Weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive Banach space

H. H. G. Hashem; A. M. A. El-Sayed; Maha A. Alenizi; H. H. G. Hashem; A. M. A. El-Sayed; Maha A. Alenizi

doi:10.3934/math.2021004

AIMS Mathematics

2021, Volume 6, Issue 1: 52-65. doi: 10.3934/math.2021004

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Research article

Weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive Banach space

1.
Department of mathematics, College of Science, Qassim University, P.O. Box 6644 Buraidah 51452, Saudi Arabia
2.
Faculty of Science, Alexandria University, Alexandria, Egypt

Received: 19 July 2020 Accepted: 17 September 2020 Published: 28 September 2020
MSC : 34A12, 47G10, 28B05

In this paper, we establish some existence results of weak solutions and pseudo-solutions for the initial value problem of the arbitrary (fractional) orders differential equation $ \frac{dx}{dt}~ = ~ f(t, D^\gamma x(t)), ~\gamma \in (0, 1), ~~t~\in [0, T]=\mathbb{I}\\ x(0) = x_0. $ in nonreflexive Banach spaces $~E, ~$ where $~D^\gamma x(\cdot)~$ is a fractional %pseudo- derivative of the function $~x(\cdot):\mathbb{I} \rightarrow E~$ of order $~\gamma.~$ The function $~f(t, x):\mathbb{I}\times E \rightarrow E~$ will be assumed to be weakly sequentially continuous in $x~$ for each $~t\in \mathbb{I}~$ and Pettis integrable in $~t~$ on $~\mathbb{I}~$ for each $~x\in C[\mathbb{I}, E].~$ Also, a weak noncompactness type condition (expressed in terms of measure of noncompactness) will be imposed.
- measure of weak noncompactness,
- weakly continuous solution,
- pseudo-solution,
- weakly relatively compact,
- fractional Pettis integral
Citation: H. H. G. Hashem, A. M. A. El-Sayed, Maha A. Alenizi. Weak and pseudo-solutions of an arbitrary (fractional) orders differential equation in nonreflexive Banach space[J]. AIMS Mathematics, 2021, 6(1): 52-65. doi: 10.3934/math.2021004

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Abstract

In this paper, we establish some existence results of weak solutions and pseudo-solutions for the initial value problem of the arbitrary (fractional) orders differential equation $ \frac{dx}{dt}~ = ~ f(t, D^\gamma x(t)), ~\gamma \in (0, 1), ~~t~\in [0, T]=\mathbb{I}\\ x(0) = x_0. $ in nonreflexive Banach spaces $~E, ~$ where $~D^\gamma x(\cdot)~$ is a fractional %pseudo- derivative of the function $~x(\cdot):\mathbb{I} \rightarrow E~$ of order $~\gamma.~$ The function $~f(t, x):\mathbb{I}\times E \rightarrow E~$ will be assumed to be weakly sequentially continuous in $x~$ for each $~t\in \mathbb{I}~$ and Pettis integrable in $~t~$ on $~\mathbb{I}~$ for each $~x\in C[\mathbb{I}, E].~$ Also, a weak noncompactness type condition (expressed in terms of measure of noncompactness) will be imposed.

