FRACTIONAL ORDER BOUNDARY |VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS INVOLVING PETTIS INTEGRAL

doi:10.1016/S0252-9602(11)60266-X

Abstract

Abstract:

In this article, we investigate the existence of Pseudo solutions for some fractional order boundary value problem with integral
boundary conditions in the Banach space of continuous function equipped with its weak topology. The class of such problems
constitute a very interesting and important class of problems. They include two, three, multi-point and nonlocal boundary-value
problems as special cases. In our investigation, the right hand side of the above problem is assumed to be Pettis integrable function. To encompass the full scope of this article, we give an example illustrating the main result.

Key words: Fractional calculus, Pseudo solution, boundary value problem, Pettis integrals

CLC Number:

26A33

Hussein A.H.Salem . FRACTIONAL ORDER BOUNDARY |VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS INVOLVING PETTIS INTEGRAL[J].Acta mathematica scientia,Series B, 2011, 31(2): 661-672.

Trendmd

References

[1]Ahmad B, Sivasundaram S. Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear
Analysis: Hybrid Systems, 2010, 4: 134--141

[2] Ahmad B, Nieto J. Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems, 2009 (2009), Article ID 708576, 11 pages

[3] Arino O, Gautier S, Penot T P. A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations. Funkcial Ekvac, 1984, 27: 273--279

[4] Bai Z, L"{u}   H. Positive solutions for boundary value problem of nonlinear fractional differential equation. J Math Anal Appl, 2005, 311: 495--505

[5] Ball J M. Weak continuity properties of mapping and semi-groups. Proc Royal Soc Edinburgh Sect, 1973/1974, 72A: 275--280

[6] Bugajewski D, Szulfa S. Kneser's theorem for weak solutions of the Darboux problem in a Banach space. Nonlinear Anal, 1993, 20(2): 169--173

[7] Chandra J, Lakshmikantham V, Mitchell A R. Existence of solutions of boundary value problems for nonlinear second order
systems in Banach spaces. Nonlinear Anal, 1978, 2: 157--168

[8] Cicho\'{n} M. Weak solutions of differential equations in Banach spaces. Discuss Math Diff Incl, 1995, 15: 5--14

[9] Diestel J, Uhl J J Jr. Vector Measures. Math Surveys 15. Providence, R I: Amer Math Soc, 1977

[10] Feng B, Ji D, Ge W. Positive solutions for a class of boundary value problem with integral boundary conditions in Banach spaces. J Comput Appl Math, 2008, 222: 351--363

[11] Feng M, Du B, Ge W. Impulsive boundary value problems with integral boundary conditions and one-dimensional p-Laplacian. Nonlinear Anal, 2009, 70: 3119--3126

[12] Gallardo J M. Second order differential operators with integral boundary conditions and generation of semigroups. Rocky
Mountain J Math, 2000, 30: 1265--1292

[13] Geitz R F. Pettis integration. Proc Amer Math Soc, 1981, 82: 81--86

[14] Geitz R F. Geomerty and the Pettis integral. Trans Amer Math Soc, 1982, 269:  535--548

[15] Gorenflo R,  Vessella S. Abel integral equations//Lecture Notes in Math, Vol 1461. Berlin: Springer, 1991

[16] Liu B. Positive solutions of a nonlinear four-point boundary value problems in Banach spaces. J Math Anal Appl, 2005, 305: 253--276

[17] Lomtatidze A, Malaguti L. On a nonlocal boundary-value problems for second rder nonlinear singular differential equations. Georgian Math J, 2000, 7: 133--154

[18] M\"{o}nch H. Boundary value problems for nonlinear ordinary differential equation of second order in Banach spaces. Nonlinear Anal, 1980, 4: 985--999

[19] O'Regan D. Integral equations in reflexive Banach spaces and the weak topologies. Proc AMS, 1998, 27(5): 607--614

[20] O'Regan D. Weak solutions of ordinary differential equations in Banach spaces. Appl Math Letters, 1999, 12: 101--105

[21] Ozdarska D, Szufla S. Weak solutions for boundary value problem for nonlinear ordinary differential quation of second order in Banach spaces. Math Slovaca, 1993, 43: 301--307

[22] Pettis B J. On integration in vector spaces. Trans Amer Math Soc, 1938, 44: 277--304

[23] Salem H A H, Väth M. An abstract Gronwall lemma and application to global existence results for functional differential and integral
equations of fractional order. J Integral Equations Appl, 2004, 16: 411--429

[24] Salem H A H, El-Sayed A M A,  Moustafa O L. A note on the fractional calculus in Banach spaces. Studia Sci Math Hungar, 2005, 42(2): 115--130

[25] Samko S, Kilbas A, Marichev O. Fractional Integrals andDrivatives. Gordon and Breach Science Publisher, 1993

[26] Satco B. Second order three boundary value problemin Banach spaces via Henstock and Henstock-Kurzweil-Pettis integral. J Math Anal Appl, 2007, 332: 919--933

[27] Szulfa S.Boundary value problems for nonlinear ordinary differential equation of second   order in Banach spaces. Nonlinear Anal, 1984, 4: 1481--1487

[28] Zhang X, Ge W. Positive solutions for a class of boundary-value problems with integral boundary conditions. Comput Math Appl, 2009, 58: 203--215

