Abstract
Let E be an infinite-dimensional Banach space and I be a compact interval of the real line. The aim of this paper is two-fold: On the one hand, we construct an example of a weakly absolutely continuous function taking its values in E that is nowhere weakly differentiable on I, but has weakly continuous fractional weak derivatives of some critical orders less than one. This also holds for (nearly) all orders less than one if E failing cotype. We believe that this results are of independent interest and discuss it in a rather general setting. On the other hand, we establish some examples of weakly continuous functions taking its values in Gauge space fail to be pseudo differentiable on I, but have fractional-pseudo derivatives of “all” order less than one. An application will be given.
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$$\begin{aligned} \psi _n(t)=\chi _{\left[ \frac{n-2^k}{2^k},\frac{n-2^k+1}{2^k}\right] }\text{, }\quad k\ge 0 \text { being the unique integer with }2^k\le n<2^{k+1}, \end{aligned}$$has this property. Actually, \(\{ \psi _n\}\) is a sequence of indicator functions of intervals of decreasing length, marching across the unit interval [0,1] over and over again.
References
Agarwal, R.P., Lupulescu, V., O’Regan, D., Rahman, G.: Weak solutions for fractional differential equations in nonreflexive Banach spaces via Riemann-Pettis integrals. Math. Nachr. 289(4), 395–409 (2016). https://doi.org/10.1002/mana.201400010
Agarwal, P., Al-Mdalla, Q., Cho, Y.J.E., Jain, S.: Fractional differential equations for the generalized Mittag-Leffler function. Adv. Differ. Equ. 2018, 58 (2018). https://doi.org/10.1186/s13662-018-1500-7
Agarwal, P., El-Sayed, A.A.: Non-standard finite difference and Chebyshev collocation methods for solving fractional diffusion equation. Physica A Stat. Mech. Appl. 500, 40–49 (2018). https://doi.org/10.1016/j.physa.2018.02.014
Baltaeva, U., Agarwa, P.: Boundary-value problems for the third-order loaded equation with noncharacteristic type-change boundaries. Math. Methods Appl. Sci. 41(9), 3307–3315 (2016)
Bartle, R.G.: A Modern Theory of Integration, Graduate Studies in Mathematics. American Mathematical Society, Providence (2001)
Benchohra, M., Mostefai, F.: Weak solutions for nonlinear fractional differential equations with integral boundary conditions in Banach spaces. Opuscula Math. 32(1), 31–40 (2012)
Calabuig, J.M., Rodríguez, J., Rueda, P., Sánchez-Pérez, E.A.: On \( p \)-Dunford integrable functions with values in Banach spaces. J. Math. Anal. Appl. 464(1), 806–822 (2018)
Diestel, J., Uhl Jr., J.J.: Vector Measures, Mathematical Surveys, vol. 15. American Mathematical Society, Providence (1977)
Dilworth, J., Girardi, M.: Nowhere weak differentiability of the Pettis integral. Quaest. Math. 18(4), 365–380 (1995)
Munroe, M.E.: A note on weak differentiability of Pettis integrals. Bull. Am. Math. Soc. 52, 167–174 (1946)
Jainm, S., Agarwal, P., Kilicman, A.: Pathway fractional integral operator associated with 3m-parametric Mittag–Leffter functions. Int. J. Appl. Comput. Math. 4, 115 (2018)
Kadets, V.: Non-differentiable indefinite Pettis integrals. Quaest. Math. 17, 137–149 (1994)
Mendel, M., Naor, A.: Metric cotype. Ann. Math. 168, 247–298 (2008)
Naralenkov, K.: On Denjoy type extension of the Pettis integral. Czechoslov. Math. J. 60(135), 737–750 (2010)
Pettis, B.J.: On integration in vector spaces. Trans. Am. Math. Soc. 44, 277–304 (1938)
Phillips, R.S.: Integration in a convex linear topological space. Trans. Am. Math. Soc. 47, 114–145 (1940)
Ross, B., Samko, S.G., Love, E.R.: Functions that have no first order derivative might have fractional derivative of all orders less than one. Real Anal. Exch. 20, 140–157 (1994/1995)
Salem, H.A.H.: On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies. J. Comput. Appl. Math. 224, 565–572 (2009)
Salem, H.A.H.: On the fractional calculus in abstract spaces and their applications to the Dirichlet-type problem of fractional order. Comput. Math. Appl. 59, 1278–1293 (2010)
Salem, H.A.H., Cichoń, M.: On solutions of fractional order boundary value problems with integral boundary conditions in Banach spaces. J. Funct. Spaces Appl. 13 (Article ID 428094) (2013)
Salem, H.A.H.: Hadamard-type fractional calculus in Banach spaces, Revista de la Real Academia de Ciencias Exactas, Fsicas y Naturales. Serie A. Matemticas (2018) https://doi.org/10.1007/s13398-018-0531-y
Salem, H.A.H.: On functions without pseudo derivatives having fractional pseudo derivatives. Quaest. Math. (2018), https://doi.org/10.2989/16073606.2018.1523247
Salem, H.A.H.: On the theory of fractional calculus in the Pettis-function spaces. J. Funct. Spaces Appl. 13 (Article ID 8746148) (2018)
Solomon, D.: Denjoy Integration in Abstract Spaces, Memories of the American Mathematical Society. American Mathematical Society, Providence (1969)
Szep, A.: Existence theorem for weak solutions of ordinary differential equations in reflexive Banach spaces. Studia Sci. Math. Hungar. 6, 197–203 (1971)
Yeong, L.T.: Henstock–Kurzweil Integration on Euclidean Spaces, vol. 12. World Scientific Publishing Co. Pte. Ltd, Singapore (2011)
Zäihle, M., Ziezold, H.: Fractional derivatives of Weierstrass-type functions. J. Comput. Appl. Math. 76, 265–275 (1996)
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Salem, H.A.H. Weakly absolutely continuous functions without weak, but fractional weak derivatives. J. Pseudo-Differ. Oper. Appl. 10, 941–954 (2019). https://doi.org/10.1007/s11868-019-00274-6
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DOI: https://doi.org/10.1007/s11868-019-00274-6