On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order

Ahmad, Bashir; Sivasundaram, S.

doi:10.1016/j.amc.2010.05.080

Cited by 140 publications

(75 citation statements)

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“…The monographs [14,19] and the papers [13,16,18,20] are excellent sources for the theory and applications of fractional calculus. Among all the topics, the existence of positive solutions of BVPs of fractional differential equations has been extensively studied by many researchers in recent years; see, for example, [1,2,3,6,7,8,9,11,21,23] and the references therein. In particular, Goodrich [9] studied the BVP consisting of the equation [9], the author first obtained some properties of the Green's function associated with the problem.…”

Section: Introductionmentioning

confidence: 99%

Positive solutions for a semipositone fractional boundary value problem with a forcing term

Graef

Kong

Yang

2011

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Section: Introductionmentioning

confidence: 99%

Positive solutions for a semipositone fractional boundary value problem with a forcing term

Graef

Kong

Yang

2011

fcaa

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“…We recall some fundamental results and definitions (Benchohra eta al., 2008;Ahmad and Sivasaundaram., 2010;Zhang, 2010;.…”

Section: Basic Materialsmentioning

confidence: 99%

Investigating at Least One Solution to a Systems of Multi-Points Boundary Value Problems

Shah¹,

Bushnaq²

2017

Matrix sci. math.

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AABSTRACTCurrent article is concerned with the investigating a coupled systems of boundary value problems(BVPs) of fractional order differential equations (FDEs). By means of classical fixed point theorems, we prove the existence and uniqueness of positive solution. We include a numerical problem to verify the investigated results. IntroductionIn recent years, the theory on existence and uniqueness of solutions to (FDEs) has got considerable attentions. In this regard, plenty of research papers and books are available in literature dealing various aspects of (FDEs) (Ahmad and Sivasaundaran, 2010;Rehman and Khan, 2010;Zhang, 2010). Existence and uniqueness of solutions for multi-point initial /(BVPs) have been studied by many researchers (Yang and Ge, 2009;El-Sayed and Bin-Taher, 2013;Cui et al., 2012;Salem, 2009;Han and Wang, 2011;El-Shahed and Nieto, 2010). On the other hand, coupled systems of (BVPs) for non-linear (FDEs) are also studied by many researchers and large numbers of papers can be found dealing with existence and uniqueness of solutions (Gafiychuk et al., 2009;Haq et al., 2016;Shah and Khan, 2015a;Shah et al., 2015b;Shoab et al., 2016). The area devoted to the study of coupled systems of both classical and arbitrary order differential equations is an active area of research. Because the concerned systems modeled variety of scientific problems of dynamical systems, psychological and biological phenomenon and chemical process (Benchohra et al., 2008;Perov, 1964;Han and Wang, 2011;. Authors some appropriate conditions for existence of solution to the given coupled systems of four point (BVPs) is the standard Caputo derivative . Some researcher has obtained enough conditions related to the existence theory of positive solutions to the given coupled systems of nonlinear three-point (BVPs)are continuous (Ahmad and Nieto, 2009).Motivated by the above studies, we develop some new existence and uniqueness results for the following coupled systems of nonlinear (BVPs) of (FDEs) . We apply Banach fixed point theorem and Leray-Schauder fixed point theorem to obtain proper conditions for existence and uniqueness results . We also provide a numerical problem to demonstrate the establish results. Basic materialWe recall some fundamental results and definitions (Benchohra eta al., 2008;Ahmad and Sivasaundaram., 2010;Zhang, 2010;.Definition 2.2. The Caputo fractional order derivative of a function 2) There existis equivalent to the following coupled system of FredHolm integralProof. By means of Definition 2.2 and Theorem 2.3, the solution ofNow applying the boundary conditionsSimilarly, we can get second equation of (2.2) by solving second equation of system of BVP (1.1) asTherefore, in view of Theorem 2.5, the considered coupled system (1.1) is equivalent to the following coupled system of Fred Holm integral equations Remark 2.6. We will usethroughout this paper. Method and discussion Define. Then, the product spacesTo proceed further the following assumptions hold:Define the operatorAlso we can easily show th...

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“…In the concerned theory, they studied existence, uniqueness, and multiplicity of solutions by using different techniques of nonlinear analysis. Therefore, theory on existence and uniqueness of solutions to nonlinear FODEs has been explored very well; see [8][9][10][11][12]. Systems of FODEs have been considered in large numbers of research articles, because most of physical, biological, and chemical phenomena can be modeled in the form of systems of FODEs.…”

Section: Introductionmentioning

confidence: 99%

“…The existence and uniqueness of solutions of FODEs are an active area of research for the last few decades. For some remarkable work, we refer the reader to [8,9,13,[16][17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Ulam Type Stability for a Coupled System of Boundary Value Problems of Nonlinear Fractional Differential Equations

Khan

Shah

et al. 2017

Journal of Function Spaces

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We discuss existence, uniqueness, and Hyers-Ulam stability of solutions for coupled nonlinear fractional order differential equations (FODEs) with boundary conditions. Using generalized metric space, we obtain some relaxed conditions for uniqueness of positive solutions for the mentioned problem by using Perov's fixed point theorem. Moreover, necessary and sufficient conditions are obtained for existence of at least one solution by Leray-Schauder-type fixed point theorem. Further, we also develop some conditions for Hyers-Ulam stability. To demonstrate our main result, we provide a proper example.

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On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order

Cited by 140 publications

References 39 publications

Positive solutions for a semipositone fractional boundary value problem with a forcing term

Positive solutions for a semipositone fractional boundary value problem with a forcing term

Investigating at Least One Solution to a Systems of Multi-Points Boundary Value Problems

Ulam Type Stability for a Coupled System of Boundary Value Problems of Nonlinear Fractional Differential Equations

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