A common solution of split equality monotone inclusion problem and split equality fixed point problem in real Banach spa
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Advances in Operator Theory https://doi.org/10.1007/s43036-020-00112-3
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ORIGINAL PAPER
A common solution of split equality monotone inclusion problem and split equality fixed point problem in real Banach spaces Chinedu Izuchukwu1 • Ferdinard Udochukwu Ogbuisi1 Oluwatosin Temitope Mewomo1
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Received: 7 July 2020 / Accepted: 29 September 2020 Ó Tusi Mathematical Research Group (TMRG) 2020
Abstract In this paper, we introduce an iterative algorithm for approximating a common solution of Split Equality Monotone Inclusion Problem (SEMIP) and Split Equality Fixed Point Problem (SEFPP) in p-uniformly convex Banach spaces which are also uniformly smooth. Under standard and mild assumptions of monotonicity and right Bregman strongly condition of the SEMIP- and SEFPP-associated mappings, we establish the strong convergence of the scheme. Finally, we applied our result to study the convex minimization problem (CMP) and Equilibrium Problem. Our result complements and extends some recent results in literature. Keywords Monotone inclusion problem Fixed point problem Right Bregman strongly nonexpansive mapping Maximal monotone mapping Resolvent operators
Mathematics Subject Classification 47H09 47H10 49J20 49J40
Communicated by Daniel Pellegrino. & Oluwatosin Temitope Mewomo [email protected] Chinedu Izuchukwu [email protected] Ferdinard Udochukwu Ogbuisi [email protected] 1
School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban, South Africa
C. Izuchukwu et al.
1 Introduction Let E be a real Banach space and f : E ! R be a convex function. Then, the domain of f is defined by dom f :¼ fx 2 E : f ðxÞ\ þ 1g: Let C be a nonempty, closed and convex subset of int dom f , where int domf means interior of domain of f. Let T : C ! C be any mapping, a point u 2 C is While u 2 C is called an asymptotic fixed point called a fixed point of T if T u ¼ u. and of T if C contains a sequence fxn g1 n¼1 which converges weakly to u limn!1 jjxn Txn jj ¼ 0. The set of fixed points of T and asymptotic fixed points of ^ T is denoted by F(T) and FðTÞ, respectively. Let 1\q 2 p\1 with 1p þ 1q ¼ 1. A mapping T : C ! C is said to be (i)
right Bregman firmly nonexpansive (see [14]) if hJpE ðTxÞ JpE ðTyÞ; Tx Tyi hJpE ðTxÞ JpE ðTyÞ; x yi; 8x; y 2 C; equivalently, Dp ðTx; TyÞ þ Dp ðTy; TxÞ þ Dp ðx; TxÞ þ Dp ðy; TyÞ Dp ðx; TyÞ þ Dp ðy; TxÞ;
(ii)
where Dp ðx; yÞ (for all x; y 2 C) is as defined in (20). right Bregman strongly nonexpansive (see [22]) with respect to a nonempty ^ FðTÞ if ^ Dp ðTx; yÞ Dp ðx; yÞ; 8x 2 C; y 2 FðTÞ ^ and if whenever fxn g C is bounded, y 2 FðTÞ and lim Dp ðxn ; yÞ Dp ðTxn ; yÞ ¼ 0; n!1
it follows that lim Dp ðxn ; Txn Þ ¼ 0:
n!1
Remark 1.1 [22]. Every right Bregman firmly nonexpansive mapping is right ^ Bregman strongly nonexpansive mapping with respect to FðTÞ ¼ FðTÞ.
The duality mapping JpE : E ! 2E is defined by JpE ðxÞ ¼ fx 2 E : hx; x i ¼ jjxjjp ; jjx jj ¼ jjxjjp1 g:
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