Volume 2: Issue 1

Volume 2: Issue 1

STABILITY, EXISTENCE AND UNIQUENESS OF A COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS WITH INTEGRAL BOUNDARY CONDITIONS

Areen Al-Khateeb^{1, }Ayman A.hazaymeh ^{2}, Mowafaq al-tahat ^{3}

^{1 }Jadara University, Irbid, Jordan; areen.k@jadara.edu.jo;

^{2 }Jadara University, Irbid, Jordan; aymanha@jadara.edu.jo;

^{ 3 }Jadara University, Irbid, Jordan; mwafag.tahat1997@gmail.com

In this article a coupled system of fractional differential equations with integral boundary conditions will be discussed. Currently we present three main result of this study: firstly, the uniqueness of solution for the given problem is established by applying contraction mapping principle. Secondly,Leray-Schauder’s alternative has been used to obtain the existence of solutions. Moreover, some necessary conditions for theHyers-Ulamtype stability to the solutions of the boundary value problem (BVPs) are developed. Finally the results are supported by examples.

- Podlubny, I: Fractional Differential Equations. Academic Press, San Diego (1999)
- Kilbas, AA, Srivastava, HM, Trujillo, JJ: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier, Amsterdam (2006)
- K. Ntouyas and M. Obaid, “A coupled system of fractional differential equations with nonlocal integral boundary conditions,” Advances in Difference Equations, vol. 2012, no. 1, pp. 130-139, 2012.
- Ahmad and S. K. Ntouyas, “A Coupled System of Nonlocal Fractional Differential Equations with Coupled and Uncoupled Slit-Strips-Type Integral Boundary Conditions,” Journal of Mathematical Sciences, vol. 226, no. 3, pp. 175-196, 2017.
- H. Alsulami, S. K. Ntouyas, R. P. Agarwal, B. Ahmad and A. Alsaedi, “A study of fractional-order coupled systems with a new concept of coupled non-separated boundary conditions,” Boundary Value Problems, vol. 2017, no. 1, pp. 68-74, 2017.
- Zhang, Y., Bai, Z., & Feng, T. Existence results for a coupled system of nonlinear fractional three-point boundary value problems at resonance. Computers & Mathematics with Applications, 2011. , 61(4), 1032-1047.
- Granas, A, Dugundji, J: Fixed Point Theory. Springer, New York (2005)/
- Hyers, Donald H. “On the stability of the linear functional equation.” Proceedings of the National Academy of Sciences of the United States of America 27.4 (1941): 222.
- Rus, Ioan A. “Ulam stabilities of ordinary differential equations in a Banach space.” Carpathian journal of Mathematics(2010): 103-107.
- Cabada, Alberto, and Guotao Wang. “Positive solutions of nonlinear fractional differential equations with integral boundary value conditions.” Journal of Mathematical Analysis and Applications 389.1 (2012): 403-411.
- Graef, John R., Lingju Kong, and Min Wang. “Existence and uniqueness of solutions for a fractional boundary value problem on a graph.” Fractional Calculus and Applied Analysis17.2 (2014): 499-510.
- Ahmad, Bashir, and Juan J. Nieto. “Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions.” Computers & Mathematics with Applications 58.9 (2009): 1838-1843.
- Su, Xinwei. “Boundary value problem for a coupled system of nonlinear fractional differential equations.” Applied Mathematics Letters 22.1 (2009): 64-69.
- Wang, Jinhua, Hongjun Xiang, and Zhigang Liu. “Positive solution to nonzero boundary values problem for a coupled system of nonlinear fractional differential equations.” International Journal of Differential Equations (2010).
- Ntouyas, Sotiris K., and Mustafa Obaid. “A coupled system of fractional differential equations with nonlocal integral boundary conditions.” Advances in Difference Equations 2012.1 (2012): 130.
- Ahmad, Bashir, Sotiris K. Ntouyas, and Ahmed Alsaedi. “On a coupled system of fractional differential equations with coupled nonlocal and integral boundary conditions.” Chaos, Solitons & Fractals 83 (2016): 234-241.
- Zhai, Chengbo, and Li Xu. “Properties of positive solutions to a class of four-point boundary value problem of Caputo fractional differential equations with a parameter.” Communications in Nonlinear Science and Numerical Simulation 19.8 (2014): 2820-2827.
- Ahmad, Bashir, and Sotiris K. Ntouyas. “Existence results for a coupled system of Caputo type sequential fractional differential equations with nonlocal integral boundary conditions. Applied Mathematics and Computation 266 (2015): 615-622.
- Tariboon, Jessada, Sotiris K. Ntouyas, and WeerawatSudsutad. “Coupled systems of Riemann-Liouville fractional differential equations with Hadamard fractional integral boundary conditions.” J. Nonlinear Sci. Appl 9.1 (2016): 295-308.

