. Recall that in 1949 M.K. Fort [19] introduced the notion of essential fixed points. In 1965 F.E. Browder [12], [13] introduced the notions of ejective and repulsive fixed points. In 1965 A.N. Sharkovsky [31] provided another classification of fixed points but only for continous mappings of subsets of the Euclidean space R n . For more information see also: [15], [18]-[22], [3], [25], [27], [31]. Note that for multivalued mappings these problems were considered only in a few papers (see: [2]-[8], [14], [23], [24], [32]) - always for admissible multivalued mappings of absolute neighbourhood retracts (ANR-s). In this paper ejective, repulsive and essential fixed points for admissible multivalued mappings of absolute neighbourhood multi retracts (ANMR-s) are studied. Let as remark that the class of MANR-s is much larger as the class of ANR-s (see: [32]). In order to study the above notions we generalize the fixed point index from the case of ANR-s onto the case of ANMR-s. Next using the above fixed point index we are able to prove several new results concerning repulsive ejective and essential fixed points of admissible multivalued mappings. Moreover, the random case is mentioned. For possible applications to differential and random di?erential inclusions see: [1], [2], [8]-[11], [16], [25], [26].
{"title":"Fixed Points of Multivalued Mappings Useful in the Theory of Differential and Random Differential Inclusions","authors":"L. Górniewicz","doi":"10.31197/atnaa.1204114","DOIUrl":"https://doi.org/10.31197/atnaa.1204114","url":null,"abstract":"Fixed point theory is very useful in nonlinear analysis, diferential equations, differential and random differen- tial inclusions. It is well known that different types of fixed points implies the existence of specific solutions of the respective problem concerning differential equations or inclusions. There are several classifications of fixed points for single valued mappings. Recall that in 1949 M.K. Fort [19] introduced the notion of essential fixed points. In 1965 F.E. Browder [12], [13] introduced the notions of ejective and repulsive fixed points. In 1965 A.N. Sharkovsky [31] provided another classification of fixed points but only for continous mappings of subsets of the Euclidean space R n . For more information see also: [15], [18]-[22], [3], [25], [27], [31]. Note that for multivalued mappings these problems were considered only in a few papers (see: [2]-[8], [14], [23], [24], [32]) - always for admissible multivalued mappings of absolute neighbourhood retracts (ANR-s). In this paper ejective, repulsive and essential fixed points for admissible multivalued mappings of absolute neighbourhood multi retracts (ANMR-s) are studied. Let as remark that the class of MANR-s is much larger as the class of ANR-s (see: [32]). In order to study the above notions we generalize the fixed point index from the case of ANR-s onto the case of ANMR-s. Next using the above fixed point index we are able to prove several new results concerning repulsive ejective and essential fixed points of admissible multivalued mappings. Moreover, the random case is mentioned. For possible applications to differential and random di?erential inclusions see: [1], [2], [8]-[11], [16], [25], [26].","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89029032","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}
Boutebba Hamza, Hakim Lakhal, Slimani Kamel, Belhadi Tahar
In this paper, we investigate the existence of nontrivial solutions in the Bessel Potential space for nonlinearfractional Schrödinger-Poisson system involving distributional Riesz fractional derivative. By using themountain pass theorem in combination with the perturbation method, we prove the existence of solutions.
{"title":"The nontrivial solutions for nonlinear fractional Schrödinger-Poisson system involving new fractional operator","authors":"Boutebba Hamza, Hakim Lakhal, Slimani Kamel, Belhadi Tahar","doi":"10.31197/atnaa.1141136","DOIUrl":"https://doi.org/10.31197/atnaa.1141136","url":null,"abstract":"In this paper, we investigate the existence of nontrivial solutions in the Bessel Potential space for nonlinearfractional Schrödinger-Poisson system involving distributional Riesz fractional derivative. By using themountain pass theorem in combination with the perturbation method, we prove the existence of solutions.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89384980","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 consider a class of abstract Cauchy problems in the framework of a generalized Caputo type fractional. We discuss the existence and uniqueness of mild solutions to such a class of fractional differential equations by using properties found in the related fractional calculus, the theory of uniformly continuous semigroups of operators and the fixed point theorem. Moreover, we discuss the continuous dependence on parameters and Ulam stability of the mild solutions. At the end of this paper, we bring forth some examples to endorse the obtained results
{"title":"On abstract Cauchy problems in the frame of a generalized Caputo type derivative","authors":"F. Jarad, T. Abdeljawad","doi":"10.31197/atnaa.1147950","DOIUrl":"https://doi.org/10.31197/atnaa.1147950","url":null,"abstract":"In this paper, we consider a class of abstract Cauchy problems in the framework of a generalized Caputo type fractional. We discuss the existence and uniqueness of mild solutions to such a class of fractional differential equations by using properties found in the related fractional calculus, the theory of uniformly continuous semigroups of operators and the fixed point theorem. Moreover, we discuss the continuous dependence on parameters and Ulam stability of the mild solutions. At the end of this paper, we bring forth some examples to endorse the obtained results","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81249004","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 our previous works, a Metatheorem in ordered fixed point theory showed that certain maximum principles can be reformulated to various types of fixed point theorems for progressive maps and conversely. Therefore, there should be the dual principles related to minimality, anti-progressive maps, and others. In the present article, we derive several minimum principles particular to Metatheorem and their applications. One of such applications is the Brøndsted-Jachymski Principle. We show that known examples due to Zorn (1935), Kasahara (1976), Brézis-Browder (1976), Taskovi¢ (1989), Zhong (1997), Khamsi (2009), Cobzas (2011) and others can be improved and strengthened by our new minimum principles.
{"title":"Applications of Several Minimum Principles","authors":"Sehie Park","doi":"10.31197/atnaa.1204381","DOIUrl":"https://doi.org/10.31197/atnaa.1204381","url":null,"abstract":"In our previous works, a Metatheorem in ordered fixed point theory showed that certain maximum principles can be reformulated to various types of fixed point theorems for progressive maps and conversely. Therefore, there should be the dual principles related to minimality, anti-progressive maps, and others. In the present article, we derive several minimum principles particular to Metatheorem and their applications. One of such applications is the Brøndsted-Jachymski Principle. We show that known examples due to Zorn (1935), Kasahara (1976), Brézis-Browder (1976), Taskovi¢ (1989), Zhong (1997), Khamsi (2009), Cobzas (2011) and others can be improved and strengthened by our new minimum principles.","PeriodicalId":7440,"journal":{"name":"Advances in the Theory of Nonlinear Analysis and its Application","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83422162","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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