Mathematics Studies. Elsevier, msterdam Cambridge Scientific Publishers, Cambridge Comput Math Appl.

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J Math Sci. Kirane, M, Malik, SA: The profile of blowing-up solutions to a nonlinear system of fractional differential equations. Nonlinear Anal. Chai, G: Existence results for boundary value problems of nonlinear fractional differential equations.

### Background

Zhang, S, Su, X: The existence of a solution for a fractional differential equation with nonlinear boundary conditions considered using upper and lower solutions in reverse order. Babakhani, A, Baleanu, D: Employing of some basic theory for class of fractional differential equations. Adv Diff Equ.

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Podlubny, I: Fractional Differential Equations. Mathematics in Science and Engineering. Academic Press, San Diego World Scientific, Singapore World Scientific Series on Nonlinear Science. World Scientific72 Adv Industr Control. Springer, New York Hilfer, R: Experimental evidence for fractional time evolution in glass forming materials. Chem Phys. Wenchang, T, Wenxiao, P, Mingyu, X: A note on unsteady flows of a viscoelastic fluid with the fractional Maxwell model between two parallel plates.

Int J Non-Linear Mech. Mainardi, F, Gorenflo, R: Time-fractional derivatives in relaxation processes: a tutorial survey. Fract Calc Appl Anal. Imperial College Press, London Abstr Appl Anal. Sandev, T, Tomovski, Z: General time fractional wave equation for a vibrating string. J Phys A Math Theor. Wiley-VCH, Weinheim J Phys D Appl Phys. Wiley, Chichester De Gruyter Studies in Mathematics. De Gruyter, Berlin43 Hilfer, R, Anton, L: Fractional master equations and fractal time random walks.

Phys Rev E. Trans Am Math Soc. Phys A. Fulger, D, Scalas, E, Germano, G: Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation. In Phys Rev E Stat, vol. Nonlinear Soft Matter Phys Lecture Notes in Electrical Engineering.

Springer, Netherlands84 Glushak, AV: Cauchy-type problem for an abstract differential equation with fractional derivative. Math Notes 77 1 —38 Translated from Matematicheskie Zametki 77 1 Glushak, AV: On the properties of a Cauchy-type problem for an abstract differential equation with fractional derivatives. Math Notes 82 5 Translated from Matematicheskie Zametki 82 5 , Glushak, AV: Correctness of Cauchy-type problems for abstract differential equations with fractional derivatives.

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Russ Math 53 9 :1 —19 Translated from Izvestiya Vysshikh Uchebnykh sZavedenii. Matematika 9 , Gordon and Breach, Amsterdam Discr Dyn Nature Soc. J Inequal Appl. Kilbas, AA, Bonilla, B, Trujillo, JJ: Fractional integrals and derivatives, and differential equations of fractional order in weighted spaces of continuous functions russian. Dokl Nats Akad Nauk Belarusi. Cite this article as: Furati: A Cauchy-type problem with a sequential fractional derivative in the space of continuous functions.

Boundary Value Problems A Cauchy-type problem with a sequential fractional derivative in the space of continuous functions Academic research paper on " Mathematics ". Continuous and integrable solutions of a nonlinear Cauchy problem of fractional order with nonlocal conditions. On the existence and uniqueness of solutions for a class of non-linear fractional boundary value problems. Other physical and engineering processes are given in [31,32] In a series of articles, [], Glushak studied the uniform well-posedness of a Cauchy-type problem with two fractional derivatives and bounded operator.

### Background

The left-sided Riemann-Liouville fractional integrals and derivatives are defined as follows. Definition 1 Letf e L a,b. The proof of Lemma 4 is given in [36]. The following lemma is proved in [37]. The following lemma is proved in [38]. Consequently, from Lemma 8 we have the following property. Later, the following observation is important. For any x e a, b] we have the following inequality. Lemma 11 leads to the left inverse operator.

The following lemma relates the fractional derivative D? By applying Lemma 13 twice we obtain JiaaD? Now, Based on the composition in Lemma 24, in the next theorem we establish an equivalence with the following fractional integro-differential equation: D? For assertion a , let y e C1-a[a, b] satisfy In the next section we use this equivalence to prove the existence and uniqueness of solutions.

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## Continuous function

This solution is also a solution for Competing interests The author declares that they have no competing interests. Elsevier, msterdam 2. Cambridge Scientific Publishers, Cambridge 3. Springer, Heidelberg 4. Any continuous but non-uniformly continuous map will do that. Sign up to join this community. The best answers are voted up and rise to the top.

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Viewed times. It's a nice little exercise to show that Cauchy-continuity is stronger than continuity. Hello Hello 41 3 3 bronze badges. Featured on Meta.