References

[1]	A. Ambrosetti, Un teorema di esistenza per le equazioni differenziali negli spazi di Banach, Rend. Semin. Mat. Univ. Padova., 39 (1967), 349-369.
[2]	R. P. Agarwal, V. Lupulescu, D. O'Regan, G. U. Rahman, Nonlinear fractional differential equations in nonreflexive Banach spaces and fractional calculus, Adv. Differ. Equ., 2015 (2015), 1-18. doi: 10.1186/s13662-014-0331-4
[3]	R. P. Agarwal, V. Lupulescu, D. O'Regan, G. U. Rahman, Weak solutions for fractional differential equations in nonreflexive Banach spaces via Riemann-Pettis integrals, Math. Nachr., 289 (2016), 395-409. doi: 10.1002/mana.201400010
[4]	W. Arendt, C. Batty, M. Hieber, F. Neubrander, Vector-Valued Laplace Transforms and Cauchy Problems, Monogr. Math., 96 (2001), Birkh?user, Basel.
[5]	J. Banaś, M. Taoudi, Fixed points and solutions of operator equations for the weak topology in Banach algebras, Taiwanese J. Math, 18 (2014), 871-893. doi: 10.11650/tjm.18.2014.3860
[6]	M. Cichoń, Weak solutions of ordinary differential equations in Banach spaces, Discuss. Differ. Inc. Control Optimal., 15 (1995), 5-14.
[7]	M. Cichoń, I. Kubiaczyk, Kneser's theorem for strong, weak and pseudo-solutions of ordinary differential equations in Banach spaces, Ann. Pol. Math., 52 (1995), 13-21.
[8]	M. Cichoń, I. Kubiaczyk, A. Sikorska-Nowak, A. Yantir, Weak solutions for dynamic Cauchy problem in Banach spaces, Nonlinear Anal., 71 (2009), 2936-2943. doi: 10.1016/j.na.2009.01.175
[9]	E. Cramer, V. Lakshmiksntham, A. R. Mitchell, On the existence of weak solutions of differential equations in nonreflexive Banach spaces, Nonlinear Anal. 2 (1978), 259-262.
[10]	K. Deimling, Ordinary Differential equations in Banach Spaces, Lecture Notes Math., 596 (1977), Springer, Berlin.
[11]	J. Diestel, J. J. Uhl, Jr, Vector Measures, Math. Surveys 15, Amer. Math. Soc., Providence, R.I., (1977).
[12]	F. S. De Blasi, On a property of the unit sphere in Banach spaces, Bull. Math. Soc. Sci. Math. R. S. Roum., 21 (1977), 259-262.
[13]	N. Dinculeanu, On Kolmogorov-Tamarkin and M. Riesz compactness criteria in function spaces over a locally compact group, J. Math. Anal. Appl., 89 (1982), 67-85.
[14]	G. A. Edgar, Measurability in Banach space, Indiana Univ. Math. J. 26 (1977), 663-677.
[15]	G. A. Edgar, Measurability in Banach space, II, Indiana Univ. Math. J. 28 (1979), 559-578.
[16]	A. M. A. El-Sayed, E. O. Bin-Taher, Nonlocal and integral conditions problems for a multi-term fractional-order differential equation, Miskolc Math. Notes, 15 (2014), 439-446.
[17]	R. F. Geitz, Pettis integration, Proc. Amer. Math. Soc. 82 (1981), 81-86.
[18]	E. Hille, R. S. Phillips, Functional analysis and semi-groups, Amer. Math. Soc. Colloq. Publ. 31 (1957).
[19]	H. Gou, B. Li, Existence of weak solutions for fractional integrodifferential equations with multipoint boundary conditions, Int. J. Differential Equations, 2018 (2018), Article ID 1203031.
[20]	W. J. Knight, Solutions of differential equations in Banach spaces, Duke Math. J. 41 (1974), 437- 442.
[21]	I. Kubiaczyk, S. Szufla, Kneser's theorem for weak solutions of ordinary differential equations in Banach spaces, Publ. Inst. Math. (Beograd), 32 (1982), 99-103.
[22]	I. Kubiaczyk, On a fixed point theorem for weakly sequentially continuous mapping, Discuss. Math. Differ. Incl., 15 (1995), 15-20.
[23]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, North-Holland, 2006.
[24]	A. Kubica, P. Rybka, K. Ryszewska, Weak solutions of fractional differential equations in non cylindrical domains, Nonlinear Analysis: Real World Appl., 36 (2017), 154-182 doi: 10.1016/j.nonrwa.2017.01.005
[25]	A. R. Mitchell, Ch. Smith, An existence theorem for weak solutions of differential equations in Banach spaces, Nonlinear Equations Abstract Spaces, (1978), 387-404.
[26]	D. O'Regan, Fixed point theory for weakly sequentially continuous mapping, Math. Comput. Model., 27 (1998), 1-14.
[27]	D. O'Regan, Weak solutions of ordinary differential equations in Banach spaces, Appl. Math. Lett. 12 (1999), 101-105.
[28]	I. Podlubny, Fractional Differential equations, San Diego-NewYork-London, 1999.
[29]	B. J. Pettis, On integration in vector spaces, Trans. Amer. Math. Soc. 44 (1938), 277-304.
[30]	B. Ross, K. S. Miller, An Introduction to Fractional Calculus and Fractional Differential Equations. John Wiley, New York, (1993).
[31]	S. Szulfa, On the existence of solutions of differential equations in Banach spaces, Bull. Acad. polan. Sci. Ser. Sci. Math., 30 (1982), 507-514.
[32]	H. A. H. Salem, A. M. A. El-Sayed, O. L. Moustafa, A note on the fractional calculus in Banach spaces, Studia Sci. Math. Hung., 42 (2005), 115-130.
[33]	H. A. H. Salem, A. M. A. El-Sayed, Weak solution for fractional order integral equations in reflexive Banach spaces, Math. Slovaca., 55 (2005), 169-181.
[34]	H. A. H. Salem, M. Cichoń, On solutions of fractional order boundary value problems with integral boundary conditions in Banach spaces, J. Function Spaces Appl., 2013 (2013), Article ID 428094.

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