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36 Shayma Adil Murad et al where As , therefore is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem). Theorem 3.2. Let ∶ [0,b] × R → R be a continuous function mapping bounded subsets of [0,b] × R into relatively compact subsets of R, and the assumptions ( H2) and ( H3) hold. Then the boundary value problem (2.1) has at least one solution on [0,b]. It is clear that is contraction mapping, Continuity of implies that the operator is continuous. Also, is uniformly bounded on as Now we prove the compactness of the operator . We define , And consequently we have Proof. Letting and consider . We define the operators and s for , we find that Which is independent of x. Thus, is equicontinuous. Using the fact that maps bounded subset into relatively compact subsets, so is relatively compact on . Hence, by the Arzelá-Ascoli Theorem, is compact on . Thus all the assumptions of Theorem 3.2 aresatisfied. So the conclusion of Theorem 3.2 implies that the initial value problem (2.1) has at least one solution on [0, b]. 4. REFERENCES Now prove that is contraction mapping 1. Kilbas, A.A., H.M. Srivastava & J.J. Trujillo. Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies Vol. 204. Elsevier Science B.V., Amsterdam, The Netherlands, p. 347-463 (2006). 2. Podlubny, I. Fractional Differential Equations, Vol. 198 of Mathematics in Science and Engineering. Academic Press, San Diego, CA, USA, p. 261-307 (1999).
Existence and Uniqueness for Solution of Differential Equation 37 3. Ibrahim, R.W. & M. Darus. Subordinati-on and superordination for analytic functions involving fractional integral operator. Complex Variables and Elliptic Equations 53: 1021-1031 (2008). 4. Ibrahim, R.W. & M. Darus. Subordi-nation and superordination for univalent solutions for fractional differential equations. J. Math. Anal. Appl. 345: 871-879 (2008). 5. Hadid S.B. Local and global existence theorems on differential equations of non-integer order. J. Fractional Calculus 7: 101-105 (1995). 6. Murad, S.M. & H.J. Zekri & S. Hadid. Existence and uniqueness theorem of fractional mixed Volterra-Fredholm integro-differential equation with integral boundary conditions. International Journal of Differential Equations 2011: 1-15 (2011). 7. Cieielski, M. & J.S. Leszczynski. Numerical solution to boundary value problem for anomalous diffusion equation with Riesz-Feller fractional operator. J. Theo. and Appl. Mech. 44: 393-403 (2006). 8. Krasnosel’skiĭ, M.A. Two remarks on the method of successive Approximati-ons. Uspekhi Matematicheskikh Nauk 63: 123–127 (1955).
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Page 5 and 6: Sukhwinder Singh Billing 3 To prove
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Page 9 and 10: Sukhwinder Singh Billing 7 zq( z)
Page 13 and 14: Inequalities for Differentiable Fun
Page 23 and 24: Supra β-connectedness on Topologic
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Page 44 and 45: 42 Rabha W. Ibrahim G( r, ks; z) =
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