Second Order Three-point Boundary Value Problems in Abstract Spaces

Mieczys?aw Cichoń^{1}, Hussein A.H. Salem^{2}

1 Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland;
2 Department of Mathematics, Faculty of Sciences, Alexandria University, Egypt

Abstract In this paper we investigate the existence of solutions of the nonhomogeneous three-point boundary value problem

The coefficient functions a and b are continuous real-valued functions on [0, 1], η and ζ are some positive constants. Denote by E a Banach space and assume, that u belongs to an Orlicz space i.e., u(·)∈L_{M}([0, 1],R), where M is an N-function and c∈E.
We search for solutions of the above problem in the Banach space of continuous functions C([0, 1],E) with the Pettis integrability assumptions imposed on f. Some classes of Pettis-integrable functions are described in the paper and exploited in the proofs of main results. We stress on a class of pseudo-solutions of considered problem. Our results extend previous results of the same type for both Bochner and Pettis integrability settings. Similar results are also proved for differential inclusions i.e. when f is a multivalued function.

Abstract：
In this paper we investigate the existence of solutions of the nonhomogeneous three-point boundary value problem

The coefficient functions a and b are continuous real-valued functions on [0, 1], η and ζ are some positive constants. Denote by E a Banach space and assume, that u belongs to an Orlicz space i.e., u(·)∈L_{M}([0, 1],R), where M is an N-function and c∈E.
We search for solutions of the above problem in the Banach space of continuous functions C([0, 1],E) with the Pettis integrability assumptions imposed on f. Some classes of Pettis-integrable functions are described in the paper and exploited in the proofs of main results. We stress on a class of pseudo-solutions of considered problem. Our results extend previous results of the same type for both Bochner and Pettis integrability settings. Similar results are also proved for differential inclusions i.e. when f is a multivalued function.

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