[29] Zhang X, Feng M, Ge W. Symmetric positive solutions for p-Laplacian fourth-order differential equations with integral boundary conditions. J Comput Appl Math, 2008, 222: 561--573

[30] Zhang X, Yang X, Ge W. Positive solutions of nth-order impulsive boundary value problems with integral boundary conditions in Banach spaces. Nonlinear Anal, 2009, 71: 5930--5945

[31] Zhao Y, Chena H. Existence of multiple positive solutions for m-point boundary value problems in Banach spaces. J Comput Appl Math, 2008, 215: 79--90

FRACTIONAL ORDER BOUNDARY |VALUE PROBLEM WITH INTEGRAL BOUNDARY CONDITIONS INVOLVING PETTIS INTEGRAL

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Alberto CABADA, Nikolay D. DIMITROV. EXISTENCE OF SOLUTIONS OF NONLOCAL PERTURBATIONS OF DIRICHLET DISCRETE NONLINEAR PROBLEMS [J]. Acta mathematica scientia,Series B, 2017, 37(4): 911-926.
[2]	Fei HOU. ON THE GLOBAL EXISTENCE OF SMOOTH SOLUTIONS TO THE MULTI-DIMENSIONAL COMPRESSIBLE EULER EQUATIONS WITH TIME-DEPENDING DAMPING IN HALF SPACE [J]. Acta mathematica scientia,Series B, 2017, 37(4): 949-964.
[3]	Yu XIAO, Engui FAN, Jian XU. THE FOKAS-LENELLS EQUATION ON THE FINITE INTERVAL [J]. Acta mathematica scientia,Series B, 2017, 37(3): 852-876.
[4]	Haibo YU. GLOBAL REGULARITY TO THE 2D INCOMPRESSIBLE MHD WITH MIXED PARTIAL DISSIPATION AND MAGNETIC DIFFUSION IN A BOUNDED DOMAIN [J]. Acta mathematica scientia,Series B, 2017, 37(2): 395-404.
[5]	Nermina MUJAKOVIC, Nelida CRNJARIC-ZIC. GLOBAL SOLUTION TO 1D MODEL OF A COMPRESSIBLE VISCOUS MICROPOLAR HEAT-CONDUCTING FLUID WITH A FREE BOUNDARY [J]. Acta mathematica scientia,Series B, 2016, 36(6): 1541-1576.
[6]	Batirkhan TURMETOV. ON SOME BOUNDARY VALUE PROBLEMS FOR NONHOMOGENOUS POLYHARMONIC EQUATION WITH BOUNDARY OPERATORS OF FRACTIONAL ORDER [J]. Acta mathematica scientia,Series B, 2016, 36(3): 831-846.
[7]	Ruxu LIAN, Jian LIU. FREE BOUNDARY VALUE PROBLEM FOR THE CYLINDRICALLY SYMMETRIC COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH DENSITY-DEPENDENT VISCOSITY [J]. Acta mathematica scientia,Series B, 2016, 36(1): 111-123.
[8]	B. J. KADIRKULOV, M. KIRANE. ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR THE POISSON EQUATION WITH A NONLOCAL BOUNDARY OPERATOR [J]. Acta mathematica scientia,Series B, 2015, 35(5): 970-980.
[9]	Hua LIU. ANALYTIC BOUNDARY VALUE PROBLEMS ON CLASSICAL DOMAINS [J]. Acta mathematica scientia,Series B, 2015, 35(5): 1037-1045.
[10]	R. K. SAXENA, D. KUMAR. GENERALIZED FRACTIONAL CALCULUS OF THE ALEPH-FUNCTION INVOLVING A GENERAL CLASS OF POLYNOMIALS [J]. Acta mathematica scientia,Series B, 2015, 35(5): 1095-1110.
[11]	Rinaldo M. COLOMBO, Elena ROSSI. RIGOROUS ESTIMATES ON BALANCE LAWS IN BOUNDED DOMAINS [J]. Acta mathematica scientia,Series B, 2015, 35(4): 906-944.
[12]	Eid H. DOHA, Waleed M. ABD-ELHAMEED, Mahmoud A. BASSUONY. ON THE COEFFICIENTS OF DIFFERENTIATED EXPANSIONS AND DERIVATIVES OF CHEBYSHEV POLYNOMIALS OF THE THIRD AND FOURTH KINDS [J]. Acta mathematica scientia,Series B, 2015, 35(2): 326-338.
[13]	A.S. BERDYSHEV, A. CABADA, B.Kh. TURMETOV. ON SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR A NONHOMOGENEOUS BIHARMONIC EQUATION WITH A BOUNDARY [J]. Acta mathematica scientia,Series B, 2014, 34(6): 1695-1706.
[14]	WEI Zhong-Li. SOME NECESSARY AND SUFFICIENT CONDITIONS FOR EXISTENCE OF POSITIVE SOLUTIONS FOR THIRD ORDER SINGULAR SUBLINEAR MULTI-POINT BOUNDARY VALUE PROBLEMS [J]. Acta mathematica scientia,Series B, 2014, 34(6): 1795-1810.
[15]	Rabha W. IBRAHIM, Janusz SOK′O L. ON A NEW CLASS OF ANALYTIC FUNCTION DERIVED BY A FRACTIONAL DIFFERENTIAL OPERATOR [J]. Acta mathematica scientia,Series B, 2014, 34(5): 1417-1426.