The Mathematical Errors of Infinity

Anthony C. Patton, San Diego, California, USA, pattonanthonyc@gmail.com

Philosophical analysis of infinity has identified multiple logical and metaphysical problems that warrant a response. However, mathematicians generally do not consider these criticisms valid or relevant and instead rely on supposed mathematical rigor to assess the validity or practical uses of infinity. Therefore, this paper will highlight three errors in the mathematics of infinity to address mathematicians on their own terms.

- Patton, Anthony C. The End of Infinity: Where Mathematics and Philosophy Meet, (New York: Algora Publishing, 2018).
- Alexander, Amir, Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (New York: Scientific American/Farrar, Straus and Giroux, 2014).
- Aristotle, The Basic Works of Aristotle, edited by Richard McKeon (New York: Random House, 1941).
- Bagaria, Joan, “Set Theory”, The Stanford Encyclopedia of Philosophy (Winter 2014 Edition), Edward N. Zalta (ed.), URL = .
- Balaguer, Mark, “Fictionalism in the Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Summer 2015 Edition), Edward N. Zalta (ed.), URL = .
- Bell, John L., “Continuity and Infinitesimals”, The Stanford Encyclopedia of Philosophy (Winter 2014 Edition), Edward N. Zalta (ed.), URL = .
- Bridges, Douglas and Palmgren, Erik, “Constructive Mathematics”, The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (ed.), URL = .
- Bueno, Otávio, “Nominalism in the Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Spring 2014 Edition), Edward N. Zalta (ed.), URL = .
- Colyvan, Mark, “Indispensability Arguments in the Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Spring 2015 Edition), Edward N. Zalta (ed.), URL = .
- Copleston, Frederick, A History of Philosophy Volume I: Greece and Rome (New York: Image Books, 1993).
- Dauben, Joseph Warren, “Georg Cantor: His Mathematics and Philosophy of the Infinite” (Cambridge: Harvard University Press, 1979).
- Franklin, James, An Aristotelian Realist Philosophy of Mathematics: Mathematics as the Science of Quantity and Structure (New York: Palgrave Macmillin, 2014).
- George, Alexander; Velleman, Daniel J., Philosophies of Mathematics (Malden, MA: Blackwell, 2002).
- Havil, Julian, The Irrationals: A Story of the Numbers You Can’t Count On (Princeton: Princeton University Press, 2012).
- Hume, David, A Treatise of Human Nature (New York: Barnes & Noble, 2005).
- Iemhoff, Rosalie, “Intuitionism in the Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Spring 2015 Edition), Edward N. Zalta (ed.), URL = .
- Linnebo, Øystein, “Platonism in the Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Winter 2013 Edition), Edward N. Zalta (ed.), URL = .
- Mancosu, Paolo, “Explanation in Mathematics”, The Stanford Encyclopedia of Philosophy (Summer 2015 Edition), Edward N. Zalta (ed.), URL = .
- Mendell, Henry, “Aristotle and Mathematics”, The Stanford Encyclopedia of Philosophy (Winter 2008 Edition), Edward N. Zalta (ed.), URL = .
- Merzbach, Uta C.; Boyer, Carl B., A History of Mathematics, third edition (Hoboken, NJ: John Wiley & Sons, 2011).
- Núñez, Rafael E., “Creating mathematical infinities: Metaphor, blending, and the beauty of transfinite cardinals” (Journal of Pragmatics, 37 (2005), 1717 – 1741).
- Paseau, Alexander, “Naturalism in the Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Summer 2013 Edition), Edward N. Zalta (ed.), URL = .
- Plato, Complete Works, edited by John M. Cooper (Indianapolis: Hackett Publishing, 1997).
- Reichenbach, Bruce, “Cosmological Argument”, The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.), URL = .
- Rodych, Victor, “Wittgenstein’s Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Summer 2011 Edition), Edward N. Zalta (ed.), URL = .
- Ross, Sir David, Aristotle, sixth edition (New York: Routledge, 1995).
- Russell, Bertrand, The History of Western Philosophy (New York: Simon & Schuster, 1972).
- —Introduction to Mathematical Philosophy (Digireads.com Publishing, 2010).
- Schopenhauer, Arthur, On the Fourfold Root of the Principle of Sufficient Reason (La Salle, IL: Open Court, 1999).
- —The World as Will and Representation (New York: Dover Publications, 1969).
- Shabel, Lisa, “Kant’s Philosophy of Mathematics”, The Stanford Encyclopedia of Philosophy (Fall 2014 Edition), Edward N. Zalta (ed.), URL = .
- Shapiro, Steward, Thinking About Mathematics: The Philosophy of Mathematics (New York: Oxford University Press, 2011).
- Zenkin, Alexander A., “Logic of Actual Infinity and G. Cantor’s Diagonal Proof of the Uncountability of the Continuum,” The Review of Modern Logic (Volume 9, Numbers 3 & 4, December 2003 – August 2004, Issue 30).